Twee---2.7.UNS-Prf.s

View TPTP
Problem Process solution in
SystemOnTSTP Download .tgz
%------------------------------------------------------------------------------
% File     : Twee---2.7
% Problem  : LCL088-1 : TPTP v9.3.1. Released v1.0.0.
% Transfm  : none
% Format   : tptp:raw
% Command  : run_twee /export/starexec/sandbox/benchmark/theBenchmark.p

% Computer : n007.cluster.edu
% Model    : x86_64 x86_64
% CPU      : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory   : 8046.5625MB
% OS       : Linux 6.8.0-71-generic
% CPULimit : 300s
% WCLimit  : 300s
% DateTime : Tue Sep 29 11:49:00 AM UTC 2026

% Result   : Unsatisfiable 0.21s 0.56s
% Output   : Proof 28.34s
% Verified : 
% SZS Type : -

% Comments : 
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.03  % Problem  : LCL088-1 : TPTP v9.3.1. Released v1.0.0.
% 0.00/0.05  % Command  : run_twee /export/starexec/sandbox/benchmark/theBenchmark.p
% 0.12/0.38  % Computer : n007.cluster.edu
% 0.12/0.38  % Model    : x86_64 x86_64
% 0.12/0.38  % CPU      : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.38  % Memory   : 8046.5625MB
% 0.12/0.38  % OS       : Linux 6.8.0-71-generic
% 0.12/0.38  % CPULimit : 300
% 0.12/0.38  % WCLimit  : 300
% 0.12/0.38  % DateTime : Sun Sep 27 15:19:09 UTC 2026
% 0.12/0.38  % CPUTime  : 
% 0.12/0.38  Running run_twee /export/starexec/sandbox/benchmark/theBenchmark.p
% 0.21/0.56  Command-line arguments: --no-flatten-goal
% 0.21/0.56  
% 0.21/0.56  % SZS status Unsatisfiable
% 0.21/0.56  
% 19.64/3.36  % SZS output start Proof
% 19.64/3.36  Axiom 1 (ifeq_axiom): ifeq(X, X, Y, Z) = Y.
% 19.64/3.36  Axiom 2 (condensed_detachment): ifeq(is_a_theorem(implies(X, Y)), true, ifeq(is_a_theorem(X), true, is_a_theorem(Y), true), true) = true.
% 20.25/3.44  Axiom 3 (ic_JLukasiewicz_4): is_a_theorem(implies(implies(implies(X, Y), implies(Z, W)), implies(V, implies(implies(W, X), implies(Z, X))))) = true.
% 20.25/3.44  
% 20.25/3.44  Goal 1 (prove_ic_3): is_a_theorem(implies(implies(implies(a, b), a), a)) = true.
% 20.25/3.44  Proof:
% 20.25/3.44    is_a_theorem(implies(implies(implies(a, b), a), a))
% 20.25/3.44  = { by axiom 1 (ifeq_axiom) R->L }
% 20.25/3.44    ifeq(true, true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 20.25/3.44  = { by axiom 2 (condensed_detachment) R->L }
% 20.25/3.44    ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a)))), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 20.25/3.44  = { by axiom 1 (ifeq_axiom) R->L }
% 20.25/3.44    ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a)))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 20.25/3.44  = { by axiom 2 (condensed_detachment) R->L }
% 20.25/3.45    ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(Z, W), implies(W, implies(Z, W))), implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a)))))), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))), true, is_a_theorem(implies(implies(implies(Z, W), implies(W, implies(Z, W))), implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))))), true), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a)))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 20.25/3.45  = { by axiom 3 (ic_JLukasiewicz_4) }
% 20.25/3.45    ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))), true, is_a_theorem(implies(implies(implies(Z, W), implies(W, implies(Z, W))), implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))))), true), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a)))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 20.25/3.45  = { by axiom 1 (ifeq_axiom) }
% 20.25/3.46    ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))), true, is_a_theorem(implies(implies(implies(Z, W), implies(W, implies(Z, W))), implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a)))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 20.25/3.46  = { by axiom 1 (ifeq_axiom) R->L }
% 20.25/3.46    ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))), true), true, is_a_theorem(implies(implies(implies(Z, W), implies(W, implies(Z, W))), implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a)))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 20.25/3.46  = { by axiom 2 (condensed_detachment) R->L }
% 20.25/3.47    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))), implies(implies(implies(V, U), implies(U, implies(V, U))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))))), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))))), true, is_a_theorem(implies(implies(implies(V, U), implies(U, implies(V, U))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))))), true), true), true, is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))), true), true, is_a_theorem(implies(implies(implies(Z, W), implies(W, implies(Z, W))), implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a)))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 20.25/3.47  = { by axiom 1 (ifeq_axiom) R->L }
% 20.25/3.48    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))), implies(implies(implies(V, U), implies(U, implies(V, U))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))))), true, is_a_theorem(implies(implies(implies(V, U), implies(U, implies(V, U))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))))), true), true), true, is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))), true), true, is_a_theorem(implies(implies(implies(Z, W), implies(W, implies(Z, W))), implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a)))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 20.25/3.49  = { by axiom 3 (ic_JLukasiewicz_4) R->L }
% 20.25/3.49    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(T, S), implies(X2, Y2)), implies(Z2, implies(implies(Y2, T), implies(X2, T))))), true, is_a_theorem(implies(implies(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))), implies(implies(implies(V, U), implies(U, implies(V, U))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))))), true, is_a_theorem(implies(implies(implies(V, U), implies(U, implies(V, U))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))))), true), true), true, is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))), true), true, is_a_theorem(implies(implies(implies(Z, W), implies(W, implies(Z, W))), implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a)))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 20.25/3.50  = { by axiom 1 (ifeq_axiom) R->L }
% 20.87/3.50    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(implies(T, S), implies(X2, Y2)), implies(Z2, implies(implies(Y2, T), implies(X2, T))))), true, is_a_theorem(implies(implies(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))), implies(implies(implies(V, U), implies(U, implies(V, U))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))))), true, is_a_theorem(implies(implies(implies(V, U), implies(U, implies(V, U))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))))), true), true), true, is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))), true), true, is_a_theorem(implies(implies(implies(Z, W), implies(W, implies(Z, W))), implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a)))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 20.87/3.51  = { by axiom 2 (condensed_detachment) R->L }
% 20.87/3.52    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))), implies(implies(implies(V, U), implies(U, implies(V, U))), implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))))), implies(implies(implies(implies(T, S), implies(X2, Y2)), implies(Z2, implies(implies(Y2, T), implies(X2, T)))), implies(implies(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))), implies(implies(implies(V, U), implies(U, implies(V, U))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))))))), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))), implies(implies(implies(V, U), implies(U, implies(V, U))), implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))))), true, is_a_theorem(implies(implies(implies(implies(T, S), implies(X2, Y2)), implies(Z2, implies(implies(Y2, T), implies(X2, T)))), implies(implies(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))), implies(implies(implies(V, U), implies(U, implies(V, U))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(T, S), implies(X2, Y2)), implies(Z2, implies(implies(Y2, T), implies(X2, T))))), true, is_a_theorem(implies(implies(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))), implies(implies(implies(V, U), implies(U, implies(V, U))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))))), true, is_a_theorem(implies(implies(implies(V, U), implies(U, implies(V, U))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))))), true), true), true, is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))), true), true, is_a_theorem(implies(implies(implies(Z, W), implies(W, implies(Z, W))), implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a)))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 20.87/3.52  = { by axiom 3 (ic_JLukasiewicz_4) }
% 20.87/3.53    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))), implies(implies(implies(V, U), implies(U, implies(V, U))), implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))))), true, is_a_theorem(implies(implies(implies(implies(T, S), implies(X2, Y2)), implies(Z2, implies(implies(Y2, T), implies(X2, T)))), implies(implies(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))), implies(implies(implies(V, U), implies(U, implies(V, U))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(T, S), implies(X2, Y2)), implies(Z2, implies(implies(Y2, T), implies(X2, T))))), true, is_a_theorem(implies(implies(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))), implies(implies(implies(V, U), implies(U, implies(V, U))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))))), true, is_a_theorem(implies(implies(implies(V, U), implies(U, implies(V, U))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))))), true), true), true, is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))), true), true, is_a_theorem(implies(implies(implies(Z, W), implies(W, implies(Z, W))), implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a)))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 20.87/3.54  = { by axiom 1 (ifeq_axiom) }
% 20.87/3.54    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))), implies(implies(implies(V, U), implies(U, implies(V, U))), implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))))), true, is_a_theorem(implies(implies(implies(implies(T, S), implies(X2, Y2)), implies(Z2, implies(implies(Y2, T), implies(X2, T)))), implies(implies(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))), implies(implies(implies(V, U), implies(U, implies(V, U))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))))))), true), true, ifeq(is_a_theorem(implies(implies(implies(T, S), implies(X2, Y2)), implies(Z2, implies(implies(Y2, T), implies(X2, T))))), true, is_a_theorem(implies(implies(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))), implies(implies(implies(V, U), implies(U, implies(V, U))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))))), true, is_a_theorem(implies(implies(implies(V, U), implies(U, implies(V, U))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))))), true), true), true, is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))), true), true, is_a_theorem(implies(implies(implies(Z, W), implies(W, implies(Z, W))), implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a)))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 20.87/3.55  = { by axiom 3 (ic_JLukasiewicz_4) }
% 20.87/3.55    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(implies(T, S), implies(X2, Y2)), implies(Z2, implies(implies(Y2, T), implies(X2, T)))), implies(implies(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))), implies(implies(implies(V, U), implies(U, implies(V, U))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))))))), true), true, ifeq(is_a_theorem(implies(implies(implies(T, S), implies(X2, Y2)), implies(Z2, implies(implies(Y2, T), implies(X2, T))))), true, is_a_theorem(implies(implies(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))), implies(implies(implies(V, U), implies(U, implies(V, U))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))))), true, is_a_theorem(implies(implies(implies(V, U), implies(U, implies(V, U))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))))), true), true), true, is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))), true), true, is_a_theorem(implies(implies(implies(Z, W), implies(W, implies(Z, W))), implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a)))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 20.87/3.56  = { by axiom 1 (ifeq_axiom) }
% 20.87/3.56    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(T, S), implies(X2, Y2)), implies(Z2, implies(implies(Y2, T), implies(X2, T)))), implies(implies(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))), implies(implies(implies(V, U), implies(U, implies(V, U))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))))))), true, ifeq(is_a_theorem(implies(implies(implies(T, S), implies(X2, Y2)), implies(Z2, implies(implies(Y2, T), implies(X2, T))))), true, is_a_theorem(implies(implies(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))), implies(implies(implies(V, U), implies(U, implies(V, U))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))))), true, is_a_theorem(implies(implies(implies(V, U), implies(U, implies(V, U))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))))), true), true), true, is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))), true), true, is_a_theorem(implies(implies(implies(Z, W), implies(W, implies(Z, W))), implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a)))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 20.87/3.57  = { by axiom 2 (condensed_detachment) }
% 20.87/3.57    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))))), true, is_a_theorem(implies(implies(implies(V, U), implies(U, implies(V, U))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))))), true), true), true, is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))), true), true, is_a_theorem(implies(implies(implies(Z, W), implies(W, implies(Z, W))), implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a)))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 20.87/3.57  = { by axiom 1 (ifeq_axiom) }
% 20.87/3.58    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))))), true, is_a_theorem(implies(implies(implies(V, U), implies(U, implies(V, U))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))), true), true, is_a_theorem(implies(implies(implies(Z, W), implies(W, implies(Z, W))), implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a)))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 20.87/3.58  = { by axiom 1 (ifeq_axiom) R->L }
% 20.87/3.58    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))))), true), true, is_a_theorem(implies(implies(implies(V, U), implies(U, implies(V, U))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))), true), true, is_a_theorem(implies(implies(implies(Z, W), implies(W, implies(Z, W))), implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a)))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 20.87/3.59  = { by axiom 2 (condensed_detachment) R->L }
% 20.87/3.59    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)))), implies(implies(implies(W2, V2), implies(V2, implies(W2, V2))), implies(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))))))), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a))))), true, is_a_theorem(implies(implies(implies(W2, V2), implies(V2, implies(W2, V2))), implies(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))))), true), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))))), true), true, is_a_theorem(implies(implies(implies(V, U), implies(U, implies(V, U))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))), true), true, is_a_theorem(implies(implies(implies(Z, W), implies(W, implies(Z, W))), implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a)))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 20.87/3.60  = { by axiom 3 (ic_JLukasiewicz_4) }
% 21.47/3.60    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a))))), true, is_a_theorem(implies(implies(implies(W2, V2), implies(V2, implies(W2, V2))), implies(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))))), true), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))))), true), true, is_a_theorem(implies(implies(implies(V, U), implies(U, implies(V, U))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))), true), true, is_a_theorem(implies(implies(implies(Z, W), implies(W, implies(Z, W))), implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a)))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 21.47/3.61  = { by axiom 1 (ifeq_axiom) }
% 21.47/3.61    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a))))), true, is_a_theorem(implies(implies(implies(W2, V2), implies(V2, implies(W2, V2))), implies(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))))), true), true, is_a_theorem(implies(implies(implies(V, U), implies(U, implies(V, U))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))), true), true, is_a_theorem(implies(implies(implies(Z, W), implies(W, implies(Z, W))), implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a)))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 21.47/3.62  = { by axiom 1 (ifeq_axiom) R->L }
% 21.47/3.62    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a))))), true), true, is_a_theorem(implies(implies(implies(W2, V2), implies(V2, implies(W2, V2))), implies(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))))), true), true, is_a_theorem(implies(implies(implies(V, U), implies(U, implies(V, U))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))), true), true, is_a_theorem(implies(implies(implies(Z, W), implies(W, implies(Z, W))), implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a)))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 21.47/3.63  = { by axiom 3 (ic_JLukasiewicz_4) R->L }
% 21.47/3.64    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(U2, T2), implies(S2, X3)), implies(Y3, implies(implies(X3, U2), implies(S2, U2))))), true, is_a_theorem(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a))))), true), true, is_a_theorem(implies(implies(implies(W2, V2), implies(V2, implies(W2, V2))), implies(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))))), true), true, is_a_theorem(implies(implies(implies(V, U), implies(U, implies(V, U))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))), true), true, is_a_theorem(implies(implies(implies(Z, W), implies(W, implies(Z, W))), implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a)))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 21.47/3.66  = { by axiom 1 (ifeq_axiom) R->L }
% 21.47/3.66    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(implies(U2, T2), implies(S2, X3)), implies(Y3, implies(implies(X3, U2), implies(S2, U2))))), true, is_a_theorem(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a))))), true), true), true, is_a_theorem(implies(implies(implies(W2, V2), implies(V2, implies(W2, V2))), implies(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))))), true), true, is_a_theorem(implies(implies(implies(V, U), implies(U, implies(V, U))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))), true), true, is_a_theorem(implies(implies(implies(Z, W), implies(W, implies(Z, W))), implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a)))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 21.47/3.68  = { by axiom 2 (condensed_detachment) R->L }
% 21.47/3.69    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(implies(Z3, implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)))), implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a))))), implies(implies(implies(implies(U2, T2), implies(S2, X3)), implies(Y3, implies(implies(X3, U2), implies(S2, U2)))), implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a))))))), true, ifeq(is_a_theorem(implies(implies(implies(implies(Z3, implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)))), implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)))))), true, is_a_theorem(implies(implies(implies(implies(U2, T2), implies(S2, X3)), implies(Y3, implies(implies(X3, U2), implies(S2, U2)))), implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(U2, T2), implies(S2, X3)), implies(Y3, implies(implies(X3, U2), implies(S2, U2))))), true, is_a_theorem(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a))))), true), true), true, is_a_theorem(implies(implies(implies(W2, V2), implies(V2, implies(W2, V2))), implies(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))))), true), true, is_a_theorem(implies(implies(implies(V, U), implies(U, implies(V, U))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))), true), true, is_a_theorem(implies(implies(implies(Z, W), implies(W, implies(Z, W))), implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a)))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 22.06/3.71  = { by axiom 1 (ifeq_axiom) R->L }
% 22.06/3.72    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(implies(implies(Z3, implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)))), implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a))))), implies(implies(implies(implies(U2, T2), implies(S2, X3)), implies(Y3, implies(implies(X3, U2), implies(S2, U2)))), implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a))))))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(Z3, implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)))), implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)))))), true, is_a_theorem(implies(implies(implies(implies(U2, T2), implies(S2, X3)), implies(Y3, implies(implies(X3, U2), implies(S2, U2)))), implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(U2, T2), implies(S2, X3)), implies(Y3, implies(implies(X3, U2), implies(S2, U2))))), true, is_a_theorem(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a))))), true), true), true, is_a_theorem(implies(implies(implies(W2, V2), implies(V2, implies(W2, V2))), implies(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))))), true), true, is_a_theorem(implies(implies(implies(V, U), implies(U, implies(V, U))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))), true), true, is_a_theorem(implies(implies(implies(Z, W), implies(W, implies(Z, W))), implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a)))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 22.06/3.75  = { by axiom 3 (ic_JLukasiewicz_4) R->L }
% 22.06/3.76    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(W3, V3), implies(U3, T3)), implies(S3, implies(implies(T3, W3), implies(U3, W3))))), true, is_a_theorem(implies(implies(implies(implies(implies(Z3, implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)))), implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a))))), implies(implies(implies(implies(U2, T2), implies(S2, X3)), implies(Y3, implies(implies(X3, U2), implies(S2, U2)))), implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a))))))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(Z3, implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)))), implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)))))), true, is_a_theorem(implies(implies(implies(implies(U2, T2), implies(S2, X3)), implies(Y3, implies(implies(X3, U2), implies(S2, U2)))), implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(U2, T2), implies(S2, X3)), implies(Y3, implies(implies(X3, U2), implies(S2, U2))))), true, is_a_theorem(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a))))), true), true), true, is_a_theorem(implies(implies(implies(W2, V2), implies(V2, implies(W2, V2))), implies(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))))), true), true, is_a_theorem(implies(implies(implies(V, U), implies(U, implies(V, U))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))), true), true, is_a_theorem(implies(implies(implies(Z, W), implies(W, implies(Z, W))), implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a)))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 22.06/3.78  = { by axiom 1 (ifeq_axiom) R->L }
% 22.06/3.79    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(implies(W3, V3), implies(U3, T3)), implies(S3, implies(implies(T3, W3), implies(U3, W3))))), true, is_a_theorem(implies(implies(implies(implies(implies(Z3, implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)))), implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a))))), implies(implies(implies(implies(U2, T2), implies(S2, X3)), implies(Y3, implies(implies(X3, U2), implies(S2, U2)))), implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a))))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(Z3, implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)))), implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)))))), true, is_a_theorem(implies(implies(implies(implies(U2, T2), implies(S2, X3)), implies(Y3, implies(implies(X3, U2), implies(S2, U2)))), implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(U2, T2), implies(S2, X3)), implies(Y3, implies(implies(X3, U2), implies(S2, U2))))), true, is_a_theorem(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a))))), true), true), true, is_a_theorem(implies(implies(implies(W2, V2), implies(V2, implies(W2, V2))), implies(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))))), true), true, is_a_theorem(implies(implies(implies(V, U), implies(U, implies(V, U))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))), true), true, is_a_theorem(implies(implies(implies(Z, W), implies(W, implies(Z, W))), implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a)))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 22.69/3.81  = { by axiom 2 (condensed_detachment) R->L }
% 22.69/3.84    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(Z3, implies(implies(implies(implies(a, b), a), a), a)))), implies(implies(implies(implies(U2, T2), implies(S2, X3)), implies(Y3, implies(implies(X3, U2), implies(S2, U2)))), implies(implies(implies(Z3, implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)))))), implies(implies(implies(implies(W3, V3), implies(U3, T3)), implies(S3, implies(implies(T3, W3), implies(U3, W3)))), implies(implies(implies(implies(implies(Z3, implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)))), implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a))))), implies(implies(implies(implies(U2, T2), implies(S2, X3)), implies(Y3, implies(implies(X3, U2), implies(S2, U2)))), implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a))))))))), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(Z3, implies(implies(implies(implies(a, b), a), a), a)))), implies(implies(implies(implies(U2, T2), implies(S2, X3)), implies(Y3, implies(implies(X3, U2), implies(S2, U2)))), implies(implies(implies(Z3, implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a))))))), true, is_a_theorem(implies(implies(implies(implies(W3, V3), implies(U3, T3)), implies(S3, implies(implies(T3, W3), implies(U3, W3)))), implies(implies(implies(implies(implies(Z3, implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)))), implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a))))), implies(implies(implies(implies(U2, T2), implies(S2, X3)), implies(Y3, implies(implies(X3, U2), implies(S2, U2)))), implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)))))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(W3, V3), implies(U3, T3)), implies(S3, implies(implies(T3, W3), implies(U3, W3))))), true, is_a_theorem(implies(implies(implies(implies(implies(Z3, implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)))), implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a))))), implies(implies(implies(implies(U2, T2), implies(S2, X3)), implies(Y3, implies(implies(X3, U2), implies(S2, U2)))), implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a))))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(Z3, implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)))), implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)))))), true, is_a_theorem(implies(implies(implies(implies(U2, T2), implies(S2, X3)), implies(Y3, implies(implies(X3, U2), implies(S2, U2)))), implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(U2, T2), implies(S2, X3)), implies(Y3, implies(implies(X3, U2), implies(S2, U2))))), true, is_a_theorem(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a))))), true), true), true, is_a_theorem(implies(implies(implies(W2, V2), implies(V2, implies(W2, V2))), implies(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))))), true), true, is_a_theorem(implies(implies(implies(V, U), implies(U, implies(V, U))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))), true), true, is_a_theorem(implies(implies(implies(Z, W), implies(W, implies(Z, W))), implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a)))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 22.69/3.86  = { by axiom 3 (ic_JLukasiewicz_4) }
% 22.69/3.88    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(Z3, implies(implies(implies(implies(a, b), a), a), a)))), implies(implies(implies(implies(U2, T2), implies(S2, X3)), implies(Y3, implies(implies(X3, U2), implies(S2, U2)))), implies(implies(implies(Z3, implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a))))))), true, is_a_theorem(implies(implies(implies(implies(W3, V3), implies(U3, T3)), implies(S3, implies(implies(T3, W3), implies(U3, W3)))), implies(implies(implies(implies(implies(Z3, implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)))), implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a))))), implies(implies(implies(implies(U2, T2), implies(S2, X3)), implies(Y3, implies(implies(X3, U2), implies(S2, U2)))), implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)))))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(W3, V3), implies(U3, T3)), implies(S3, implies(implies(T3, W3), implies(U3, W3))))), true, is_a_theorem(implies(implies(implies(implies(implies(Z3, implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)))), implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a))))), implies(implies(implies(implies(U2, T2), implies(S2, X3)), implies(Y3, implies(implies(X3, U2), implies(S2, U2)))), implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a))))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(Z3, implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)))), implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)))))), true, is_a_theorem(implies(implies(implies(implies(U2, T2), implies(S2, X3)), implies(Y3, implies(implies(X3, U2), implies(S2, U2)))), implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(U2, T2), implies(S2, X3)), implies(Y3, implies(implies(X3, U2), implies(S2, U2))))), true, is_a_theorem(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a))))), true), true), true, is_a_theorem(implies(implies(implies(W2, V2), implies(V2, implies(W2, V2))), implies(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))))), true), true, is_a_theorem(implies(implies(implies(V, U), implies(U, implies(V, U))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))), true), true, is_a_theorem(implies(implies(implies(Z, W), implies(W, implies(Z, W))), implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a)))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 22.69/3.89  = { by axiom 1 (ifeq_axiom) }
