Twee---2.7.THM-Prf.s

View TPTP
Problem Process solution in
SystemOnTSTP Download .tgz
%------------------------------------------------------------------------------
% File     : Twee---2.7
% Problem  : LCL982+1 : TPTP v9.3.1. Released v9.1.0.
% Transfm  : none
% Format   : tptp:raw
% Command  : run_twee /export/starexec/sandbox2/benchmark/theBenchmark.p

% Computer : n018.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 11:50:53 AM UTC 2026

% Result   : Theorem 0.21s 0.55s
% Output   : Proof 6.52s
% Verified : 
% SZS Type : -

% Comments : 
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.03  % Problem  : LCL982+1 : TPTP v9.3.1. Released v9.1.0.
% 0.00/0.04  % Command  : run_twee /export/starexec/sandbox2/benchmark/theBenchmark.p
% 0.14/0.41  % Computer : n018.cluster.edu
% 0.14/0.41  % Model    : x86_64 x86_64
% 0.14/0.41  % CPU      : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.14/0.41  % Memory   : 8046.5625MB
% 0.14/0.41  % OS       : Linux 6.8.0-71-generic
% 0.14/0.41  % CPULimit : 300
% 0.14/0.41  % WCLimit  : 300
% 0.14/0.41  % DateTime : Sun Sep 27 17:12:38 UTC 2026
% 0.14/0.41  % CPUTime  : 
% 0.14/0.41  Running run_twee /export/starexec/sandbox2/benchmark/theBenchmark.p
% 0.21/0.55  Command-line arguments: --no-flatten-goal
% 0.21/0.55  
% 0.21/0.55  % SZS status Theorem
% 0.21/0.55  
% 2.35/0.85  % SZS output start Proof
% 2.35/0.85  Axiom 1 (ifeq_axiom): ifeq(X, X, Y, Z) = Y.
% 2.35/0.85  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.
% 2.35/0.86  Axiom 3 (a5): is_a_theorem(implies(implies(X, Y), implies(implies(implies(Z, implies(W, V)), implies(Y, implies(not(W), not(X)))), implies(X, W)))) = true.
% 2.35/0.86  
% 2.35/0.86  Goal 1 (prove_cn_1): is_a_theorem(implies(implies(a, b), implies(implies(b, c), implies(a, c)))) = true.
% 2.35/0.86  Proof:
% 2.35/0.86    is_a_theorem(implies(implies(a, b), implies(implies(b, c), implies(a, c))))
% 2.35/0.87  = { by axiom 1 (ifeq_axiom) R->L }
% 2.35/0.87    ifeq(true, true, is_a_theorem(implies(implies(a, b), implies(implies(b, c), implies(a, c)))), true)
% 2.35/0.88  = { by axiom 2 (condensed_detachment) R->L }
% 2.35/0.89    ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(X, implies(implies(not(implies(implies(b, c), implies(a, c))), not(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)))), W)), implies(implies(implies(b, c), implies(a, c)), implies(not(implies(not(implies(implies(b, c), implies(a, c))), not(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c))))), not(implies(implies(implies(V, implies(implies(not(implies(a, c)), not(implies(b, c))), U)), implies(implies(a, c), implies(not(implies(not(implies(a, c)), not(implies(b, c)))), not(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))))))), implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(not(implies(a, c)), not(implies(b, c))))))))), implies(implies(implies(implies(V, implies(implies(not(implies(a, c)), not(implies(b, c))), U)), implies(implies(a, c), implies(not(implies(not(implies(a, c)), not(implies(b, c)))), not(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))))))), implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(not(implies(a, c)), not(implies(b, c))))), implies(not(implies(implies(b, c), implies(a, c))), not(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)))))), implies(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)), implies(implies(b, c), implies(a, c))))), true, ifeq(is_a_theorem(implies(implies(implies(X, implies(implies(not(implies(implies(b, c), implies(a, c))), not(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)))), W)), implies(implies(implies(b, c), implies(a, c)), implies(not(implies(not(implies(implies(b, c), implies(a, c))), not(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c))))), not(implies(implies(implies(V, implies(implies(not(implies(a, c)), not(implies(b, c))), U)), implies(implies(a, c), implies(not(implies(not(implies(a, c)), not(implies(b, c)))), not(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))))))), implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(not(implies(a, c)), not(implies(b, c))))))))), implies(implies(implies(implies(V, implies(implies(not(implies(a, c)), not(implies(b, c))), U)), implies(implies(a, c), implies(not(implies(not(implies(a, c)), not(implies(b, c)))), not(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))))))), implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(not(implies(a, c)), not(implies(b, c))))), implies(not(implies(implies(b, c), implies(a, c))), not(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c))))))), true, is_a_theorem(implies(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)), implies(implies(b, c), implies(a, c)))), true), true), true, is_a_theorem(implies(implies(a, b), implies(implies(b, c), implies(a, c)))), true)
% 2.35/0.90  = { by axiom 1 (ifeq_axiom) R->L }
% 3.02/0.91    ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(implies(X, implies(implies(not(implies(implies(b, c), implies(a, c))), not(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)))), W)), implies(implies(implies(b, c), implies(a, c)), implies(not(implies(not(implies(implies(b, c), implies(a, c))), not(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c))))), not(implies(implies(implies(V, implies(implies(not(implies(a, c)), not(implies(b, c))), U)), implies(implies(a, c), implies(not(implies(not(implies(a, c)), not(implies(b, c)))), not(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))))))), implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(not(implies(a, c)), not(implies(b, c))))))))), implies(implies(implies(implies(V, implies(implies(not(implies(a, c)), not(implies(b, c))), U)), implies(implies(a, c), implies(not(implies(not(implies(a, c)), not(implies(b, c)))), not(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))))))), implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(not(implies(a, c)), not(implies(b, c))))), implies(not(implies(implies(b, c), implies(a, c))), not(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)))))), implies(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)), implies(implies(b, c), implies(a, c))))), true), true, ifeq(is_a_theorem(implies(implies(implies(X, implies(implies(not(implies(implies(b, c), implies(a, c))), not(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)))), W)), implies(implies(implies(b, c), implies(a, c)), implies(not(implies(not(implies(implies(b, c), implies(a, c))), not(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c))))), not(implies(implies(implies(V, implies(implies(not(implies(a, c)), not(implies(b, c))), U)), implies(implies(a, c), implies(not(implies(not(implies(a, c)), not(implies(b, c)))), not(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))))))), implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(not(implies(a, c)), not(implies(b, c))))))))), implies(implies(implies(implies(V, implies(implies(not(implies(a, c)), not(implies(b, c))), U)), implies(implies(a, c), implies(not(implies(not(implies(a, c)), not(implies(b, c)))), not(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))))))), implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(not(implies(a, c)), not(implies(b, c))))), implies(not(implies(implies(b, c), implies(a, c))), not(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c))))))), true, is_a_theorem(implies(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)), implies(implies(b, c), implies(a, c)))), true), true), true, is_a_theorem(implies(implies(a, b), implies(implies(b, c), implies(a, c)))), true)
% 3.02/0.92  = { by axiom 3 (a5) R->L }
% 3.02/0.93    ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)), implies(implies(implies(V, implies(implies(not(implies(a, c)), not(implies(b, c))), U)), implies(implies(a, c), implies(not(implies(not(implies(a, c)), not(implies(b, c)))), not(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))))))), implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(not(implies(a, c)), not(implies(b, c))))))), true, is_a_theorem(implies(implies(implies(implies(X, implies(implies(not(implies(implies(b, c), implies(a, c))), not(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)))), W)), implies(implies(implies(b, c), implies(a, c)), implies(not(implies(not(implies(implies(b, c), implies(a, c))), not(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c))))), not(implies(implies(implies(V, implies(implies(not(implies(a, c)), not(implies(b, c))), U)), implies(implies(a, c), implies(not(implies(not(implies(a, c)), not(implies(b, c)))), not(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))))))), implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(not(implies(a, c)), not(implies(b, c))))))))), implies(implies(implies(implies(V, implies(implies(not(implies(a, c)), not(implies(b, c))), U)), implies(implies(a, c), implies(not(implies(not(implies(a, c)), not(implies(b, c)))), not(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))))))), implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(not(implies(a, c)), not(implies(b, c))))), implies(not(implies(implies(b, c), implies(a, c))), not(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)))))), implies(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)), implies(implies(b, c), implies(a, c))))), true), true, ifeq(is_a_theorem(implies(implies(implies(X, implies(implies(not(implies(implies(b, c), implies(a, c))), not(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)))), W)), implies(implies(implies(b, c), implies(a, c)), implies(not(implies(not(implies(implies(b, c), implies(a, c))), not(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c))))), not(implies(implies(implies(V, implies(implies(not(implies(a, c)), not(implies(b, c))), U)), implies(implies(a, c), implies(not(implies(not(implies(a, c)), not(implies(b, c)))), not(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))))))), implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(not(implies(a, c)), not(implies(b, c))))))))), implies(implies(implies(implies(V, implies(implies(not(implies(a, c)), not(implies(b, c))), U)), implies(implies(a, c), implies(not(implies(not(implies(a, c)), not(implies(b, c)))), not(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))))))), implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(not(implies(a, c)), not(implies(b, c))))), implies(not(implies(implies(b, c), implies(a, c))), not(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c))))))), true, is_a_theorem(implies(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)), implies(implies(b, c), implies(a, c)))), true), true), true, is_a_theorem(implies(implies(a, b), implies(implies(b, c), implies(a, c)))), true)
% 3.02/0.94  = { by axiom 1 (ifeq_axiom) R->L }
% 3.02/0.95    ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)), implies(implies(implies(V, implies(implies(not(implies(a, c)), not(implies(b, c))), U)), implies(implies(a, c), implies(not(implies(not(implies(a, c)), not(implies(b, c)))), not(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))))))), implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(not(implies(a, c)), not(implies(b, c))))))), true, is_a_theorem(implies(implies(implies(implies(X, implies(implies(not(implies(implies(b, c), implies(a, c))), not(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)))), W)), implies(implies(implies(b, c), implies(a, c)), implies(not(implies(not(implies(implies(b, c), implies(a, c))), not(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c))))), not(implies(implies(implies(V, implies(implies(not(implies(a, c)), not(implies(b, c))), U)), implies(implies(a, c), implies(not(implies(not(implies(a, c)), not(implies(b, c)))), not(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))))))), implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(not(implies(a, c)), 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not(a))), not(b)))), implies(b, implies(not(c), not(a)))))))), implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(not(implies(a, c)), not(implies(b, c))))))))), implies(implies(implies(implies(V, implies(implies(not(implies(a, c)), not(implies(b, c))), U)), implies(implies(a, c), implies(not(implies(not(implies(a, c)), not(implies(b, c)))), not(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))))))), implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(not(implies(a, c)), not(implies(b, c))))), implies(not(implies(implies(b, c), implies(a, c))), not(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), 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% 3.02/0.95  = { by axiom 3 (a5) R->L }
% 3.02/0.98    ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)), implies(implies(implies(V, implies(implies(not(implies(a, c)), not(implies(b, c))), U)), implies(implies(a, c), implies(not(implies(not(implies(a, c)), not(implies(b, c)))), not(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))))))), implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(not(implies(a, c)), not(implies(b, c)))))), implies(implies(implies(implies(X, implies(implies(not(implies(implies(b, c), implies(a, c))), not(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, 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% 3.02/0.98  = { by axiom 2 (condensed_detachment) }
% 3.02/0.99    ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(implies(X, implies(implies(not(implies(implies(b, c), implies(a, c))), not(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)))), W)), implies(implies(implies(b, c), implies(a, c)), implies(not(implies(not(implies(implies(b, c), implies(a, c))), not(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c))))), not(implies(implies(implies(V, implies(implies(not(implies(a, c)), not(implies(b, c))), U)), implies(implies(a, c), implies(not(implies(not(implies(a, c)), not(implies(b, c)))), not(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))))))), implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(not(implies(a, c)), not(implies(b, c))))))))), implies(implies(implies(implies(V, implies(implies(not(implies(a, c)), not(implies(b, c))), U)), implies(implies(a, c), implies(not(implies(not(implies(a, c)), not(implies(b, c)))), not(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))))))), implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(not(implies(a, c)), not(implies(b, c))))), implies(not(implies(implies(b, c), implies(a, c))), not(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c))))))), true, is_a_theorem(implies(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)), implies(implies(b, c), implies(a, c)))), true), true), true, is_a_theorem(implies(implies(a, b), implies(implies(b, c), implies(a, c)))), true)
% 3.02/0.99  = { by axiom 1 (ifeq_axiom) }
% 3.02/0.99    ifeq(ifeq(is_a_theorem(implies(implies(implies(X, implies(implies(not(implies(implies(b, c), implies(a, c))), not(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)))), W)), implies(implies(implies(b, c), implies(a, c)), implies(not(implies(not(implies(implies(b, c), implies(a, c))), not(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c))))), not(implies(implies(implies(V, implies(implies(not(implies(a, c)), not(implies(b, c))), U)), implies(implies(a, c), implies(not(implies(not(implies(a, c)), not(implies(b, c)))), not(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))))))), implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(not(implies(a, c)), not(implies(b, c))))))))), implies(implies(implies(implies(V, implies(implies(not(implies(a, c)), not(implies(b, c))), U)), implies(implies(a, c), implies(not(implies(not(implies(a, c)), not(implies(b, c)))), not(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))))))), implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(not(implies(a, c)), not(implies(b, c))))), implies(not(implies(implies(b, c), implies(a, c))), not(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c))))))), true, is_a_theorem(implies(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)), implies(implies(b, c), implies(a, c)))), true), true, is_a_theorem(implies(implies(a, b), implies(implies(b, c), implies(a, c)))), true)
% 3.02/1.00  = { by axiom 1 (ifeq_axiom) R->L }
% 3.02/1.01    ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(X, implies(implies(not(implies(implies(b, c), implies(a, c))), not(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)))), W)), implies(implies(implies(b, c), implies(a, c)), implies(not(implies(not(implies(implies(b, c), implies(a, c))), not(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c))))), not(implies(implies(implies(V, implies(implies(not(implies(a, c)), not(implies(b, c))), U)), implies(implies(a, c), implies(not(implies(not(implies(a, c)), not(implies(b, c)))), not(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))))))), implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(not(implies(a, c)), not(implies(b, c))))))))), implies(implies(implies(implies(V, implies(implies(not(implies(a, c)), not(implies(b, c))), U)), implies(implies(a, c), implies(not(implies(not(implies(a, c)), not(implies(b, c)))), not(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))))))), implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(not(implies(a, c)), not(implies(b, c))))), implies(not(implies(implies(b, c), implies(a, c))), not(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c))))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)), implies(implies(b, c), implies(a, c)))), true), true, is_a_theorem(implies(implies(a, b), implies(implies(b, c), implies(a, c)))), true)
% 3.02/1.01  = { by axiom 2 (condensed_detachment) R->L }
