%------------------------------------------------------------------------------
% File : Twee---2.7
% Problem : LCL216-3 : TPTP v9.3.1. Released v2.3.0.
% Transfm : none
% Format : tptp:raw
% Command : run_twee /export/starexec/sandbox/benchmark/theBenchmark.p
% Computer : n019.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8046.5625MB
% OS : Linux 6.8.0-71-generic
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Tue Sep 29 11:49:22 AM UTC 2026
% Result : Unsatisfiable 10.69s 1.81s
% Output : Proof 13.84s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.03 % Problem : LCL216-3 : TPTP v9.3.1. Released v2.3.0.
% 0.00/0.04 % Command : run_twee /export/starexec/sandbox/benchmark/theBenchmark.p
% 0.10/0.37 % Computer : n019.cluster.edu
% 0.10/0.37 % Model : x86_64 x86_64
% 0.10/0.37 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.10/0.37 % Memory : 8046.5625MB
% 0.10/0.37 % OS : Linux 6.8.0-71-generic
% 0.10/0.37 % CPULimit : 300
% 0.10/0.37 % WCLimit : 300
% 0.10/0.37 % DateTime : Sun Sep 27 15:27:18 UTC 2026
% 0.10/0.38 % CPUTime :
% 0.10/0.38 Running run_twee /export/starexec/sandbox/benchmark/theBenchmark.p
% 10.69/1.81 Command-line arguments: --lhs-weight 1 --flip-ordering --normalise-queue-percent 10 --cp-renormalise-threshold 10 --complete-subsets --ground-joining-incomplete-limit 15 --flatten-every 2
% 10.69/1.81
% 10.69/1.81 % SZS status Unsatisfiable
% 10.69/1.81
% 11.50/2.07 % SZS output start Proof
% 11.50/2.07 Axiom 1 (implies_definition): implies(X, Y) = or(not(X), Y).
% 11.50/2.07 Axiom 2 (ifeq_axiom): ifeq(X, X, Y, Z) = Y.
% 11.50/2.07 Axiom 3 (axiom_1_3): axiom(implies(X, or(Y, X))) = true.
% 11.50/2.07 Axiom 4 (axiom_1_2): axiom(implies(or(X, X), X)) = true.
% 11.50/2.07 Axiom 5 (rule_1): ifeq(axiom(X), true, theorem(X), true) = true.
% 11.50/2.07 Axiom 6 (axiom_1_4): axiom(implies(or(X, Y), or(Y, X))) = true.
% 11.50/2.07 Axiom 7 (axiom_1_6): axiom(implies(implies(X, Y), implies(or(Z, X), or(Z, Y)))) = true.
% 11.50/2.07 Axiom 8 (axiom_1_5): axiom(implies(or(X, or(Y, Z)), or(Y, or(X, Z)))) = true.
% 12.82/2.08 Axiom 9 (rule_2): ifeq(theorem(implies(X, Y)), true, ifeq(theorem(X), true, theorem(Y), true), true) = true.
% 12.82/2.08
% 12.82/2.08 Goal 1 (prove_this): theorem(implies(or(p, q), implies(or(p, not(q)), p))) = true.