% 23.28/3.91    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(Z3, implies(implies(implies(implies(a, b), a), a), a)))), implies(implies(implies(implies(U2, T2), implies(S2, X3)), implies(Y3, implies(implies(X3, U2), implies(S2, U2)))), implies(implies(implies(Z3, implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a))))))), true, is_a_theorem(implies(implies(implies(implies(W3, V3), implies(U3, T3)), implies(S3, implies(implies(T3, W3), implies(U3, W3)))), implies(implies(implies(implies(implies(Z3, implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)))), implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a))))), implies(implies(implies(implies(U2, T2), implies(S2, X3)), implies(Y3, implies(implies(X3, U2), implies(S2, U2)))), implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)))))))), true), true, ifeq(is_a_theorem(implies(implies(implies(W3, V3), implies(U3, T3)), implies(S3, implies(implies(T3, W3), implies(U3, W3))))), true, is_a_theorem(implies(implies(implies(implies(implies(Z3, implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)))), implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a))))), implies(implies(implies(implies(U2, T2), implies(S2, X3)), implies(Y3, implies(implies(X3, U2), implies(S2, U2)))), implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a))))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(Z3, implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)))), implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)))))), true, is_a_theorem(implies(implies(implies(implies(U2, T2), implies(S2, X3)), implies(Y3, implies(implies(X3, U2), implies(S2, U2)))), implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(U2, T2), implies(S2, X3)), implies(Y3, implies(implies(X3, U2), implies(S2, U2))))), true, is_a_theorem(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a))))), true), true), true, is_a_theorem(implies(implies(implies(W2, V2), implies(V2, implies(W2, V2))), implies(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))))), true), true, is_a_theorem(implies(implies(implies(V, U), implies(U, implies(V, U))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))), true), true, is_a_theorem(implies(implies(implies(Z, W), implies(W, implies(Z, W))), implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a)))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 23.28/3.92  = { by axiom 3 (ic_JLukasiewicz_4) }
% 23.28/3.93    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(implies(W3, V3), implies(U3, T3)), implies(S3, implies(implies(T3, W3), implies(U3, W3)))), implies(implies(implies(implies(implies(Z3, implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)))), implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a))))), implies(implies(implies(implies(U2, T2), implies(S2, X3)), implies(Y3, implies(implies(X3, U2), implies(S2, U2)))), implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)))))))), true), true, ifeq(is_a_theorem(implies(implies(implies(W3, V3), implies(U3, T3)), implies(S3, implies(implies(T3, W3), implies(U3, W3))))), true, is_a_theorem(implies(implies(implies(implies(implies(Z3, implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)))), implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a))))), implies(implies(implies(implies(U2, T2), implies(S2, X3)), implies(Y3, implies(implies(X3, U2), implies(S2, U2)))), implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a))))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(Z3, implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)))), implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)))))), true, is_a_theorem(implies(implies(implies(implies(U2, T2), implies(S2, X3)), implies(Y3, implies(implies(X3, U2), implies(S2, U2)))), implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(U2, T2), implies(S2, X3)), implies(Y3, implies(implies(X3, U2), implies(S2, U2))))), true, is_a_theorem(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a))))), true), true), true, is_a_theorem(implies(implies(implies(W2, V2), implies(V2, implies(W2, V2))), implies(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))))), true), true, is_a_theorem(implies(implies(implies(V, U), implies(U, implies(V, U))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))), true), true, is_a_theorem(implies(implies(implies(Z, W), implies(W, implies(Z, W))), implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a)))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 23.28/3.94  = { by axiom 1 (ifeq_axiom) }
% 23.28/3.96    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(W3, V3), implies(U3, T3)), implies(S3, implies(implies(T3, W3), implies(U3, W3)))), implies(implies(implies(implies(implies(Z3, implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)))), implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a))))), implies(implies(implies(implies(U2, T2), implies(S2, X3)), implies(Y3, implies(implies(X3, U2), implies(S2, U2)))), implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)))))))), true, ifeq(is_a_theorem(implies(implies(implies(W3, V3), implies(U3, T3)), implies(S3, implies(implies(T3, W3), implies(U3, W3))))), true, is_a_theorem(implies(implies(implies(implies(implies(Z3, implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)))), implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a))))), implies(implies(implies(implies(U2, T2), implies(S2, X3)), implies(Y3, implies(implies(X3, U2), implies(S2, U2)))), implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a))))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(Z3, implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)))), implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)))))), true, is_a_theorem(implies(implies(implies(implies(U2, T2), implies(S2, X3)), implies(Y3, implies(implies(X3, U2), implies(S2, U2)))), implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(U2, T2), implies(S2, X3)), implies(Y3, implies(implies(X3, U2), implies(S2, U2))))), true, is_a_theorem(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a))))), true), true), true, is_a_theorem(implies(implies(implies(W2, V2), implies(V2, implies(W2, V2))), implies(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))))), true), true, is_a_theorem(implies(implies(implies(V, U), implies(U, implies(V, U))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))), true), true, is_a_theorem(implies(implies(implies(Z, W), implies(W, implies(Z, W))), implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a)))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 23.28/3.97  = { by axiom 2 (condensed_detachment) }
% 23.28/3.97    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(implies(implies(Z3, implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)))), implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)))))), true, is_a_theorem(implies(implies(implies(implies(U2, T2), implies(S2, X3)), implies(Y3, implies(implies(X3, U2), implies(S2, U2)))), implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(U2, T2), implies(S2, X3)), implies(Y3, implies(implies(X3, U2), implies(S2, U2))))), true, is_a_theorem(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a))))), true), true), true, is_a_theorem(implies(implies(implies(W2, V2), implies(V2, implies(W2, V2))), implies(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))))), true), true, is_a_theorem(implies(implies(implies(V, U), implies(U, implies(V, U))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))), true), true, is_a_theorem(implies(implies(implies(Z, W), implies(W, implies(Z, W))), implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a)))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 23.28/3.98  = { by axiom 1 (ifeq_axiom) }
% 23.28/3.99    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(Z3, implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)))), implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)))))), true, is_a_theorem(implies(implies(implies(implies(U2, T2), implies(S2, X3)), implies(Y3, implies(implies(X3, U2), implies(S2, U2)))), implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(U2, T2), implies(S2, X3)), implies(Y3, implies(implies(X3, U2), implies(S2, U2))))), true, is_a_theorem(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a))))), true), true), true, is_a_theorem(implies(implies(implies(W2, V2), implies(V2, implies(W2, V2))), implies(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))))), true), true, is_a_theorem(implies(implies(implies(V, U), implies(U, implies(V, U))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))), true), true, is_a_theorem(implies(implies(implies(Z, W), implies(W, implies(Z, W))), implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a)))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 23.28/4.01  = { by axiom 1 (ifeq_axiom) R->L }
% 23.28/4.02    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(implies(Z3, implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)))), implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)))))), true), true, is_a_theorem(implies(implies(implies(implies(U2, T2), implies(S2, X3)), implies(Y3, implies(implies(X3, U2), implies(S2, U2)))), implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(U2, T2), implies(S2, X3)), implies(Y3, implies(implies(X3, U2), implies(S2, U2))))), true, is_a_theorem(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a))))), true), true), true, is_a_theorem(implies(implies(implies(W2, V2), implies(V2, implies(W2, V2))), implies(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))))), true), true, is_a_theorem(implies(implies(implies(V, U), implies(U, implies(V, U))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))), true), true, is_a_theorem(implies(implies(implies(Z, W), implies(W, implies(Z, W))), implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a)))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 23.28/4.03  = { by axiom 3 (ic_JLukasiewicz_4) R->L }
% 23.28/4.04    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(X4, Y4), implies(Z4, W4)), implies(V4, implies(implies(W4, X4), implies(Z4, X4))))), true, is_a_theorem(implies(implies(implies(implies(Z3, implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)))), implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)))))), true), true, is_a_theorem(implies(implies(implies(implies(U2, T2), implies(S2, X3)), implies(Y3, implies(implies(X3, U2), implies(S2, U2)))), implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(U2, T2), implies(S2, X3)), implies(Y3, implies(implies(X3, U2), implies(S2, U2))))), true, is_a_theorem(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a))))), true), true), true, is_a_theorem(implies(implies(implies(W2, V2), implies(V2, implies(W2, V2))), implies(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))))), true), true, is_a_theorem(implies(implies(implies(V, U), implies(U, implies(V, U))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))), true), true, is_a_theorem(implies(implies(implies(Z, W), implies(W, implies(Z, W))), implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a)))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 23.28/4.05  = { by axiom 1 (ifeq_axiom) R->L }
% 23.28/4.06    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(implies(X4, Y4), implies(Z4, W4)), implies(V4, implies(implies(W4, X4), implies(Z4, X4))))), true, is_a_theorem(implies(implies(implies(implies(Z3, implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)))), implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)))))), true), true), true, is_a_theorem(implies(implies(implies(implies(U2, T2), implies(S2, X3)), implies(Y3, implies(implies(X3, U2), implies(S2, U2)))), implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(U2, T2), implies(S2, X3)), implies(Y3, implies(implies(X3, U2), implies(S2, U2))))), true, is_a_theorem(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a))))), true), true), true, is_a_theorem(implies(implies(implies(W2, V2), implies(V2, implies(W2, V2))), implies(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))))), true), true, is_a_theorem(implies(implies(implies(V, U), implies(U, implies(V, U))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))), true), true, is_a_theorem(implies(implies(implies(Z, W), implies(W, implies(Z, W))), implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a)))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 24.28/4.07  = { by axiom 2 (condensed_detachment) R->L }
% 24.28/4.09    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), Z3)), implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(Z3, implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a))))), implies(implies(implies(implies(X4, Y4), implies(Z4, W4)), implies(V4, implies(implies(W4, X4), implies(Z4, X4)))), implies(implies(implies(implies(Z3, implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)))), implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)))))))), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), Z3)), implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(Z3, implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)))))), true, is_a_theorem(implies(implies(implies(implies(X4, Y4), implies(Z4, W4)), implies(V4, implies(implies(W4, X4), implies(Z4, X4)))), implies(implies(implies(implies(Z3, implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)))), implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a))))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(X4, Y4), implies(Z4, W4)), implies(V4, implies(implies(W4, X4), implies(Z4, X4))))), true, is_a_theorem(implies(implies(implies(implies(Z3, implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)))), implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)))))), true), true), true, is_a_theorem(implies(implies(implies(implies(U2, T2), implies(S2, X3)), implies(Y3, implies(implies(X3, U2), implies(S2, U2)))), implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(U2, T2), implies(S2, X3)), implies(Y3, implies(implies(X3, U2), implies(S2, U2))))), true, is_a_theorem(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a))))), true), true), true, is_a_theorem(implies(implies(implies(W2, V2), implies(V2, implies(W2, V2))), implies(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))))), true), true, is_a_theorem(implies(implies(implies(V, U), implies(U, implies(V, U))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))), true), true, is_a_theorem(implies(implies(implies(Z, W), implies(W, implies(Z, W))), implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a)))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 24.28/4.10  = { by axiom 3 (ic_JLukasiewicz_4) }
% 24.28/4.12    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), Z3)), implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(Z3, implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)))))), true, is_a_theorem(implies(implies(implies(implies(X4, Y4), implies(Z4, W4)), implies(V4, implies(implies(W4, X4), implies(Z4, X4)))), implies(implies(implies(implies(Z3, implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)))), implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a))))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(X4, Y4), implies(Z4, W4)), implies(V4, implies(implies(W4, X4), implies(Z4, X4))))), true, is_a_theorem(implies(implies(implies(implies(Z3, implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)))), implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)))))), true), true), true, is_a_theorem(implies(implies(implies(implies(U2, T2), implies(S2, X3)), implies(Y3, implies(implies(X3, U2), implies(S2, U2)))), implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(U2, T2), implies(S2, X3)), implies(Y3, implies(implies(X3, U2), implies(S2, U2))))), true, is_a_theorem(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a))))), true), true), true, is_a_theorem(implies(implies(implies(W2, V2), implies(V2, implies(W2, V2))), implies(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))))), true), true, is_a_theorem(implies(implies(implies(V, U), implies(U, implies(V, U))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))), true), true, is_a_theorem(implies(implies(implies(Z, W), implies(W, implies(Z, W))), implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a)))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 24.28/4.12  = { by axiom 1 (ifeq_axiom) }
% 24.28/4.14    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), Z3)), implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(Z3, implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)))))), true, is_a_theorem(implies(implies(implies(implies(X4, Y4), implies(Z4, W4)), implies(V4, implies(implies(W4, X4), implies(Z4, X4)))), implies(implies(implies(implies(Z3, implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)))), implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a))))))), true), true, ifeq(is_a_theorem(implies(implies(implies(X4, Y4), implies(Z4, W4)), implies(V4, implies(implies(W4, X4), implies(Z4, X4))))), true, is_a_theorem(implies(implies(implies(implies(Z3, implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)))), implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)))))), true), true), true, is_a_theorem(implies(implies(implies(implies(U2, T2), implies(S2, X3)), implies(Y3, implies(implies(X3, U2), implies(S2, U2)))), implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(U2, T2), implies(S2, X3)), implies(Y3, implies(implies(X3, U2), implies(S2, U2))))), true, is_a_theorem(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a))))), true), true), true, is_a_theorem(implies(implies(implies(W2, V2), implies(V2, implies(W2, V2))), implies(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))))), true), true, is_a_theorem(implies(implies(implies(V, U), implies(U, implies(V, U))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))), true), true, is_a_theorem(implies(implies(implies(Z, W), implies(W, implies(Z, W))), implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a)))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 24.28/4.14  = { by axiom 3 (ic_JLukasiewicz_4) }
% 24.28/4.15    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(implies(X4, Y4), implies(Z4, W4)), implies(V4, implies(implies(W4, X4), implies(Z4, X4)))), implies(implies(implies(implies(Z3, implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)))), implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a))))))), true), true, ifeq(is_a_theorem(implies(implies(implies(X4, Y4), implies(Z4, W4)), implies(V4, implies(implies(W4, X4), implies(Z4, X4))))), true, is_a_theorem(implies(implies(implies(implies(Z3, implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)))), implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)))))), true), true), true, is_a_theorem(implies(implies(implies(implies(U2, T2), implies(S2, X3)), implies(Y3, implies(implies(X3, U2), implies(S2, U2)))), implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(U2, T2), implies(S2, X3)), implies(Y3, implies(implies(X3, U2), implies(S2, U2))))), true, is_a_theorem(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a))))), true), true), true, is_a_theorem(implies(implies(implies(W2, V2), implies(V2, implies(W2, V2))), implies(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))))), true), true, is_a_theorem(implies(implies(implies(V, U), implies(U, implies(V, U))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))), true), true, is_a_theorem(implies(implies(implies(Z, W), implies(W, implies(Z, W))), implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a)))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 24.28/4.16  = { by axiom 1 (ifeq_axiom) }
% 24.28/4.17    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(X4, Y4), implies(Z4, W4)), implies(V4, implies(implies(W4, X4), implies(Z4, X4)))), implies(implies(implies(implies(Z3, implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)))), implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a))))))), true, ifeq(is_a_theorem(implies(implies(implies(X4, Y4), implies(Z4, W4)), implies(V4, implies(implies(W4, X4), implies(Z4, X4))))), true, is_a_theorem(implies(implies(implies(implies(Z3, implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)))), implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)))))), true), true), true, is_a_theorem(implies(implies(implies(implies(U2, T2), implies(S2, X3)), implies(Y3, implies(implies(X3, U2), implies(S2, U2)))), implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(U2, T2), implies(S2, X3)), implies(Y3, implies(implies(X3, U2), implies(S2, U2))))), true, is_a_theorem(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a))))), true), true), true, is_a_theorem(implies(implies(implies(W2, V2), implies(V2, implies(W2, V2))), implies(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))))), true), true, is_a_theorem(implies(implies(implies(V, U), implies(U, implies(V, U))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))), true), true, is_a_theorem(implies(implies(implies(Z, W), implies(W, implies(Z, W))), implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a)))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 24.87/4.17  = { by axiom 2 (condensed_detachment) }
% 24.87/4.18    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(implies(U2, T2), implies(S2, X3)), implies(Y3, implies(implies(X3, U2), implies(S2, U2)))), implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(U2, T2), implies(S2, X3)), implies(Y3, implies(implies(X3, U2), implies(S2, U2))))), true, is_a_theorem(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a))))), true), true), true, is_a_theorem(implies(implies(implies(W2, V2), implies(V2, implies(W2, V2))), implies(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))))), true), true, is_a_theorem(implies(implies(implies(V, U), implies(U, implies(V, U))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))), true), true, is_a_theorem(implies(implies(implies(Z, W), implies(W, implies(Z, W))), implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a)))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 24.87/4.18  = { by axiom 1 (ifeq_axiom) }
% 24.87/4.19    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(U2, T2), implies(S2, X3)), implies(Y3, implies(implies(X3, U2), implies(S2, U2)))), implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)))))), true, ifeq(is_a_theorem(implies(implies(implies(U2, T2), implies(S2, X3)), implies(Y3, implies(implies(X3, U2), implies(S2, U2))))), true, is_a_theorem(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a))))), true), true), true, is_a_theorem(implies(implies(implies(W2, V2), implies(V2, implies(W2, V2))), implies(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))))), true), true, is_a_theorem(implies(implies(implies(V, U), implies(U, implies(V, U))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))), true), true, is_a_theorem(implies(implies(implies(Z, W), implies(W, implies(Z, W))), implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a)))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 24.87/4.19  = { by axiom 2 (condensed_detachment) }
% 24.87/4.19    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(W2, V2), implies(V2, implies(W2, V2))), implies(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))))), true), true, is_a_theorem(implies(implies(implies(V, U), implies(U, implies(V, U))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))), true), true, is_a_theorem(implies(implies(implies(Z, W), implies(W, implies(Z, W))), implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a)))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 24.87/4.20  = { by axiom 1 (ifeq_axiom) }
% 24.87/4.20    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(W2, V2), implies(V2, implies(W2, V2))), implies(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))))), true, is_a_theorem(implies(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))))), true), true, is_a_theorem(implies(implies(implies(V, U), implies(U, implies(V, U))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))), true), true, is_a_theorem(implies(implies(implies(Z, W), implies(W, implies(Z, W))), implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a)))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 25.08/4.21  = { by axiom 1 (ifeq_axiom) R->L }
% 25.08/4.22    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(W2, V2), implies(V2, implies(W2, V2))), implies(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))))), true, ifeq(true, true, is_a_theorem(implies(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))))), true), true), true, is_a_theorem(implies(implies(implies(V, U), implies(U, implies(V, U))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))), true), true, is_a_theorem(implies(implies(implies(Z, W), implies(W, implies(Z, W))), implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a)))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 25.08/4.22  = { by axiom 2 (condensed_detachment) R->L }
% 25.08/4.22    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(W2, V2), implies(V2, implies(W2, V2))), implies(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))))), true, ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(U4, T4), implies(S4, X5)), implies(Y5, implies(implies(X5, U4), implies(S4, U4)))), implies(implies(W2, V2), implies(V2, implies(W2, V2))))), true, ifeq(is_a_theorem(implies(implies(implies(U4, T4), implies(S4, X5)), implies(Y5, implies(implies(X5, U4), implies(S4, U4))))), true, is_a_theorem(implies(implies(W2, V2), implies(V2, implies(W2, V2)))), true), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))))), true), true), true, is_a_theorem(implies(implies(implies(V, U), implies(U, implies(V, U))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))), true), true, is_a_theorem(implies(implies(implies(Z, W), implies(W, implies(Z, W))), implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a)))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 25.08/4.23  = { by axiom 1 (ifeq_axiom) R->L }
% 25.08/4.23    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(W2, V2), implies(V2, implies(W2, V2))), implies(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))))), true, ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(implies(U4, T4), implies(S4, X5)), implies(Y5, implies(implies(X5, U4), implies(S4, U4)))), implies(implies(W2, V2), implies(V2, implies(W2, V2))))), true), true, ifeq(is_a_theorem(implies(implies(implies(U4, T4), implies(S4, X5)), implies(Y5, implies(implies(X5, U4), implies(S4, U4))))), true, is_a_theorem(implies(implies(W2, V2), implies(V2, implies(W2, V2)))), true), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))))), true), true), true, is_a_theorem(implies(implies(implies(V, U), implies(U, implies(V, U))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))), true), true, is_a_theorem(implies(implies(implies(Z, W), implies(W, implies(Z, W))), implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a)))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 25.08/4.24  = { by axiom 2 (condensed_detachment) R->L }
% 25.08/4.24    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(W2, V2), implies(V2, implies(W2, V2))), implies(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))))), true, ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(Z5, W5), implies(V5, U5)), implies(T5, implies(implies(U5, Z5), implies(V5, Z5)))), implies(implies(implies(implies(S5, V2), implies(W2, V2)), implies(V2, implies(W2, V2))), implies(implies(W2, V2), implies(V2, implies(W2, V2)))))), true, ifeq(is_a_theorem(implies(implies(implies(Z5, W5), implies(V5, U5)), implies(T5, implies(implies(U5, Z5), implies(V5, Z5))))), true, is_a_theorem(implies(implies(implies(implies(S5, V2), implies(W2, V2)), implies(V2, implies(W2, V2))), implies(implies(W2, V2), implies(V2, implies(W2, V2))))), true), true), true, is_a_theorem(implies(implies(implies(implies(U4, T4), implies(S4, X5)), implies(Y5, implies(implies(X5, U4), implies(S4, U4)))), implies(implies(W2, V2), implies(V2, implies(W2, V2))))), true), true, ifeq(is_a_theorem(implies(implies(implies(U4, T4), implies(S4, X5)), implies(Y5, implies(implies(X5, U4), implies(S4, U4))))), true, is_a_theorem(implies(implies(W2, V2), implies(V2, implies(W2, V2)))), true), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))))), true), true), true, is_a_theorem(implies(implies(implies(V, U), implies(U, implies(V, U))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))), true), true, is_a_theorem(implies(implies(implies(Z, W), implies(W, implies(Z, W))), implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a)))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 25.08/4.25  = { by axiom 1 (ifeq_axiom) R->L }
% 25.08/4.26    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(W2, V2), implies(V2, implies(W2, V2))), implies(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))))), true, ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(implies(Z5, W5), implies(V5, U5)), implies(T5, implies(implies(U5, Z5), implies(V5, Z5)))), implies(implies(implies(implies(S5, V2), implies(W2, V2)), implies(V2, implies(W2, V2))), implies(implies(W2, V2), implies(V2, implies(W2, V2)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(Z5, W5), implies(V5, U5)), implies(T5, implies(implies(U5, Z5), implies(V5, Z5))))), true, is_a_theorem(implies(implies(implies(implies(S5, V2), implies(W2, V2)), implies(V2, implies(W2, V2))), implies(implies(W2, V2), implies(V2, implies(W2, V2))))), true), true), true, is_a_theorem(implies(implies(implies(implies(U4, T4), implies(S4, X5)), implies(Y5, implies(implies(X5, U4), implies(S4, U4)))), implies(implies(W2, V2), implies(V2, implies(W2, V2))))), true), true, ifeq(is_a_theorem(implies(implies(implies(U4, T4), implies(S4, X5)), implies(Y5, implies(implies(X5, U4), implies(S4, U4))))), true, is_a_theorem(implies(implies(W2, V2), implies(V2, implies(W2, V2)))), true), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))))), true), true), true, is_a_theorem(implies(implies(implies(V, U), implies(U, implies(V, U))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))), true), true, is_a_theorem(implies(implies(implies(Z, W), implies(W, implies(Z, W))), implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a)))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 25.08/4.27  = { by axiom 3 (ic_JLukasiewicz_4) R->L }
% 25.08/4.27    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(W2, V2), implies(V2, implies(W2, V2))), implies(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))))), true, ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(V2, implies(W2, V2)), implies(W2, S5)), implies(implies(W2, V2), implies(implies(S5, V2), implies(W2, V2))))), true, is_a_theorem(implies(implies(implies(implies(Z5, W5), implies(V5, U5)), implies(T5, implies(implies(U5, Z5), implies(V5, Z5)))), implies(implies(implies(implies(S5, V2), implies(W2, V2)), implies(V2, implies(W2, V2))), implies(implies(W2, V2), implies(V2, implies(W2, V2)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(Z5, W5), implies(V5, U5)), implies(T5, implies(implies(U5, Z5), implies(V5, Z5))))), true, is_a_theorem(implies(implies(implies(implies(S5, V2), implies(W2, V2)), implies(V2, implies(W2, V2))), implies(implies(W2, V2), implies(V2, implies(W2, V2))))), true), true), true, is_a_theorem(implies(implies(implies(implies(U4, T4), implies(S4, X5)), implies(Y5, implies(implies(X5, U4), implies(S4, U4)))), implies(implies(W2, V2), implies(V2, implies(W2, V2))))), true), true, ifeq(is_a_theorem(implies(implies(implies(U4, T4), implies(S4, X5)), implies(Y5, implies(implies(X5, U4), implies(S4, U4))))), true, is_a_theorem(implies(implies(W2, V2), implies(V2, implies(W2, V2)))), true), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))))), true), true), true, is_a_theorem(implies(implies(implies(V, U), implies(U, implies(V, U))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))), true), true, is_a_theorem(implies(implies(implies(Z, W), implies(W, implies(Z, W))), implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a)))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 25.08/4.28  = { by axiom 1 (ifeq_axiom) R->L }