% 3.02/1.02    ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(b, c), implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a))))), implies(implies(implies(implies(V, implies(implies(not(implies(a, c)), not(implies(b, c))), U)), implies(implies(a, c), implies(not(implies(not(implies(a, c)), not(implies(b, c)))), not(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))))))), implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(not(implies(a, c)), not(implies(b, c))))), implies(implies(b, c), implies(a, c))))), true, ifeq(is_a_theorem(implies(implies(b, c), implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))))), true, is_a_theorem(implies(implies(implies(implies(V, implies(implies(not(implies(a, c)), not(implies(b, c))), U)), implies(implies(a, c), implies(not(implies(not(implies(a, c)), not(implies(b, c)))), not(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))))))), implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(not(implies(a, c)), not(implies(b, c))))), implies(implies(b, c), implies(a, c)))), true), true), true, is_a_theorem(implies(implies(implies(X, implies(implies(not(implies(implies(b, c), implies(a, c))), not(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)))), W)), implies(implies(implies(b, c), implies(a, c)), implies(not(implies(not(implies(implies(b, c), implies(a, c))), not(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c))))), not(implies(implies(implies(V, implies(implies(not(implies(a, c)), not(implies(b, c))), U)), implies(implies(a, c), implies(not(implies(not(implies(a, c)), not(implies(b, c)))), not(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))))))), implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(not(implies(a, c)), not(implies(b, c))))))))), implies(implies(implies(implies(V, implies(implies(not(implies(a, c)), not(implies(b, c))), U)), implies(implies(a, c), implies(not(implies(not(implies(a, c)), not(implies(b, c)))), not(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))))))), implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(not(implies(a, c)), not(implies(b, c))))), implies(not(implies(implies(b, c), implies(a, c))), not(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c))))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)), implies(implies(b, c), implies(a, c)))), true), true, is_a_theorem(implies(implies(a, b), implies(implies(b, c), implies(a, c)))), true)
% 3.02/1.03  = { by axiom 3 (a5) }
% 3.02/1.03    ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(b, c), implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))))), true, is_a_theorem(implies(implies(implies(implies(V, implies(implies(not(implies(a, c)), not(implies(b, c))), U)), implies(implies(a, c), implies(not(implies(not(implies(a, c)), not(implies(b, c)))), not(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))))))), implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(not(implies(a, c)), not(implies(b, c))))), implies(implies(b, c), implies(a, c)))), true), true), true, is_a_theorem(implies(implies(implies(X, implies(implies(not(implies(implies(b, c), implies(a, c))), not(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)))), W)), implies(implies(implies(b, c), implies(a, c)), implies(not(implies(not(implies(implies(b, c), implies(a, c))), not(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c))))), not(implies(implies(implies(V, implies(implies(not(implies(a, c)), not(implies(b, c))), U)), implies(implies(a, c), implies(not(implies(not(implies(a, c)), not(implies(b, c)))), not(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))))))), implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(not(implies(a, c)), not(implies(b, c))))))))), implies(implies(implies(implies(V, implies(implies(not(implies(a, c)), not(implies(b, c))), U)), implies(implies(a, c), implies(not(implies(not(implies(a, c)), not(implies(b, c)))), not(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))))))), implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(not(implies(a, c)), not(implies(b, c))))), implies(not(implies(implies(b, c), implies(a, c))), not(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c))))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)), implies(implies(b, c), implies(a, c)))), true), true, is_a_theorem(implies(implies(a, b), implies(implies(b, c), implies(a, c)))), true)
% 3.02/1.04  = { by axiom 1 (ifeq_axiom) }
% 3.02/1.04    ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(b, c), implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))))), true, is_a_theorem(implies(implies(implies(implies(V, implies(implies(not(implies(a, c)), not(implies(b, c))), U)), implies(implies(a, c), implies(not(implies(not(implies(a, c)), not(implies(b, c)))), not(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))))))), implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(not(implies(a, c)), not(implies(b, c))))), implies(implies(b, c), implies(a, c)))), true), true, is_a_theorem(implies(implies(implies(X, implies(implies(not(implies(implies(b, c), implies(a, c))), not(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)))), W)), implies(implies(implies(b, c), implies(a, c)), implies(not(implies(not(implies(implies(b, c), implies(a, c))), not(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c))))), not(implies(implies(implies(V, implies(implies(not(implies(a, c)), not(implies(b, c))), U)), implies(implies(a, c), implies(not(implies(not(implies(a, c)), not(implies(b, c)))), not(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))))))), implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(not(implies(a, c)), not(implies(b, c))))))))), implies(implies(implies(implies(V, implies(implies(not(implies(a, c)), not(implies(b, c))), U)), implies(implies(a, c), implies(not(implies(not(implies(a, c)), not(implies(b, c)))), not(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))))))), implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(not(implies(a, c)), not(implies(b, c))))), implies(not(implies(implies(b, c), implies(a, c))), not(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c))))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)), implies(implies(b, c), implies(a, c)))), true), true, is_a_theorem(implies(implies(a, b), implies(implies(b, c), implies(a, c)))), true)
% 3.02/1.05  = { by axiom 3 (a5) }
% 3.02/1.05    ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(implies(V, implies(implies(not(implies(a, c)), not(implies(b, c))), U)), implies(implies(a, c), implies(not(implies(not(implies(a, c)), not(implies(b, c)))), not(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))))))), implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(not(implies(a, c)), not(implies(b, c))))), implies(implies(b, c), implies(a, c)))), true), true, is_a_theorem(implies(implies(implies(X, implies(implies(not(implies(implies(b, c), implies(a, c))), not(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)))), W)), implies(implies(implies(b, c), implies(a, c)), implies(not(implies(not(implies(implies(b, c), implies(a, c))), not(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c))))), not(implies(implies(implies(V, implies(implies(not(implies(a, c)), not(implies(b, c))), U)), implies(implies(a, c), implies(not(implies(not(implies(a, c)), not(implies(b, c)))), not(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))))))), implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(not(implies(a, c)), not(implies(b, c))))))))), implies(implies(implies(implies(V, implies(implies(not(implies(a, c)), not(implies(b, c))), U)), implies(implies(a, c), implies(not(implies(not(implies(a, c)), not(implies(b, c)))), not(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))))))), implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(not(implies(a, c)), not(implies(b, c))))), implies(not(implies(implies(b, c), implies(a, c))), not(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c))))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)), implies(implies(b, c), implies(a, c)))), true), true, is_a_theorem(implies(implies(a, b), implies(implies(b, c), implies(a, c)))), true)
% 3.02/1.06  = { by axiom 1 (ifeq_axiom) }
% 3.02/1.06    ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(V, implies(implies(not(implies(a, c)), not(implies(b, c))), U)), implies(implies(a, c), implies(not(implies(not(implies(a, c)), not(implies(b, c)))), not(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))))))), implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(not(implies(a, c)), not(implies(b, c))))), implies(implies(b, c), implies(a, c)))), true, is_a_theorem(implies(implies(implies(X, implies(implies(not(implies(implies(b, c), implies(a, c))), not(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)))), W)), implies(implies(implies(b, c), implies(a, c)), implies(not(implies(not(implies(implies(b, c), implies(a, c))), not(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c))))), not(implies(implies(implies(V, implies(implies(not(implies(a, c)), not(implies(b, c))), U)), implies(implies(a, c), implies(not(implies(not(implies(a, c)), not(implies(b, c)))), not(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))))))), implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(not(implies(a, c)), not(implies(b, c))))))))), implies(implies(implies(implies(V, implies(implies(not(implies(a, c)), not(implies(b, c))), U)), implies(implies(a, c), implies(not(implies(not(implies(a, c)), not(implies(b, c)))), not(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))))))), implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(not(implies(a, c)), not(implies(b, c))))), implies(not(implies(implies(b, c), implies(a, c))), not(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c))))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)), implies(implies(b, c), implies(a, c)))), true), true, is_a_theorem(implies(implies(a, b), implies(implies(b, c), implies(a, c)))), true)