% 12.82/2.08 Proof:
% 12.82/2.08 theorem(implies(or(p, q), implies(or(p, not(q)), p)))
% 12.82/2.08 = { by axiom 2 (ifeq_axiom) R->L }
% 12.82/2.08 ifeq(true, true, theorem(implies(or(p, q), implies(or(p, not(q)), p))), true)
% 12.82/2.08 = { by axiom 9 (rule_2) R->L }
% 12.82/2.08 ifeq(ifeq(theorem(implies(or(not(or(p, not(q))), or(implies(or(p, q), implies(or(p, not(q)), p)), p)), or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p)))), true, ifeq(theorem(or(not(or(p, not(q))), or(implies(or(p, q), implies(or(p, not(q)), p)), p))), true, theorem(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), true), true), true, theorem(implies(or(p, q), implies(or(p, not(q)), p))), true)
% 12.82/2.08 = { by axiom 1 (implies_definition) }
% 12.82/2.08 ifeq(ifeq(theorem(or(not(or(not(or(p, not(q))), or(implies(or(p, q), implies(or(p, not(q)), p)), p))), or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p)))), true, ifeq(theorem(or(not(or(p, not(q))), or(implies(or(p, q), implies(or(p, not(q)), p)), p))), true, theorem(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), true), true), true, theorem(implies(or(p, q), implies(or(p, not(q)), p))), true)
% 12.82/2.08 = { by axiom 2 (ifeq_axiom) R->L }
% 12.82/2.08 ifeq(ifeq(ifeq(true, true, theorem(or(not(or(not(or(p, not(q))), or(implies(or(p, q), implies(or(p, not(q)), p)), p))), or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p)))), true), true, ifeq(theorem(or(not(or(p, not(q))), or(implies(or(p, q), implies(or(p, not(q)), p)), p))), true, theorem(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), true), true), true, theorem(implies(or(p, q), implies(or(p, not(q)), p))), true)
% 12.82/2.08 = { by axiom 8 (axiom_1_5) R->L }
% 12.82/2.08 ifeq(ifeq(ifeq(axiom(implies(or(not(or(p, not(q))), or(implies(or(p, q), implies(or(p, not(q)), p)), p)), or(implies(or(p, q), implies(or(p, not(q)), p)), or(not(or(p, not(q))), p)))), true, theorem(or(not(or(not(or(p, not(q))), or(implies(or(p, q), implies(or(p, not(q)), p)), p))), or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p)))), true), true, ifeq(theorem(or(not(or(p, not(q))), or(implies(or(p, q), implies(or(p, not(q)), p)), p))), true, theorem(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), true), true), true, theorem(implies(or(p, q), implies(or(p, not(q)), p))), true)
% 12.82/2.08 = { by axiom 1 (implies_definition) }
% 12.82/2.08 ifeq(ifeq(ifeq(axiom(or(not(or(not(or(p, not(q))), or(implies(or(p, q), implies(or(p, not(q)), p)), p))), or(implies(or(p, q), implies(or(p, not(q)), p)), or(not(or(p, not(q))), p)))), true, theorem(or(not(or(not(or(p, not(q))), or(implies(or(p, q), implies(or(p, not(q)), p)), p))), or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p)))), true), true, ifeq(theorem(or(not(or(p, not(q))), or(implies(or(p, q), implies(or(p, not(q)), p)), p))), true, theorem(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), true), true), true, theorem(implies(or(p, q), implies(or(p, not(q)), p))), true)
% 12.82/2.08 = { by axiom 1 (implies_definition) R->L }
% 12.82/2.09 ifeq(ifeq(ifeq(axiom(or(not(or(not(or(p, not(q))), or(implies(or(p, q), implies(or(p, not(q)), p)), p))), or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p)))), true, theorem(or(not(or(not(or(p, not(q))), or(implies(or(p, q), implies(or(p, not(q)), p)), p))), or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p)))), true), true, ifeq(theorem(or(not(or(p, not(q))), or(implies(or(p, q), implies(or(p, not(q)), p)), p))), true, theorem(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), true), true), true, theorem(implies(or(p, q), implies(or(p, not(q)), p))), true)