% 25.08/4.28    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(W2, V2), implies(V2, implies(W2, V2))), implies(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))))), true, ifeq(ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(implies(V2, implies(W2, V2)), implies(W2, S5)), implies(implies(W2, V2), implies(implies(S5, V2), implies(W2, V2))))), true, is_a_theorem(implies(implies(implies(implies(Z5, W5), implies(V5, U5)), implies(T5, implies(implies(U5, Z5), implies(V5, Z5)))), implies(implies(implies(implies(S5, V2), implies(W2, V2)), implies(V2, implies(W2, V2))), implies(implies(W2, V2), implies(V2, implies(W2, V2)))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(Z5, W5), implies(V5, U5)), implies(T5, implies(implies(U5, Z5), implies(V5, Z5))))), true, is_a_theorem(implies(implies(implies(implies(S5, V2), implies(W2, V2)), implies(V2, implies(W2, V2))), implies(implies(W2, V2), implies(V2, implies(W2, V2))))), true), true), true, is_a_theorem(implies(implies(implies(implies(U4, T4), implies(S4, X5)), implies(Y5, implies(implies(X5, U4), implies(S4, U4)))), implies(implies(W2, V2), implies(V2, implies(W2, V2))))), true), true, ifeq(is_a_theorem(implies(implies(implies(U4, T4), implies(S4, X5)), implies(Y5, implies(implies(X5, U4), implies(S4, U4))))), true, is_a_theorem(implies(implies(W2, V2), implies(V2, implies(W2, V2)))), true), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))))), true), true), true, is_a_theorem(implies(implies(implies(V, U), implies(U, implies(V, U))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))), true), true, is_a_theorem(implies(implies(implies(Z, W), implies(W, implies(Z, W))), implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a)))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 25.08/4.29  = { by axiom 3 (ic_JLukasiewicz_4) R->L }
% 25.08/4.30    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(W2, V2), implies(V2, implies(W2, V2))), implies(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))))), true, ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(V2, implies(W2, V2)), implies(W2, S5)), implies(implies(W2, V2), implies(implies(S5, V2), implies(W2, V2)))), implies(implies(implies(implies(Z5, W5), implies(V5, U5)), implies(T5, implies(implies(U5, Z5), implies(V5, Z5)))), implies(implies(implies(implies(S5, V2), implies(W2, V2)), implies(V2, implies(W2, V2))), implies(implies(W2, V2), implies(V2, implies(W2, V2))))))), true, ifeq(is_a_theorem(implies(implies(implies(V2, implies(W2, V2)), implies(W2, S5)), implies(implies(W2, V2), implies(implies(S5, V2), implies(W2, V2))))), true, is_a_theorem(implies(implies(implies(implies(Z5, W5), implies(V5, U5)), implies(T5, implies(implies(U5, Z5), implies(V5, Z5)))), implies(implies(implies(implies(S5, V2), implies(W2, V2)), implies(V2, implies(W2, V2))), implies(implies(W2, V2), implies(V2, implies(W2, V2)))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(Z5, W5), implies(V5, U5)), implies(T5, implies(implies(U5, Z5), implies(V5, Z5))))), true, is_a_theorem(implies(implies(implies(implies(S5, V2), implies(W2, V2)), implies(V2, implies(W2, V2))), implies(implies(W2, V2), implies(V2, implies(W2, V2))))), true), true), true, is_a_theorem(implies(implies(implies(implies(U4, T4), implies(S4, X5)), implies(Y5, implies(implies(X5, U4), implies(S4, U4)))), implies(implies(W2, V2), implies(V2, implies(W2, V2))))), true), true, ifeq(is_a_theorem(implies(implies(implies(U4, T4), implies(S4, X5)), implies(Y5, implies(implies(X5, U4), implies(S4, U4))))), true, is_a_theorem(implies(implies(W2, V2), implies(V2, implies(W2, V2)))), true), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))))), true), true), true, is_a_theorem(implies(implies(implies(V, U), implies(U, implies(V, U))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))), true), true, is_a_theorem(implies(implies(implies(Z, W), implies(W, implies(Z, W))), implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a)))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 25.57/4.30  = { by axiom 2 (condensed_detachment) }
% 25.57/4.31    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(W2, V2), implies(V2, implies(W2, V2))), implies(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))))), true, ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(implies(Z5, W5), implies(V5, U5)), implies(T5, implies(implies(U5, Z5), implies(V5, Z5))))), true, is_a_theorem(implies(implies(implies(implies(S5, V2), implies(W2, V2)), implies(V2, implies(W2, V2))), implies(implies(W2, V2), implies(V2, implies(W2, V2))))), true), true), true, is_a_theorem(implies(implies(implies(implies(U4, T4), implies(S4, X5)), implies(Y5, implies(implies(X5, U4), implies(S4, U4)))), implies(implies(W2, V2), implies(V2, implies(W2, V2))))), true), true, ifeq(is_a_theorem(implies(implies(implies(U4, T4), implies(S4, X5)), implies(Y5, implies(implies(X5, U4), implies(S4, U4))))), true, is_a_theorem(implies(implies(W2, V2), implies(V2, implies(W2, V2)))), true), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))))), true), true), true, is_a_theorem(implies(implies(implies(V, U), implies(U, implies(V, U))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))), true), true, is_a_theorem(implies(implies(implies(Z, W), implies(W, implies(Z, W))), implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a)))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 25.57/4.31  = { by axiom 1 (ifeq_axiom) }
% 25.57/4.32    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(W2, V2), implies(V2, implies(W2, V2))), implies(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))))), true, ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(Z5, W5), implies(V5, U5)), implies(T5, implies(implies(U5, Z5), implies(V5, Z5))))), true, is_a_theorem(implies(implies(implies(implies(S5, V2), implies(W2, V2)), implies(V2, implies(W2, V2))), implies(implies(W2, V2), implies(V2, implies(W2, V2))))), true), true, is_a_theorem(implies(implies(implies(implies(U4, T4), implies(S4, X5)), implies(Y5, implies(implies(X5, U4), implies(S4, U4)))), implies(implies(W2, V2), implies(V2, implies(W2, V2))))), true), true, ifeq(is_a_theorem(implies(implies(implies(U4, T4), implies(S4, X5)), implies(Y5, implies(implies(X5, U4), implies(S4, U4))))), true, is_a_theorem(implies(implies(W2, V2), implies(V2, implies(W2, V2)))), true), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))))), true), true), true, is_a_theorem(implies(implies(implies(V, U), implies(U, implies(V, U))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))), true), true, is_a_theorem(implies(implies(implies(Z, W), implies(W, implies(Z, W))), implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a)))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 25.57/4.32  = { by axiom 3 (ic_JLukasiewicz_4) }
% 25.57/4.33    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(W2, V2), implies(V2, implies(W2, V2))), implies(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))))), true, ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(implies(S5, V2), implies(W2, V2)), implies(V2, implies(W2, V2))), implies(implies(W2, V2), implies(V2, implies(W2, V2))))), true), true, is_a_theorem(implies(implies(implies(implies(U4, T4), implies(S4, X5)), implies(Y5, implies(implies(X5, U4), implies(S4, U4)))), implies(implies(W2, V2), implies(V2, implies(W2, V2))))), true), true, ifeq(is_a_theorem(implies(implies(implies(U4, T4), implies(S4, X5)), implies(Y5, implies(implies(X5, U4), implies(S4, U4))))), true, is_a_theorem(implies(implies(W2, V2), implies(V2, implies(W2, V2)))), true), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))))), true), true), true, is_a_theorem(implies(implies(implies(V, U), implies(U, implies(V, U))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))), true), true, is_a_theorem(implies(implies(implies(Z, W), implies(W, implies(Z, W))), implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a)))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 25.57/4.34  = { by axiom 1 (ifeq_axiom) }
% 25.57/4.34    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(W2, V2), implies(V2, implies(W2, V2))), implies(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))))), true, ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(S5, V2), implies(W2, V2)), implies(V2, implies(W2, V2))), implies(implies(W2, V2), implies(V2, implies(W2, V2))))), true, is_a_theorem(implies(implies(implies(implies(U4, T4), implies(S4, X5)), implies(Y5, implies(implies(X5, U4), implies(S4, U4)))), implies(implies(W2, V2), implies(V2, implies(W2, V2))))), true), true, ifeq(is_a_theorem(implies(implies(implies(U4, T4), implies(S4, X5)), implies(Y5, implies(implies(X5, U4), implies(S4, U4))))), true, is_a_theorem(implies(implies(W2, V2), implies(V2, implies(W2, V2)))), true), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))))), true), true), true, is_a_theorem(implies(implies(implies(V, U), implies(U, implies(V, U))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))), true), true, is_a_theorem(implies(implies(implies(Z, W), implies(W, implies(Z, W))), implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a)))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 25.57/4.35  = { by axiom 1 (ifeq_axiom) R->L }
% 25.57/4.35    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(W2, V2), implies(V2, implies(W2, V2))), implies(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))))), true, ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(implies(implies(S5, V2), implies(W2, V2)), implies(V2, implies(W2, V2))), implies(implies(W2, V2), implies(V2, implies(W2, V2))))), true, is_a_theorem(implies(implies(implies(implies(U4, T4), implies(S4, X5)), implies(Y5, implies(implies(X5, U4), implies(S4, U4)))), implies(implies(W2, V2), implies(V2, implies(W2, V2))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(U4, T4), implies(S4, X5)), implies(Y5, implies(implies(X5, U4), implies(S4, U4))))), true, is_a_theorem(implies(implies(W2, V2), implies(V2, implies(W2, V2)))), true), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))))), true), true), true, is_a_theorem(implies(implies(implies(V, U), implies(U, implies(V, U))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))), true), true, is_a_theorem(implies(implies(implies(Z, W), implies(W, implies(Z, W))), implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a)))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 25.57/4.37  = { by axiom 2 (condensed_detachment) R->L }
% 25.57/4.38    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(W2, V2), implies(V2, implies(W2, V2))), implies(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))))), true, ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(X6, Y6), implies(Z6, W6)), implies(V6, implies(implies(W6, X6), implies(Z6, X6)))), implies(implies(implies(implies(implies(S5, V2), implies(W2, V2)), implies(V2, implies(W2, V2))), implies(implies(W2, V2), implies(V2, implies(W2, V2)))), implies(implies(implies(implies(U4, T4), implies(S4, X5)), implies(Y5, implies(implies(X5, U4), implies(S4, U4)))), implies(implies(W2, V2), implies(V2, implies(W2, V2))))))), true, ifeq(is_a_theorem(implies(implies(implies(X6, Y6), implies(Z6, W6)), implies(V6, implies(implies(W6, X6), implies(Z6, X6))))), true, is_a_theorem(implies(implies(implies(implies(implies(S5, V2), implies(W2, V2)), implies(V2, implies(W2, V2))), implies(implies(W2, V2), implies(V2, implies(W2, V2)))), implies(implies(implies(implies(U4, T4), implies(S4, X5)), implies(Y5, implies(implies(X5, U4), implies(S4, U4)))), implies(implies(W2, V2), implies(V2, implies(W2, V2)))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(S5, V2), implies(W2, V2)), implies(V2, implies(W2, V2))), implies(implies(W2, V2), implies(V2, implies(W2, V2))))), true, is_a_theorem(implies(implies(implies(implies(U4, T4), implies(S4, X5)), implies(Y5, implies(implies(X5, U4), implies(S4, U4)))), implies(implies(W2, V2), implies(V2, implies(W2, V2))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(U4, T4), implies(S4, X5)), implies(Y5, implies(implies(X5, U4), implies(S4, U4))))), true, is_a_theorem(implies(implies(W2, V2), implies(V2, implies(W2, V2)))), true), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))))), true), true), true, is_a_theorem(implies(implies(implies(V, U), implies(U, implies(V, U))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))), true), true, is_a_theorem(implies(implies(implies(Z, W), implies(W, implies(Z, W))), implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a)))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 25.57/4.38  = { by axiom 1 (ifeq_axiom) R->L }
% 25.57/4.39    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(W2, V2), implies(V2, implies(W2, V2))), implies(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))))), true, ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(implies(X6, Y6), implies(Z6, W6)), implies(V6, implies(implies(W6, X6), implies(Z6, X6)))), implies(implies(implies(implies(implies(S5, V2), implies(W2, V2)), implies(V2, implies(W2, V2))), implies(implies(W2, V2), implies(V2, implies(W2, V2)))), implies(implies(implies(implies(U4, T4), implies(S4, X5)), implies(Y5, implies(implies(X5, U4), implies(S4, U4)))), implies(implies(W2, V2), implies(V2, implies(W2, V2))))))), true), true, ifeq(is_a_theorem(implies(implies(implies(X6, Y6), implies(Z6, W6)), implies(V6, implies(implies(W6, X6), implies(Z6, X6))))), true, is_a_theorem(implies(implies(implies(implies(implies(S5, V2), implies(W2, V2)), implies(V2, implies(W2, V2))), implies(implies(W2, V2), implies(V2, implies(W2, V2)))), implies(implies(implies(implies(U4, T4), implies(S4, X5)), implies(Y5, implies(implies(X5, U4), implies(S4, U4)))), implies(implies(W2, V2), implies(V2, implies(W2, V2)))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(S5, V2), implies(W2, V2)), implies(V2, implies(W2, V2))), implies(implies(W2, V2), implies(V2, implies(W2, V2))))), true, is_a_theorem(implies(implies(implies(implies(U4, T4), implies(S4, X5)), implies(Y5, implies(implies(X5, U4), implies(S4, U4)))), implies(implies(W2, V2), implies(V2, implies(W2, V2))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(U4, T4), implies(S4, X5)), implies(Y5, implies(implies(X5, U4), implies(S4, U4))))), true, is_a_theorem(implies(implies(W2, V2), implies(V2, implies(W2, V2)))), true), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))))), true), true), true, is_a_theorem(implies(implies(implies(V, U), implies(U, implies(V, U))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))), true), true, is_a_theorem(implies(implies(implies(Z, W), implies(W, implies(Z, W))), implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a)))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 25.57/4.40  = { by axiom 3 (ic_JLukasiewicz_4) R->L }
% 26.06/4.41    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(W2, V2), implies(V2, implies(W2, V2))), implies(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))))), true, ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(W2, V2), implies(V2, implies(W2, V2))), implies(V2, implies(S5, V2))), implies(implies(implies(implies(U4, T4), implies(S4, X5)), implies(Y5, implies(implies(X5, U4), implies(S4, U4)))), implies(implies(implies(S5, V2), implies(W2, V2)), implies(V2, implies(W2, V2)))))), true, is_a_theorem(implies(implies(implies(implies(X6, Y6), implies(Z6, W6)), implies(V6, implies(implies(W6, X6), implies(Z6, X6)))), implies(implies(implies(implies(implies(S5, V2), implies(W2, V2)), implies(V2, implies(W2, V2))), implies(implies(W2, V2), implies(V2, implies(W2, V2)))), implies(implies(implies(implies(U4, T4), implies(S4, X5)), implies(Y5, implies(implies(X5, U4), implies(S4, U4)))), implies(implies(W2, V2), implies(V2, implies(W2, V2))))))), true), true, ifeq(is_a_theorem(implies(implies(implies(X6, Y6), implies(Z6, W6)), implies(V6, implies(implies(W6, X6), implies(Z6, X6))))), true, is_a_theorem(implies(implies(implies(implies(implies(S5, V2), implies(W2, V2)), implies(V2, implies(W2, V2))), implies(implies(W2, V2), implies(V2, implies(W2, V2)))), implies(implies(implies(implies(U4, T4), implies(S4, X5)), implies(Y5, implies(implies(X5, U4), implies(S4, U4)))), implies(implies(W2, V2), implies(V2, implies(W2, V2)))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(S5, V2), implies(W2, V2)), implies(V2, implies(W2, V2))), implies(implies(W2, V2), implies(V2, implies(W2, V2))))), true, is_a_theorem(implies(implies(implies(implies(U4, T4), implies(S4, X5)), implies(Y5, implies(implies(X5, U4), implies(S4, U4)))), implies(implies(W2, V2), implies(V2, implies(W2, V2))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(U4, T4), implies(S4, X5)), implies(Y5, implies(implies(X5, U4), implies(S4, U4))))), true, is_a_theorem(implies(implies(W2, V2), implies(V2, implies(W2, V2)))), true), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))))), true), true), true, is_a_theorem(implies(implies(implies(V, U), implies(U, implies(V, U))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))), true), true, is_a_theorem(implies(implies(implies(Z, W), implies(W, implies(Z, W))), implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a)))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 26.06/4.41  = { by axiom 1 (ifeq_axiom) R->L }
% 26.06/4.42    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(W2, V2), implies(V2, implies(W2, V2))), implies(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))))), true, ifeq(ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(implies(implies(W2, V2), implies(V2, implies(W2, V2))), implies(V2, implies(S5, V2))), implies(implies(implies(implies(U4, T4), implies(S4, X5)), implies(Y5, implies(implies(X5, U4), implies(S4, U4)))), implies(implies(implies(S5, V2), implies(W2, V2)), implies(V2, implies(W2, V2)))))), true, is_a_theorem(implies(implies(implies(implies(X6, Y6), implies(Z6, W6)), implies(V6, implies(implies(W6, X6), implies(Z6, X6)))), implies(implies(implies(implies(implies(S5, V2), implies(W2, V2)), implies(V2, implies(W2, V2))), implies(implies(W2, V2), implies(V2, implies(W2, V2)))), implies(implies(implies(implies(U4, T4), implies(S4, X5)), implies(Y5, implies(implies(X5, U4), implies(S4, U4)))), implies(implies(W2, V2), implies(V2, implies(W2, V2))))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(X6, Y6), implies(Z6, W6)), implies(V6, implies(implies(W6, X6), implies(Z6, X6))))), true, is_a_theorem(implies(implies(implies(implies(implies(S5, V2), implies(W2, V2)), implies(V2, implies(W2, V2))), implies(implies(W2, V2), implies(V2, implies(W2, V2)))), implies(implies(implies(implies(U4, T4), implies(S4, X5)), implies(Y5, implies(implies(X5, U4), implies(S4, U4)))), implies(implies(W2, V2), implies(V2, implies(W2, V2)))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(S5, V2), implies(W2, V2)), implies(V2, implies(W2, V2))), implies(implies(W2, V2), implies(V2, implies(W2, V2))))), true, is_a_theorem(implies(implies(implies(implies(U4, T4), implies(S4, X5)), implies(Y5, implies(implies(X5, U4), implies(S4, U4)))), implies(implies(W2, V2), implies(V2, implies(W2, V2))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(U4, T4), implies(S4, X5)), implies(Y5, implies(implies(X5, U4), implies(S4, U4))))), true, is_a_theorem(implies(implies(W2, V2), implies(V2, implies(W2, V2)))), true), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))))), true), true), true, is_a_theorem(implies(implies(implies(V, U), implies(U, implies(V, U))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))), true), true, is_a_theorem(implies(implies(implies(Z, W), implies(W, implies(Z, W))), implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a)))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 26.06/4.43  = { by axiom 3 (ic_JLukasiewicz_4) R->L }
% 26.06/4.44    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(W2, V2), implies(V2, implies(W2, V2))), implies(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))))), true, ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(implies(W2, V2), implies(V2, implies(W2, V2))), implies(V2, implies(S5, V2))), implies(implies(implies(implies(U4, T4), implies(S4, X5)), implies(Y5, implies(implies(X5, U4), implies(S4, U4)))), implies(implies(implies(S5, V2), implies(W2, V2)), implies(V2, implies(W2, V2))))), implies(implies(implies(implies(X6, Y6), implies(Z6, W6)), implies(V6, implies(implies(W6, X6), implies(Z6, X6)))), implies(implies(implies(implies(implies(S5, V2), implies(W2, V2)), implies(V2, implies(W2, V2))), implies(implies(W2, V2), implies(V2, implies(W2, V2)))), implies(implies(implies(implies(U4, T4), implies(S4, X5)), implies(Y5, implies(implies(X5, U4), implies(S4, U4)))), implies(implies(W2, V2), implies(V2, implies(W2, V2)))))))), true, ifeq(is_a_theorem(implies(implies(implies(implies(W2, V2), implies(V2, implies(W2, V2))), implies(V2, implies(S5, V2))), implies(implies(implies(implies(U4, T4), implies(S4, X5)), implies(Y5, implies(implies(X5, U4), implies(S4, U4)))), implies(implies(implies(S5, V2), implies(W2, V2)), implies(V2, implies(W2, V2)))))), true, is_a_theorem(implies(implies(implies(implies(X6, Y6), implies(Z6, W6)), implies(V6, implies(implies(W6, X6), implies(Z6, X6)))), implies(implies(implies(implies(implies(S5, V2), implies(W2, V2)), implies(V2, implies(W2, V2))), implies(implies(W2, V2), implies(V2, implies(W2, V2)))), implies(implies(implies(implies(U4, T4), implies(S4, X5)), implies(Y5, implies(implies(X5, U4), implies(S4, U4)))), implies(implies(W2, V2), implies(V2, implies(W2, V2))))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(X6, Y6), implies(Z6, W6)), implies(V6, implies(implies(W6, X6), implies(Z6, X6))))), true, is_a_theorem(implies(implies(implies(implies(implies(S5, V2), implies(W2, V2)), implies(V2, implies(W2, V2))), implies(implies(W2, V2), implies(V2, implies(W2, V2)))), implies(implies(implies(implies(U4, T4), implies(S4, X5)), implies(Y5, implies(implies(X5, U4), implies(S4, U4)))), implies(implies(W2, V2), implies(V2, implies(W2, V2)))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(S5, V2), implies(W2, V2)), implies(V2, implies(W2, V2))), implies(implies(W2, V2), implies(V2, implies(W2, V2))))), true, is_a_theorem(implies(implies(implies(implies(U4, T4), implies(S4, X5)), implies(Y5, implies(implies(X5, U4), implies(S4, U4)))), implies(implies(W2, V2), implies(V2, implies(W2, V2))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(U4, T4), implies(S4, X5)), implies(Y5, implies(implies(X5, U4), implies(S4, U4))))), true, is_a_theorem(implies(implies(W2, V2), implies(V2, implies(W2, V2)))), true), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))))), true), true), true, is_a_theorem(implies(implies(implies(V, U), implies(U, implies(V, U))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))), true), true, is_a_theorem(implies(implies(implies(Z, W), implies(W, implies(Z, W))), implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a)))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 26.06/4.44  = { by axiom 2 (condensed_detachment) }
% 26.06/4.45    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(W2, V2), implies(V2, implies(W2, V2))), implies(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))))), true, ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(implies(X6, Y6), implies(Z6, W6)), implies(V6, implies(implies(W6, X6), implies(Z6, X6))))), true, is_a_theorem(implies(implies(implies(implies(implies(S5, V2), implies(W2, V2)), implies(V2, implies(W2, V2))), implies(implies(W2, V2), implies(V2, implies(W2, V2)))), implies(implies(implies(implies(U4, T4), implies(S4, X5)), implies(Y5, implies(implies(X5, U4), implies(S4, U4)))), implies(implies(W2, V2), implies(V2, implies(W2, V2)))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(S5, V2), implies(W2, V2)), implies(V2, implies(W2, V2))), implies(implies(W2, V2), implies(V2, implies(W2, V2))))), true, is_a_theorem(implies(implies(implies(implies(U4, T4), implies(S4, X5)), implies(Y5, implies(implies(X5, U4), implies(S4, U4)))), implies(implies(W2, V2), implies(V2, implies(W2, V2))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(U4, T4), implies(S4, X5)), implies(Y5, implies(implies(X5, U4), implies(S4, U4))))), true, is_a_theorem(implies(implies(W2, V2), implies(V2, implies(W2, V2)))), true), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))))), true), true), true, is_a_theorem(implies(implies(implies(V, U), implies(U, implies(V, U))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))), true), true, is_a_theorem(implies(implies(implies(Z, W), implies(W, implies(Z, W))), implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a)))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 26.06/4.45  = { by axiom 1 (ifeq_axiom) }
% 26.06/4.46    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(W2, V2), implies(V2, implies(W2, V2))), implies(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))))), true, ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(X6, Y6), implies(Z6, W6)), implies(V6, implies(implies(W6, X6), implies(Z6, X6))))), true, is_a_theorem(implies(implies(implies(implies(implies(S5, V2), implies(W2, V2)), implies(V2, implies(W2, V2))), implies(implies(W2, V2), implies(V2, implies(W2, V2)))), implies(implies(implies(implies(U4, T4), implies(S4, X5)), implies(Y5, implies(implies(X5, U4), implies(S4, U4)))), implies(implies(W2, V2), implies(V2, implies(W2, V2)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(S5, V2), implies(W2, V2)), implies(V2, implies(W2, V2))), implies(implies(W2, V2), implies(V2, implies(W2, V2))))), true, is_a_theorem(implies(implies(implies(implies(U4, T4), implies(S4, X5)), implies(Y5, implies(implies(X5, U4), implies(S4, U4)))), implies(implies(W2, V2), implies(V2, implies(W2, V2))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(U4, T4), implies(S4, X5)), implies(Y5, implies(implies(X5, U4), implies(S4, U4))))), true, is_a_theorem(implies(implies(W2, V2), implies(V2, implies(W2, V2)))), true), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))))), true), true), true, is_a_theorem(implies(implies(implies(V, U), implies(U, implies(V, U))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))), true), true, is_a_theorem(implies(implies(implies(Z, W), implies(W, implies(Z, W))), implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a)))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 26.06/4.46  = { by axiom 3 (ic_JLukasiewicz_4) }
% 26.06/4.47    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(W2, V2), implies(V2, implies(W2, V2))), implies(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))))), true, ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(implies(implies(S5, V2), implies(W2, V2)), implies(V2, implies(W2, V2))), implies(implies(W2, V2), implies(V2, implies(W2, V2)))), implies(implies(implies(implies(U4, T4), implies(S4, X5)), implies(Y5, implies(implies(X5, U4), implies(S4, U4)))), implies(implies(W2, V2), implies(V2, implies(W2, V2)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(S5, V2), implies(W2, V2)), implies(V2, implies(W2, V2))), implies(implies(W2, V2), implies(V2, implies(W2, V2))))), true, is_a_theorem(implies(implies(implies(implies(U4, T4), implies(S4, X5)), implies(Y5, implies(implies(X5, U4), implies(S4, U4)))), implies(implies(W2, V2), implies(V2, implies(W2, V2))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(U4, T4), implies(S4, X5)), implies(Y5, implies(implies(X5, U4), implies(S4, U4))))), true, is_a_theorem(implies(implies(W2, V2), implies(V2, implies(W2, V2)))), true), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))))), true), true), true, is_a_theorem(implies(implies(implies(V, U), implies(U, implies(V, U))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))), true), true, is_a_theorem(implies(implies(implies(Z, W), implies(W, implies(Z, W))), implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a)))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 26.06/4.48  = { by axiom 1 (ifeq_axiom) }
% 26.06/4.48    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(W2, V2), implies(V2, implies(W2, V2))), implies(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))))), true, ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(implies(S5, V2), implies(W2, V2)), implies(V2, implies(W2, V2))), implies(implies(W2, V2), implies(V2, implies(W2, V2)))), implies(implies(implies(implies(U4, T4), implies(S4, X5)), implies(Y5, implies(implies(X5, U4), implies(S4, U4)))), implies(implies(W2, V2), implies(V2, implies(W2, V2)))))), true, ifeq(is_a_theorem(implies(implies(implies(implies(S5, V2), implies(W2, V2)), implies(V2, implies(W2, V2))), implies(implies(W2, V2), implies(V2, implies(W2, V2))))), true, is_a_theorem(implies(implies(implies(implies(U4, T4), implies(S4, X5)), implies(Y5, implies(implies(X5, U4), implies(S4, U4)))), implies(implies(W2, V2), implies(V2, implies(W2, V2))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(U4, T4), implies(S4, X5)), implies(Y5, implies(implies(X5, U4), implies(S4, U4))))), true, is_a_theorem(implies(implies(W2, V2), implies(V2, implies(W2, V2)))), true), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))))), true), true), true, is_a_theorem(implies(implies(implies(V, U), implies(U, implies(V, U))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))), true), true, is_a_theorem(implies(implies(implies(Z, W), implies(W, implies(Z, W))), implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a)))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 26.06/4.48  = { by axiom 2 (condensed_detachment) }
% 26.06/4.49    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(W2, V2), implies(V2, implies(W2, V2))), implies(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))))), true, ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(implies(U4, T4), implies(S4, X5)), implies(Y5, implies(implies(X5, U4), implies(S4, U4))))), true, is_a_theorem(implies(implies(W2, V2), implies(V2, implies(W2, V2)))), true), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))))), true), true), true, is_a_theorem(implies(implies(implies(V, U), implies(U, implies(V, U))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))), true), true, is_a_theorem(implies(implies(implies(Z, W), implies(W, implies(Z, W))), implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a)))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 26.06/4.49  = { by axiom 1 (ifeq_axiom) }
% 26.06/4.49    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(W2, V2), implies(V2, implies(W2, V2))), implies(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))))), true, ifeq(ifeq(is_a_theorem(implies(implies(implies(U4, T4), implies(S4, X5)), implies(Y5, implies(implies(X5, U4), implies(S4, U4))))), true, is_a_theorem(implies(implies(W2, V2), implies(V2, implies(W2, V2)))), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))))), true), true), true, is_a_theorem(implies(implies(implies(V, U), implies(U, implies(V, U))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))), true), true, is_a_theorem(implies(implies(implies(Z, W), implies(W, implies(Z, W))), implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a)))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 26.06/4.50  = { by axiom 3 (ic_JLukasiewicz_4) }
% 26.06/4.50    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(W2, V2), implies(V2, implies(W2, V2))), implies(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))))), true, ifeq(ifeq(true, true, is_a_theorem(implies(implies(W2, V2), implies(V2, implies(W2, V2)))), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))))), true), true), true, is_a_theorem(implies(implies(implies(V, U), implies(U, implies(V, U))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))), true), true, is_a_theorem(implies(implies(implies(Z, W), implies(W, implies(Z, W))), implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a)))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 26.06/4.50  = { by axiom 1 (ifeq_axiom) }