% 3.02/1.06  = { by axiom 1 (ifeq_axiom) R->L }
% 3.02/1.07    ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(implies(implies(V, implies(implies(not(implies(a, c)), not(implies(b, c))), U)), implies(implies(a, c), implies(not(implies(not(implies(a, c)), not(implies(b, c)))), not(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))))))), implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(not(implies(a, c)), not(implies(b, c))))), implies(implies(b, c), implies(a, c)))), true, is_a_theorem(implies(implies(implies(X, implies(implies(not(implies(implies(b, c), implies(a, c))), not(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)))), W)), implies(implies(implies(b, c), implies(a, c)), implies(not(implies(not(implies(implies(b, c), implies(a, c))), not(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c))))), not(implies(implies(implies(V, implies(implies(not(implies(a, c)), not(implies(b, c))), U)), implies(implies(a, c), implies(not(implies(not(implies(a, c)), not(implies(b, c)))), not(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))))))), implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(not(implies(a, c)), not(implies(b, c))))))))), implies(implies(implies(implies(V, implies(implies(not(implies(a, c)), not(implies(b, c))), U)), implies(implies(a, c), implies(not(implies(not(implies(a, c)), not(implies(b, c)))), not(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))))))), implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(not(implies(a, c)), not(implies(b, c))))), implies(not(implies(implies(b, c), implies(a, c))), not(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c))))))), true), true), true, is_a_theorem(implies(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)), implies(implies(b, c), implies(a, c)))), true), true, is_a_theorem(implies(implies(a, b), implies(implies(b, c), implies(a, c)))), true)
% 3.02/1.07  = { by axiom 3 (a5) R->L }
% 4.70/1.09    ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(implies(V, implies(implies(not(implies(a, c)), not(implies(b, c))), U)), implies(implies(a, c), implies(not(implies(not(implies(a, c)), not(implies(b, c)))), not(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))))))), implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(not(implies(a, c)), not(implies(b, c))))), implies(implies(b, c), implies(a, c))), implies(implies(implies(X, implies(implies(not(implies(implies(b, c), implies(a, c))), not(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)))), W)), implies(implies(implies(b, c), implies(a, c)), implies(not(implies(not(implies(implies(b, c), implies(a, c))), not(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c))))), not(implies(implies(implies(V, implies(implies(not(implies(a, c)), not(implies(b, c))), U)), implies(implies(a, c), implies(not(implies(not(implies(a, c)), not(implies(b, c)))), not(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))))))), implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(not(implies(a, c)), not(implies(b, c))))))))), implies(implies(implies(implies(V, implies(implies(not(implies(a, c)), not(implies(b, c))), U)), implies(implies(a, c), implies(not(implies(not(implies(a, c)), not(implies(b, c)))), not(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))))))), implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(not(implies(a, c)), not(implies(b, c))))), implies(not(implies(implies(b, c), implies(a, c))), not(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)))))))), true, ifeq(is_a_theorem(implies(implies(implies(implies(V, implies(implies(not(implies(a, c)), not(implies(b, c))), U)), implies(implies(a, c), implies(not(implies(not(implies(a, c)), not(implies(b, c)))), not(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))))))), implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(not(implies(a, c)), not(implies(b, c))))), implies(implies(b, c), implies(a, c)))), true, is_a_theorem(implies(implies(implies(X, implies(implies(not(implies(implies(b, c), implies(a, c))), not(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)))), W)), implies(implies(implies(b, c), implies(a, c)), implies(not(implies(not(implies(implies(b, c), implies(a, c))), not(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c))))), not(implies(implies(implies(V, implies(implies(not(implies(a, c)), not(implies(b, c))), U)), implies(implies(a, c), implies(not(implies(not(implies(a, c)), not(implies(b, c)))), not(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))))))), implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(not(implies(a, c)), not(implies(b, c))))))))), implies(implies(implies(implies(V, implies(implies(not(implies(a, c)), not(implies(b, c))), U)), implies(implies(a, c), implies(not(implies(not(implies(a, c)), not(implies(b, c)))), not(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))))))), implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(not(implies(a, c)), not(implies(b, c))))), implies(not(implies(implies(b, c), implies(a, c))), not(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c))))))), true), true), true, is_a_theorem(implies(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)), implies(implies(b, c), implies(a, c)))), true), true, is_a_theorem(implies(implies(a, b), implies(implies(b, c), implies(a, c)))), true)
% 4.70/1.09  = { by axiom 2 (condensed_detachment) }
% 4.70/1.09    ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)), implies(implies(b, c), implies(a, c)))), true), true, is_a_theorem(implies(implies(a, b), implies(implies(b, c), implies(a, c)))), true)
% 4.70/1.10  = { by axiom 1 (ifeq_axiom) }
% 4.70/1.10    ifeq(is_a_theorem(implies(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)), implies(implies(b, c), implies(a, c)))), true, is_a_theorem(implies(implies(a, b), implies(implies(b, c), implies(a, c)))), true)
% 4.70/1.11  = { by axiom 1 (ifeq_axiom) R->L }
% 4.70/1.11    ifeq(true, true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)), implies(implies(b, c), implies(a, c)))), true, is_a_theorem(implies(implies(a, b), implies(implies(b, c), implies(a, c)))), true), true)
% 4.70/1.12  = { by axiom 2 (condensed_detachment) R->L }
% 4.70/1.13    ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(T, implies(implies(not(implies(implies(a, b), implies(implies(b, c), implies(a, c)))), not(implies(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)), implies(implies(b, c), implies(a, c))))), S)), implies(implies(implies(a, b), implies(implies(b, c), implies(a, c))), implies(not(implies(not(implies(implies(a, b), implies(implies(b, c), implies(a, c)))), not(implies(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)), implies(implies(b, c), implies(a, c)))))), not(implies(implies(implies(X2, implies(implies(not(implies(implies(b, c), implies(a, c))), not(implies(a, b))), Y2)), implies(implies(implies(b, c), implies(a, c)), 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c)))))), implies(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)), implies(not(implies(implies(b, c), implies(a, c))), not(implies(a, b))))), implies(not(implies(implies(a, b), implies(implies(b, c), implies(a, c)))), not(implies(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)), implies(implies(b, c), implies(a, c))))))), implies(implies(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)), implies(implies(b, c), implies(a, c))), implies(implies(a, b), implies(implies(b, c), implies(a, c)))))), true, ifeq(is_a_theorem(implies(implies(implies(T, implies(implies(not(implies(implies(a, b), 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not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)), implies(not(implies(implies(b, c), implies(a, c))), not(implies(a, b))))), implies(not(implies(implies(a, b), implies(implies(b, c), implies(a, c)))), not(implies(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)), implies(implies(b, c), implies(a, c)))))))), true, is_a_theorem(implies(implies(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)), implies(implies(b, c), implies(a, c))), implies(implies(a, b), implies(implies(b, c), implies(a, c))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, 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% 4.70/1.14  = { by axiom 1 (ifeq_axiom) R->L }