% 12.82/2.09 = { by axiom 5 (rule_1) }
% 12.82/2.09 ifeq(ifeq(true, true, ifeq(theorem(or(not(or(p, not(q))), or(implies(or(p, q), implies(or(p, not(q)), p)), p))), true, theorem(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), true), true), true, theorem(implies(or(p, q), implies(or(p, not(q)), p))), true)
% 12.82/2.09 = { by axiom 2 (ifeq_axiom) }
% 12.82/2.09 ifeq(ifeq(theorem(or(not(or(p, not(q))), or(implies(or(p, q), implies(or(p, not(q)), p)), p))), true, theorem(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), true), true, theorem(implies(or(p, q), implies(or(p, not(q)), p))), true)
% 12.82/2.09 = { by axiom 2 (ifeq_axiom) R->L }
% 12.82/2.09 ifeq(ifeq(ifeq(true, true, theorem(or(not(or(p, not(q))), or(implies(or(p, q), implies(or(p, not(q)), p)), p))), true), true, theorem(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), true), true, theorem(implies(or(p, q), implies(or(p, not(q)), p))), true)
% 12.82/2.09 = { by axiom 9 (rule_2) R->L }
% 12.82/2.09 ifeq(ifeq(ifeq(ifeq(theorem(implies(or(p, implies(or(p, q), implies(or(p, not(q)), p))), or(implies(or(p, q), implies(or(p, not(q)), p)), p))), true, ifeq(theorem(or(p, implies(or(p, q), implies(or(p, not(q)), p)))), true, theorem(or(implies(or(p, q), implies(or(p, not(q)), p)), p)), true), true), true, theorem(or(not(or(p, not(q))), or(implies(or(p, q), implies(or(p, not(q)), p)), p))), true), true, theorem(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), true), true, theorem(implies(or(p, q), implies(or(p, not(q)), p))), true)
% 12.82/2.09 = { by axiom 1 (implies_definition) }
% 12.82/2.10 ifeq(ifeq(ifeq(ifeq(theorem(or(not(or(p, implies(or(p, q), implies(or(p, not(q)), p)))), or(implies(or(p, q), implies(or(p, not(q)), p)), p))), true, ifeq(theorem(or(p, implies(or(p, q), implies(or(p, not(q)), p)))), true, theorem(or(implies(or(p, q), implies(or(p, not(q)), p)), p)), true), true), true, theorem(or(not(or(p, not(q))), or(implies(or(p, q), implies(or(p, not(q)), p)), p))), true), true, theorem(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), true), true, theorem(implies(or(p, q), implies(or(p, not(q)), p))), true)
% 12.82/2.10 = { by axiom 2 (ifeq_axiom) R->L }
% 12.82/2.10 ifeq(ifeq(ifeq(ifeq(ifeq(true, true, theorem(or(not(or(p, implies(or(p, q), implies(or(p, not(q)), p)))), or(implies(or(p, q), implies(or(p, not(q)), p)), p))), true), true, ifeq(theorem(or(p, implies(or(p, q), implies(or(p, not(q)), p)))), true, theorem(or(implies(or(p, q), implies(or(p, not(q)), p)), p)), true), true), true, theorem(or(not(or(p, not(q))), or(implies(or(p, q), implies(or(p, not(q)), p)), p))), true), true, theorem(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), true), true, theorem(implies(or(p, q), implies(or(p, not(q)), p))), true)
% 12.82/2.10 = { by axiom 6 (axiom_1_4) R->L }
% 12.82/2.10 ifeq(ifeq(ifeq(ifeq(ifeq(axiom(implies(or(p, implies(or(p, q), implies(or(p, not(q)), p))), or(implies(or(p, q), implies(or(p, not(q)), p)), p))), true, theorem(or(not(or(p, implies(or(p, q), implies(or(p, not(q)), p)))), or(implies(or(p, q), implies(or(p, not(q)), p)), p))), true), true, ifeq(theorem(or(p, implies(or(p, q), implies(or(p, not(q)), p)))), true, theorem(or(implies(or(p, q), implies(or(p, not(q)), p)), p)), true), true), true, theorem(or(not(or(p, not(q))), or(implies(or(p, q), implies(or(p, not(q)), p)), p))), true), true, theorem(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), true), true, theorem(implies(or(p, q), implies(or(p, not(q)), p))), true)