% 26.60/4.50    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(W2, V2), implies(V2, implies(W2, V2))), implies(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))))), true, ifeq(is_a_theorem(implies(implies(W2, V2), implies(V2, implies(W2, V2)))), true, is_a_theorem(implies(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(implies(a, b), a), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))))), true), true), true, is_a_theorem(implies(implies(implies(V, U), implies(U, implies(V, U))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))), true), true, is_a_theorem(implies(implies(implies(Z, W), implies(W, implies(Z, W))), implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a)))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 26.60/4.51  = { by axiom 2 (condensed_detachment) }
% 26.60/4.51    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(V, U), implies(U, implies(V, U))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))), true), true, is_a_theorem(implies(implies(implies(Z, W), implies(W, implies(Z, W))), implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a)))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 26.60/4.51  = { by axiom 1 (ifeq_axiom) }
% 26.60/4.51    ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(V, U), implies(U, implies(V, U))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))))), true, is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))), true), true, is_a_theorem(implies(implies(implies(Z, W), implies(W, implies(Z, W))), implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a)))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 26.60/4.52  = { by axiom 1 (ifeq_axiom) R->L }
% 26.60/4.52    ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(V, U), implies(U, implies(V, U))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))))), true, ifeq(true, true, is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))), true), true), true, is_a_theorem(implies(implies(implies(Z, W), implies(W, implies(Z, W))), implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a)))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 26.60/4.52  = { by axiom 2 (condensed_detachment) R->L }
% 26.60/4.52    ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(V, U), implies(U, implies(V, U))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))))), true, ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(U6, T6), implies(S6, X7)), implies(Y7, implies(implies(X7, U6), implies(S6, U6)))), implies(implies(V, U), implies(U, implies(V, U))))), true, ifeq(is_a_theorem(implies(implies(implies(U6, T6), implies(S6, X7)), implies(Y7, implies(implies(X7, U6), implies(S6, U6))))), true, is_a_theorem(implies(implies(V, U), implies(U, implies(V, U)))), true), true), true, is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))), true), true), true, is_a_theorem(implies(implies(implies(Z, W), implies(W, implies(Z, W))), implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a)))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 26.60/4.53  = { by axiom 1 (ifeq_axiom) R->L }
% 26.60/4.53    ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(V, U), implies(U, implies(V, U))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))))), true, ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(implies(U6, T6), implies(S6, X7)), implies(Y7, implies(implies(X7, U6), implies(S6, U6)))), implies(implies(V, U), implies(U, implies(V, U))))), true), true, ifeq(is_a_theorem(implies(implies(implies(U6, T6), implies(S6, X7)), implies(Y7, implies(implies(X7, U6), implies(S6, U6))))), true, is_a_theorem(implies(implies(V, U), implies(U, implies(V, U)))), true), true), true, is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))), true), true), true, is_a_theorem(implies(implies(implies(Z, W), implies(W, implies(Z, W))), implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a)))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 26.60/4.53  = { by axiom 2 (condensed_detachment) R->L }
% 26.60/4.54    ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(V, U), implies(U, implies(V, U))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))))), true, ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(Z7, W7), implies(V7, U7)), implies(T7, implies(implies(U7, Z7), implies(V7, Z7)))), implies(implies(implies(implies(S7, U), implies(V, U)), implies(U, implies(V, U))), implies(implies(V, U), implies(U, implies(V, U)))))), true, ifeq(is_a_theorem(implies(implies(implies(Z7, W7), implies(V7, U7)), implies(T7, implies(implies(U7, Z7), implies(V7, Z7))))), true, is_a_theorem(implies(implies(implies(implies(S7, U), implies(V, U)), implies(U, implies(V, U))), implies(implies(V, U), implies(U, implies(V, U))))), true), true), true, is_a_theorem(implies(implies(implies(implies(U6, T6), implies(S6, X7)), implies(Y7, implies(implies(X7, U6), implies(S6, U6)))), implies(implies(V, U), implies(U, implies(V, U))))), true), true, ifeq(is_a_theorem(implies(implies(implies(U6, T6), implies(S6, X7)), implies(Y7, implies(implies(X7, U6), implies(S6, U6))))), true, is_a_theorem(implies(implies(V, U), implies(U, implies(V, U)))), true), true), true, is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))), true), true), true, is_a_theorem(implies(implies(implies(Z, W), implies(W, implies(Z, W))), implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a)))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 26.60/4.54  = { by axiom 1 (ifeq_axiom) R->L }
% 26.60/4.54    ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(V, U), implies(U, implies(V, U))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))))), true, ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(implies(Z7, W7), implies(V7, U7)), implies(T7, implies(implies(U7, Z7), implies(V7, Z7)))), implies(implies(implies(implies(S7, U), implies(V, U)), implies(U, implies(V, U))), implies(implies(V, U), implies(U, implies(V, U)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(Z7, W7), implies(V7, U7)), implies(T7, implies(implies(U7, Z7), implies(V7, Z7))))), true, is_a_theorem(implies(implies(implies(implies(S7, U), implies(V, U)), implies(U, implies(V, U))), implies(implies(V, U), implies(U, implies(V, U))))), true), true), true, is_a_theorem(implies(implies(implies(implies(U6, T6), implies(S6, X7)), implies(Y7, implies(implies(X7, U6), implies(S6, U6)))), implies(implies(V, U), implies(U, implies(V, U))))), true), true, ifeq(is_a_theorem(implies(implies(implies(U6, T6), implies(S6, X7)), implies(Y7, implies(implies(X7, U6), implies(S6, U6))))), true, is_a_theorem(implies(implies(V, U), implies(U, implies(V, U)))), true), true), true, is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))), true), true), true, is_a_theorem(implies(implies(implies(Z, W), implies(W, implies(Z, W))), implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a)))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 26.60/4.55  = { by axiom 3 (ic_JLukasiewicz_4) R->L }
% 26.60/4.55    ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(V, U), implies(U, implies(V, U))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))))), true, ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(U, implies(V, U)), implies(V, S7)), implies(implies(V, U), implies(implies(S7, U), implies(V, U))))), true, is_a_theorem(implies(implies(implies(implies(Z7, W7), implies(V7, U7)), implies(T7, implies(implies(U7, Z7), implies(V7, Z7)))), implies(implies(implies(implies(S7, U), implies(V, U)), implies(U, implies(V, U))), implies(implies(V, U), implies(U, implies(V, U)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(Z7, W7), implies(V7, U7)), implies(T7, implies(implies(U7, Z7), implies(V7, Z7))))), true, is_a_theorem(implies(implies(implies(implies(S7, U), implies(V, U)), implies(U, implies(V, U))), implies(implies(V, U), implies(U, implies(V, U))))), true), true), true, is_a_theorem(implies(implies(implies(implies(U6, T6), implies(S6, X7)), implies(Y7, implies(implies(X7, U6), implies(S6, U6)))), implies(implies(V, U), implies(U, implies(V, U))))), true), true, ifeq(is_a_theorem(implies(implies(implies(U6, T6), implies(S6, X7)), implies(Y7, implies(implies(X7, U6), implies(S6, U6))))), true, is_a_theorem(implies(implies(V, U), implies(U, implies(V, U)))), true), true), true, is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))), true), true), true, is_a_theorem(implies(implies(implies(Z, W), implies(W, implies(Z, W))), implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a)))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 26.60/4.55  = { by axiom 1 (ifeq_axiom) R->L }
% 26.60/4.56    ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(V, U), implies(U, implies(V, U))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))))), true, ifeq(ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(implies(U, implies(V, U)), implies(V, S7)), implies(implies(V, U), implies(implies(S7, U), implies(V, U))))), true, is_a_theorem(implies(implies(implies(implies(Z7, W7), implies(V7, U7)), implies(T7, implies(implies(U7, Z7), implies(V7, Z7)))), implies(implies(implies(implies(S7, U), implies(V, U)), implies(U, implies(V, U))), implies(implies(V, U), implies(U, implies(V, U)))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(Z7, W7), implies(V7, U7)), implies(T7, implies(implies(U7, Z7), implies(V7, Z7))))), true, is_a_theorem(implies(implies(implies(implies(S7, U), implies(V, U)), implies(U, implies(V, U))), implies(implies(V, U), implies(U, implies(V, U))))), true), true), true, is_a_theorem(implies(implies(implies(implies(U6, T6), implies(S6, X7)), implies(Y7, implies(implies(X7, U6), implies(S6, U6)))), implies(implies(V, U), implies(U, implies(V, U))))), true), true, ifeq(is_a_theorem(implies(implies(implies(U6, T6), implies(S6, X7)), implies(Y7, implies(implies(X7, U6), implies(S6, U6))))), true, is_a_theorem(implies(implies(V, U), implies(U, implies(V, U)))), true), true), true, is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))), true), true), true, is_a_theorem(implies(implies(implies(Z, W), implies(W, implies(Z, W))), implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a)))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 26.60/4.56  = { by axiom 3 (ic_JLukasiewicz_4) R->L }
% 26.60/4.56    ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(V, U), implies(U, implies(V, U))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))))), true, ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(U, implies(V, U)), implies(V, S7)), implies(implies(V, U), implies(implies(S7, U), implies(V, U)))), implies(implies(implies(implies(Z7, W7), implies(V7, U7)), implies(T7, implies(implies(U7, Z7), implies(V7, Z7)))), implies(implies(implies(implies(S7, U), implies(V, U)), implies(U, implies(V, U))), implies(implies(V, U), implies(U, implies(V, U))))))), true, ifeq(is_a_theorem(implies(implies(implies(U, implies(V, U)), implies(V, S7)), implies(implies(V, U), implies(implies(S7, U), implies(V, U))))), true, is_a_theorem(implies(implies(implies(implies(Z7, W7), implies(V7, U7)), implies(T7, implies(implies(U7, Z7), implies(V7, Z7)))), implies(implies(implies(implies(S7, U), implies(V, U)), implies(U, implies(V, U))), implies(implies(V, U), implies(U, implies(V, U)))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(Z7, W7), implies(V7, U7)), implies(T7, implies(implies(U7, Z7), implies(V7, Z7))))), true, is_a_theorem(implies(implies(implies(implies(S7, U), implies(V, U)), implies(U, implies(V, U))), implies(implies(V, U), implies(U, implies(V, U))))), true), true), true, is_a_theorem(implies(implies(implies(implies(U6, T6), implies(S6, X7)), implies(Y7, implies(implies(X7, U6), implies(S6, U6)))), implies(implies(V, U), implies(U, implies(V, U))))), true), true, ifeq(is_a_theorem(implies(implies(implies(U6, T6), implies(S6, X7)), implies(Y7, implies(implies(X7, U6), implies(S6, U6))))), true, is_a_theorem(implies(implies(V, U), implies(U, implies(V, U)))), true), true), true, is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))), true), true), true, is_a_theorem(implies(implies(implies(Z, W), implies(W, implies(Z, W))), implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a)))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 26.60/4.57  = { by axiom 2 (condensed_detachment) }
% 26.60/4.57    ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(V, U), implies(U, implies(V, U))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))))), true, ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(implies(Z7, W7), implies(V7, U7)), implies(T7, implies(implies(U7, Z7), implies(V7, Z7))))), true, is_a_theorem(implies(implies(implies(implies(S7, U), implies(V, U)), implies(U, implies(V, U))), implies(implies(V, U), implies(U, implies(V, U))))), true), true), true, is_a_theorem(implies(implies(implies(implies(U6, T6), implies(S6, X7)), implies(Y7, implies(implies(X7, U6), implies(S6, U6)))), implies(implies(V, U), implies(U, implies(V, U))))), true), true, ifeq(is_a_theorem(implies(implies(implies(U6, T6), implies(S6, X7)), implies(Y7, implies(implies(X7, U6), implies(S6, U6))))), true, is_a_theorem(implies(implies(V, U), implies(U, implies(V, U)))), true), true), true, is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))), true), true), true, is_a_theorem(implies(implies(implies(Z, W), implies(W, implies(Z, W))), implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a)))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 26.60/4.57  = { by axiom 1 (ifeq_axiom) }
% 26.60/4.58    ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(V, U), implies(U, implies(V, U))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))))), true, ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(Z7, W7), implies(V7, U7)), implies(T7, implies(implies(U7, Z7), implies(V7, Z7))))), true, is_a_theorem(implies(implies(implies(implies(S7, U), implies(V, U)), implies(U, implies(V, U))), implies(implies(V, U), implies(U, implies(V, U))))), true), true, is_a_theorem(implies(implies(implies(implies(U6, T6), implies(S6, X7)), implies(Y7, implies(implies(X7, U6), implies(S6, U6)))), implies(implies(V, U), implies(U, implies(V, U))))), true), true, ifeq(is_a_theorem(implies(implies(implies(U6, T6), implies(S6, X7)), implies(Y7, implies(implies(X7, U6), implies(S6, U6))))), true, is_a_theorem(implies(implies(V, U), implies(U, implies(V, U)))), true), true), true, is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))), true), true), true, is_a_theorem(implies(implies(implies(Z, W), implies(W, implies(Z, W))), implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a)))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 26.60/4.58  = { by axiom 3 (ic_JLukasiewicz_4) }
% 26.60/4.58    ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(V, U), implies(U, implies(V, U))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))))), true, ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(implies(S7, U), implies(V, U)), implies(U, implies(V, U))), implies(implies(V, U), implies(U, implies(V, U))))), true), true, is_a_theorem(implies(implies(implies(implies(U6, T6), implies(S6, X7)), implies(Y7, implies(implies(X7, U6), implies(S6, U6)))), implies(implies(V, U), implies(U, implies(V, U))))), true), true, ifeq(is_a_theorem(implies(implies(implies(U6, T6), implies(S6, X7)), implies(Y7, implies(implies(X7, U6), implies(S6, U6))))), true, is_a_theorem(implies(implies(V, U), implies(U, implies(V, U)))), true), true), true, is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))), true), true), true, is_a_theorem(implies(implies(implies(Z, W), implies(W, implies(Z, W))), implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a)))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 26.60/4.59  = { by axiom 1 (ifeq_axiom) }
% 26.60/4.59    ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(V, U), implies(U, implies(V, U))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))))), true, ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(S7, U), implies(V, U)), implies(U, implies(V, U))), implies(implies(V, U), implies(U, implies(V, U))))), true, is_a_theorem(implies(implies(implies(implies(U6, T6), implies(S6, X7)), implies(Y7, implies(implies(X7, U6), implies(S6, U6)))), implies(implies(V, U), implies(U, implies(V, U))))), true), true, ifeq(is_a_theorem(implies(implies(implies(U6, T6), implies(S6, X7)), implies(Y7, implies(implies(X7, U6), implies(S6, U6))))), true, is_a_theorem(implies(implies(V, U), implies(U, implies(V, U)))), true), true), true, is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))), true), true), true, is_a_theorem(implies(implies(implies(Z, W), implies(W, implies(Z, W))), implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a)))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 26.60/4.59  = { by axiom 1 (ifeq_axiom) R->L }
% 26.60/4.60    ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(V, U), implies(U, implies(V, U))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))))), true, ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(implies(implies(S7, U), implies(V, U)), implies(U, implies(V, U))), implies(implies(V, U), implies(U, implies(V, U))))), true, is_a_theorem(implies(implies(implies(implies(U6, T6), implies(S6, X7)), implies(Y7, implies(implies(X7, U6), implies(S6, U6)))), implies(implies(V, U), implies(U, implies(V, U))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(U6, T6), implies(S6, X7)), implies(Y7, implies(implies(X7, U6), implies(S6, U6))))), true, is_a_theorem(implies(implies(V, U), implies(U, implies(V, U)))), true), true), true, is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))), true), true), true, is_a_theorem(implies(implies(implies(Z, W), implies(W, implies(Z, W))), implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a)))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 26.60/4.60  = { by axiom 2 (condensed_detachment) R->L }
% 27.10/4.60    ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(V, U), implies(U, implies(V, U))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))))), true, ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(X8, Y8), implies(Z8, W8)), implies(V8, implies(implies(W8, X8), implies(Z8, X8)))), implies(implies(implies(implies(implies(S7, U), implies(V, U)), implies(U, implies(V, U))), implies(implies(V, U), implies(U, implies(V, U)))), implies(implies(implies(implies(U6, T6), implies(S6, X7)), implies(Y7, implies(implies(X7, U6), implies(S6, U6)))), implies(implies(V, U), implies(U, implies(V, U))))))), true, ifeq(is_a_theorem(implies(implies(implies(X8, Y8), implies(Z8, W8)), implies(V8, implies(implies(W8, X8), implies(Z8, X8))))), true, is_a_theorem(implies(implies(implies(implies(implies(S7, U), implies(V, U)), implies(U, implies(V, U))), implies(implies(V, U), implies(U, implies(V, U)))), implies(implies(implies(implies(U6, T6), implies(S6, X7)), implies(Y7, implies(implies(X7, U6), implies(S6, U6)))), implies(implies(V, U), implies(U, implies(V, U)))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(S7, U), implies(V, U)), implies(U, implies(V, U))), implies(implies(V, U), implies(U, implies(V, U))))), true, is_a_theorem(implies(implies(implies(implies(U6, T6), implies(S6, X7)), implies(Y7, implies(implies(X7, U6), implies(S6, U6)))), implies(implies(V, U), implies(U, implies(V, U))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(U6, T6), implies(S6, X7)), implies(Y7, implies(implies(X7, U6), implies(S6, U6))))), true, is_a_theorem(implies(implies(V, U), implies(U, implies(V, U)))), true), true), true, is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))), true), true), true, is_a_theorem(implies(implies(implies(Z, W), implies(W, implies(Z, W))), implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a)))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 27.10/4.61  = { by axiom 1 (ifeq_axiom) R->L }
% 27.10/4.61    ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(V, U), implies(U, implies(V, U))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))))), true, ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(implies(X8, Y8), implies(Z8, W8)), implies(V8, implies(implies(W8, X8), implies(Z8, X8)))), implies(implies(implies(implies(implies(S7, U), implies(V, U)), implies(U, implies(V, U))), implies(implies(V, U), implies(U, implies(V, U)))), implies(implies(implies(implies(U6, T6), implies(S6, X7)), implies(Y7, implies(implies(X7, U6), implies(S6, U6)))), implies(implies(V, U), implies(U, implies(V, U))))))), true), true, ifeq(is_a_theorem(implies(implies(implies(X8, Y8), implies(Z8, W8)), implies(V8, implies(implies(W8, X8), implies(Z8, X8))))), true, is_a_theorem(implies(implies(implies(implies(implies(S7, U), implies(V, U)), implies(U, implies(V, U))), implies(implies(V, U), implies(U, implies(V, U)))), implies(implies(implies(implies(U6, T6), implies(S6, X7)), implies(Y7, implies(implies(X7, U6), implies(S6, U6)))), implies(implies(V, U), implies(U, implies(V, U)))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(S7, U), implies(V, U)), implies(U, implies(V, U))), implies(implies(V, U), implies(U, implies(V, U))))), true, is_a_theorem(implies(implies(implies(implies(U6, T6), implies(S6, X7)), implies(Y7, implies(implies(X7, U6), implies(S6, U6)))), implies(implies(V, U), implies(U, implies(V, U))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(U6, T6), implies(S6, X7)), implies(Y7, implies(implies(X7, U6), implies(S6, U6))))), true, is_a_theorem(implies(implies(V, U), implies(U, implies(V, U)))), true), true), true, is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))), true), true), true, is_a_theorem(implies(implies(implies(Z, W), implies(W, implies(Z, W))), implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a)))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 27.10/4.62  = { by axiom 3 (ic_JLukasiewicz_4) R->L }
% 27.10/4.63    ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(V, U), implies(U, implies(V, U))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))))), true, ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(V, U), implies(U, implies(V, U))), implies(U, implies(S7, U))), implies(implies(implies(implies(U6, T6), implies(S6, X7)), implies(Y7, implies(implies(X7, U6), implies(S6, U6)))), implies(implies(implies(S7, U), implies(V, U)), implies(U, implies(V, U)))))), true, is_a_theorem(implies(implies(implies(implies(X8, Y8), implies(Z8, W8)), implies(V8, implies(implies(W8, X8), implies(Z8, X8)))), implies(implies(implies(implies(implies(S7, U), implies(V, U)), implies(U, implies(V, U))), implies(implies(V, U), implies(U, implies(V, U)))), implies(implies(implies(implies(U6, T6), implies(S6, X7)), implies(Y7, implies(implies(X7, U6), implies(S6, U6)))), implies(implies(V, U), implies(U, implies(V, U))))))), true), true, ifeq(is_a_theorem(implies(implies(implies(X8, Y8), implies(Z8, W8)), implies(V8, implies(implies(W8, X8), implies(Z8, X8))))), true, is_a_theorem(implies(implies(implies(implies(implies(S7, U), implies(V, U)), implies(U, implies(V, U))), implies(implies(V, U), implies(U, implies(V, U)))), implies(implies(implies(implies(U6, T6), implies(S6, X7)), implies(Y7, implies(implies(X7, U6), implies(S6, U6)))), implies(implies(V, U), implies(U, implies(V, U)))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(S7, U), implies(V, U)), implies(U, implies(V, U))), implies(implies(V, U), implies(U, implies(V, U))))), true, is_a_theorem(implies(implies(implies(implies(U6, T6), implies(S6, X7)), implies(Y7, implies(implies(X7, U6), implies(S6, U6)))), implies(implies(V, U), implies(U, implies(V, U))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(U6, T6), implies(S6, X7)), implies(Y7, implies(implies(X7, U6), implies(S6, U6))))), true, is_a_theorem(implies(implies(V, U), implies(U, implies(V, U)))), true), true), true, is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))), true), true), true, is_a_theorem(implies(implies(implies(Z, W), implies(W, implies(Z, W))), implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a)))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 27.10/4.63  = { by axiom 1 (ifeq_axiom) R->L }
% 27.10/4.63    ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(V, U), implies(U, implies(V, U))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))))), true, ifeq(ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(implies(implies(V, U), implies(U, implies(V, U))), implies(U, implies(S7, U))), implies(implies(implies(implies(U6, T6), implies(S6, X7)), implies(Y7, implies(implies(X7, U6), implies(S6, U6)))), implies(implies(implies(S7, U), implies(V, U)), implies(U, implies(V, U)))))), true, is_a_theorem(implies(implies(implies(implies(X8, Y8), implies(Z8, W8)), implies(V8, implies(implies(W8, X8), implies(Z8, X8)))), implies(implies(implies(implies(implies(S7, U), implies(V, U)), implies(U, implies(V, U))), implies(implies(V, U), implies(U, implies(V, U)))), implies(implies(implies(implies(U6, T6), implies(S6, X7)), implies(Y7, implies(implies(X7, U6), implies(S6, U6)))), implies(implies(V, U), implies(U, implies(V, U))))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(X8, Y8), implies(Z8, W8)), implies(V8, implies(implies(W8, X8), implies(Z8, X8))))), true, is_a_theorem(implies(implies(implies(implies(implies(S7, U), implies(V, U)), implies(U, implies(V, U))), implies(implies(V, U), implies(U, implies(V, U)))), implies(implies(implies(implies(U6, T6), implies(S6, X7)), implies(Y7, implies(implies(X7, U6), implies(S6, U6)))), implies(implies(V, U), implies(U, implies(V, U)))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(S7, U), implies(V, U)), implies(U, implies(V, U))), implies(implies(V, U), implies(U, implies(V, U))))), true, is_a_theorem(implies(implies(implies(implies(U6, T6), implies(S6, X7)), implies(Y7, implies(implies(X7, U6), implies(S6, U6)))), implies(implies(V, U), implies(U, implies(V, U))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(U6, T6), implies(S6, X7)), implies(Y7, implies(implies(X7, U6), implies(S6, U6))))), true, is_a_theorem(implies(implies(V, U), implies(U, implies(V, U)))), true), true), true, is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))), true), true), true, is_a_theorem(implies(implies(implies(Z, W), implies(W, implies(Z, W))), implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a)))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 27.10/4.64  = { by axiom 3 (ic_JLukasiewicz_4) R->L }
% 27.10/4.64    ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(V, U), implies(U, implies(V, U))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))))), true, ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(implies(V, U), implies(U, implies(V, U))), implies(U, implies(S7, U))), implies(implies(implies(implies(U6, T6), implies(S6, X7)), implies(Y7, implies(implies(X7, U6), implies(S6, U6)))), implies(implies(implies(S7, U), implies(V, U)), implies(U, implies(V, U))))), implies(implies(implies(implies(X8, Y8), implies(Z8, W8)), implies(V8, implies(implies(W8, X8), implies(Z8, X8)))), implies(implies(implies(implies(implies(S7, U), implies(V, U)), implies(U, implies(V, U))), implies(implies(V, U), implies(U, implies(V, U)))), implies(implies(implies(implies(U6, T6), implies(S6, X7)), implies(Y7, implies(implies(X7, U6), implies(S6, U6)))), implies(implies(V, U), implies(U, implies(V, U)))))))), true, ifeq(is_a_theorem(implies(implies(implies(implies(V, U), implies(U, implies(V, U))), implies(U, implies(S7, U))), implies(implies(implies(implies(U6, T6), implies(S6, X7)), implies(Y7, implies(implies(X7, U6), implies(S6, U6)))), implies(implies(implies(S7, U), implies(V, U)), implies(U, implies(V, U)))))), true, is_a_theorem(implies(implies(implies(implies(X8, Y8), implies(Z8, W8)), implies(V8, implies(implies(W8, X8), implies(Z8, X8)))), implies(implies(implies(implies(implies(S7, U), implies(V, U)), implies(U, implies(V, U))), implies(implies(V, U), implies(U, implies(V, U)))), implies(implies(implies(implies(U6, T6), implies(S6, X7)), implies(Y7, implies(implies(X7, U6), implies(S6, U6)))), implies(implies(V, U), implies(U, implies(V, U))))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(X8, Y8), implies(Z8, W8)), implies(V8, implies(implies(W8, X8), implies(Z8, X8))))), true, is_a_theorem(implies(implies(implies(implies(implies(S7, U), implies(V, U)), implies(U, implies(V, U))), implies(implies(V, U), implies(U, implies(V, U)))), implies(implies(implies(implies(U6, T6), implies(S6, X7)), implies(Y7, implies(implies(X7, U6), implies(S6, U6)))), implies(implies(V, U), implies(U, implies(V, U)))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(S7, U), implies(V, U)), implies(U, implies(V, U))), implies(implies(V, U), implies(U, implies(V, U))))), true, is_a_theorem(implies(implies(implies(implies(U6, T6), implies(S6, X7)), implies(Y7, implies(implies(X7, U6), implies(S6, U6)))), implies(implies(V, U), implies(U, implies(V, U))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(U6, T6), implies(S6, X7)), implies(Y7, implies(implies(X7, U6), implies(S6, U6))))), true, is_a_theorem(implies(implies(V, U), implies(U, implies(V, U)))), true), true), true, is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))), true), true), true, is_a_theorem(implies(implies(implies(Z, W), implies(W, implies(Z, W))), implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a)))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 27.10/4.65  = { by axiom 2 (condensed_detachment) }
% 27.10/4.65    ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(V, U), implies(U, implies(V, U))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))))), true, ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(implies(X8, Y8), implies(Z8, W8)), implies(V8, implies(implies(W8, X8), implies(Z8, X8))))), true, is_a_theorem(implies(implies(implies(implies(implies(S7, U), implies(V, U)), implies(U, implies(V, U))), implies(implies(V, U), implies(U, implies(V, U)))), implies(implies(implies(implies(U6, T6), implies(S6, X7)), implies(Y7, implies(implies(X7, U6), implies(S6, U6)))), implies(implies(V, U), implies(U, implies(V, U)))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(S7, U), implies(V, U)), implies(U, implies(V, U))), implies(implies(V, U), implies(U, implies(V, U))))), true, is_a_theorem(implies(implies(implies(implies(U6, T6), implies(S6, X7)), implies(Y7, implies(implies(X7, U6), implies(S6, U6)))), implies(implies(V, U), implies(U, implies(V, U))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(U6, T6), implies(S6, X7)), implies(Y7, implies(implies(X7, U6), implies(S6, U6))))), true, is_a_theorem(implies(implies(V, U), implies(U, implies(V, U)))), true), true), true, is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))), true), true), true, is_a_theorem(implies(implies(implies(Z, W), implies(W, implies(Z, W))), implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a)))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 27.10/4.66  = { by axiom 1 (ifeq_axiom) }
% 27.10/4.66    ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(V, U), implies(U, implies(V, U))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))))), true, ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(X8, Y8), implies(Z8, W8)), implies(V8, implies(implies(W8, X8), implies(Z8, X8))))), true, is_a_theorem(implies(implies(implies(implies(implies(S7, U), implies(V, U)), implies(U, implies(V, U))), implies(implies(V, U), implies(U, implies(V, U)))), implies(implies(implies(implies(U6, T6), implies(S6, X7)), implies(Y7, implies(implies(X7, U6), implies(S6, U6)))), implies(implies(V, U), implies(U, implies(V, U)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(S7, U), implies(V, U)), implies(U, implies(V, U))), implies(implies(V, U), implies(U, implies(V, U))))), true, is_a_theorem(implies(implies(implies(implies(U6, T6), implies(S6, X7)), implies(Y7, implies(implies(X7, U6), implies(S6, U6)))), implies(implies(V, U), implies(U, implies(V, U))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(U6, T6), implies(S6, X7)), implies(Y7, implies(implies(X7, U6), implies(S6, U6))))), true, is_a_theorem(implies(implies(V, U), implies(U, implies(V, U)))), true), true), true, is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))), true), true), true, is_a_theorem(implies(implies(implies(Z, W), implies(W, implies(Z, W))), implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a)))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 27.10/4.67  = { by axiom 3 (ic_JLukasiewicz_4) }