% 4.70/1.16    ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(implies(T, implies(implies(not(implies(implies(a, b), implies(implies(b, c), implies(a, c)))), not(implies(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)), implies(implies(b, c), implies(a, c))))), S)), implies(implies(implies(a, b), implies(implies(b, c), implies(a, c))), implies(not(implies(not(implies(implies(a, b), implies(implies(b, c), implies(a, c)))), not(implies(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)), implies(implies(b, c), implies(a, c)))))), not(implies(implies(implies(X2, implies(implies(not(implies(implies(b, c), implies(a, c))), not(implies(a, b))), Y2)), implies(implies(implies(b, c), implies(a, c)), 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c)))))), implies(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)), implies(not(implies(implies(b, c), implies(a, c))), not(implies(a, b))))), implies(not(implies(implies(a, b), implies(implies(b, c), implies(a, c)))), not(implies(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)), implies(implies(b, c), implies(a, c))))))), implies(implies(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)), implies(implies(b, c), implies(a, c))), implies(implies(a, b), implies(implies(b, c), implies(a, c)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(T, 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% 4.70/1.17  = { by axiom 3 (a5) R->L }
% 5.53/1.19    ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)), implies(implies(b, c), implies(a, c))), implies(implies(implies(X2, implies(implies(not(implies(implies(b, c), implies(a, c))), not(implies(a, b))), Y2)), implies(implies(implies(b, c), implies(a, c)), implies(not(implies(not(implies(implies(b, c), implies(a, c))), not(implies(a, b)))), not(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)))))), implies(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)), implies(not(implies(implies(b, c), implies(a, c))), not(implies(a, b))))))), true, is_a_theorem(implies(implies(implies(implies(T, implies(implies(not(implies(implies(a, b), implies(implies(b, c), implies(a, c)))), not(implies(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)), implies(implies(b, c), implies(a, c))))), S)), implies(implies(implies(a, b), implies(implies(b, c), implies(a, c))), implies(not(implies(not(implies(implies(a, b), implies(implies(b, c), implies(a, c)))), not(implies(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)), implies(implies(b, c), implies(a, c)))))), not(implies(implies(implies(X2, implies(implies(not(implies(implies(b, c), implies(a, c))), not(implies(a, b))), Y2)), implies(implies(implies(b, c), implies(a, c)), implies(not(implies(not(implies(implies(b, c), 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implies(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)), implies(not(implies(implies(b, c), implies(a, c))), not(implies(a, b))))), implies(not(implies(implies(a, b), implies(implies(b, c), implies(a, c)))), not(implies(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)), implies(implies(b, c), implies(a, c))))))), implies(implies(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)), implies(implies(b, c), implies(a, c))), implies(implies(a, b), implies(implies(b, c), implies(a, c)))))), true), true, ifeq(is_a_theorem(implies(implies(implies(T, implies(implies(not(implies(implies(a, b), 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not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)), implies(not(implies(implies(b, c), implies(a, c))), not(implies(a, b))))), implies(not(implies(implies(a, b), implies(implies(b, c), implies(a, c)))), not(implies(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)), implies(implies(b, c), implies(a, c)))))))), true, is_a_theorem(implies(implies(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)), implies(implies(b, c), implies(a, c))), implies(implies(a, b), implies(implies(b, c), implies(a, c))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)), implies(implies(b, c), implies(a, c)))), true, is_a_theorem(implies(implies(a, b), implies(implies(b, c), implies(a, c)))), true), true)
% 5.53/1.19  = { by axiom 1 (ifeq_axiom) R->L }
% 5.74/1.21    ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)), implies(implies(b, c), implies(a, c))), implies(implies(implies(X2, implies(implies(not(implies(implies(b, c), implies(a, c))), not(implies(a, b))), Y2)), implies(implies(implies(b, c), implies(a, c)), implies(not(implies(not(implies(implies(b, c), implies(a, c))), not(implies(a, b)))), not(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)))))), implies(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)), implies(not(implies(implies(b, c), implies(a, c))), 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c)))))), implies(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)), implies(not(implies(implies(b, c), implies(a, c))), not(implies(a, b))))), implies(not(implies(implies(a, b), implies(implies(b, c), implies(a, c)))), not(implies(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)), implies(implies(b, c), implies(a, c))))))), implies(implies(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)), implies(implies(b, c), implies(a, c))), implies(implies(a, b), implies(implies(b, c), implies(a, c)))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(T, 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not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)), implies(implies(b, c), implies(a, c)))), true, is_a_theorem(implies(implies(a, b), implies(implies(b, c), implies(a, c)))), true), true)
% 5.74/1.22  = { by axiom 3 (a5) R->L }
% 5.74/1.24    ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)), implies(implies(b, c), implies(a, c))), implies(implies(implies(X2, implies(implies(not(implies(implies(b, c), implies(a, c))), not(implies(a, b))), Y2)), implies(implies(implies(b, c), implies(a, c)), implies(not(implies(not(implies(implies(b, c), implies(a, c))), not(implies(a, b)))), not(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)))))), implies(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)), implies(not(implies(implies(b, c), implies(a, c))), not(implies(a, b)))))), implies(implies(implies(implies(T, implies(implies(not(implies(implies(a, b), implies(implies(b, c), implies(a, c)))), not(implies(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)), implies(implies(b, c), implies(a, c))))), S)), implies(implies(implies(a, b), implies(implies(b, c), implies(a, c))), implies(not(implies(not(implies(implies(a, b), implies(implies(b, c), implies(a, c)))), not(implies(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)), implies(implies(b, c), implies(a, c)))))), not(implies(implies(implies(X2, implies(implies(not(implies(implies(b, c), implies(a, c))), not(implies(a, b))), Y2)), implies(implies(implies(b, c), implies(a, c)), implies(not(implies(not(implies(implies(b, c), implies(a, 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c)))), true), true)
% 5.74/1.25  = { by axiom 2 (condensed_detachment) }
% 5.74/1.26    ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(implies(T, implies(implies(not(implies(implies(a, b), implies(implies(b, c), implies(a, c)))), not(implies(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)), implies(implies(b, c), implies(a, c))))), S)), implies(implies(implies(a, b), implies(implies(b, c), implies(a, c))), implies(not(implies(not(implies(implies(a, b), implies(implies(b, c), implies(a, c)))), not(implies(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)), implies(implies(b, c), implies(a, c)))))), not(implies(implies(implies(X2, implies(implies(not(implies(implies(b, c), implies(a, c))), not(implies(a, b))), Y2)), implies(implies(implies(b, c), implies(a, c)), implies(not(implies(not(implies(implies(b, c), implies(a, c))), not(implies(a, b)))), not(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)))))), implies(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)), implies(not(implies(implies(b, c), implies(a, c))), not(implies(a, b))))))))), implies(implies(implies(implies(X2, implies(implies(not(implies(implies(b, c), implies(a, c))), not(implies(a, b))), Y2)), implies(implies(implies(b, c), implies(a, c)), implies(not(implies(not(implies(implies(b, c), implies(a, c))), not(implies(a, b)))), not(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)))))), implies(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)), implies(not(implies(implies(b, c), implies(a, c))), not(implies(a, b))))), implies(not(implies(implies(a, b), implies(implies(b, c), implies(a, c)))), not(implies(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)), implies(implies(b, c), implies(a, c)))))))), true, is_a_theorem(implies(implies(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)), implies(implies(b, c), implies(a, c))), implies(implies(a, b), implies(implies(b, c), implies(a, c))))), true), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)), implies(implies(b, c), implies(a, c)))), true, is_a_theorem(implies(implies(a, b), implies(implies(b, c), implies(a, c)))), true), true)