% 12.82/2.10 = { by axiom 1 (implies_definition) }
% 12.82/2.10 ifeq(ifeq(ifeq(ifeq(ifeq(axiom(or(not(or(p, implies(or(p, q), implies(or(p, not(q)), p)))), or(implies(or(p, q), implies(or(p, not(q)), p)), p))), true, theorem(or(not(or(p, implies(or(p, q), implies(or(p, not(q)), p)))), or(implies(or(p, q), implies(or(p, not(q)), p)), p))), true), true, ifeq(theorem(or(p, implies(or(p, q), implies(or(p, not(q)), p)))), true, theorem(or(implies(or(p, q), implies(or(p, not(q)), p)), p)), true), true), true, theorem(or(not(or(p, not(q))), or(implies(or(p, q), implies(or(p, not(q)), p)), p))), true), true, theorem(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), true), true, theorem(implies(or(p, q), implies(or(p, not(q)), p))), true)
% 12.82/2.10 = { by axiom 5 (rule_1) }
% 12.82/2.10 ifeq(ifeq(ifeq(ifeq(true, true, ifeq(theorem(or(p, implies(or(p, q), implies(or(p, not(q)), p)))), true, theorem(or(implies(or(p, q), implies(or(p, not(q)), p)), p)), true), true), true, theorem(or(not(or(p, not(q))), or(implies(or(p, q), implies(or(p, not(q)), p)), p))), true), true, theorem(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), true), true, theorem(implies(or(p, q), implies(or(p, not(q)), p))), true)
% 12.82/2.10 = { by axiom 2 (ifeq_axiom) }
% 12.82/2.10 ifeq(ifeq(ifeq(ifeq(theorem(or(p, implies(or(p, q), implies(or(p, not(q)), p)))), true, theorem(or(implies(or(p, q), implies(or(p, not(q)), p)), p)), true), true, theorem(or(not(or(p, not(q))), or(implies(or(p, q), implies(or(p, not(q)), p)), p))), true), true, theorem(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), true), true, theorem(implies(or(p, q), implies(or(p, not(q)), p))), true)
% 12.82/2.10 = { by axiom 1 (implies_definition) }
% 12.82/2.10 ifeq(ifeq(ifeq(ifeq(theorem(or(p, or(not(or(p, q)), implies(or(p, not(q)), p)))), true, theorem(or(implies(or(p, q), implies(or(p, not(q)), p)), p)), true), true, theorem(or(not(or(p, not(q))), or(implies(or(p, q), implies(or(p, not(q)), p)), p))), true), true, theorem(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), true), true, theorem(implies(or(p, q), implies(or(p, not(q)), p))), true)
% 12.82/2.11 = { by axiom 2 (ifeq_axiom) R->L }
% 12.82/2.11 ifeq(ifeq(ifeq(ifeq(ifeq(true, true, theorem(or(p, or(not(or(p, q)), implies(or(p, not(q)), p)))), true), true, theorem(or(implies(or(p, q), implies(or(p, not(q)), p)), p)), true), true, theorem(or(not(or(p, not(q))), or(implies(or(p, q), implies(or(p, not(q)), p)), p))), true), true, theorem(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), true), true, theorem(implies(or(p, q), implies(or(p, not(q)), p))), true)
% 12.82/2.11 = { by axiom 9 (rule_2) R->L }
% 12.82/2.11 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(theorem(implies(or(not(q), implies(or(p, not(q)), p)), or(not(or(p, q)), or(p, implies(or(p, not(q)), p))))), true, ifeq(theorem(or(not(q), implies(or(p, not(q)), p))), true, theorem(or(not(or(p, q)), or(p, implies(or(p, not(q)), p)))), true), true), true, theorem(or(p, or(not(or(p, q)), implies(or(p, not(q)), p)))), true), true, theorem(or(implies(or(p, q), implies(or(p, not(q)), p)), p)), true), true, theorem(or(not(or(p, not(q))), or(implies(or(p, q), implies(or(p, not(q)), p)), p))), true), true, theorem(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), true), true, theorem(implies(or(p, q), implies(or(p, not(q)), p))), true)
% 12.82/2.11 = { by axiom 1 (implies_definition) }