% 27.10/4.67    ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(V, U), implies(U, implies(V, U))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))))), true, ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(implies(implies(S7, U), implies(V, U)), implies(U, implies(V, U))), implies(implies(V, U), implies(U, implies(V, U)))), implies(implies(implies(implies(U6, T6), implies(S6, X7)), implies(Y7, implies(implies(X7, U6), implies(S6, U6)))), implies(implies(V, U), implies(U, implies(V, U)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(S7, U), implies(V, U)), implies(U, implies(V, U))), implies(implies(V, U), implies(U, implies(V, U))))), true, is_a_theorem(implies(implies(implies(implies(U6, T6), implies(S6, X7)), implies(Y7, implies(implies(X7, U6), implies(S6, U6)))), implies(implies(V, U), implies(U, implies(V, U))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(U6, T6), implies(S6, X7)), implies(Y7, implies(implies(X7, U6), implies(S6, U6))))), true, is_a_theorem(implies(implies(V, U), implies(U, implies(V, U)))), true), true), true, is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))), true), true), true, is_a_theorem(implies(implies(implies(Z, W), implies(W, implies(Z, W))), implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a)))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 27.10/4.67  = { by axiom 1 (ifeq_axiom) }
% 27.10/4.67    ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(V, U), implies(U, implies(V, U))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))))), true, ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(implies(S7, U), implies(V, U)), implies(U, implies(V, U))), implies(implies(V, U), implies(U, implies(V, U)))), implies(implies(implies(implies(U6, T6), implies(S6, X7)), implies(Y7, implies(implies(X7, U6), implies(S6, U6)))), implies(implies(V, U), implies(U, implies(V, U)))))), true, ifeq(is_a_theorem(implies(implies(implies(implies(S7, U), implies(V, U)), implies(U, implies(V, U))), implies(implies(V, U), implies(U, implies(V, U))))), true, is_a_theorem(implies(implies(implies(implies(U6, T6), implies(S6, X7)), implies(Y7, implies(implies(X7, U6), implies(S6, U6)))), implies(implies(V, U), implies(U, implies(V, U))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(U6, T6), implies(S6, X7)), implies(Y7, implies(implies(X7, U6), implies(S6, U6))))), true, is_a_theorem(implies(implies(V, U), implies(U, implies(V, U)))), true), true), true, is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))), true), true), true, is_a_theorem(implies(implies(implies(Z, W), implies(W, implies(Z, W))), implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a)))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 27.10/4.68  = { by axiom 2 (condensed_detachment) }
% 27.10/4.68    ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(V, U), implies(U, implies(V, U))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))))), true, ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(implies(U6, T6), implies(S6, X7)), implies(Y7, implies(implies(X7, U6), implies(S6, U6))))), true, is_a_theorem(implies(implies(V, U), implies(U, implies(V, U)))), true), true), true, is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))), true), true), true, is_a_theorem(implies(implies(implies(Z, W), implies(W, implies(Z, W))), implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a)))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 27.10/4.68  = { by axiom 1 (ifeq_axiom) }
% 27.10/4.68    ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(V, U), implies(U, implies(V, U))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))))), true, ifeq(ifeq(is_a_theorem(implies(implies(implies(U6, T6), implies(S6, X7)), implies(Y7, implies(implies(X7, U6), implies(S6, U6))))), true, is_a_theorem(implies(implies(V, U), implies(U, implies(V, U)))), true), true, is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))), true), true), true, is_a_theorem(implies(implies(implies(Z, W), implies(W, implies(Z, W))), implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a)))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 27.10/4.68  = { by axiom 3 (ic_JLukasiewicz_4) }
% 27.10/4.68    ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(V, U), implies(U, implies(V, U))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))))), true, ifeq(ifeq(true, true, is_a_theorem(implies(implies(V, U), implies(U, implies(V, U)))), true), true, is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))), true), true), true, is_a_theorem(implies(implies(implies(Z, W), implies(W, implies(Z, W))), implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a)))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 27.10/4.68  = { by axiom 1 (ifeq_axiom) }
% 27.10/4.68    ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(V, U), implies(U, implies(V, U))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a))))), true, ifeq(is_a_theorem(implies(implies(V, U), implies(U, implies(V, U)))), true, is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(implies(a, b), a), a), a)))), true), true), true, is_a_theorem(implies(implies(implies(Z, W), implies(W, implies(Z, W))), implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a)))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 27.10/4.69  = { by axiom 2 (condensed_detachment) }
% 27.10/4.69    ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(Z, W), implies(W, implies(Z, W))), implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a)))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 27.10/4.69  = { by axiom 1 (ifeq_axiom) }
% 27.10/4.69    ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(Z, W), implies(W, implies(Z, W))), implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))))), true, is_a_theorem(implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a)))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 27.10/4.69  = { by axiom 1 (ifeq_axiom) R->L }
% 27.10/4.69    ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(Z, W), implies(W, implies(Z, W))), implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))))), true, ifeq(true, true, is_a_theorem(implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a)))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 27.10/4.69  = { by axiom 2 (condensed_detachment) R->L }
% 27.10/4.69    ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(Z, W), implies(W, implies(Z, W))), implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))))), true, ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(U8, T8), implies(S8, X9)), implies(Y9, implies(implies(X9, U8), implies(S8, U8)))), implies(implies(Z, W), implies(W, implies(Z, W))))), true, ifeq(is_a_theorem(implies(implies(implies(U8, T8), implies(S8, X9)), implies(Y9, implies(implies(X9, U8), implies(S8, U8))))), true, is_a_theorem(implies(implies(Z, W), implies(W, implies(Z, W)))), true), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a)))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 27.10/4.70  = { by axiom 1 (ifeq_axiom) R->L }
% 27.10/4.70    ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(Z, W), implies(W, implies(Z, W))), implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))))), true, ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(implies(U8, T8), implies(S8, X9)), implies(Y9, implies(implies(X9, U8), implies(S8, U8)))), implies(implies(Z, W), implies(W, implies(Z, W))))), true), true, ifeq(is_a_theorem(implies(implies(implies(U8, T8), implies(S8, X9)), implies(Y9, implies(implies(X9, U8), implies(S8, U8))))), true, is_a_theorem(implies(implies(Z, W), implies(W, implies(Z, W)))), true), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a)))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 27.54/4.70  = { by axiom 2 (condensed_detachment) R->L }
% 27.54/4.70    ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(Z, W), implies(W, implies(Z, W))), implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))))), true, ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(Z9, W9), implies(V9, U9)), implies(T9, implies(implies(U9, Z9), implies(V9, Z9)))), implies(implies(implies(implies(S9, W), implies(Z, W)), implies(W, implies(Z, W))), implies(implies(Z, W), implies(W, implies(Z, W)))))), true, ifeq(is_a_theorem(implies(implies(implies(Z9, W9), implies(V9, U9)), implies(T9, implies(implies(U9, Z9), implies(V9, Z9))))), true, is_a_theorem(implies(implies(implies(implies(S9, W), implies(Z, W)), implies(W, implies(Z, W))), implies(implies(Z, W), implies(W, implies(Z, W))))), true), true), true, is_a_theorem(implies(implies(implies(implies(U8, T8), implies(S8, X9)), implies(Y9, implies(implies(X9, U8), implies(S8, U8)))), implies(implies(Z, W), implies(W, implies(Z, W))))), true), true, ifeq(is_a_theorem(implies(implies(implies(U8, T8), implies(S8, X9)), implies(Y9, implies(implies(X9, U8), implies(S8, U8))))), true, is_a_theorem(implies(implies(Z, W), implies(W, implies(Z, W)))), true), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a)))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 27.54/4.71  = { by axiom 1 (ifeq_axiom) R->L }
% 27.54/4.71    ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(Z, W), implies(W, implies(Z, W))), implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))))), true, ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(implies(Z9, W9), implies(V9, U9)), implies(T9, implies(implies(U9, Z9), implies(V9, Z9)))), implies(implies(implies(implies(S9, W), implies(Z, W)), implies(W, implies(Z, W))), implies(implies(Z, W), implies(W, implies(Z, W)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(Z9, W9), implies(V9, U9)), implies(T9, implies(implies(U9, Z9), implies(V9, Z9))))), true, is_a_theorem(implies(implies(implies(implies(S9, W), implies(Z, W)), implies(W, implies(Z, W))), implies(implies(Z, W), implies(W, implies(Z, W))))), true), true), true, is_a_theorem(implies(implies(implies(implies(U8, T8), implies(S8, X9)), implies(Y9, implies(implies(X9, U8), implies(S8, U8)))), implies(implies(Z, W), implies(W, implies(Z, W))))), true), true, ifeq(is_a_theorem(implies(implies(implies(U8, T8), implies(S8, X9)), implies(Y9, implies(implies(X9, U8), implies(S8, U8))))), true, is_a_theorem(implies(implies(Z, W), implies(W, implies(Z, W)))), true), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a)))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 27.54/4.71  = { by axiom 3 (ic_JLukasiewicz_4) R->L }
% 27.54/4.71    ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(Z, W), implies(W, implies(Z, W))), implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))))), true, ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(W, implies(Z, W)), implies(Z, S9)), implies(implies(Z, W), implies(implies(S9, W), implies(Z, W))))), true, is_a_theorem(implies(implies(implies(implies(Z9, W9), implies(V9, U9)), implies(T9, implies(implies(U9, Z9), implies(V9, Z9)))), implies(implies(implies(implies(S9, W), implies(Z, W)), implies(W, implies(Z, W))), implies(implies(Z, W), implies(W, implies(Z, W)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(Z9, W9), implies(V9, U9)), implies(T9, implies(implies(U9, Z9), implies(V9, Z9))))), true, is_a_theorem(implies(implies(implies(implies(S9, W), implies(Z, W)), implies(W, implies(Z, W))), implies(implies(Z, W), implies(W, implies(Z, W))))), true), true), true, is_a_theorem(implies(implies(implies(implies(U8, T8), implies(S8, X9)), implies(Y9, implies(implies(X9, U8), implies(S8, U8)))), implies(implies(Z, W), implies(W, implies(Z, W))))), true), true, ifeq(is_a_theorem(implies(implies(implies(U8, T8), implies(S8, X9)), implies(Y9, implies(implies(X9, U8), implies(S8, U8))))), true, is_a_theorem(implies(implies(Z, W), implies(W, implies(Z, W)))), true), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a)))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 27.54/4.71  = { by axiom 1 (ifeq_axiom) R->L }
% 27.54/4.72    ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(Z, W), implies(W, implies(Z, W))), implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))))), true, ifeq(ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(implies(W, implies(Z, W)), implies(Z, S9)), implies(implies(Z, W), implies(implies(S9, W), implies(Z, W))))), true, is_a_theorem(implies(implies(implies(implies(Z9, W9), implies(V9, U9)), implies(T9, implies(implies(U9, Z9), implies(V9, Z9)))), implies(implies(implies(implies(S9, W), implies(Z, W)), implies(W, implies(Z, W))), implies(implies(Z, W), implies(W, implies(Z, W)))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(Z9, W9), implies(V9, U9)), implies(T9, implies(implies(U9, Z9), implies(V9, Z9))))), true, is_a_theorem(implies(implies(implies(implies(S9, W), implies(Z, W)), implies(W, implies(Z, W))), implies(implies(Z, W), implies(W, implies(Z, W))))), true), true), true, is_a_theorem(implies(implies(implies(implies(U8, T8), implies(S8, X9)), implies(Y9, implies(implies(X9, U8), implies(S8, U8)))), implies(implies(Z, W), implies(W, implies(Z, W))))), true), true, ifeq(is_a_theorem(implies(implies(implies(U8, T8), implies(S8, X9)), implies(Y9, implies(implies(X9, U8), implies(S8, U8))))), true, is_a_theorem(implies(implies(Z, W), implies(W, implies(Z, W)))), true), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a)))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 27.54/4.72  = { by axiom 3 (ic_JLukasiewicz_4) R->L }
% 27.54/4.72    ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(Z, W), implies(W, implies(Z, W))), implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))))), true, ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(W, implies(Z, W)), implies(Z, S9)), implies(implies(Z, W), implies(implies(S9, W), implies(Z, W)))), implies(implies(implies(implies(Z9, W9), implies(V9, U9)), implies(T9, implies(implies(U9, Z9), implies(V9, Z9)))), implies(implies(implies(implies(S9, W), implies(Z, W)), implies(W, implies(Z, W))), implies(implies(Z, W), implies(W, implies(Z, W))))))), true, ifeq(is_a_theorem(implies(implies(implies(W, implies(Z, W)), implies(Z, S9)), implies(implies(Z, W), implies(implies(S9, W), implies(Z, W))))), true, is_a_theorem(implies(implies(implies(implies(Z9, W9), implies(V9, U9)), implies(T9, implies(implies(U9, Z9), implies(V9, Z9)))), implies(implies(implies(implies(S9, W), implies(Z, W)), implies(W, implies(Z, W))), implies(implies(Z, W), implies(W, implies(Z, W)))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(Z9, W9), implies(V9, U9)), implies(T9, implies(implies(U9, Z9), implies(V9, Z9))))), true, is_a_theorem(implies(implies(implies(implies(S9, W), implies(Z, W)), implies(W, implies(Z, W))), implies(implies(Z, W), implies(W, implies(Z, W))))), true), true), true, is_a_theorem(implies(implies(implies(implies(U8, T8), implies(S8, X9)), implies(Y9, implies(implies(X9, U8), implies(S8, U8)))), implies(implies(Z, W), implies(W, implies(Z, W))))), true), true, ifeq(is_a_theorem(implies(implies(implies(U8, T8), implies(S8, X9)), implies(Y9, implies(implies(X9, U8), implies(S8, U8))))), true, is_a_theorem(implies(implies(Z, W), implies(W, implies(Z, W)))), true), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a)))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 27.54/4.72  = { by axiom 2 (condensed_detachment) }
% 27.54/4.73    ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(Z, W), implies(W, implies(Z, W))), implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))))), true, ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(implies(Z9, W9), implies(V9, U9)), implies(T9, implies(implies(U9, Z9), implies(V9, Z9))))), true, is_a_theorem(implies(implies(implies(implies(S9, W), implies(Z, W)), implies(W, implies(Z, W))), implies(implies(Z, W), implies(W, implies(Z, W))))), true), true), true, is_a_theorem(implies(implies(implies(implies(U8, T8), implies(S8, X9)), implies(Y9, implies(implies(X9, U8), implies(S8, U8)))), implies(implies(Z, W), implies(W, implies(Z, W))))), true), true, ifeq(is_a_theorem(implies(implies(implies(U8, T8), implies(S8, X9)), implies(Y9, implies(implies(X9, U8), implies(S8, U8))))), true, is_a_theorem(implies(implies(Z, W), implies(W, implies(Z, W)))), true), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a)))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 27.54/4.73  = { by axiom 1 (ifeq_axiom) }
% 27.54/4.73    ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(Z, W), implies(W, implies(Z, W))), implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))))), true, ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(Z9, W9), implies(V9, U9)), implies(T9, implies(implies(U9, Z9), implies(V9, Z9))))), true, is_a_theorem(implies(implies(implies(implies(S9, W), implies(Z, W)), implies(W, implies(Z, W))), implies(implies(Z, W), implies(W, implies(Z, W))))), true), true, is_a_theorem(implies(implies(implies(implies(U8, T8), implies(S8, X9)), implies(Y9, implies(implies(X9, U8), implies(S8, U8)))), implies(implies(Z, W), implies(W, implies(Z, W))))), true), true, ifeq(is_a_theorem(implies(implies(implies(U8, T8), implies(S8, X9)), implies(Y9, implies(implies(X9, U8), implies(S8, U8))))), true, is_a_theorem(implies(implies(Z, W), implies(W, implies(Z, W)))), true), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a)))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 27.54/4.73  = { by axiom 3 (ic_JLukasiewicz_4) }
% 27.54/4.73    ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(Z, W), implies(W, implies(Z, W))), implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))))), true, ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(implies(S9, W), implies(Z, W)), implies(W, implies(Z, W))), implies(implies(Z, W), implies(W, implies(Z, W))))), true), true, is_a_theorem(implies(implies(implies(implies(U8, T8), implies(S8, X9)), implies(Y9, implies(implies(X9, U8), implies(S8, U8)))), implies(implies(Z, W), implies(W, implies(Z, W))))), true), true, ifeq(is_a_theorem(implies(implies(implies(U8, T8), implies(S8, X9)), implies(Y9, implies(implies(X9, U8), implies(S8, U8))))), true, is_a_theorem(implies(implies(Z, W), implies(W, implies(Z, W)))), true), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a)))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 27.54/4.74  = { by axiom 1 (ifeq_axiom) }
% 27.54/4.74    ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(Z, W), implies(W, implies(Z, W))), implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))))), true, ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(S9, W), implies(Z, W)), implies(W, implies(Z, W))), implies(implies(Z, W), implies(W, implies(Z, W))))), true, is_a_theorem(implies(implies(implies(implies(U8, T8), implies(S8, X9)), implies(Y9, implies(implies(X9, U8), implies(S8, U8)))), implies(implies(Z, W), implies(W, implies(Z, W))))), true), true, ifeq(is_a_theorem(implies(implies(implies(U8, T8), implies(S8, X9)), implies(Y9, implies(implies(X9, U8), implies(S8, U8))))), true, is_a_theorem(implies(implies(Z, W), implies(W, implies(Z, W)))), true), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a)))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 27.54/4.74  = { by axiom 1 (ifeq_axiom) R->L }
% 27.54/4.74    ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(Z, W), implies(W, implies(Z, W))), implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))))), true, ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(implies(implies(S9, W), implies(Z, W)), implies(W, implies(Z, W))), implies(implies(Z, W), implies(W, implies(Z, W))))), true, is_a_theorem(implies(implies(implies(implies(U8, T8), implies(S8, X9)), implies(Y9, implies(implies(X9, U8), implies(S8, U8)))), implies(implies(Z, W), implies(W, implies(Z, W))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(U8, T8), implies(S8, X9)), implies(Y9, implies(implies(X9, U8), implies(S8, U8))))), true, is_a_theorem(implies(implies(Z, W), implies(W, implies(Z, W)))), true), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a)))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 27.54/4.75  = { by axiom 2 (condensed_detachment) R->L }
% 27.54/4.75    ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(Z, W), implies(W, implies(Z, W))), implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))))), true, ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(X10, Y10), implies(Z10, W10)), implies(V10, implies(implies(W10, X10), implies(Z10, X10)))), implies(implies(implies(implies(implies(S9, W), implies(Z, W)), implies(W, implies(Z, W))), implies(implies(Z, W), implies(W, implies(Z, W)))), implies(implies(implies(implies(U8, T8), implies(S8, X9)), implies(Y9, implies(implies(X9, U8), implies(S8, U8)))), implies(implies(Z, W), implies(W, implies(Z, W))))))), true, ifeq(is_a_theorem(implies(implies(implies(X10, Y10), implies(Z10, W10)), implies(V10, implies(implies(W10, X10), implies(Z10, X10))))), true, is_a_theorem(implies(implies(implies(implies(implies(S9, W), implies(Z, W)), implies(W, implies(Z, W))), implies(implies(Z, W), implies(W, implies(Z, W)))), implies(implies(implies(implies(U8, T8), implies(S8, X9)), implies(Y9, implies(implies(X9, U8), implies(S8, U8)))), implies(implies(Z, W), implies(W, implies(Z, W)))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(S9, W), implies(Z, W)), implies(W, implies(Z, W))), implies(implies(Z, W), implies(W, implies(Z, W))))), true, is_a_theorem(implies(implies(implies(implies(U8, T8), implies(S8, X9)), implies(Y9, implies(implies(X9, U8), implies(S8, U8)))), implies(implies(Z, W), implies(W, implies(Z, W))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(U8, T8), implies(S8, X9)), implies(Y9, implies(implies(X9, U8), implies(S8, U8))))), true, is_a_theorem(implies(implies(Z, W), implies(W, implies(Z, W)))), true), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a)))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 27.54/4.75  = { by axiom 1 (ifeq_axiom) R->L }
% 27.54/4.76    ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(Z, W), implies(W, implies(Z, W))), implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))))), true, ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(implies(X10, Y10), implies(Z10, W10)), implies(V10, implies(implies(W10, X10), implies(Z10, X10)))), implies(implies(implies(implies(implies(S9, W), implies(Z, W)), implies(W, implies(Z, W))), implies(implies(Z, W), implies(W, implies(Z, W)))), implies(implies(implies(implies(U8, T8), implies(S8, X9)), implies(Y9, implies(implies(X9, U8), implies(S8, U8)))), implies(implies(Z, W), implies(W, implies(Z, W))))))), true), true, ifeq(is_a_theorem(implies(implies(implies(X10, Y10), implies(Z10, W10)), implies(V10, implies(implies(W10, X10), implies(Z10, X10))))), true, is_a_theorem(implies(implies(implies(implies(implies(S9, W), implies(Z, W)), implies(W, implies(Z, W))), implies(implies(Z, W), implies(W, implies(Z, W)))), implies(implies(implies(implies(U8, T8), implies(S8, X9)), implies(Y9, implies(implies(X9, U8), implies(S8, U8)))), implies(implies(Z, W), implies(W, implies(Z, W)))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(S9, W), implies(Z, W)), implies(W, implies(Z, W))), implies(implies(Z, W), implies(W, implies(Z, W))))), true, is_a_theorem(implies(implies(implies(implies(U8, T8), implies(S8, X9)), implies(Y9, implies(implies(X9, U8), implies(S8, U8)))), implies(implies(Z, W), implies(W, implies(Z, W))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(U8, T8), implies(S8, X9)), implies(Y9, implies(implies(X9, U8), implies(S8, U8))))), true, is_a_theorem(implies(implies(Z, W), implies(W, implies(Z, W)))), true), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a)))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 27.54/4.76  = { by axiom 3 (ic_JLukasiewicz_4) R->L }
% 27.54/4.76    ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(Z, W), implies(W, implies(Z, W))), implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))))), true, ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(Z, W), implies(W, implies(Z, W))), implies(W, implies(S9, W))), implies(implies(implies(implies(U8, T8), implies(S8, X9)), implies(Y9, implies(implies(X9, U8), implies(S8, U8)))), implies(implies(implies(S9, W), implies(Z, W)), implies(W, implies(Z, W)))))), true, is_a_theorem(implies(implies(implies(implies(X10, Y10), implies(Z10, W10)), implies(V10, implies(implies(W10, X10), implies(Z10, X10)))), implies(implies(implies(implies(implies(S9, W), implies(Z, W)), implies(W, implies(Z, W))), implies(implies(Z, W), implies(W, implies(Z, W)))), implies(implies(implies(implies(U8, T8), implies(S8, X9)), implies(Y9, implies(implies(X9, U8), implies(S8, U8)))), implies(implies(Z, W), implies(W, implies(Z, W))))))), true), true, ifeq(is_a_theorem(implies(implies(implies(X10, Y10), implies(Z10, W10)), implies(V10, implies(implies(W10, X10), implies(Z10, X10))))), true, is_a_theorem(implies(implies(implies(implies(implies(S9, W), implies(Z, W)), implies(W, implies(Z, W))), implies(implies(Z, W), implies(W, implies(Z, W)))), implies(implies(implies(implies(U8, T8), implies(S8, X9)), implies(Y9, implies(implies(X9, U8), implies(S8, U8)))), implies(implies(Z, W), implies(W, implies(Z, W)))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(S9, W), implies(Z, W)), implies(W, implies(Z, W))), implies(implies(Z, W), implies(W, implies(Z, W))))), true, is_a_theorem(implies(implies(implies(implies(U8, T8), implies(S8, X9)), implies(Y9, implies(implies(X9, U8), implies(S8, U8)))), implies(implies(Z, W), implies(W, implies(Z, W))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(U8, T8), implies(S8, X9)), implies(Y9, implies(implies(X9, U8), implies(S8, U8))))), true, is_a_theorem(implies(implies(Z, W), implies(W, implies(Z, W)))), true), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a)))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 27.54/4.77  = { by axiom 1 (ifeq_axiom) R->L }
% 27.54/4.77    ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(Z, W), implies(W, implies(Z, W))), implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))))), true, ifeq(ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(implies(implies(Z, W), implies(W, implies(Z, W))), implies(W, implies(S9, W))), implies(implies(implies(implies(U8, T8), implies(S8, X9)), implies(Y9, implies(implies(X9, U8), implies(S8, U8)))), implies(implies(implies(S9, W), implies(Z, W)), implies(W, implies(Z, W)))))), true, is_a_theorem(implies(implies(implies(implies(X10, Y10), implies(Z10, W10)), implies(V10, implies(implies(W10, X10), implies(Z10, X10)))), implies(implies(implies(implies(implies(S9, W), implies(Z, W)), implies(W, implies(Z, W))), implies(implies(Z, W), implies(W, implies(Z, W)))), implies(implies(implies(implies(U8, T8), implies(S8, X9)), implies(Y9, implies(implies(X9, U8), implies(S8, U8)))), implies(implies(Z, W), implies(W, implies(Z, W))))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(X10, Y10), implies(Z10, W10)), implies(V10, implies(implies(W10, X10), implies(Z10, X10))))), true, is_a_theorem(implies(implies(implies(implies(implies(S9, W), implies(Z, W)), implies(W, implies(Z, W))), implies(implies(Z, W), implies(W, implies(Z, W)))), implies(implies(implies(implies(U8, T8), implies(S8, X9)), implies(Y9, implies(implies(X9, U8), implies(S8, U8)))), implies(implies(Z, W), implies(W, implies(Z, W)))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(S9, W), implies(Z, W)), implies(W, implies(Z, W))), implies(implies(Z, W), implies(W, implies(Z, W))))), true, is_a_theorem(implies(implies(implies(implies(U8, T8), implies(S8, X9)), implies(Y9, implies(implies(X9, U8), implies(S8, U8)))), implies(implies(Z, W), implies(W, implies(Z, W))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(U8, T8), implies(S8, X9)), implies(Y9, implies(implies(X9, U8), implies(S8, U8))))), true, is_a_theorem(implies(implies(Z, W), implies(W, implies(Z, W)))), true), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a)))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 27.54/4.77  = { by axiom 3 (ic_JLukasiewicz_4) R->L }
% 27.54/4.78    ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(Z, W), implies(W, implies(Z, W))), implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))))), true, ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(implies(Z, W), implies(W, implies(Z, W))), implies(W, implies(S9, W))), implies(implies(implies(implies(U8, T8), implies(S8, X9)), implies(Y9, implies(implies(X9, U8), implies(S8, U8)))), implies(implies(implies(S9, W), implies(Z, W)), implies(W, implies(Z, W))))), implies(implies(implies(implies(X10, Y10), implies(Z10, W10)), implies(V10, implies(implies(W10, X10), implies(Z10, X10)))), implies(implies(implies(implies(implies(S9, W), implies(Z, W)), implies(W, implies(Z, W))), implies(implies(Z, W), implies(W, implies(Z, W)))), implies(implies(implies(implies(U8, T8), implies(S8, X9)), implies(Y9, implies(implies(X9, U8), implies(S8, U8)))), implies(implies(Z, W), implies(W, implies(Z, W)))))))), true, ifeq(is_a_theorem(implies(implies(implies(implies(Z, W), implies(W, implies(Z, W))), implies(W, implies(S9, W))), implies(implies(implies(implies(U8, T8), implies(S8, X9)), implies(Y9, implies(implies(X9, U8), implies(S8, U8)))), implies(implies(implies(S9, W), implies(Z, W)), implies(W, implies(Z, W)))))), true, is_a_theorem(implies(implies(implies(implies(X10, Y10), implies(Z10, W10)), implies(V10, implies(implies(W10, X10), implies(Z10, X10)))), implies(implies(implies(implies(implies(S9, W), implies(Z, W)), implies(W, implies(Z, W))), implies(implies(Z, W), implies(W, implies(Z, W)))), implies(implies(implies(implies(U8, T8), implies(S8, X9)), implies(Y9, implies(implies(X9, U8), implies(S8, U8)))), implies(implies(Z, W), implies(W, implies(Z, W))))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(X10, Y10), implies(Z10, W10)), implies(V10, implies(implies(W10, X10), implies(Z10, X10))))), true, is_a_theorem(implies(implies(implies(implies(implies(S9, W), implies(Z, W)), implies(W, implies(Z, W))), implies(implies(Z, W), implies(W, implies(Z, W)))), implies(implies(implies(implies(U8, T8), implies(S8, X9)), implies(Y9, implies(implies(X9, U8), implies(S8, U8)))), implies(implies(Z, W), implies(W, implies(Z, W)))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(S9, W), implies(Z, W)), implies(W, implies(Z, W))), implies(implies(Z, W), implies(W, implies(Z, W))))), true, is_a_theorem(implies(implies(implies(implies(U8, T8), implies(S8, X9)), implies(Y9, implies(implies(X9, U8), implies(S8, U8)))), implies(implies(Z, W), implies(W, implies(Z, W))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(U8, T8), implies(S8, X9)), implies(Y9, implies(implies(X9, U8), implies(S8, U8))))), true, is_a_theorem(implies(implies(Z, W), implies(W, implies(Z, W)))), true), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a)))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 27.54/4.78  = { by axiom 2 (condensed_detachment) }
% 27.54/4.78    ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(Z, W), implies(W, implies(Z, W))), implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))))), true, ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(implies(X10, Y10), implies(Z10, W10)), implies(V10, implies(implies(W10, X10), implies(Z10, X10))))), true, is_a_theorem(implies(implies(implies(implies(implies(S9, W), implies(Z, W)), implies(W, implies(Z, W))), implies(implies(Z, W), implies(W, implies(Z, W)))), implies(implies(implies(implies(U8, T8), implies(S8, X9)), implies(Y9, implies(implies(X9, U8), implies(S8, U8)))), implies(implies(Z, W), implies(W, implies(Z, W)))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(S9, W), implies(Z, W)), implies(W, implies(Z, W))), implies(implies(Z, W), implies(W, implies(Z, W))))), true, is_a_theorem(implies(implies(implies(implies(U8, T8), implies(S8, X9)), implies(Y9, implies(implies(X9, U8), implies(S8, U8)))), implies(implies(Z, W), implies(W, implies(Z, W))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(U8, T8), implies(S8, X9)), implies(Y9, implies(implies(X9, U8), implies(S8, U8))))), true, is_a_theorem(implies(implies(Z, W), implies(W, implies(Z, W)))), true), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a)))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 27.54/4.78  = { by axiom 1 (ifeq_axiom) }