% 5.74/1.27  = { by axiom 1 (ifeq_axiom) }
% 5.74/1.27    ifeq(ifeq(is_a_theorem(implies(implies(implies(T, implies(implies(not(implies(implies(a, b), implies(implies(b, c), implies(a, c)))), not(implies(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)), implies(implies(b, c), implies(a, c))))), S)), implies(implies(implies(a, b), implies(implies(b, c), implies(a, c))), implies(not(implies(not(implies(implies(a, b), implies(implies(b, c), implies(a, c)))), not(implies(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)), implies(implies(b, c), implies(a, c)))))), not(implies(implies(implies(X2, implies(implies(not(implies(implies(b, c), implies(a, c))), not(implies(a, b))), Y2)), implies(implies(implies(b, c), implies(a, c)), implies(not(implies(not(implies(implies(b, c), implies(a, c))), not(implies(a, b)))), not(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)))))), implies(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)), implies(not(implies(implies(b, c), implies(a, c))), not(implies(a, b))))))))), implies(implies(implies(implies(X2, implies(implies(not(implies(implies(b, c), implies(a, c))), not(implies(a, b))), Y2)), implies(implies(implies(b, c), implies(a, c)), implies(not(implies(not(implies(implies(b, c), implies(a, c))), not(implies(a, b)))), not(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)))))), implies(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)), implies(not(implies(implies(b, c), implies(a, c))), not(implies(a, b))))), implies(not(implies(implies(a, b), implies(implies(b, c), implies(a, c)))), not(implies(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)), implies(implies(b, c), implies(a, c)))))))), true, is_a_theorem(implies(implies(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)), implies(implies(b, c), implies(a, c))), implies(implies(a, b), implies(implies(b, c), implies(a, c))))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)), implies(implies(b, c), implies(a, c)))), true, is_a_theorem(implies(implies(a, b), implies(implies(b, c), implies(a, c)))), true), true)
% 5.74/1.28  = { by axiom 1 (ifeq_axiom) R->L }
% 5.74/1.29    ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(T, implies(implies(not(implies(implies(a, b), implies(implies(b, c), implies(a, c)))), not(implies(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)), implies(implies(b, c), implies(a, c))))), S)), implies(implies(implies(a, b), implies(implies(b, c), implies(a, c))), implies(not(implies(not(implies(implies(a, b), implies(implies(b, c), implies(a, c)))), not(implies(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)), implies(implies(b, c), implies(a, c)))))), not(implies(implies(implies(X2, implies(implies(not(implies(implies(b, c), implies(a, c))), not(implies(a, b))), Y2)), implies(implies(implies(b, c), implies(a, c)), implies(not(implies(not(implies(implies(b, c), implies(a, c))), not(implies(a, b)))), not(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)))))), implies(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)), implies(not(implies(implies(b, c), implies(a, c))), not(implies(a, b))))))))), implies(implies(implies(implies(X2, implies(implies(not(implies(implies(b, c), implies(a, c))), not(implies(a, b))), Y2)), implies(implies(implies(b, c), implies(a, c)), implies(not(implies(not(implies(implies(b, c), implies(a, c))), not(implies(a, b)))), not(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)))))), implies(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)), implies(not(implies(implies(b, c), implies(a, c))), not(implies(a, b))))), implies(not(implies(implies(a, b), implies(implies(b, c), implies(a, c)))), not(implies(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)), implies(implies(b, c), implies(a, c)))))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)), implies(implies(b, c), implies(a, c))), implies(implies(a, b), implies(implies(b, c), implies(a, c))))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)), implies(implies(b, c), implies(a, c)))), true, is_a_theorem(implies(implies(a, b), implies(implies(b, c), implies(a, c)))), true), true)
% 5.74/1.29  = { by axiom 2 (condensed_detachment) R->L }
% 5.74/1.30    ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(a, b), implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c))), implies(implies(implies(implies(X2, implies(implies(not(implies(implies(b, c), implies(a, c))), not(implies(a, b))), Y2)), implies(implies(implies(b, c), implies(a, c)), implies(not(implies(not(implies(implies(b, c), implies(a, c))), not(implies(a, b)))), not(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)))))), implies(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)), implies(not(implies(implies(b, c), implies(a, c))), not(implies(a, b))))), implies(implies(a, b), implies(implies(b, c), implies(a, c)))))), true, ifeq(is_a_theorem(implies(implies(a, b), implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)))), true, is_a_theorem(implies(implies(implies(implies(X2, implies(implies(not(implies(implies(b, c), implies(a, c))), not(implies(a, b))), Y2)), implies(implies(implies(b, c), implies(a, c)), implies(not(implies(not(implies(implies(b, c), implies(a, c))), not(implies(a, b)))), not(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)))))), implies(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)), implies(not(implies(implies(b, c), implies(a, c))), not(implies(a, b))))), implies(implies(a, b), implies(implies(b, c), implies(a, c))))), true), true), true, is_a_theorem(implies(implies(implies(T, implies(implies(not(implies(implies(a, b), implies(implies(b, c), implies(a, c)))), not(implies(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)), implies(implies(b, c), implies(a, c))))), S)), implies(implies(implies(a, b), implies(implies(b, c), implies(a, c))), implies(not(implies(not(implies(implies(a, b), implies(implies(b, c), implies(a, c)))), not(implies(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)), implies(implies(b, c), implies(a, c)))))), not(implies(implies(implies(X2, implies(implies(not(implies(implies(b, c), implies(a, c))), not(implies(a, b))), Y2)), implies(implies(implies(b, c), implies(a, c)), implies(not(implies(not(implies(implies(b, c), implies(a, c))), not(implies(a, b)))), not(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)))))), implies(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)), implies(not(implies(implies(b, c), implies(a, c))), not(implies(a, b))))))))), implies(implies(implies(implies(X2, implies(implies(not(implies(implies(b, c), implies(a, c))), not(implies(a, b))), Y2)), implies(implies(implies(b, c), implies(a, c)), implies(not(implies(not(implies(implies(b, c), implies(a, c))), not(implies(a, b)))), not(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)))))), implies(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)), implies(not(implies(implies(b, c), implies(a, c))), not(implies(a, b))))), implies(not(implies(implies(a, b), implies(implies(b, c), implies(a, c)))), not(implies(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)), implies(implies(b, c), implies(a, c)))))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)), implies(implies(b, c), implies(a, c))), implies(implies(a, b), implies(implies(b, c), implies(a, c))))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)), implies(implies(b, c), implies(a, c)))), true, is_a_theorem(implies(implies(a, b), implies(implies(b, c), implies(a, c)))), true), true)
% 6.52/1.31  = { by axiom 3 (a5) }