% 12.82/2.11 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(theorem(or(not(or(not(q), implies(or(p, not(q)), p))), or(not(or(p, q)), or(p, implies(or(p, not(q)), p))))), true, ifeq(theorem(or(not(q), implies(or(p, not(q)), p))), true, theorem(or(not(or(p, q)), or(p, implies(or(p, not(q)), p)))), true), true), true, theorem(or(p, or(not(or(p, q)), implies(or(p, not(q)), p)))), true), true, theorem(or(implies(or(p, q), implies(or(p, not(q)), p)), p)), true), true, theorem(or(not(or(p, not(q))), or(implies(or(p, q), implies(or(p, not(q)), p)), p))), true), true, theorem(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), true), true, theorem(implies(or(p, q), implies(or(p, not(q)), p))), true)
% 12.82/2.11 = { by axiom 2 (ifeq_axiom) R->L }
% 12.82/2.11 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, theorem(or(not(or(not(q), implies(or(p, not(q)), p))), or(not(or(p, q)), or(p, implies(or(p, not(q)), p))))), true), true, ifeq(theorem(or(not(q), implies(or(p, not(q)), p))), true, theorem(or(not(or(p, q)), or(p, implies(or(p, not(q)), p)))), true), true), true, theorem(or(p, or(not(or(p, q)), implies(or(p, not(q)), p)))), true), true, theorem(or(implies(or(p, q), implies(or(p, not(q)), p)), p)), true), true, theorem(or(not(or(p, not(q))), or(implies(or(p, q), implies(or(p, not(q)), p)), p))), true), true, theorem(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), true), true, theorem(implies(or(p, q), implies(or(p, not(q)), p))), true)
% 12.82/2.11 = { by axiom 7 (axiom_1_6) R->L }
% 12.82/2.11 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(axiom(implies(implies(q, implies(or(p, not(q)), p)), implies(or(p, q), or(p, implies(or(p, not(q)), p))))), true, theorem(or(not(or(not(q), implies(or(p, not(q)), p))), or(not(or(p, q)), or(p, implies(or(p, not(q)), p))))), true), true, ifeq(theorem(or(not(q), implies(or(p, not(q)), p))), true, theorem(or(not(or(p, q)), or(p, implies(or(p, not(q)), p)))), true), true), true, theorem(or(p, or(not(or(p, q)), implies(or(p, not(q)), p)))), true), true, theorem(or(implies(or(p, q), implies(or(p, not(q)), p)), p)), true), true, theorem(or(not(or(p, not(q))), or(implies(or(p, q), implies(or(p, not(q)), p)), p))), true), true, theorem(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), true), true, theorem(implies(or(p, q), implies(or(p, not(q)), p))), true)
% 12.82/2.12 = { by axiom 1 (implies_definition) }
% 12.82/2.12 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(axiom(or(not(implies(q, implies(or(p, not(q)), p))), implies(or(p, q), or(p, implies(or(p, not(q)), p))))), true, theorem(or(not(or(not(q), implies(or(p, not(q)), p))), or(not(or(p, q)), or(p, implies(or(p, not(q)), p))))), true), true, ifeq(theorem(or(not(q), implies(or(p, not(q)), p))), true, theorem(or(not(or(p, q)), or(p, implies(or(p, not(q)), p)))), true), true), true, theorem(or(p, or(not(or(p, q)), implies(or(p, not(q)), p)))), true), true, theorem(or(implies(or(p, q), implies(or(p, not(q)), p)), p)), true), true, theorem(or(not(or(p, not(q))), or(implies(or(p, q), implies(or(p, not(q)), p)), p))), true), true, theorem(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), true), true, theorem(implies(or(p, q), implies(or(p, not(q)), p))), true)
% 12.82/2.12 = { by axiom 1 (implies_definition) }