% 27.54/4.79    ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(Z, W), implies(W, implies(Z, W))), implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))))), true, ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(X10, Y10), implies(Z10, W10)), implies(V10, implies(implies(W10, X10), implies(Z10, X10))))), true, is_a_theorem(implies(implies(implies(implies(implies(S9, W), implies(Z, W)), implies(W, implies(Z, W))), implies(implies(Z, W), implies(W, implies(Z, W)))), implies(implies(implies(implies(U8, T8), implies(S8, X9)), implies(Y9, implies(implies(X9, U8), implies(S8, U8)))), implies(implies(Z, W), implies(W, implies(Z, W)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(S9, W), implies(Z, W)), implies(W, implies(Z, W))), implies(implies(Z, W), implies(W, implies(Z, W))))), true, is_a_theorem(implies(implies(implies(implies(U8, T8), implies(S8, X9)), implies(Y9, implies(implies(X9, U8), implies(S8, U8)))), implies(implies(Z, W), implies(W, implies(Z, W))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(U8, T8), implies(S8, X9)), implies(Y9, implies(implies(X9, U8), implies(S8, U8))))), true, is_a_theorem(implies(implies(Z, W), implies(W, implies(Z, W)))), true), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a)))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 27.54/4.79  = { by axiom 3 (ic_JLukasiewicz_4) }
% 27.54/4.79    ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(Z, W), implies(W, implies(Z, W))), implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))))), true, ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(implies(implies(S9, W), implies(Z, W)), implies(W, implies(Z, W))), implies(implies(Z, W), implies(W, implies(Z, W)))), implies(implies(implies(implies(U8, T8), implies(S8, X9)), implies(Y9, implies(implies(X9, U8), implies(S8, U8)))), implies(implies(Z, W), implies(W, implies(Z, W)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(S9, W), implies(Z, W)), implies(W, implies(Z, W))), implies(implies(Z, W), implies(W, implies(Z, W))))), true, is_a_theorem(implies(implies(implies(implies(U8, T8), implies(S8, X9)), implies(Y9, implies(implies(X9, U8), implies(S8, U8)))), implies(implies(Z, W), implies(W, implies(Z, W))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(U8, T8), implies(S8, X9)), implies(Y9, implies(implies(X9, U8), implies(S8, U8))))), true, is_a_theorem(implies(implies(Z, W), implies(W, implies(Z, W)))), true), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a)))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 27.54/4.79  = { by axiom 1 (ifeq_axiom) }
% 27.54/4.79    ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(Z, W), implies(W, implies(Z, W))), implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))))), true, ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(implies(S9, W), implies(Z, W)), implies(W, implies(Z, W))), implies(implies(Z, W), implies(W, implies(Z, W)))), implies(implies(implies(implies(U8, T8), implies(S8, X9)), implies(Y9, implies(implies(X9, U8), implies(S8, U8)))), implies(implies(Z, W), implies(W, implies(Z, W)))))), true, ifeq(is_a_theorem(implies(implies(implies(implies(S9, W), implies(Z, W)), implies(W, implies(Z, W))), implies(implies(Z, W), implies(W, implies(Z, W))))), true, is_a_theorem(implies(implies(implies(implies(U8, T8), implies(S8, X9)), implies(Y9, implies(implies(X9, U8), implies(S8, U8)))), implies(implies(Z, W), implies(W, implies(Z, W))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(U8, T8), implies(S8, X9)), implies(Y9, implies(implies(X9, U8), implies(S8, U8))))), true, is_a_theorem(implies(implies(Z, W), implies(W, implies(Z, W)))), true), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a)))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 27.54/4.80  = { by axiom 2 (condensed_detachment) }
% 27.54/4.80    ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(Z, W), implies(W, implies(Z, W))), implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))))), true, ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(implies(U8, T8), implies(S8, X9)), implies(Y9, implies(implies(X9, U8), implies(S8, U8))))), true, is_a_theorem(implies(implies(Z, W), implies(W, implies(Z, W)))), true), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a)))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 27.54/4.80  = { by axiom 1 (ifeq_axiom) }
% 27.54/4.80    ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(Z, W), implies(W, implies(Z, W))), implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))))), true, ifeq(ifeq(is_a_theorem(implies(implies(implies(U8, T8), implies(S8, X9)), implies(Y9, implies(implies(X9, U8), implies(S8, U8))))), true, is_a_theorem(implies(implies(Z, W), implies(W, implies(Z, W)))), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a)))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 27.54/4.80  = { by axiom 3 (ic_JLukasiewicz_4) }
% 27.54/4.80    ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(Z, W), implies(W, implies(Z, W))), implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))))), true, ifeq(ifeq(true, true, is_a_theorem(implies(implies(Z, W), implies(W, implies(Z, W)))), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a)))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 27.54/4.80  = { by axiom 1 (ifeq_axiom) }
% 27.54/4.80    ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(Z, W), implies(W, implies(Z, W))), implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))))), true, ifeq(is_a_theorem(implies(implies(Z, W), implies(W, implies(Z, W)))), true, is_a_theorem(implies(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a)))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 27.54/4.80  = { by axiom 2 (condensed_detachment) }
% 27.54/4.80    ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 27.54/4.80  = { by axiom 1 (ifeq_axiom) }
% 27.54/4.80    ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 27.95/4.80  = { by axiom 1 (ifeq_axiom) R->L }
% 27.95/4.80    ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 27.95/4.80  = { by axiom 2 (condensed_detachment) R->L }
% 27.95/4.80    ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(a, b), implies(implies(implies(a, b), a), implies(implies(implies(a, b), a), a))), implies(implies(implies(U10, T10), implies(T10, implies(U10, T10))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))))), true, ifeq(is_a_theorem(implies(implies(a, b), implies(implies(implies(a, b), a), implies(implies(implies(a, b), a), a)))), true, is_a_theorem(implies(implies(implies(U10, T10), implies(T10, implies(U10, T10))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)))), true), true), true, is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 27.95/4.80  = { by axiom 3 (ic_JLukasiewicz_4) }
% 27.95/4.81    ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(a, b), implies(implies(implies(a, b), a), implies(implies(implies(a, b), a), a)))), true, is_a_theorem(implies(implies(implies(U10, T10), implies(T10, implies(U10, T10))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)))), true), true), true, is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 27.95/4.81  = { by axiom 1 (ifeq_axiom) }
% 27.95/4.81    ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(a, b), implies(implies(implies(a, b), a), implies(implies(implies(a, b), a), a)))), true, is_a_theorem(implies(implies(implies(U10, T10), implies(T10, implies(U10, T10))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)))), true), true, is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 27.95/4.81  = { by axiom 1 (ifeq_axiom) R->L }
% 27.95/4.81    ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(a, b), implies(implies(implies(a, b), a), implies(implies(implies(a, b), a), a)))), true), true, is_a_theorem(implies(implies(implies(U10, T10), implies(T10, implies(U10, T10))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)))), true), true, is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 27.95/4.81  = { by axiom 3 (ic_JLukasiewicz_4) R->L }
% 27.95/4.81    ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(S10, X11), implies(Y11, Z11)), implies(W11, implies(implies(Z11, S10), implies(Y11, S10))))), true, is_a_theorem(implies(implies(a, b), implies(implies(implies(a, b), a), implies(implies(implies(a, b), a), a)))), true), true, is_a_theorem(implies(implies(implies(U10, T10), implies(T10, implies(U10, T10))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)))), true), true, is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 27.95/4.81  = { by axiom 1 (ifeq_axiom) R->L }
% 27.95/4.81    ifeq(ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(implies(S10, X11), implies(Y11, Z11)), implies(W11, implies(implies(Z11, S10), implies(Y11, S10))))), true, is_a_theorem(implies(implies(a, b), implies(implies(implies(a, b), a), implies(implies(implies(a, b), a), a)))), true), true), true, is_a_theorem(implies(implies(implies(U10, T10), implies(T10, implies(U10, T10))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)))), true), true, is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 27.95/4.82  = { by axiom 2 (condensed_detachment) R->L }
% 27.95/4.82    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(a, implies(a, b)), implies(implies(implies(a, b), a), implies(a, b))), implies(implies(a, b), implies(implies(implies(a, b), a), implies(implies(implies(a, b), a), a)))), implies(implies(implies(implies(S10, X11), implies(Y11, Z11)), implies(W11, implies(implies(Z11, S10), implies(Y11, S10)))), implies(implies(a, b), implies(implies(implies(a, b), a), implies(implies(implies(a, b), a), a)))))), true, ifeq(is_a_theorem(implies(implies(implies(a, implies(a, b)), implies(implies(implies(a, b), a), implies(a, b))), implies(implies(a, b), implies(implies(implies(a, b), a), implies(implies(implies(a, b), a), a))))), true, is_a_theorem(implies(implies(implies(implies(S10, X11), implies(Y11, Z11)), implies(W11, implies(implies(Z11, S10), implies(Y11, S10)))), implies(implies(a, b), implies(implies(implies(a, b), a), implies(implies(implies(a, b), a), a))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(S10, X11), implies(Y11, Z11)), implies(W11, implies(implies(Z11, S10), implies(Y11, S10))))), true, is_a_theorem(implies(implies(a, b), implies(implies(implies(a, b), a), implies(implies(implies(a, b), a), a)))), true), true), true, is_a_theorem(implies(implies(implies(U10, T10), implies(T10, implies(U10, T10))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)))), true), true, is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 27.95/4.82  = { by axiom 1 (ifeq_axiom) R->L }
% 27.95/4.82    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(implies(a, implies(a, b)), implies(implies(implies(a, b), a), implies(a, b))), implies(implies(a, b), implies(implies(implies(a, b), a), implies(implies(implies(a, b), a), a)))), implies(implies(implies(implies(S10, X11), implies(Y11, Z11)), implies(W11, implies(implies(Z11, S10), implies(Y11, S10)))), implies(implies(a, b), implies(implies(implies(a, b), a), implies(implies(implies(a, b), a), a)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(a, implies(a, b)), implies(implies(implies(a, b), a), implies(a, b))), implies(implies(a, b), implies(implies(implies(a, b), a), implies(implies(implies(a, b), a), a))))), true, is_a_theorem(implies(implies(implies(implies(S10, X11), implies(Y11, Z11)), implies(W11, implies(implies(Z11, S10), implies(Y11, S10)))), implies(implies(a, b), implies(implies(implies(a, b), a), implies(implies(implies(a, b), a), a))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(S10, X11), implies(Y11, Z11)), implies(W11, implies(implies(Z11, S10), implies(Y11, S10))))), true, is_a_theorem(implies(implies(a, b), implies(implies(implies(a, b), a), implies(implies(implies(a, b), a), a)))), true), true), true, is_a_theorem(implies(implies(implies(U10, T10), implies(T10, implies(U10, T10))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)))), true), true, is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 27.95/4.83  = { by axiom 3 (ic_JLukasiewicz_4) R->L }
% 27.95/4.83    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(V11, U11), implies(T11, S11)), implies(X12, implies(implies(S11, V11), implies(T11, V11))))), true, is_a_theorem(implies(implies(implies(implies(a, implies(a, b)), implies(implies(implies(a, b), a), implies(a, b))), implies(implies(a, b), implies(implies(implies(a, b), a), implies(implies(implies(a, b), a), a)))), implies(implies(implies(implies(S10, X11), implies(Y11, Z11)), implies(W11, implies(implies(Z11, S10), implies(Y11, S10)))), implies(implies(a, b), implies(implies(implies(a, b), a), implies(implies(implies(a, b), a), a)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(a, implies(a, b)), implies(implies(implies(a, b), a), implies(a, b))), implies(implies(a, b), implies(implies(implies(a, b), a), implies(implies(implies(a, b), a), a))))), true, is_a_theorem(implies(implies(implies(implies(S10, X11), implies(Y11, Z11)), implies(W11, implies(implies(Z11, S10), implies(Y11, S10)))), implies(implies(a, b), implies(implies(implies(a, b), a), implies(implies(implies(a, b), a), a))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(S10, X11), implies(Y11, Z11)), implies(W11, implies(implies(Z11, S10), implies(Y11, S10))))), true, is_a_theorem(implies(implies(a, b), implies(implies(implies(a, b), a), implies(implies(implies(a, b), a), a)))), true), true), true, is_a_theorem(implies(implies(implies(U10, T10), implies(T10, implies(U10, T10))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)))), true), true, is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 27.95/4.83  = { by axiom 1 (ifeq_axiom) R->L }
% 27.95/4.84    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(implies(V11, U11), implies(T11, S11)), implies(X12, implies(implies(S11, V11), implies(T11, V11))))), true, is_a_theorem(implies(implies(implies(implies(a, implies(a, b)), implies(implies(implies(a, b), a), implies(a, b))), implies(implies(a, b), implies(implies(implies(a, b), a), implies(implies(implies(a, b), a), a)))), implies(implies(implies(implies(S10, X11), implies(Y11, Z11)), implies(W11, implies(implies(Z11, S10), implies(Y11, S10)))), implies(implies(a, b), implies(implies(implies(a, b), a), implies(implies(implies(a, b), a), a)))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(a, implies(a, b)), implies(implies(implies(a, b), a), implies(a, b))), implies(implies(a, b), implies(implies(implies(a, b), a), implies(implies(implies(a, b), a), a))))), true, is_a_theorem(implies(implies(implies(implies(S10, X11), implies(Y11, Z11)), implies(W11, implies(implies(Z11, S10), implies(Y11, S10)))), implies(implies(a, b), implies(implies(implies(a, b), a), implies(implies(implies(a, b), a), a))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(S10, X11), implies(Y11, Z11)), implies(W11, implies(implies(Z11, S10), implies(Y11, S10))))), true, is_a_theorem(implies(implies(a, b), implies(implies(implies(a, b), a), implies(implies(implies(a, b), a), a)))), true), true), true, is_a_theorem(implies(implies(implies(U10, T10), implies(T10, implies(U10, T10))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)))), true), true, is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 27.95/4.84  = { by axiom 2 (condensed_detachment) R->L }
% 27.95/4.84    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(implies(a, b), implies(implies(implies(a, b), a), implies(implies(implies(a, b), a), a))), implies(implies(implies(a, b), a), a)), implies(implies(implies(implies(S10, X11), implies(Y11, Z11)), implies(W11, implies(implies(Z11, S10), implies(Y11, S10)))), implies(implies(a, implies(a, b)), implies(implies(implies(a, b), a), implies(a, b))))), implies(implies(implies(implies(V11, U11), implies(T11, S11)), implies(X12, implies(implies(S11, V11), implies(T11, V11)))), implies(implies(implies(implies(a, implies(a, b)), implies(implies(implies(a, b), a), implies(a, b))), implies(implies(a, b), implies(implies(implies(a, b), a), implies(implies(implies(a, b), a), a)))), implies(implies(implies(implies(S10, X11), implies(Y11, Z11)), implies(W11, implies(implies(Z11, S10), implies(Y11, S10)))), implies(implies(a, b), implies(implies(implies(a, b), a), implies(implies(implies(a, b), a), a)))))))), true, ifeq(is_a_theorem(implies(implies(implies(implies(a, b), implies(implies(implies(a, b), a), implies(implies(implies(a, b), a), a))), implies(implies(implies(a, b), a), a)), implies(implies(implies(implies(S10, X11), implies(Y11, Z11)), implies(W11, implies(implies(Z11, S10), implies(Y11, S10)))), implies(implies(a, implies(a, b)), implies(implies(implies(a, b), a), implies(a, b)))))), true, is_a_theorem(implies(implies(implies(implies(V11, U11), implies(T11, S11)), implies(X12, implies(implies(S11, V11), implies(T11, V11)))), implies(implies(implies(implies(a, implies(a, b)), implies(implies(implies(a, b), a), implies(a, b))), implies(implies(a, b), implies(implies(implies(a, b), a), implies(implies(implies(a, b), a), a)))), implies(implies(implies(implies(S10, X11), implies(Y11, Z11)), implies(W11, implies(implies(Z11, S10), implies(Y11, S10)))), implies(implies(a, b), implies(implies(implies(a, b), a), implies(implies(implies(a, b), a), a))))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(V11, U11), implies(T11, S11)), implies(X12, implies(implies(S11, V11), implies(T11, V11))))), true, is_a_theorem(implies(implies(implies(implies(a, implies(a, b)), implies(implies(implies(a, b), a), implies(a, b))), implies(implies(a, b), implies(implies(implies(a, b), a), implies(implies(implies(a, b), a), a)))), implies(implies(implies(implies(S10, X11), implies(Y11, Z11)), implies(W11, implies(implies(Z11, S10), implies(Y11, S10)))), implies(implies(a, b), implies(implies(implies(a, b), a), implies(implies(implies(a, b), a), a)))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(a, implies(a, b)), implies(implies(implies(a, b), a), implies(a, b))), implies(implies(a, b), implies(implies(implies(a, b), a), implies(implies(implies(a, b), a), a))))), true, is_a_theorem(implies(implies(implies(implies(S10, X11), implies(Y11, Z11)), implies(W11, implies(implies(Z11, S10), implies(Y11, S10)))), implies(implies(a, b), implies(implies(implies(a, b), a), implies(implies(implies(a, b), a), a))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(S10, X11), implies(Y11, Z11)), implies(W11, implies(implies(Z11, S10), implies(Y11, S10))))), true, is_a_theorem(implies(implies(a, b), implies(implies(implies(a, b), a), implies(implies(implies(a, b), a), a)))), true), true), true, is_a_theorem(implies(implies(implies(U10, T10), implies(T10, implies(U10, T10))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)))), true), true, is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 27.95/4.85  = { by axiom 3 (ic_JLukasiewicz_4) }
% 27.95/4.85    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(implies(implies(a, b), implies(implies(implies(a, b), a), implies(implies(implies(a, b), a), a))), implies(implies(implies(a, b), a), a)), implies(implies(implies(implies(S10, X11), implies(Y11, Z11)), implies(W11, implies(implies(Z11, S10), implies(Y11, S10)))), implies(implies(a, implies(a, b)), implies(implies(implies(a, b), a), implies(a, b)))))), true, is_a_theorem(implies(implies(implies(implies(V11, U11), implies(T11, S11)), implies(X12, implies(implies(S11, V11), implies(T11, V11)))), implies(implies(implies(implies(a, implies(a, b)), implies(implies(implies(a, b), a), implies(a, b))), implies(implies(a, b), implies(implies(implies(a, b), a), implies(implies(implies(a, b), a), a)))), implies(implies(implies(implies(S10, X11), implies(Y11, Z11)), implies(W11, implies(implies(Z11, S10), implies(Y11, S10)))), implies(implies(a, b), implies(implies(implies(a, b), a), implies(implies(implies(a, b), a), a))))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(V11, U11), implies(T11, S11)), implies(X12, implies(implies(S11, V11), implies(T11, V11))))), true, is_a_theorem(implies(implies(implies(implies(a, implies(a, b)), implies(implies(implies(a, b), a), implies(a, b))), implies(implies(a, b), implies(implies(implies(a, b), a), implies(implies(implies(a, b), a), a)))), implies(implies(implies(implies(S10, X11), implies(Y11, Z11)), implies(W11, implies(implies(Z11, S10), implies(Y11, S10)))), implies(implies(a, b), implies(implies(implies(a, b), a), implies(implies(implies(a, b), a), a)))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(a, implies(a, b)), implies(implies(implies(a, b), a), implies(a, b))), implies(implies(a, b), implies(implies(implies(a, b), a), implies(implies(implies(a, b), a), a))))), true, is_a_theorem(implies(implies(implies(implies(S10, X11), implies(Y11, Z11)), implies(W11, implies(implies(Z11, S10), implies(Y11, S10)))), implies(implies(a, b), implies(implies(implies(a, b), a), implies(implies(implies(a, b), a), a))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(S10, X11), implies(Y11, Z11)), implies(W11, implies(implies(Z11, S10), implies(Y11, S10))))), true, is_a_theorem(implies(implies(a, b), implies(implies(implies(a, b), a), implies(implies(implies(a, b), a), a)))), true), true), true, is_a_theorem(implies(implies(implies(U10, T10), implies(T10, implies(U10, T10))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)))), true), true, is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 27.95/4.85  = { by axiom 1 (ifeq_axiom) }
% 27.95/4.86    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(a, b), implies(implies(implies(a, b), a), implies(implies(implies(a, b), a), a))), implies(implies(implies(a, b), a), a)), implies(implies(implies(implies(S10, X11), implies(Y11, Z11)), implies(W11, implies(implies(Z11, S10), implies(Y11, S10)))), implies(implies(a, implies(a, b)), implies(implies(implies(a, b), a), implies(a, b)))))), true, is_a_theorem(implies(implies(implies(implies(V11, U11), implies(T11, S11)), implies(X12, implies(implies(S11, V11), implies(T11, V11)))), implies(implies(implies(implies(a, implies(a, b)), implies(implies(implies(a, b), a), implies(a, b))), implies(implies(a, b), implies(implies(implies(a, b), a), implies(implies(implies(a, b), a), a)))), implies(implies(implies(implies(S10, X11), implies(Y11, Z11)), implies(W11, implies(implies(Z11, S10), implies(Y11, S10)))), implies(implies(a, b), implies(implies(implies(a, b), a), implies(implies(implies(a, b), a), a))))))), true), true, ifeq(is_a_theorem(implies(implies(implies(V11, U11), implies(T11, S11)), implies(X12, implies(implies(S11, V11), implies(T11, V11))))), true, is_a_theorem(implies(implies(implies(implies(a, implies(a, b)), implies(implies(implies(a, b), a), implies(a, b))), implies(implies(a, b), implies(implies(implies(a, b), a), implies(implies(implies(a, b), a), a)))), implies(implies(implies(implies(S10, X11), implies(Y11, Z11)), implies(W11, implies(implies(Z11, S10), implies(Y11, S10)))), implies(implies(a, b), implies(implies(implies(a, b), a), implies(implies(implies(a, b), a), a)))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(a, implies(a, b)), implies(implies(implies(a, b), a), implies(a, b))), implies(implies(a, b), implies(implies(implies(a, b), a), implies(implies(implies(a, b), a), a))))), true, is_a_theorem(implies(implies(implies(implies(S10, X11), implies(Y11, Z11)), implies(W11, implies(implies(Z11, S10), implies(Y11, S10)))), implies(implies(a, b), implies(implies(implies(a, b), a), implies(implies(implies(a, b), a), a))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(S10, X11), implies(Y11, Z11)), implies(W11, implies(implies(Z11, S10), implies(Y11, S10))))), true, is_a_theorem(implies(implies(a, b), implies(implies(implies(a, b), a), implies(implies(implies(a, b), a), a)))), true), true), true, is_a_theorem(implies(implies(implies(U10, T10), implies(T10, implies(U10, T10))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)))), true), true, is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 27.95/4.86  = { by axiom 3 (ic_JLukasiewicz_4) }
% 27.95/4.86    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(implies(V11, U11), implies(T11, S11)), implies(X12, implies(implies(S11, V11), implies(T11, V11)))), implies(implies(implies(implies(a, implies(a, b)), implies(implies(implies(a, b), a), implies(a, b))), implies(implies(a, b), implies(implies(implies(a, b), a), implies(implies(implies(a, b), a), a)))), implies(implies(implies(implies(S10, X11), implies(Y11, Z11)), implies(W11, implies(implies(Z11, S10), implies(Y11, S10)))), implies(implies(a, b), implies(implies(implies(a, b), a), implies(implies(implies(a, b), a), a))))))), true), true, ifeq(is_a_theorem(implies(implies(implies(V11, U11), implies(T11, S11)), implies(X12, implies(implies(S11, V11), implies(T11, V11))))), true, is_a_theorem(implies(implies(implies(implies(a, implies(a, b)), implies(implies(implies(a, b), a), implies(a, b))), implies(implies(a, b), implies(implies(implies(a, b), a), implies(implies(implies(a, b), a), a)))), implies(implies(implies(implies(S10, X11), implies(Y11, Z11)), implies(W11, implies(implies(Z11, S10), implies(Y11, S10)))), implies(implies(a, b), implies(implies(implies(a, b), a), implies(implies(implies(a, b), a), a)))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(a, implies(a, b)), implies(implies(implies(a, b), a), implies(a, b))), implies(implies(a, b), implies(implies(implies(a, b), a), implies(implies(implies(a, b), a), a))))), true, is_a_theorem(implies(implies(implies(implies(S10, X11), implies(Y11, Z11)), implies(W11, implies(implies(Z11, S10), implies(Y11, S10)))), implies(implies(a, b), implies(implies(implies(a, b), a), implies(implies(implies(a, b), a), a))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(S10, X11), implies(Y11, Z11)), implies(W11, implies(implies(Z11, S10), implies(Y11, S10))))), true, is_a_theorem(implies(implies(a, b), implies(implies(implies(a, b), a), implies(implies(implies(a, b), a), a)))), true), true), true, is_a_theorem(implies(implies(implies(U10, T10), implies(T10, implies(U10, T10))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)))), true), true, is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 27.95/4.86  = { by axiom 1 (ifeq_axiom) }
% 27.95/4.87    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(V11, U11), implies(T11, S11)), implies(X12, implies(implies(S11, V11), implies(T11, V11)))), implies(implies(implies(implies(a, implies(a, b)), implies(implies(implies(a, b), a), implies(a, b))), implies(implies(a, b), implies(implies(implies(a, b), a), implies(implies(implies(a, b), a), a)))), implies(implies(implies(implies(S10, X11), implies(Y11, Z11)), implies(W11, implies(implies(Z11, S10), implies(Y11, S10)))), implies(implies(a, b), implies(implies(implies(a, b), a), implies(implies(implies(a, b), a), a))))))), true, ifeq(is_a_theorem(implies(implies(implies(V11, U11), implies(T11, S11)), implies(X12, implies(implies(S11, V11), implies(T11, V11))))), true, is_a_theorem(implies(implies(implies(implies(a, implies(a, b)), implies(implies(implies(a, b), a), implies(a, b))), implies(implies(a, b), implies(implies(implies(a, b), a), implies(implies(implies(a, b), a), a)))), implies(implies(implies(implies(S10, X11), implies(Y11, Z11)), implies(W11, implies(implies(Z11, S10), implies(Y11, S10)))), implies(implies(a, b), implies(implies(implies(a, b), a), implies(implies(implies(a, b), a), a)))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(a, implies(a, b)), implies(implies(implies(a, b), a), implies(a, b))), implies(implies(a, b), implies(implies(implies(a, b), a), implies(implies(implies(a, b), a), a))))), true, is_a_theorem(implies(implies(implies(implies(S10, X11), implies(Y11, Z11)), implies(W11, implies(implies(Z11, S10), implies(Y11, S10)))), implies(implies(a, b), implies(implies(implies(a, b), a), implies(implies(implies(a, b), a), a))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(S10, X11), implies(Y11, Z11)), implies(W11, implies(implies(Z11, S10), implies(Y11, S10))))), true, is_a_theorem(implies(implies(a, b), implies(implies(implies(a, b), a), implies(implies(implies(a, b), a), a)))), true), true), true, is_a_theorem(implies(implies(implies(U10, T10), implies(T10, implies(U10, T10))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)))), true), true, is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 27.95/4.87  = { by axiom 2 (condensed_detachment) }
% 27.95/4.87    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(implies(a, implies(a, b)), implies(implies(implies(a, b), a), implies(a, b))), implies(implies(a, b), implies(implies(implies(a, b), a), implies(implies(implies(a, b), a), a))))), true, is_a_theorem(implies(implies(implies(implies(S10, X11), implies(Y11, Z11)), implies(W11, implies(implies(Z11, S10), implies(Y11, S10)))), implies(implies(a, b), implies(implies(implies(a, b), a), implies(implies(implies(a, b), a), a))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(S10, X11), implies(Y11, Z11)), implies(W11, implies(implies(Z11, S10), implies(Y11, S10))))), true, is_a_theorem(implies(implies(a, b), implies(implies(implies(a, b), a), implies(implies(implies(a, b), a), a)))), true), true), true, is_a_theorem(implies(implies(implies(U10, T10), implies(T10, implies(U10, T10))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)))), true), true, is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 27.95/4.87  = { by axiom 1 (ifeq_axiom) }
% 27.95/4.87    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(a, implies(a, b)), implies(implies(implies(a, b), a), implies(a, b))), implies(implies(a, b), implies(implies(implies(a, b), a), implies(implies(implies(a, b), a), a))))), true, is_a_theorem(implies(implies(implies(implies(S10, X11), implies(Y11, Z11)), implies(W11, implies(implies(Z11, S10), implies(Y11, S10)))), implies(implies(a, b), implies(implies(implies(a, b), a), implies(implies(implies(a, b), a), a))))), true), true, ifeq(is_a_theorem(implies(implies(implies(S10, X11), implies(Y11, Z11)), implies(W11, implies(implies(Z11, S10), implies(Y11, S10))))), true, is_a_theorem(implies(implies(a, b), implies(implies(implies(a, b), a), implies(implies(implies(a, b), a), a)))), true), true), true, is_a_theorem(implies(implies(implies(U10, T10), implies(T10, implies(U10, T10))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)))), true), true, is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 27.95/4.87  = { by axiom 3 (ic_JLukasiewicz_4) }
% 27.95/4.87    ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(implies(S10, X11), implies(Y11, Z11)), implies(W11, implies(implies(Z11, S10), implies(Y11, S10)))), implies(implies(a, b), implies(implies(implies(a, b), a), implies(implies(implies(a, b), a), a))))), true), true, ifeq(is_a_theorem(implies(implies(implies(S10, X11), implies(Y11, Z11)), implies(W11, implies(implies(Z11, S10), implies(Y11, S10))))), true, is_a_theorem(implies(implies(a, b), implies(implies(implies(a, b), a), implies(implies(implies(a, b), a), a)))), true), true), true, is_a_theorem(implies(implies(implies(U10, T10), implies(T10, implies(U10, T10))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)))), true), true, is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 27.95/4.87  = { by axiom 1 (ifeq_axiom) }