% 6.52/1.32    ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(a, b), implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)))), true, is_a_theorem(implies(implies(implies(implies(X2, implies(implies(not(implies(implies(b, c), implies(a, c))), not(implies(a, b))), Y2)), implies(implies(implies(b, c), implies(a, c)), implies(not(implies(not(implies(implies(b, c), implies(a, c))), not(implies(a, b)))), not(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)))))), implies(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)), implies(not(implies(implies(b, c), implies(a, c))), not(implies(a, b))))), implies(implies(a, b), implies(implies(b, c), implies(a, c))))), true), true), true, is_a_theorem(implies(implies(implies(T, implies(implies(not(implies(implies(a, b), implies(implies(b, c), implies(a, c)))), not(implies(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)), implies(implies(b, c), implies(a, c))))), S)), implies(implies(implies(a, b), implies(implies(b, c), implies(a, c))), implies(not(implies(not(implies(implies(a, b), implies(implies(b, c), implies(a, c)))), not(implies(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)), implies(implies(b, c), implies(a, c)))))), not(implies(implies(implies(X2, implies(implies(not(implies(implies(b, c), implies(a, c))), not(implies(a, b))), Y2)), implies(implies(implies(b, c), implies(a, c)), implies(not(implies(not(implies(implies(b, c), implies(a, c))), not(implies(a, b)))), not(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)))))), implies(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)), implies(not(implies(implies(b, c), implies(a, c))), not(implies(a, b))))))))), implies(implies(implies(implies(X2, implies(implies(not(implies(implies(b, c), implies(a, c))), not(implies(a, b))), Y2)), implies(implies(implies(b, c), implies(a, c)), implies(not(implies(not(implies(implies(b, c), implies(a, c))), not(implies(a, b)))), not(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)))))), implies(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)), implies(not(implies(implies(b, c), implies(a, c))), not(implies(a, b))))), implies(not(implies(implies(a, b), implies(implies(b, c), implies(a, c)))), not(implies(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)), implies(implies(b, c), implies(a, c)))))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)), implies(implies(b, c), implies(a, c))), implies(implies(a, b), implies(implies(b, c), implies(a, c))))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)), implies(implies(b, c), implies(a, c)))), true, is_a_theorem(implies(implies(a, b), implies(implies(b, c), implies(a, c)))), true), true)
% 6.52/1.33  = { by axiom 1 (ifeq_axiom) }
% 6.52/1.33    ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(a, b), implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)))), true, is_a_theorem(implies(implies(implies(implies(X2, implies(implies(not(implies(implies(b, c), implies(a, c))), not(implies(a, b))), Y2)), implies(implies(implies(b, c), implies(a, c)), implies(not(implies(not(implies(implies(b, c), implies(a, c))), not(implies(a, b)))), not(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)))))), implies(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)), implies(not(implies(implies(b, c), implies(a, c))), not(implies(a, b))))), implies(implies(a, b), implies(implies(b, c), implies(a, c))))), true), true, is_a_theorem(implies(implies(implies(T, implies(implies(not(implies(implies(a, b), implies(implies(b, c), implies(a, c)))), not(implies(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)), implies(implies(b, c), implies(a, c))))), S)), implies(implies(implies(a, b), implies(implies(b, c), implies(a, c))), implies(not(implies(not(implies(implies(a, b), implies(implies(b, c), implies(a, c)))), not(implies(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)), implies(implies(b, c), implies(a, c)))))), not(implies(implies(implies(X2, implies(implies(not(implies(implies(b, c), implies(a, c))), not(implies(a, b))), Y2)), implies(implies(implies(b, c), implies(a, c)), implies(not(implies(not(implies(implies(b, c), implies(a, c))), not(implies(a, b)))), not(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)))))), implies(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)), implies(not(implies(implies(b, c), implies(a, c))), not(implies(a, b))))))))), implies(implies(implies(implies(X2, implies(implies(not(implies(implies(b, c), implies(a, c))), not(implies(a, b))), Y2)), implies(implies(implies(b, c), implies(a, c)), implies(not(implies(not(implies(implies(b, c), implies(a, c))), not(implies(a, b)))), not(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)))))), implies(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)), implies(not(implies(implies(b, c), implies(a, c))), not(implies(a, b))))), implies(not(implies(implies(a, b), implies(implies(b, c), implies(a, c)))), not(implies(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)), implies(implies(b, c), implies(a, c)))))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)), implies(implies(b, c), implies(a, c))), implies(implies(a, b), implies(implies(b, c), implies(a, c))))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)), implies(implies(b, c), implies(a, c)))), true, is_a_theorem(implies(implies(a, b), implies(implies(b, c), implies(a, c)))), true), true)
% 6.52/1.34  = { by axiom 3 (a5) }
% 6.52/1.35    ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(implies(X2, implies(implies(not(implies(implies(b, c), implies(a, c))), not(implies(a, b))), Y2)), implies(implies(implies(b, c), implies(a, c)), implies(not(implies(not(implies(implies(b, c), implies(a, c))), not(implies(a, b)))), not(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)))))), implies(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)), implies(not(implies(implies(b, c), implies(a, c))), not(implies(a, b))))), implies(implies(a, b), implies(implies(b, c), implies(a, c))))), true), true, is_a_theorem(implies(implies(implies(T, implies(implies(not(implies(implies(a, b), implies(implies(b, c), implies(a, c)))), not(implies(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)), implies(implies(b, c), implies(a, c))))), S)), implies(implies(implies(a, b), implies(implies(b, c), implies(a, c))), implies(not(implies(not(implies(implies(a, b), implies(implies(b, c), implies(a, c)))), not(implies(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)), implies(implies(b, c), implies(a, c)))))), not(implies(implies(implies(X2, implies(implies(not(implies(implies(b, c), implies(a, c))), not(implies(a, b))), Y2)), implies(implies(implies(b, c), implies(a, c)), implies(not(implies(not(implies(implies(b, c), implies(a, c))), not(implies(a, b)))), not(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)))))), implies(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)), implies(not(implies(implies(b, c), implies(a, c))), not(implies(a, b))))))))), implies(implies(implies(implies(X2, implies(implies(not(implies(implies(b, c), implies(a, c))), not(implies(a, b))), Y2)), implies(implies(implies(b, c), implies(a, c)), implies(not(implies(not(implies(implies(b, c), implies(a, c))), not(implies(a, b)))), not(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)))))), implies(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)), implies(not(implies(implies(b, c), implies(a, c))), not(implies(a, b))))), implies(not(implies(implies(a, b), implies(implies(b, c), implies(a, c)))), not(implies(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)), implies(implies(b, c), implies(a, c)))))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)), implies(implies(b, c), implies(a, c))), implies(implies(a, b), implies(implies(b, c), implies(a, c))))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)), implies(implies(b, c), implies(a, c)))), true, is_a_theorem(implies(implies(a, b), implies(implies(b, c), implies(a, c)))), true), true)
% 6.52/1.36  = { by axiom 1 (ifeq_axiom) }
% 6.52/1.36    ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(X2, implies(implies(not(implies(implies(b, c), implies(a, c))), not(implies(a, b))), Y2)), implies(implies(implies(b, c), implies(a, c)), implies(not(implies(not(implies(implies(b, c), implies(a, c))), not(implies(a, b)))), not(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)))))), implies(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)), implies(not(implies(implies(b, c), implies(a, c))), not(implies(a, b))))), implies(implies(a, b), implies(implies(b, c), implies(a, c))))), true, is_a_theorem(implies(implies(implies(T, implies(implies(not(implies(implies(a, b), implies(implies(b, c), implies(a, c)))), not(implies(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)), implies(implies(b, c), implies(a, c))))), S)), implies(implies(implies(a, b), implies(implies(b, c), implies(a, c))), implies(not(implies(not(implies(implies(a, b), implies(implies(b, c), implies(a, c)))), not(implies(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)), implies(implies(b, c), implies(a, c)))))), not(implies(implies(implies(X2, implies(implies(not(implies(implies(b, c), implies(a, c))), not(implies(a, b))), Y2)), implies(implies(implies(b, c), implies(a, c)), implies(not(implies(not(implies(implies(b, c), implies(a, c))), not(implies(a, b)))), not(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)))))), implies(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)), implies(not(implies(implies(b, c), implies(a, c))), not(implies(a, b))))))))), implies(implies(implies(implies(X2, implies(implies(not(implies(implies(b, c), implies(a, c))), not(implies(a, b))), Y2)), implies(implies(implies(b, c), implies(a, c)), implies(not(implies(not(implies(implies(b, c), implies(a, c))), not(implies(a, b)))), not(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)))))), implies(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)), implies(not(implies(implies(b, c), implies(a, c))), not(implies(a, b))))), implies(not(implies(implies(a, b), implies(implies(b, c), implies(a, c)))), not(implies(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)), implies(implies(b, c), implies(a, c)))))))), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)), implies(implies(b, c), implies(a, c))), implies(implies(a, b), implies(implies(b, c), implies(a, c))))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)), implies(implies(b, c), implies(a, c)))), true, is_a_theorem(implies(implies(a, b), implies(implies(b, c), implies(a, c)))), true), true)