% 12.82/2.12 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(axiom(or(not(or(not(q), implies(or(p, not(q)), p))), implies(or(p, q), or(p, implies(or(p, not(q)), p))))), true, theorem(or(not(or(not(q), implies(or(p, not(q)), p))), or(not(or(p, q)), or(p, implies(or(p, not(q)), p))))), true), true, ifeq(theorem(or(not(q), implies(or(p, not(q)), p))), true, theorem(or(not(or(p, q)), or(p, implies(or(p, not(q)), p)))), true), true), true, theorem(or(p, or(not(or(p, q)), implies(or(p, not(q)), p)))), true), true, theorem(or(implies(or(p, q), implies(or(p, not(q)), p)), p)), true), true, theorem(or(not(or(p, not(q))), or(implies(or(p, q), implies(or(p, not(q)), p)), p))), true), true, theorem(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), true), true, theorem(implies(or(p, q), implies(or(p, not(q)), p))), true)
% 12.82/2.12 = { by axiom 1 (implies_definition) }
% 12.82/2.12 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(axiom(or(not(or(not(q), implies(or(p, not(q)), p))), or(not(or(p, q)), or(p, implies(or(p, not(q)), p))))), true, theorem(or(not(or(not(q), implies(or(p, not(q)), p))), or(not(or(p, q)), or(p, implies(or(p, not(q)), p))))), true), true, ifeq(theorem(or(not(q), implies(or(p, not(q)), p))), true, theorem(or(not(or(p, q)), or(p, implies(or(p, not(q)), p)))), true), true), true, theorem(or(p, or(not(or(p, q)), implies(or(p, not(q)), p)))), true), true, theorem(or(implies(or(p, q), implies(or(p, not(q)), p)), p)), true), true, theorem(or(not(or(p, not(q))), or(implies(or(p, q), implies(or(p, not(q)), p)), p))), true), true, theorem(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), true), true, theorem(implies(or(p, q), implies(or(p, not(q)), p))), true)
% 12.82/2.12 = { by axiom 5 (rule_1) }
% 12.82/2.12 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, ifeq(theorem(or(not(q), implies(or(p, not(q)), p))), true, theorem(or(not(or(p, q)), or(p, implies(or(p, not(q)), p)))), true), true), true, theorem(or(p, or(not(or(p, q)), implies(or(p, not(q)), p)))), true), true, theorem(or(implies(or(p, q), implies(or(p, not(q)), p)), p)), true), true, theorem(or(not(or(p, not(q))), or(implies(or(p, q), implies(or(p, not(q)), p)), p))), true), true, theorem(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), true), true, theorem(implies(or(p, q), implies(or(p, not(q)), p))), true)
% 12.82/2.12 = { by axiom 2 (ifeq_axiom) }
% 12.82/2.12 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(theorem(or(not(q), implies(or(p, not(q)), p))), true, theorem(or(not(or(p, q)), or(p, implies(or(p, not(q)), p)))), true), true, theorem(or(p, or(not(or(p, q)), implies(or(p, not(q)), p)))), true), true, theorem(or(implies(or(p, q), implies(or(p, not(q)), p)), p)), true), true, theorem(or(not(or(p, not(q))), or(implies(or(p, q), implies(or(p, not(q)), p)), p))), true), true, theorem(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), true), true, theorem(implies(or(p, q), implies(or(p, not(q)), p))), true)
% 12.82/2.13 = { by axiom 1 (implies_definition) }
% 12.82/2.13 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(theorem(or(not(q), or(not(or(p, not(q))), p))), true, theorem(or(not(or(p, q)), or(p, implies(or(p, not(q)), p)))), true), true, theorem(or(p, or(not(or(p, q)), implies(or(p, not(q)), p)))), true), true, theorem(or(implies(or(p, q), implies(or(p, not(q)), p)), p)), true), true, theorem(or(not(or(p, not(q))), or(implies(or(p, q), implies(or(p, not(q)), p)), p))), true), true, theorem(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), true), true, theorem(implies(or(p, q), implies(or(p, not(q)), p))), true)
% 12.82/2.13 = { by axiom 2 (ifeq_axiom) R->L }