% 27.95/4.87    ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(S10, X11), implies(Y11, Z11)), implies(W11, implies(implies(Z11, S10), implies(Y11, S10)))), implies(implies(a, b), implies(implies(implies(a, b), a), implies(implies(implies(a, b), a), a))))), true, ifeq(is_a_theorem(implies(implies(implies(S10, X11), implies(Y11, Z11)), implies(W11, implies(implies(Z11, S10), implies(Y11, S10))))), true, is_a_theorem(implies(implies(a, b), implies(implies(implies(a, b), a), implies(implies(implies(a, b), a), a)))), true), true), true, is_a_theorem(implies(implies(implies(U10, T10), implies(T10, implies(U10, T10))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)))), true), true, is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 27.95/4.87  = { by axiom 2 (condensed_detachment) }
% 27.95/4.87    ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(U10, T10), implies(T10, implies(U10, T10))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)))), true), true, is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 27.95/4.88  = { by axiom 1 (ifeq_axiom) }
% 27.95/4.88    ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(U10, T10), implies(T10, implies(U10, T10))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)))), true, is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 27.95/4.88  = { by axiom 1 (ifeq_axiom) R->L }
% 27.95/4.88    ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(U10, T10), implies(T10, implies(U10, T10))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)))), true, ifeq(true, true, is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 27.95/4.88  = { by axiom 2 (condensed_detachment) R->L }
% 27.95/4.88    ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(U10, T10), implies(T10, implies(U10, T10))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)))), true, ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(Y12, Z12), implies(W12, V12)), implies(U12, implies(implies(V12, Y12), implies(W12, Y12)))), implies(implies(U10, T10), implies(T10, implies(U10, T10))))), true, ifeq(is_a_theorem(implies(implies(implies(Y12, Z12), implies(W12, V12)), implies(U12, implies(implies(V12, Y12), implies(W12, Y12))))), true, is_a_theorem(implies(implies(U10, T10), implies(T10, implies(U10, T10)))), true), true), true, is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 27.95/4.88  = { by axiom 1 (ifeq_axiom) R->L }
% 27.95/4.88    ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(U10, T10), implies(T10, implies(U10, T10))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)))), true, ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(implies(Y12, Z12), implies(W12, V12)), implies(U12, implies(implies(V12, Y12), implies(W12, Y12)))), implies(implies(U10, T10), implies(T10, implies(U10, T10))))), true), true, ifeq(is_a_theorem(implies(implies(implies(Y12, Z12), implies(W12, V12)), implies(U12, implies(implies(V12, Y12), implies(W12, Y12))))), true, is_a_theorem(implies(implies(U10, T10), implies(T10, implies(U10, T10)))), true), true), true, is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 27.95/4.89  = { by axiom 2 (condensed_detachment) R->L }
% 27.95/4.89    ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(U10, T10), implies(T10, implies(U10, T10))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)))), true, ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(T12, S12), implies(X13, Y13)), implies(Z13, implies(implies(Y13, T12), implies(X13, T12)))), implies(implies(implies(implies(W13, T10), implies(U10, T10)), implies(T10, implies(U10, T10))), implies(implies(U10, T10), implies(T10, implies(U10, T10)))))), true, ifeq(is_a_theorem(implies(implies(implies(T12, S12), implies(X13, Y13)), implies(Z13, implies(implies(Y13, T12), implies(X13, T12))))), true, is_a_theorem(implies(implies(implies(implies(W13, T10), implies(U10, T10)), implies(T10, implies(U10, T10))), implies(implies(U10, T10), implies(T10, implies(U10, T10))))), true), true), true, is_a_theorem(implies(implies(implies(implies(Y12, Z12), implies(W12, V12)), implies(U12, implies(implies(V12, Y12), implies(W12, Y12)))), implies(implies(U10, T10), implies(T10, implies(U10, T10))))), true), true, ifeq(is_a_theorem(implies(implies(implies(Y12, Z12), implies(W12, V12)), implies(U12, implies(implies(V12, Y12), implies(W12, Y12))))), true, is_a_theorem(implies(implies(U10, T10), implies(T10, implies(U10, T10)))), true), true), true, is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 27.95/4.89  = { by axiom 1 (ifeq_axiom) R->L }
% 27.95/4.89    ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(U10, T10), implies(T10, implies(U10, T10))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)))), true, ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(implies(T12, S12), implies(X13, Y13)), implies(Z13, implies(implies(Y13, T12), implies(X13, T12)))), implies(implies(implies(implies(W13, T10), implies(U10, T10)), implies(T10, implies(U10, T10))), implies(implies(U10, T10), implies(T10, implies(U10, T10)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(T12, S12), implies(X13, Y13)), implies(Z13, implies(implies(Y13, T12), implies(X13, T12))))), true, is_a_theorem(implies(implies(implies(implies(W13, T10), implies(U10, T10)), implies(T10, implies(U10, T10))), implies(implies(U10, T10), implies(T10, implies(U10, T10))))), true), true), true, is_a_theorem(implies(implies(implies(implies(Y12, Z12), implies(W12, V12)), implies(U12, implies(implies(V12, Y12), implies(W12, Y12)))), implies(implies(U10, T10), implies(T10, implies(U10, T10))))), true), true, ifeq(is_a_theorem(implies(implies(implies(Y12, Z12), implies(W12, V12)), implies(U12, implies(implies(V12, Y12), implies(W12, Y12))))), true, is_a_theorem(implies(implies(U10, T10), implies(T10, implies(U10, T10)))), true), true), true, is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 27.95/4.89  = { by axiom 3 (ic_JLukasiewicz_4) R->L }
% 27.95/4.90    ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(U10, T10), implies(T10, implies(U10, T10))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)))), true, ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(T10, implies(U10, T10)), implies(U10, W13)), implies(implies(U10, T10), implies(implies(W13, T10), implies(U10, T10))))), true, is_a_theorem(implies(implies(implies(implies(T12, S12), implies(X13, Y13)), implies(Z13, implies(implies(Y13, T12), implies(X13, T12)))), implies(implies(implies(implies(W13, T10), implies(U10, T10)), implies(T10, implies(U10, T10))), implies(implies(U10, T10), implies(T10, implies(U10, T10)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(T12, S12), implies(X13, Y13)), implies(Z13, implies(implies(Y13, T12), implies(X13, T12))))), true, is_a_theorem(implies(implies(implies(implies(W13, T10), implies(U10, T10)), implies(T10, implies(U10, T10))), implies(implies(U10, T10), implies(T10, implies(U10, T10))))), true), true), true, is_a_theorem(implies(implies(implies(implies(Y12, Z12), implies(W12, V12)), implies(U12, implies(implies(V12, Y12), implies(W12, Y12)))), implies(implies(U10, T10), implies(T10, implies(U10, T10))))), true), true, ifeq(is_a_theorem(implies(implies(implies(Y12, Z12), implies(W12, V12)), implies(U12, implies(implies(V12, Y12), implies(W12, Y12))))), true, is_a_theorem(implies(implies(U10, T10), implies(T10, implies(U10, T10)))), true), true), true, is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 27.95/4.90  = { by axiom 1 (ifeq_axiom) R->L }
% 28.34/4.90    ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(U10, T10), implies(T10, implies(U10, T10))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)))), true, ifeq(ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(implies(T10, implies(U10, T10)), implies(U10, W13)), implies(implies(U10, T10), implies(implies(W13, T10), implies(U10, T10))))), true, is_a_theorem(implies(implies(implies(implies(T12, S12), implies(X13, Y13)), implies(Z13, implies(implies(Y13, T12), implies(X13, T12)))), implies(implies(implies(implies(W13, T10), implies(U10, T10)), implies(T10, implies(U10, T10))), implies(implies(U10, T10), implies(T10, implies(U10, T10)))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(T12, S12), implies(X13, Y13)), implies(Z13, implies(implies(Y13, T12), implies(X13, T12))))), true, is_a_theorem(implies(implies(implies(implies(W13, T10), implies(U10, T10)), implies(T10, implies(U10, T10))), implies(implies(U10, T10), implies(T10, implies(U10, T10))))), true), true), true, is_a_theorem(implies(implies(implies(implies(Y12, Z12), implies(W12, V12)), implies(U12, implies(implies(V12, Y12), implies(W12, Y12)))), implies(implies(U10, T10), implies(T10, implies(U10, T10))))), true), true, ifeq(is_a_theorem(implies(implies(implies(Y12, Z12), implies(W12, V12)), implies(U12, implies(implies(V12, Y12), implies(W12, Y12))))), true, is_a_theorem(implies(implies(U10, T10), implies(T10, implies(U10, T10)))), true), true), true, is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 28.34/4.90  = { by axiom 3 (ic_JLukasiewicz_4) R->L }
% 28.34/4.90    ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(U10, T10), implies(T10, implies(U10, T10))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)))), true, ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(T10, implies(U10, T10)), implies(U10, W13)), implies(implies(U10, T10), implies(implies(W13, T10), implies(U10, T10)))), implies(implies(implies(implies(T12, S12), implies(X13, Y13)), implies(Z13, implies(implies(Y13, T12), implies(X13, T12)))), implies(implies(implies(implies(W13, T10), implies(U10, T10)), implies(T10, implies(U10, T10))), implies(implies(U10, T10), implies(T10, implies(U10, T10))))))), true, ifeq(is_a_theorem(implies(implies(implies(T10, implies(U10, T10)), implies(U10, W13)), implies(implies(U10, T10), implies(implies(W13, T10), implies(U10, T10))))), true, is_a_theorem(implies(implies(implies(implies(T12, S12), implies(X13, Y13)), implies(Z13, implies(implies(Y13, T12), implies(X13, T12)))), implies(implies(implies(implies(W13, T10), implies(U10, T10)), implies(T10, implies(U10, T10))), implies(implies(U10, T10), implies(T10, implies(U10, T10)))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(T12, S12), implies(X13, Y13)), implies(Z13, implies(implies(Y13, T12), implies(X13, T12))))), true, is_a_theorem(implies(implies(implies(implies(W13, T10), implies(U10, T10)), implies(T10, implies(U10, T10))), implies(implies(U10, T10), implies(T10, implies(U10, T10))))), true), true), true, is_a_theorem(implies(implies(implies(implies(Y12, Z12), implies(W12, V12)), implies(U12, implies(implies(V12, Y12), implies(W12, Y12)))), implies(implies(U10, T10), implies(T10, implies(U10, T10))))), true), true, ifeq(is_a_theorem(implies(implies(implies(Y12, Z12), implies(W12, V12)), implies(U12, implies(implies(V12, Y12), implies(W12, Y12))))), true, is_a_theorem(implies(implies(U10, T10), implies(T10, implies(U10, T10)))), true), true), true, is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 28.34/4.91  = { by axiom 2 (condensed_detachment) }
% 28.34/4.91    ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(U10, T10), implies(T10, implies(U10, T10))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)))), true, ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(implies(T12, S12), implies(X13, Y13)), implies(Z13, implies(implies(Y13, T12), implies(X13, T12))))), true, is_a_theorem(implies(implies(implies(implies(W13, T10), implies(U10, T10)), implies(T10, implies(U10, T10))), implies(implies(U10, T10), implies(T10, implies(U10, T10))))), true), true), true, is_a_theorem(implies(implies(implies(implies(Y12, Z12), implies(W12, V12)), implies(U12, implies(implies(V12, Y12), implies(W12, Y12)))), implies(implies(U10, T10), implies(T10, implies(U10, T10))))), true), true, ifeq(is_a_theorem(implies(implies(implies(Y12, Z12), implies(W12, V12)), implies(U12, implies(implies(V12, Y12), implies(W12, Y12))))), true, is_a_theorem(implies(implies(U10, T10), implies(T10, implies(U10, T10)))), true), true), true, is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 28.34/4.91  = { by axiom 1 (ifeq_axiom) }
% 28.34/4.91    ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(U10, T10), implies(T10, implies(U10, T10))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)))), true, ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(T12, S12), implies(X13, Y13)), implies(Z13, implies(implies(Y13, T12), implies(X13, T12))))), true, is_a_theorem(implies(implies(implies(implies(W13, T10), implies(U10, T10)), implies(T10, implies(U10, T10))), implies(implies(U10, T10), implies(T10, implies(U10, T10))))), true), true, is_a_theorem(implies(implies(implies(implies(Y12, Z12), implies(W12, V12)), implies(U12, implies(implies(V12, Y12), implies(W12, Y12)))), implies(implies(U10, T10), implies(T10, implies(U10, T10))))), true), true, ifeq(is_a_theorem(implies(implies(implies(Y12, Z12), implies(W12, V12)), implies(U12, implies(implies(V12, Y12), implies(W12, Y12))))), true, is_a_theorem(implies(implies(U10, T10), implies(T10, implies(U10, T10)))), true), true), true, is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 28.34/4.91  = { by axiom 3 (ic_JLukasiewicz_4) }
% 28.34/4.91    ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(U10, T10), implies(T10, implies(U10, T10))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)))), true, ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(implies(W13, T10), implies(U10, T10)), implies(T10, implies(U10, T10))), implies(implies(U10, T10), implies(T10, implies(U10, T10))))), true), true, is_a_theorem(implies(implies(implies(implies(Y12, Z12), implies(W12, V12)), implies(U12, implies(implies(V12, Y12), implies(W12, Y12)))), implies(implies(U10, T10), implies(T10, implies(U10, T10))))), true), true, ifeq(is_a_theorem(implies(implies(implies(Y12, Z12), implies(W12, V12)), implies(U12, implies(implies(V12, Y12), implies(W12, Y12))))), true, is_a_theorem(implies(implies(U10, T10), implies(T10, implies(U10, T10)))), true), true), true, is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 28.34/4.91  = { by axiom 1 (ifeq_axiom) }
% 28.34/4.92    ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(U10, T10), implies(T10, implies(U10, T10))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)))), true, ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(W13, T10), implies(U10, T10)), implies(T10, implies(U10, T10))), implies(implies(U10, T10), implies(T10, implies(U10, T10))))), true, is_a_theorem(implies(implies(implies(implies(Y12, Z12), implies(W12, V12)), implies(U12, implies(implies(V12, Y12), implies(W12, Y12)))), implies(implies(U10, T10), implies(T10, implies(U10, T10))))), true), true, ifeq(is_a_theorem(implies(implies(implies(Y12, Z12), implies(W12, V12)), implies(U12, implies(implies(V12, Y12), implies(W12, Y12))))), true, is_a_theorem(implies(implies(U10, T10), implies(T10, implies(U10, T10)))), true), true), true, is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 28.34/4.92  = { by axiom 1 (ifeq_axiom) R->L }
% 28.34/4.92    ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(U10, T10), implies(T10, implies(U10, T10))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)))), true, ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(implies(implies(W13, T10), implies(U10, T10)), implies(T10, implies(U10, T10))), implies(implies(U10, T10), implies(T10, implies(U10, T10))))), true, is_a_theorem(implies(implies(implies(implies(Y12, Z12), implies(W12, V12)), implies(U12, implies(implies(V12, Y12), implies(W12, Y12)))), implies(implies(U10, T10), implies(T10, implies(U10, T10))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(Y12, Z12), implies(W12, V12)), implies(U12, implies(implies(V12, Y12), implies(W12, Y12))))), true, is_a_theorem(implies(implies(U10, T10), implies(T10, implies(U10, T10)))), true), true), true, is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 28.34/4.92  = { by axiom 2 (condensed_detachment) R->L }
% 28.34/4.93    ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(U10, T10), implies(T10, implies(U10, T10))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)))), true, ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(V13, U13), implies(T13, S13)), implies(X14, implies(implies(S13, V13), implies(T13, V13)))), implies(implies(implies(implies(implies(W13, T10), implies(U10, T10)), implies(T10, implies(U10, T10))), implies(implies(U10, T10), implies(T10, implies(U10, T10)))), implies(implies(implies(implies(Y12, Z12), implies(W12, V12)), implies(U12, implies(implies(V12, Y12), implies(W12, Y12)))), implies(implies(U10, T10), implies(T10, implies(U10, T10))))))), true, ifeq(is_a_theorem(implies(implies(implies(V13, U13), implies(T13, S13)), implies(X14, implies(implies(S13, V13), implies(T13, V13))))), true, is_a_theorem(implies(implies(implies(implies(implies(W13, T10), implies(U10, T10)), implies(T10, implies(U10, T10))), implies(implies(U10, T10), implies(T10, implies(U10, T10)))), implies(implies(implies(implies(Y12, Z12), implies(W12, V12)), implies(U12, implies(implies(V12, Y12), implies(W12, Y12)))), implies(implies(U10, T10), implies(T10, implies(U10, T10)))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(W13, T10), implies(U10, T10)), implies(T10, implies(U10, T10))), implies(implies(U10, T10), implies(T10, implies(U10, T10))))), true, is_a_theorem(implies(implies(implies(implies(Y12, Z12), implies(W12, V12)), implies(U12, implies(implies(V12, Y12), implies(W12, Y12)))), implies(implies(U10, T10), implies(T10, implies(U10, T10))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(Y12, Z12), implies(W12, V12)), implies(U12, implies(implies(V12, Y12), implies(W12, Y12))))), true, is_a_theorem(implies(implies(U10, T10), implies(T10, implies(U10, T10)))), true), true), true, is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 28.34/4.93  = { by axiom 1 (ifeq_axiom) R->L }
% 28.34/4.93    ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(U10, T10), implies(T10, implies(U10, T10))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)))), true, ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(implies(V13, U13), implies(T13, S13)), implies(X14, implies(implies(S13, V13), implies(T13, V13)))), implies(implies(implies(implies(implies(W13, T10), implies(U10, T10)), implies(T10, implies(U10, T10))), implies(implies(U10, T10), implies(T10, implies(U10, T10)))), implies(implies(implies(implies(Y12, Z12), implies(W12, V12)), implies(U12, implies(implies(V12, Y12), implies(W12, Y12)))), implies(implies(U10, T10), implies(T10, implies(U10, T10))))))), true), true, ifeq(is_a_theorem(implies(implies(implies(V13, U13), implies(T13, S13)), implies(X14, implies(implies(S13, V13), implies(T13, V13))))), true, is_a_theorem(implies(implies(implies(implies(implies(W13, T10), implies(U10, T10)), implies(T10, implies(U10, T10))), implies(implies(U10, T10), implies(T10, implies(U10, T10)))), implies(implies(implies(implies(Y12, Z12), implies(W12, V12)), implies(U12, implies(implies(V12, Y12), implies(W12, Y12)))), implies(implies(U10, T10), implies(T10, implies(U10, T10)))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(W13, T10), implies(U10, T10)), implies(T10, implies(U10, T10))), implies(implies(U10, T10), implies(T10, implies(U10, T10))))), true, is_a_theorem(implies(implies(implies(implies(Y12, Z12), implies(W12, V12)), implies(U12, implies(implies(V12, Y12), implies(W12, Y12)))), implies(implies(U10, T10), implies(T10, implies(U10, T10))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(Y12, Z12), implies(W12, V12)), implies(U12, implies(implies(V12, Y12), implies(W12, Y12))))), true, is_a_theorem(implies(implies(U10, T10), implies(T10, implies(U10, T10)))), true), true), true, is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 28.34/4.93  = { by axiom 3 (ic_JLukasiewicz_4) R->L }
% 28.34/4.94    ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(U10, T10), implies(T10, implies(U10, T10))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)))), true, ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(U10, T10), implies(T10, implies(U10, T10))), implies(T10, implies(W13, T10))), implies(implies(implies(implies(Y12, Z12), implies(W12, V12)), implies(U12, implies(implies(V12, Y12), implies(W12, Y12)))), implies(implies(implies(W13, T10), implies(U10, T10)), implies(T10, implies(U10, T10)))))), true, is_a_theorem(implies(implies(implies(implies(V13, U13), implies(T13, S13)), implies(X14, implies(implies(S13, V13), implies(T13, V13)))), implies(implies(implies(implies(implies(W13, T10), implies(U10, T10)), implies(T10, implies(U10, T10))), implies(implies(U10, T10), implies(T10, implies(U10, T10)))), implies(implies(implies(implies(Y12, Z12), implies(W12, V12)), implies(U12, implies(implies(V12, Y12), implies(W12, Y12)))), implies(implies(U10, T10), implies(T10, implies(U10, T10))))))), true), true, ifeq(is_a_theorem(implies(implies(implies(V13, U13), implies(T13, S13)), implies(X14, implies(implies(S13, V13), implies(T13, V13))))), true, is_a_theorem(implies(implies(implies(implies(implies(W13, T10), implies(U10, T10)), implies(T10, implies(U10, T10))), implies(implies(U10, T10), implies(T10, implies(U10, T10)))), implies(implies(implies(implies(Y12, Z12), implies(W12, V12)), implies(U12, implies(implies(V12, Y12), implies(W12, Y12)))), implies(implies(U10, T10), implies(T10, implies(U10, T10)))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(W13, T10), implies(U10, T10)), implies(T10, implies(U10, T10))), implies(implies(U10, T10), implies(T10, implies(U10, T10))))), true, is_a_theorem(implies(implies(implies(implies(Y12, Z12), implies(W12, V12)), implies(U12, implies(implies(V12, Y12), implies(W12, Y12)))), implies(implies(U10, T10), implies(T10, implies(U10, T10))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(Y12, Z12), implies(W12, V12)), implies(U12, implies(implies(V12, Y12), implies(W12, Y12))))), true, is_a_theorem(implies(implies(U10, T10), implies(T10, implies(U10, T10)))), true), true), true, is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 28.34/4.94  = { by axiom 1 (ifeq_axiom) R->L }
% 28.34/4.94    ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(U10, T10), implies(T10, implies(U10, T10))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)))), true, ifeq(ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(implies(implies(U10, T10), implies(T10, implies(U10, T10))), implies(T10, implies(W13, T10))), implies(implies(implies(implies(Y12, Z12), implies(W12, V12)), implies(U12, implies(implies(V12, Y12), implies(W12, Y12)))), implies(implies(implies(W13, T10), implies(U10, T10)), implies(T10, implies(U10, T10)))))), true, is_a_theorem(implies(implies(implies(implies(V13, U13), implies(T13, S13)), implies(X14, implies(implies(S13, V13), implies(T13, V13)))), implies(implies(implies(implies(implies(W13, T10), implies(U10, T10)), implies(T10, implies(U10, T10))), implies(implies(U10, T10), implies(T10, implies(U10, T10)))), implies(implies(implies(implies(Y12, Z12), implies(W12, V12)), implies(U12, implies(implies(V12, Y12), implies(W12, Y12)))), implies(implies(U10, T10), implies(T10, implies(U10, T10))))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(V13, U13), implies(T13, S13)), implies(X14, implies(implies(S13, V13), implies(T13, V13))))), true, is_a_theorem(implies(implies(implies(implies(implies(W13, T10), implies(U10, T10)), implies(T10, implies(U10, T10))), implies(implies(U10, T10), implies(T10, implies(U10, T10)))), implies(implies(implies(implies(Y12, Z12), implies(W12, V12)), implies(U12, implies(implies(V12, Y12), implies(W12, Y12)))), implies(implies(U10, T10), implies(T10, implies(U10, T10)))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(W13, T10), implies(U10, T10)), implies(T10, implies(U10, T10))), implies(implies(U10, T10), implies(T10, implies(U10, T10))))), true, is_a_theorem(implies(implies(implies(implies(Y12, Z12), implies(W12, V12)), implies(U12, implies(implies(V12, Y12), implies(W12, Y12)))), implies(implies(U10, T10), implies(T10, implies(U10, T10))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(Y12, Z12), implies(W12, V12)), implies(U12, implies(implies(V12, Y12), implies(W12, Y12))))), true, is_a_theorem(implies(implies(U10, T10), implies(T10, implies(U10, T10)))), true), true), true, is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 28.34/4.94  = { by axiom 3 (ic_JLukasiewicz_4) R->L }
% 28.34/4.95    ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(U10, T10), implies(T10, implies(U10, T10))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)))), true, ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(implies(U10, T10), implies(T10, implies(U10, T10))), implies(T10, implies(W13, T10))), implies(implies(implies(implies(Y12, Z12), implies(W12, V12)), implies(U12, implies(implies(V12, Y12), implies(W12, Y12)))), implies(implies(implies(W13, T10), implies(U10, T10)), implies(T10, implies(U10, T10))))), implies(implies(implies(implies(V13, U13), implies(T13, S13)), implies(X14, implies(implies(S13, V13), implies(T13, V13)))), implies(implies(implies(implies(implies(W13, T10), implies(U10, T10)), implies(T10, implies(U10, T10))), implies(implies(U10, T10), implies(T10, implies(U10, T10)))), implies(implies(implies(implies(Y12, Z12), implies(W12, V12)), implies(U12, implies(implies(V12, Y12), implies(W12, Y12)))), implies(implies(U10, T10), implies(T10, implies(U10, T10)))))))), true, ifeq(is_a_theorem(implies(implies(implies(implies(U10, T10), implies(T10, implies(U10, T10))), implies(T10, implies(W13, T10))), implies(implies(implies(implies(Y12, Z12), implies(W12, V12)), implies(U12, implies(implies(V12, Y12), implies(W12, Y12)))), implies(implies(implies(W13, T10), implies(U10, T10)), implies(T10, implies(U10, T10)))))), true, is_a_theorem(implies(implies(implies(implies(V13, U13), implies(T13, S13)), implies(X14, implies(implies(S13, V13), implies(T13, V13)))), implies(implies(implies(implies(implies(W13, T10), implies(U10, T10)), implies(T10, implies(U10, T10))), implies(implies(U10, T10), implies(T10, implies(U10, T10)))), implies(implies(implies(implies(Y12, Z12), implies(W12, V12)), implies(U12, implies(implies(V12, Y12), implies(W12, Y12)))), implies(implies(U10, T10), implies(T10, implies(U10, T10))))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(V13, U13), implies(T13, S13)), implies(X14, implies(implies(S13, V13), implies(T13, V13))))), true, is_a_theorem(implies(implies(implies(implies(implies(W13, T10), implies(U10, T10)), implies(T10, implies(U10, T10))), implies(implies(U10, T10), implies(T10, implies(U10, T10)))), implies(implies(implies(implies(Y12, Z12), implies(W12, V12)), implies(U12, implies(implies(V12, Y12), implies(W12, Y12)))), implies(implies(U10, T10), implies(T10, implies(U10, T10)))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(W13, T10), implies(U10, T10)), implies(T10, implies(U10, T10))), implies(implies(U10, T10), implies(T10, implies(U10, T10))))), true, is_a_theorem(implies(implies(implies(implies(Y12, Z12), implies(W12, V12)), implies(U12, implies(implies(V12, Y12), implies(W12, Y12)))), implies(implies(U10, T10), implies(T10, implies(U10, T10))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(Y12, Z12), implies(W12, V12)), implies(U12, implies(implies(V12, Y12), implies(W12, Y12))))), true, is_a_theorem(implies(implies(U10, T10), implies(T10, implies(U10, T10)))), true), true), true, is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 28.34/4.95  = { by axiom 2 (condensed_detachment) }
% 28.34/4.95    ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(U10, T10), implies(T10, implies(U10, T10))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)))), true, ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(implies(V13, U13), implies(T13, S13)), implies(X14, implies(implies(S13, V13), implies(T13, V13))))), true, is_a_theorem(implies(implies(implies(implies(implies(W13, T10), implies(U10, T10)), implies(T10, implies(U10, T10))), implies(implies(U10, T10), implies(T10, implies(U10, T10)))), implies(implies(implies(implies(Y12, Z12), implies(W12, V12)), implies(U12, implies(implies(V12, Y12), implies(W12, Y12)))), implies(implies(U10, T10), implies(T10, implies(U10, T10)))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(W13, T10), implies(U10, T10)), implies(T10, implies(U10, T10))), implies(implies(U10, T10), implies(T10, implies(U10, T10))))), true, is_a_theorem(implies(implies(implies(implies(Y12, Z12), implies(W12, V12)), implies(U12, implies(implies(V12, Y12), implies(W12, Y12)))), implies(implies(U10, T10), implies(T10, implies(U10, T10))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(Y12, Z12), implies(W12, V12)), implies(U12, implies(implies(V12, Y12), implies(W12, Y12))))), true, is_a_theorem(implies(implies(U10, T10), implies(T10, implies(U10, T10)))), true), true), true, is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 28.34/4.95  = { by axiom 1 (ifeq_axiom) }
% 28.34/4.96    ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(U10, T10), implies(T10, implies(U10, T10))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)))), true, ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(V13, U13), implies(T13, S13)), implies(X14, implies(implies(S13, V13), implies(T13, V13))))), true, is_a_theorem(implies(implies(implies(implies(implies(W13, T10), implies(U10, T10)), implies(T10, implies(U10, T10))), implies(implies(U10, T10), implies(T10, implies(U10, T10)))), implies(implies(implies(implies(Y12, Z12), implies(W12, V12)), implies(U12, implies(implies(V12, Y12), implies(W12, Y12)))), implies(implies(U10, T10), implies(T10, implies(U10, T10)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(W13, T10), implies(U10, T10)), implies(T10, implies(U10, T10))), implies(implies(U10, T10), implies(T10, implies(U10, T10))))), true, is_a_theorem(implies(implies(implies(implies(Y12, Z12), implies(W12, V12)), implies(U12, implies(implies(V12, Y12), implies(W12, Y12)))), implies(implies(U10, T10), implies(T10, implies(U10, T10))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(Y12, Z12), implies(W12, V12)), implies(U12, implies(implies(V12, Y12), implies(W12, Y12))))), true, is_a_theorem(implies(implies(U10, T10), implies(T10, implies(U10, T10)))), true), true), true, is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 28.34/4.96  = { by axiom 3 (ic_JLukasiewicz_4) }
% 28.34/4.96    ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(U10, T10), implies(T10, implies(U10, T10))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)))), true, ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(implies(implies(W13, T10), implies(U10, T10)), implies(T10, implies(U10, T10))), implies(implies(U10, T10), implies(T10, implies(U10, T10)))), implies(implies(implies(implies(Y12, Z12), implies(W12, V12)), implies(U12, implies(implies(V12, Y12), implies(W12, Y12)))), implies(implies(U10, T10), implies(T10, implies(U10, T10)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(W13, T10), implies(U10, T10)), implies(T10, implies(U10, T10))), implies(implies(U10, T10), implies(T10, implies(U10, T10))))), true, is_a_theorem(implies(implies(implies(implies(Y12, Z12), implies(W12, V12)), implies(U12, implies(implies(V12, Y12), implies(W12, Y12)))), implies(implies(U10, T10), implies(T10, implies(U10, T10))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(Y12, Z12), implies(W12, V12)), implies(U12, implies(implies(V12, Y12), implies(W12, Y12))))), true, is_a_theorem(implies(implies(U10, T10), implies(T10, implies(U10, T10)))), true), true), true, is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 28.34/4.96  = { by axiom 1 (ifeq_axiom) }