% 6.52/1.36  = { by axiom 1 (ifeq_axiom) R->L }
% 6.52/1.38    ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(implies(implies(X2, implies(implies(not(implies(implies(b, c), implies(a, c))), not(implies(a, b))), Y2)), implies(implies(implies(b, c), implies(a, c)), implies(not(implies(not(implies(implies(b, c), implies(a, c))), not(implies(a, b)))), not(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)))))), implies(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)), implies(not(implies(implies(b, c), implies(a, c))), not(implies(a, b))))), implies(implies(a, b), implies(implies(b, c), implies(a, c))))), true, is_a_theorem(implies(implies(implies(T, implies(implies(not(implies(implies(a, b), implies(implies(b, c), implies(a, c)))), not(implies(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)), implies(implies(b, c), implies(a, c))))), S)), implies(implies(implies(a, b), implies(implies(b, c), implies(a, c))), implies(not(implies(not(implies(implies(a, b), implies(implies(b, c), implies(a, c)))), not(implies(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)), implies(implies(b, c), implies(a, c)))))), not(implies(implies(implies(X2, implies(implies(not(implies(implies(b, c), implies(a, c))), not(implies(a, b))), Y2)), implies(implies(implies(b, c), implies(a, c)), implies(not(implies(not(implies(implies(b, c), implies(a, c))), not(implies(a, b)))), not(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)))))), implies(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)), implies(not(implies(implies(b, c), implies(a, c))), not(implies(a, b))))))))), implies(implies(implies(implies(X2, implies(implies(not(implies(implies(b, c), implies(a, c))), not(implies(a, b))), Y2)), implies(implies(implies(b, c), implies(a, c)), implies(not(implies(not(implies(implies(b, c), implies(a, c))), not(implies(a, b)))), not(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)))))), implies(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)), implies(not(implies(implies(b, c), implies(a, c))), not(implies(a, b))))), implies(not(implies(implies(a, b), implies(implies(b, c), implies(a, c)))), not(implies(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)), implies(implies(b, c), implies(a, c)))))))), true), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)), implies(implies(b, c), implies(a, c))), implies(implies(a, b), implies(implies(b, c), implies(a, c))))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)), implies(implies(b, c), implies(a, c)))), true, is_a_theorem(implies(implies(a, b), implies(implies(b, c), implies(a, c)))), true), true)
% 6.52/1.38  = { by axiom 3 (a5) R->L }
% 6.52/1.39    ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(implies(X2, implies(implies(not(implies(implies(b, c), implies(a, c))), not(implies(a, b))), Y2)), implies(implies(implies(b, c), implies(a, c)), implies(not(implies(not(implies(implies(b, c), implies(a, c))), not(implies(a, b)))), not(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)))))), implies(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)), implies(not(implies(implies(b, c), implies(a, c))), not(implies(a, b))))), implies(implies(a, b), implies(implies(b, c), implies(a, c)))), implies(implies(implies(T, implies(implies(not(implies(implies(a, b), implies(implies(b, c), implies(a, c)))), not(implies(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)), implies(implies(b, c), implies(a, c))))), S)), implies(implies(implies(a, b), implies(implies(b, c), implies(a, c))), implies(not(implies(not(implies(implies(a, b), implies(implies(b, c), implies(a, c)))), not(implies(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)), implies(implies(b, c), implies(a, c)))))), not(implies(implies(implies(X2, implies(implies(not(implies(implies(b, c), implies(a, c))), not(implies(a, b))), Y2)), implies(implies(implies(b, c), implies(a, c)), implies(not(implies(not(implies(implies(b, c), implies(a, c))), not(implies(a, b)))), not(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)))))), implies(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)), implies(not(implies(implies(b, c), implies(a, c))), not(implies(a, b))))))))), implies(implies(implies(implies(X2, implies(implies(not(implies(implies(b, c), implies(a, c))), not(implies(a, b))), Y2)), implies(implies(implies(b, c), implies(a, c)), implies(not(implies(not(implies(implies(b, c), implies(a, c))), not(implies(a, b)))), not(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)))))), implies(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)), implies(not(implies(implies(b, c), implies(a, c))), not(implies(a, b))))), implies(not(implies(implies(a, b), implies(implies(b, c), implies(a, c)))), not(implies(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)), implies(implies(b, c), implies(a, c))))))))), true, ifeq(is_a_theorem(implies(implies(implies(implies(X2, implies(implies(not(implies(implies(b, c), implies(a, c))), not(implies(a, b))), Y2)), implies(implies(implies(b, c), implies(a, c)), implies(not(implies(not(implies(implies(b, c), implies(a, c))), not(implies(a, b)))), not(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)))))), implies(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)), implies(not(implies(implies(b, c), implies(a, c))), not(implies(a, b))))), implies(implies(a, b), implies(implies(b, c), implies(a, c))))), true, is_a_theorem(implies(implies(implies(T, implies(implies(not(implies(implies(a, b), implies(implies(b, c), implies(a, c)))), not(implies(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)), implies(implies(b, c), implies(a, c))))), S)), implies(implies(implies(a, b), implies(implies(b, c), implies(a, c))), implies(not(implies(not(implies(implies(a, b), implies(implies(b, c), implies(a, c)))), not(implies(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)), implies(implies(b, c), implies(a, c)))))), not(implies(implies(implies(X2, implies(implies(not(implies(implies(b, c), implies(a, c))), not(implies(a, b))), Y2)), implies(implies(implies(b, c), implies(a, c)), implies(not(implies(not(implies(implies(b, c), implies(a, c))), not(implies(a, b)))), not(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)))))), implies(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)), implies(not(implies(implies(b, c), implies(a, c))), not(implies(a, b))))))))), implies(implies(implies(implies(X2, implies(implies(not(implies(implies(b, c), implies(a, c))), not(implies(a, b))), Y2)), implies(implies(implies(b, c), implies(a, c)), implies(not(implies(not(implies(implies(b, c), implies(a, c))), not(implies(a, b)))), not(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)))))), implies(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)), implies(not(implies(implies(b, c), implies(a, c))), not(implies(a, b))))), implies(not(implies(implies(a, b), implies(implies(b, c), implies(a, c)))), not(implies(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)), implies(implies(b, c), implies(a, c)))))))), true), true), true, is_a_theorem(implies(implies(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)), implies(implies(b, c), implies(a, c))), implies(implies(a, b), implies(implies(b, c), implies(a, c))))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)), implies(implies(b, c), implies(a, c)))), true, is_a_theorem(implies(implies(a, b), implies(implies(b, c), implies(a, c)))), true), true)
% 6.52/1.39  = { by axiom 2 (condensed_detachment) }
% 6.52/1.39    ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)), implies(implies(b, c), implies(a, c))), implies(implies(a, b), implies(implies(b, c), implies(a, c))))), true), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)), implies(implies(b, c), implies(a, c)))), true, is_a_theorem(implies(implies(a, b), implies(implies(b, c), implies(a, c)))), true), true)
% 6.52/1.39  = { by axiom 1 (ifeq_axiom) }
% 6.52/1.40    ifeq(is_a_theorem(implies(implies(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)), implies(implies(b, c), implies(a, c))), implies(implies(a, b), implies(implies(b, c), implies(a, c))))), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(Y, implies(implies(not(c), not(a)), Z)), implies(c, implies(not(implies(not(c), not(a))), not(b)))), implies(b, implies(not(c), not(a)))), implies(a, c)), implies(implies(b, c), implies(a, c)))), true, is_a_theorem(implies(implies(a, b), implies(implies(b, c), implies(a, c)))), true), true)
% 6.52/1.40  = { by axiom 2 (condensed_detachment) }
% 6.52/1.40    true
% 6.52/1.40  % SZS output end Proof
% 6.52/1.40  
% 6.52/1.40  RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------