% 28.34/4.96    ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(U10, T10), implies(T10, implies(U10, T10))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)))), true, ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(implies(W13, T10), implies(U10, T10)), implies(T10, implies(U10, T10))), implies(implies(U10, T10), implies(T10, implies(U10, T10)))), implies(implies(implies(implies(Y12, Z12), implies(W12, V12)), implies(U12, implies(implies(V12, Y12), implies(W12, Y12)))), implies(implies(U10, T10), implies(T10, implies(U10, T10)))))), true, ifeq(is_a_theorem(implies(implies(implies(implies(W13, T10), implies(U10, T10)), implies(T10, implies(U10, T10))), implies(implies(U10, T10), implies(T10, implies(U10, T10))))), true, is_a_theorem(implies(implies(implies(implies(Y12, Z12), implies(W12, V12)), implies(U12, implies(implies(V12, Y12), implies(W12, Y12)))), implies(implies(U10, T10), implies(T10, implies(U10, T10))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(Y12, Z12), implies(W12, V12)), implies(U12, implies(implies(V12, Y12), implies(W12, Y12))))), true, is_a_theorem(implies(implies(U10, T10), implies(T10, implies(U10, T10)))), true), true), true, is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 28.34/4.96  = { by axiom 2 (condensed_detachment) }
% 28.34/4.96    ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(U10, T10), implies(T10, implies(U10, T10))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)))), true, ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(implies(Y12, Z12), implies(W12, V12)), implies(U12, implies(implies(V12, Y12), implies(W12, Y12))))), true, is_a_theorem(implies(implies(U10, T10), implies(T10, implies(U10, T10)))), true), true), true, is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 28.34/4.96  = { by axiom 1 (ifeq_axiom) }
% 28.34/4.96    ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(U10, T10), implies(T10, implies(U10, T10))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)))), true, ifeq(ifeq(is_a_theorem(implies(implies(implies(Y12, Z12), implies(W12, V12)), implies(U12, implies(implies(V12, Y12), implies(W12, Y12))))), true, is_a_theorem(implies(implies(U10, T10), implies(T10, implies(U10, T10)))), true), true, is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 28.34/4.96  = { by axiom 3 (ic_JLukasiewicz_4) }
% 28.34/4.96    ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(U10, T10), implies(T10, implies(U10, T10))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)))), true, ifeq(ifeq(true, true, is_a_theorem(implies(implies(U10, T10), implies(T10, implies(U10, T10)))), true), true, is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 28.34/4.97  = { by axiom 1 (ifeq_axiom) }
% 28.34/4.97    ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(U10, T10), implies(T10, implies(U10, T10))), implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a)))), true, ifeq(is_a_theorem(implies(implies(U10, T10), implies(T10, implies(U10, T10)))), true, is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), a), implies(implies(implies(a, b), a), a))), true), true), true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 28.34/4.97  = { by axiom 2 (condensed_detachment) }
% 28.34/4.97    ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 28.34/4.97  = { by axiom 1 (ifeq_axiom) }
% 28.34/4.97    ifeq(is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true)
% 28.34/4.97  = { by axiom 1 (ifeq_axiom) R->L }
% 28.34/4.97    ifeq(is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true, ifeq(true, true, is_a_theorem(implies(implies(implies(a, b), a), a)), true), true)
% 28.34/4.97  = { by axiom 2 (condensed_detachment) R->L }
% 28.34/4.97    ifeq(is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true, ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(Y14, Z14), implies(W14, V14)), implies(U14, implies(implies(V14, Y14), implies(W14, Y14)))), implies(implies(X, Y), implies(Y, implies(X, Y))))), true, ifeq(is_a_theorem(implies(implies(implies(Y14, Z14), implies(W14, V14)), implies(U14, implies(implies(V14, Y14), implies(W14, Y14))))), true, is_a_theorem(implies(implies(X, Y), implies(Y, implies(X, Y)))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true), true)
% 28.34/4.97  = { by axiom 1 (ifeq_axiom) R->L }
% 28.34/4.97    ifeq(is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true, ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(implies(Y14, Z14), implies(W14, V14)), implies(U14, implies(implies(V14, Y14), implies(W14, Y14)))), implies(implies(X, Y), implies(Y, implies(X, Y))))), true), true, ifeq(is_a_theorem(implies(implies(implies(Y14, Z14), implies(W14, V14)), implies(U14, implies(implies(V14, Y14), implies(W14, Y14))))), true, is_a_theorem(implies(implies(X, Y), implies(Y, implies(X, Y)))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true), true)
% 28.34/4.97  = { by axiom 2 (condensed_detachment) R->L }
% 28.34/4.97    ifeq(is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true, ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(T14, S14), implies(X15, Y15)), implies(Z15, implies(implies(Y15, T14), implies(X15, T14)))), implies(implies(implies(implies(W15, Y), implies(X, Y)), implies(Y, implies(X, Y))), implies(implies(X, Y), implies(Y, implies(X, Y)))))), true, ifeq(is_a_theorem(implies(implies(implies(T14, S14), implies(X15, Y15)), implies(Z15, implies(implies(Y15, T14), implies(X15, T14))))), true, is_a_theorem(implies(implies(implies(implies(W15, Y), implies(X, Y)), implies(Y, implies(X, Y))), implies(implies(X, Y), implies(Y, implies(X, Y))))), true), true), true, is_a_theorem(implies(implies(implies(implies(Y14, Z14), implies(W14, V14)), implies(U14, implies(implies(V14, Y14), implies(W14, Y14)))), implies(implies(X, Y), implies(Y, implies(X, Y))))), true), true, ifeq(is_a_theorem(implies(implies(implies(Y14, Z14), implies(W14, V14)), implies(U14, implies(implies(V14, Y14), implies(W14, Y14))))), true, is_a_theorem(implies(implies(X, Y), implies(Y, implies(X, Y)))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true), true)
% 28.34/4.97  = { by axiom 1 (ifeq_axiom) R->L }
% 28.34/4.98    ifeq(is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true, ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(implies(T14, S14), implies(X15, Y15)), implies(Z15, implies(implies(Y15, T14), implies(X15, T14)))), implies(implies(implies(implies(W15, Y), implies(X, Y)), implies(Y, implies(X, Y))), implies(implies(X, Y), implies(Y, implies(X, Y)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(T14, S14), implies(X15, Y15)), implies(Z15, implies(implies(Y15, T14), implies(X15, T14))))), true, is_a_theorem(implies(implies(implies(implies(W15, Y), implies(X, Y)), implies(Y, implies(X, Y))), implies(implies(X, Y), implies(Y, implies(X, Y))))), true), true), true, is_a_theorem(implies(implies(implies(implies(Y14, Z14), implies(W14, V14)), implies(U14, implies(implies(V14, Y14), implies(W14, Y14)))), implies(implies(X, Y), implies(Y, implies(X, Y))))), true), true, ifeq(is_a_theorem(implies(implies(implies(Y14, Z14), implies(W14, V14)), implies(U14, implies(implies(V14, Y14), implies(W14, Y14))))), true, is_a_theorem(implies(implies(X, Y), implies(Y, implies(X, Y)))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true), true)
% 28.34/4.98  = { by axiom 3 (ic_JLukasiewicz_4) R->L }
% 28.34/4.98    ifeq(is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true, ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(Y, implies(X, Y)), implies(X, W15)), implies(implies(X, Y), implies(implies(W15, Y), implies(X, Y))))), true, is_a_theorem(implies(implies(implies(implies(T14, S14), implies(X15, Y15)), implies(Z15, implies(implies(Y15, T14), implies(X15, T14)))), implies(implies(implies(implies(W15, Y), implies(X, Y)), implies(Y, implies(X, Y))), implies(implies(X, Y), implies(Y, implies(X, Y)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(T14, S14), implies(X15, Y15)), implies(Z15, implies(implies(Y15, T14), implies(X15, T14))))), true, is_a_theorem(implies(implies(implies(implies(W15, Y), implies(X, Y)), implies(Y, implies(X, Y))), implies(implies(X, Y), implies(Y, implies(X, Y))))), true), true), true, is_a_theorem(implies(implies(implies(implies(Y14, Z14), implies(W14, V14)), implies(U14, implies(implies(V14, Y14), implies(W14, Y14)))), implies(implies(X, Y), implies(Y, implies(X, Y))))), true), true, ifeq(is_a_theorem(implies(implies(implies(Y14, Z14), implies(W14, V14)), implies(U14, implies(implies(V14, Y14), implies(W14, Y14))))), true, is_a_theorem(implies(implies(X, Y), implies(Y, implies(X, Y)))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true), true)
% 28.34/4.98  = { by axiom 1 (ifeq_axiom) R->L }
% 28.34/4.98    ifeq(is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true, ifeq(ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(implies(Y, implies(X, Y)), implies(X, W15)), implies(implies(X, Y), implies(implies(W15, Y), implies(X, Y))))), true, is_a_theorem(implies(implies(implies(implies(T14, S14), implies(X15, Y15)), implies(Z15, implies(implies(Y15, T14), implies(X15, T14)))), implies(implies(implies(implies(W15, Y), implies(X, Y)), implies(Y, implies(X, Y))), implies(implies(X, Y), implies(Y, implies(X, Y)))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(T14, S14), implies(X15, Y15)), implies(Z15, implies(implies(Y15, T14), implies(X15, T14))))), true, is_a_theorem(implies(implies(implies(implies(W15, Y), implies(X, Y)), implies(Y, implies(X, Y))), implies(implies(X, Y), implies(Y, implies(X, Y))))), true), true), true, is_a_theorem(implies(implies(implies(implies(Y14, Z14), implies(W14, V14)), implies(U14, implies(implies(V14, Y14), implies(W14, Y14)))), implies(implies(X, Y), implies(Y, implies(X, Y))))), true), true, ifeq(is_a_theorem(implies(implies(implies(Y14, Z14), implies(W14, V14)), implies(U14, implies(implies(V14, Y14), implies(W14, Y14))))), true, is_a_theorem(implies(implies(X, Y), implies(Y, implies(X, Y)))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true), true)
% 28.34/4.98  = { by axiom 3 (ic_JLukasiewicz_4) R->L }
% 28.34/4.98    ifeq(is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true, ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(Y, implies(X, Y)), implies(X, W15)), implies(implies(X, Y), implies(implies(W15, Y), implies(X, Y)))), implies(implies(implies(implies(T14, S14), implies(X15, Y15)), implies(Z15, implies(implies(Y15, T14), implies(X15, T14)))), implies(implies(implies(implies(W15, Y), implies(X, Y)), implies(Y, implies(X, Y))), implies(implies(X, Y), implies(Y, implies(X, Y))))))), true, ifeq(is_a_theorem(implies(implies(implies(Y, implies(X, Y)), implies(X, W15)), implies(implies(X, Y), implies(implies(W15, Y), implies(X, Y))))), true, is_a_theorem(implies(implies(implies(implies(T14, S14), implies(X15, Y15)), implies(Z15, implies(implies(Y15, T14), implies(X15, T14)))), implies(implies(implies(implies(W15, Y), implies(X, Y)), implies(Y, implies(X, Y))), implies(implies(X, Y), implies(Y, implies(X, Y)))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(T14, S14), implies(X15, Y15)), implies(Z15, implies(implies(Y15, T14), implies(X15, T14))))), true, is_a_theorem(implies(implies(implies(implies(W15, Y), implies(X, Y)), implies(Y, implies(X, Y))), implies(implies(X, Y), implies(Y, implies(X, Y))))), true), true), true, is_a_theorem(implies(implies(implies(implies(Y14, Z14), implies(W14, V14)), implies(U14, implies(implies(V14, Y14), implies(W14, Y14)))), implies(implies(X, Y), implies(Y, implies(X, Y))))), true), true, ifeq(is_a_theorem(implies(implies(implies(Y14, Z14), implies(W14, V14)), implies(U14, implies(implies(V14, Y14), implies(W14, Y14))))), true, is_a_theorem(implies(implies(X, Y), implies(Y, implies(X, Y)))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true), true)
% 28.34/4.98  = { by axiom 2 (condensed_detachment) }
% 28.34/4.99    ifeq(is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true, ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(implies(T14, S14), implies(X15, Y15)), implies(Z15, implies(implies(Y15, T14), implies(X15, T14))))), true, is_a_theorem(implies(implies(implies(implies(W15, Y), implies(X, Y)), implies(Y, implies(X, Y))), implies(implies(X, Y), implies(Y, implies(X, Y))))), true), true), true, is_a_theorem(implies(implies(implies(implies(Y14, Z14), implies(W14, V14)), implies(U14, implies(implies(V14, Y14), implies(W14, Y14)))), implies(implies(X, Y), implies(Y, implies(X, Y))))), true), true, ifeq(is_a_theorem(implies(implies(implies(Y14, Z14), implies(W14, V14)), implies(U14, implies(implies(V14, Y14), implies(W14, Y14))))), true, is_a_theorem(implies(implies(X, Y), implies(Y, implies(X, Y)))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true), true)
% 28.34/4.99  = { by axiom 1 (ifeq_axiom) }
% 28.34/4.99    ifeq(is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true, ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(T14, S14), implies(X15, Y15)), implies(Z15, implies(implies(Y15, T14), implies(X15, T14))))), true, is_a_theorem(implies(implies(implies(implies(W15, Y), implies(X, Y)), implies(Y, implies(X, Y))), implies(implies(X, Y), implies(Y, implies(X, Y))))), true), true, is_a_theorem(implies(implies(implies(implies(Y14, Z14), implies(W14, V14)), implies(U14, implies(implies(V14, Y14), implies(W14, Y14)))), implies(implies(X, Y), implies(Y, implies(X, Y))))), true), true, ifeq(is_a_theorem(implies(implies(implies(Y14, Z14), implies(W14, V14)), implies(U14, implies(implies(V14, Y14), implies(W14, Y14))))), true, is_a_theorem(implies(implies(X, Y), implies(Y, implies(X, Y)))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true), true)
% 28.34/4.99  = { by axiom 3 (ic_JLukasiewicz_4) }
% 28.34/4.99    ifeq(is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true, ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(implies(W15, Y), implies(X, Y)), implies(Y, implies(X, Y))), implies(implies(X, Y), implies(Y, implies(X, Y))))), true), true, is_a_theorem(implies(implies(implies(implies(Y14, Z14), implies(W14, V14)), implies(U14, implies(implies(V14, Y14), implies(W14, Y14)))), implies(implies(X, Y), implies(Y, implies(X, Y))))), true), true, ifeq(is_a_theorem(implies(implies(implies(Y14, Z14), implies(W14, V14)), implies(U14, implies(implies(V14, Y14), implies(W14, Y14))))), true, is_a_theorem(implies(implies(X, Y), implies(Y, implies(X, Y)))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true), true)
% 28.34/4.99  = { by axiom 1 (ifeq_axiom) }
% 28.34/4.99    ifeq(is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true, ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(W15, Y), implies(X, Y)), implies(Y, implies(X, Y))), implies(implies(X, Y), implies(Y, implies(X, Y))))), true, is_a_theorem(implies(implies(implies(implies(Y14, Z14), implies(W14, V14)), implies(U14, implies(implies(V14, Y14), implies(W14, Y14)))), implies(implies(X, Y), implies(Y, implies(X, Y))))), true), true, ifeq(is_a_theorem(implies(implies(implies(Y14, Z14), implies(W14, V14)), implies(U14, implies(implies(V14, Y14), implies(W14, Y14))))), true, is_a_theorem(implies(implies(X, Y), implies(Y, implies(X, Y)))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true), true)
% 28.34/5.00  = { by axiom 1 (ifeq_axiom) R->L }
% 28.34/5.00    ifeq(is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true, ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(implies(implies(W15, Y), implies(X, Y)), implies(Y, implies(X, Y))), implies(implies(X, Y), implies(Y, implies(X, Y))))), true, is_a_theorem(implies(implies(implies(implies(Y14, Z14), implies(W14, V14)), implies(U14, implies(implies(V14, Y14), implies(W14, Y14)))), implies(implies(X, Y), implies(Y, implies(X, Y))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(Y14, Z14), implies(W14, V14)), implies(U14, implies(implies(V14, Y14), implies(W14, Y14))))), true, is_a_theorem(implies(implies(X, Y), implies(Y, implies(X, Y)))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true), true)
% 28.34/5.00  = { by axiom 2 (condensed_detachment) R->L }
% 28.34/5.00    ifeq(is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true, ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(V15, U15), implies(T15, S15)), implies(X16, implies(implies(S15, V15), implies(T15, V15)))), implies(implies(implies(implies(implies(W15, Y), implies(X, Y)), implies(Y, implies(X, Y))), implies(implies(X, Y), implies(Y, implies(X, Y)))), implies(implies(implies(implies(Y14, Z14), implies(W14, V14)), implies(U14, implies(implies(V14, Y14), implies(W14, Y14)))), implies(implies(X, Y), implies(Y, implies(X, Y))))))), true, ifeq(is_a_theorem(implies(implies(implies(V15, U15), implies(T15, S15)), implies(X16, implies(implies(S15, V15), implies(T15, V15))))), true, is_a_theorem(implies(implies(implies(implies(implies(W15, Y), implies(X, Y)), implies(Y, implies(X, Y))), implies(implies(X, Y), implies(Y, implies(X, Y)))), implies(implies(implies(implies(Y14, Z14), implies(W14, V14)), implies(U14, implies(implies(V14, Y14), implies(W14, Y14)))), implies(implies(X, Y), implies(Y, implies(X, Y)))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(W15, Y), implies(X, Y)), implies(Y, implies(X, Y))), implies(implies(X, Y), implies(Y, implies(X, Y))))), true, is_a_theorem(implies(implies(implies(implies(Y14, Z14), implies(W14, V14)), implies(U14, implies(implies(V14, Y14), implies(W14, Y14)))), implies(implies(X, Y), implies(Y, implies(X, Y))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(Y14, Z14), implies(W14, V14)), implies(U14, implies(implies(V14, Y14), implies(W14, Y14))))), true, is_a_theorem(implies(implies(X, Y), implies(Y, implies(X, Y)))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true), true)
% 28.34/5.00  = { by axiom 1 (ifeq_axiom) R->L }
% 28.34/5.01    ifeq(is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true, ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(implies(V15, U15), implies(T15, S15)), implies(X16, implies(implies(S15, V15), implies(T15, V15)))), implies(implies(implies(implies(implies(W15, Y), implies(X, Y)), implies(Y, implies(X, Y))), implies(implies(X, Y), implies(Y, implies(X, Y)))), implies(implies(implies(implies(Y14, Z14), implies(W14, V14)), implies(U14, implies(implies(V14, Y14), implies(W14, Y14)))), implies(implies(X, Y), implies(Y, implies(X, Y))))))), true), true, ifeq(is_a_theorem(implies(implies(implies(V15, U15), implies(T15, S15)), implies(X16, implies(implies(S15, V15), implies(T15, V15))))), true, is_a_theorem(implies(implies(implies(implies(implies(W15, Y), implies(X, Y)), implies(Y, implies(X, Y))), implies(implies(X, Y), implies(Y, implies(X, Y)))), implies(implies(implies(implies(Y14, Z14), implies(W14, V14)), implies(U14, implies(implies(V14, Y14), implies(W14, Y14)))), implies(implies(X, Y), implies(Y, implies(X, Y)))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(W15, Y), implies(X, Y)), implies(Y, implies(X, Y))), implies(implies(X, Y), implies(Y, implies(X, Y))))), true, is_a_theorem(implies(implies(implies(implies(Y14, Z14), implies(W14, V14)), implies(U14, implies(implies(V14, Y14), implies(W14, Y14)))), implies(implies(X, Y), implies(Y, implies(X, Y))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(Y14, Z14), implies(W14, V14)), implies(U14, implies(implies(V14, Y14), implies(W14, Y14))))), true, is_a_theorem(implies(implies(X, Y), implies(Y, implies(X, Y)))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true), true)
% 28.34/5.01  = { by axiom 3 (ic_JLukasiewicz_4) R->L }
% 28.34/5.01    ifeq(is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true, ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(Y, implies(W15, Y))), implies(implies(implies(implies(Y14, Z14), implies(W14, V14)), implies(U14, implies(implies(V14, Y14), implies(W14, Y14)))), implies(implies(implies(W15, Y), implies(X, Y)), implies(Y, implies(X, Y)))))), true, is_a_theorem(implies(implies(implies(implies(V15, U15), implies(T15, S15)), implies(X16, implies(implies(S15, V15), implies(T15, V15)))), implies(implies(implies(implies(implies(W15, Y), implies(X, Y)), implies(Y, implies(X, Y))), implies(implies(X, Y), implies(Y, implies(X, Y)))), implies(implies(implies(implies(Y14, Z14), implies(W14, V14)), implies(U14, implies(implies(V14, Y14), implies(W14, Y14)))), implies(implies(X, Y), implies(Y, implies(X, Y))))))), true), true, ifeq(is_a_theorem(implies(implies(implies(V15, U15), implies(T15, S15)), implies(X16, implies(implies(S15, V15), implies(T15, V15))))), true, is_a_theorem(implies(implies(implies(implies(implies(W15, Y), implies(X, Y)), implies(Y, implies(X, Y))), implies(implies(X, Y), implies(Y, implies(X, Y)))), implies(implies(implies(implies(Y14, Z14), implies(W14, V14)), implies(U14, implies(implies(V14, Y14), implies(W14, Y14)))), implies(implies(X, Y), implies(Y, implies(X, Y)))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(W15, Y), implies(X, Y)), implies(Y, implies(X, Y))), implies(implies(X, Y), implies(Y, implies(X, Y))))), true, is_a_theorem(implies(implies(implies(implies(Y14, Z14), implies(W14, V14)), implies(U14, implies(implies(V14, Y14), implies(W14, Y14)))), implies(implies(X, Y), implies(Y, implies(X, Y))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(Y14, Z14), implies(W14, V14)), implies(U14, implies(implies(V14, Y14), implies(W14, Y14))))), true, is_a_theorem(implies(implies(X, Y), implies(Y, implies(X, Y)))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true), true)
% 28.34/5.01  = { by axiom 1 (ifeq_axiom) R->L }
% 28.34/5.01    ifeq(is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true, ifeq(ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(Y, implies(W15, Y))), implies(implies(implies(implies(Y14, Z14), implies(W14, V14)), implies(U14, implies(implies(V14, Y14), implies(W14, Y14)))), implies(implies(implies(W15, Y), implies(X, Y)), implies(Y, implies(X, Y)))))), true, is_a_theorem(implies(implies(implies(implies(V15, U15), implies(T15, S15)), implies(X16, implies(implies(S15, V15), implies(T15, V15)))), implies(implies(implies(implies(implies(W15, Y), implies(X, Y)), implies(Y, implies(X, Y))), implies(implies(X, Y), implies(Y, implies(X, Y)))), implies(implies(implies(implies(Y14, Z14), implies(W14, V14)), implies(U14, implies(implies(V14, Y14), implies(W14, Y14)))), implies(implies(X, Y), implies(Y, implies(X, Y))))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(V15, U15), implies(T15, S15)), implies(X16, implies(implies(S15, V15), implies(T15, V15))))), true, is_a_theorem(implies(implies(implies(implies(implies(W15, Y), implies(X, Y)), implies(Y, implies(X, Y))), implies(implies(X, Y), implies(Y, implies(X, Y)))), implies(implies(implies(implies(Y14, Z14), implies(W14, V14)), implies(U14, implies(implies(V14, Y14), implies(W14, Y14)))), implies(implies(X, Y), implies(Y, implies(X, Y)))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(W15, Y), implies(X, Y)), implies(Y, implies(X, Y))), implies(implies(X, Y), implies(Y, implies(X, Y))))), true, is_a_theorem(implies(implies(implies(implies(Y14, Z14), implies(W14, V14)), implies(U14, implies(implies(V14, Y14), implies(W14, Y14)))), implies(implies(X, Y), implies(Y, implies(X, Y))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(Y14, Z14), implies(W14, V14)), implies(U14, implies(implies(V14, Y14), implies(W14, Y14))))), true, is_a_theorem(implies(implies(X, Y), implies(Y, implies(X, Y)))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true), true)
% 28.34/5.01  = { by axiom 3 (ic_JLukasiewicz_4) R->L }
% 28.34/5.02    ifeq(is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true, ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(Y, implies(W15, Y))), implies(implies(implies(implies(Y14, Z14), implies(W14, V14)), implies(U14, implies(implies(V14, Y14), implies(W14, Y14)))), implies(implies(implies(W15, Y), implies(X, Y)), implies(Y, implies(X, Y))))), implies(implies(implies(implies(V15, U15), implies(T15, S15)), implies(X16, implies(implies(S15, V15), implies(T15, V15)))), implies(implies(implies(implies(implies(W15, Y), implies(X, Y)), implies(Y, implies(X, Y))), implies(implies(X, Y), implies(Y, implies(X, Y)))), implies(implies(implies(implies(Y14, Z14), implies(W14, V14)), implies(U14, implies(implies(V14, Y14), implies(W14, Y14)))), implies(implies(X, Y), implies(Y, implies(X, Y)))))))), true, ifeq(is_a_theorem(implies(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(Y, implies(W15, Y))), implies(implies(implies(implies(Y14, Z14), implies(W14, V14)), implies(U14, implies(implies(V14, Y14), implies(W14, Y14)))), implies(implies(implies(W15, Y), implies(X, Y)), implies(Y, implies(X, Y)))))), true, is_a_theorem(implies(implies(implies(implies(V15, U15), implies(T15, S15)), implies(X16, implies(implies(S15, V15), implies(T15, V15)))), implies(implies(implies(implies(implies(W15, Y), implies(X, Y)), implies(Y, implies(X, Y))), implies(implies(X, Y), implies(Y, implies(X, Y)))), implies(implies(implies(implies(Y14, Z14), implies(W14, V14)), implies(U14, implies(implies(V14, Y14), implies(W14, Y14)))), implies(implies(X, Y), implies(Y, implies(X, Y))))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(V15, U15), implies(T15, S15)), implies(X16, implies(implies(S15, V15), implies(T15, V15))))), true, is_a_theorem(implies(implies(implies(implies(implies(W15, Y), implies(X, Y)), implies(Y, implies(X, Y))), implies(implies(X, Y), implies(Y, implies(X, Y)))), implies(implies(implies(implies(Y14, Z14), implies(W14, V14)), implies(U14, implies(implies(V14, Y14), implies(W14, Y14)))), implies(implies(X, Y), implies(Y, implies(X, Y)))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(W15, Y), implies(X, Y)), implies(Y, implies(X, Y))), implies(implies(X, Y), implies(Y, implies(X, Y))))), true, is_a_theorem(implies(implies(implies(implies(Y14, Z14), implies(W14, V14)), implies(U14, implies(implies(V14, Y14), implies(W14, Y14)))), implies(implies(X, Y), implies(Y, implies(X, Y))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(Y14, Z14), implies(W14, V14)), implies(U14, implies(implies(V14, Y14), implies(W14, Y14))))), true, is_a_theorem(implies(implies(X, Y), implies(Y, implies(X, Y)))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true), true)
% 28.34/5.02  = { by axiom 2 (condensed_detachment) }
% 28.34/5.02    ifeq(is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true, ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(implies(V15, U15), implies(T15, S15)), implies(X16, implies(implies(S15, V15), implies(T15, V15))))), true, is_a_theorem(implies(implies(implies(implies(implies(W15, Y), implies(X, Y)), implies(Y, implies(X, Y))), implies(implies(X, Y), implies(Y, implies(X, Y)))), implies(implies(implies(implies(Y14, Z14), implies(W14, V14)), implies(U14, implies(implies(V14, Y14), implies(W14, Y14)))), implies(implies(X, Y), implies(Y, implies(X, Y)))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(W15, Y), implies(X, Y)), implies(Y, implies(X, Y))), implies(implies(X, Y), implies(Y, implies(X, Y))))), true, is_a_theorem(implies(implies(implies(implies(Y14, Z14), implies(W14, V14)), implies(U14, implies(implies(V14, Y14), implies(W14, Y14)))), implies(implies(X, Y), implies(Y, implies(X, Y))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(Y14, Z14), implies(W14, V14)), implies(U14, implies(implies(V14, Y14), implies(W14, Y14))))), true, is_a_theorem(implies(implies(X, Y), implies(Y, implies(X, Y)))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true), true)
% 28.34/5.02  = { by axiom 1 (ifeq_axiom) }
% 28.34/5.02    ifeq(is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true, ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(V15, U15), implies(T15, S15)), implies(X16, implies(implies(S15, V15), implies(T15, V15))))), true, is_a_theorem(implies(implies(implies(implies(implies(W15, Y), implies(X, Y)), implies(Y, implies(X, Y))), implies(implies(X, Y), implies(Y, implies(X, Y)))), implies(implies(implies(implies(Y14, Z14), implies(W14, V14)), implies(U14, implies(implies(V14, Y14), implies(W14, Y14)))), implies(implies(X, Y), implies(Y, implies(X, Y)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(W15, Y), implies(X, Y)), implies(Y, implies(X, Y))), implies(implies(X, Y), implies(Y, implies(X, Y))))), true, is_a_theorem(implies(implies(implies(implies(Y14, Z14), implies(W14, V14)), implies(U14, implies(implies(V14, Y14), implies(W14, Y14)))), implies(implies(X, Y), implies(Y, implies(X, Y))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(Y14, Z14), implies(W14, V14)), implies(U14, implies(implies(V14, Y14), implies(W14, Y14))))), true, is_a_theorem(implies(implies(X, Y), implies(Y, implies(X, Y)))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true), true)
% 28.34/5.02  = { by axiom 3 (ic_JLukasiewicz_4) }
% 28.34/5.02    ifeq(is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true, ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(implies(implies(W15, Y), implies(X, Y)), implies(Y, implies(X, Y))), implies(implies(X, Y), implies(Y, implies(X, Y)))), implies(implies(implies(implies(Y14, Z14), implies(W14, V14)), implies(U14, implies(implies(V14, Y14), implies(W14, Y14)))), implies(implies(X, Y), implies(Y, implies(X, Y)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(W15, Y), implies(X, Y)), implies(Y, implies(X, Y))), implies(implies(X, Y), implies(Y, implies(X, Y))))), true, is_a_theorem(implies(implies(implies(implies(Y14, Z14), implies(W14, V14)), implies(U14, implies(implies(V14, Y14), implies(W14, Y14)))), implies(implies(X, Y), implies(Y, implies(X, Y))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(Y14, Z14), implies(W14, V14)), implies(U14, implies(implies(V14, Y14), implies(W14, Y14))))), true, is_a_theorem(implies(implies(X, Y), implies(Y, implies(X, Y)))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true), true)
% 28.34/5.02  = { by axiom 1 (ifeq_axiom) }
% 28.34/5.03    ifeq(is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true, ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(implies(W15, Y), implies(X, Y)), implies(Y, implies(X, Y))), implies(implies(X, Y), implies(Y, implies(X, Y)))), implies(implies(implies(implies(Y14, Z14), implies(W14, V14)), implies(U14, implies(implies(V14, Y14), implies(W14, Y14)))), implies(implies(X, Y), implies(Y, implies(X, Y)))))), true, ifeq(is_a_theorem(implies(implies(implies(implies(W15, Y), implies(X, Y)), implies(Y, implies(X, Y))), implies(implies(X, Y), implies(Y, implies(X, Y))))), true, is_a_theorem(implies(implies(implies(implies(Y14, Z14), implies(W14, V14)), implies(U14, implies(implies(V14, Y14), implies(W14, Y14)))), implies(implies(X, Y), implies(Y, implies(X, Y))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(Y14, Z14), implies(W14, V14)), implies(U14, implies(implies(V14, Y14), implies(W14, Y14))))), true, is_a_theorem(implies(implies(X, Y), implies(Y, implies(X, Y)))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true), true)
% 28.34/5.03  = { by axiom 2 (condensed_detachment) }
% 28.34/5.03    ifeq(is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true, ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(implies(Y14, Z14), implies(W14, V14)), implies(U14, implies(implies(V14, Y14), implies(W14, Y14))))), true, is_a_theorem(implies(implies(X, Y), implies(Y, implies(X, Y)))), true), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true), true)
% 28.34/5.03  = { by axiom 1 (ifeq_axiom) }
% 28.34/5.03    ifeq(is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true, ifeq(ifeq(is_a_theorem(implies(implies(implies(Y14, Z14), implies(W14, V14)), implies(U14, implies(implies(V14, Y14), implies(W14, Y14))))), true, is_a_theorem(implies(implies(X, Y), implies(Y, implies(X, Y)))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true), true)
% 28.34/5.03  = { by axiom 3 (ic_JLukasiewicz_4) }
% 28.34/5.03    ifeq(is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true, ifeq(ifeq(true, true, is_a_theorem(implies(implies(X, Y), implies(Y, implies(X, Y)))), true), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true), true)
% 28.34/5.03  = { by axiom 1 (ifeq_axiom) }
% 28.34/5.03    ifeq(is_a_theorem(implies(implies(implies(X, Y), implies(Y, implies(X, Y))), implies(implies(implies(a, b), a), a))), true, ifeq(is_a_theorem(implies(implies(X, Y), implies(Y, implies(X, Y)))), true, is_a_theorem(implies(implies(implies(a, b), a), a)), true), true)
% 28.34/5.03  = { by axiom 2 (condensed_detachment) }
% 28.34/5.03    true
% 28.34/5.03  % SZS output end Proof
% 28.34/5.03  
% 28.34/5.03  RESULT: Unsatisfiable (the axioms are contradictory).
%------------------------------------------------------------------------------