% 12.82/2.13 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, theorem(or(not(q), or(not(or(p, not(q))), p))), true), true, theorem(or(not(or(p, q)), or(p, implies(or(p, not(q)), p)))), true), true, theorem(or(p, or(not(or(p, q)), implies(or(p, not(q)), p)))), true), true, theorem(or(implies(or(p, q), implies(or(p, not(q)), p)), p)), true), true, theorem(or(not(or(p, not(q))), or(implies(or(p, q), implies(or(p, not(q)), p)), p))), true), true, theorem(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), true), true, theorem(implies(or(p, q), implies(or(p, not(q)), p))), true)
% 12.82/2.13 = { by axiom 5 (rule_1) R->L }
% 12.82/2.13 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(axiom(or(not(or(p, not(q))), or(not(q), p))), true, theorem(or(not(or(p, not(q))), or(not(q), p))), true), true, theorem(or(not(q), or(not(or(p, not(q))), p))), true), true, theorem(or(not(or(p, q)), or(p, implies(or(p, not(q)), p)))), true), true, theorem(or(p, or(not(or(p, q)), implies(or(p, not(q)), p)))), true), true, theorem(or(implies(or(p, q), implies(or(p, not(q)), p)), p)), true), true, theorem(or(not(or(p, not(q))), or(implies(or(p, q), implies(or(p, not(q)), p)), p))), true), true, theorem(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), true), true, theorem(implies(or(p, q), implies(or(p, not(q)), p))), true)
% 12.82/2.13 = { by axiom 1 (implies_definition) R->L }
% 12.82/2.13 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(axiom(implies(or(p, not(q)), or(not(q), p))), true, theorem(or(not(or(p, not(q))), or(not(q), p))), true), true, theorem(or(not(q), or(not(or(p, not(q))), p))), true), true, theorem(or(not(or(p, q)), or(p, implies(or(p, not(q)), p)))), true), true, theorem(or(p, or(not(or(p, q)), implies(or(p, not(q)), p)))), true), true, theorem(or(implies(or(p, q), implies(or(p, not(q)), p)), p)), true), true, theorem(or(not(or(p, not(q))), or(implies(or(p, q), implies(or(p, not(q)), p)), p))), true), true, theorem(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), true), true, theorem(implies(or(p, q), implies(or(p, not(q)), p))), true)
% 12.82/2.13 = { by axiom 6 (axiom_1_4) }
% 12.82/2.13 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, theorem(or(not(or(p, not(q))), or(not(q), p))), true), true, theorem(or(not(q), or(not(or(p, not(q))), p))), true), true, theorem(or(not(or(p, q)), or(p, implies(or(p, not(q)), p)))), true), true, theorem(or(p, or(not(or(p, q)), implies(or(p, not(q)), p)))), true), true, theorem(or(implies(or(p, q), implies(or(p, not(q)), p)), p)), true), true, theorem(or(not(or(p, not(q))), or(implies(or(p, q), implies(or(p, not(q)), p)), p))), true), true, theorem(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), true), true, theorem(implies(or(p, q), implies(or(p, not(q)), p))), true)
% 12.82/2.14 = { by axiom 2 (ifeq_axiom) }
% 12.82/2.14 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(theorem(or(not(or(p, not(q))), or(not(q), p))), true, theorem(or(not(q), or(not(or(p, not(q))), p))), true), true, theorem(or(not(or(p, q)), or(p, implies(or(p, not(q)), p)))), true), true, theorem(or(p, or(not(or(p, q)), implies(or(p, not(q)), p)))), true), true, theorem(or(implies(or(p, q), implies(or(p, not(q)), p)), p)), true), true, theorem(or(not(or(p, not(q))), or(implies(or(p, q), implies(or(p, not(q)), p)), p))), true), true, theorem(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), true), true, theorem(implies(or(p, q), implies(or(p, not(q)), p))), true)
% 12.82/2.14 = { by axiom 2 (ifeq_axiom) R->L }
% 12.82/2.14 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, ifeq(theorem(or(not(or(p, not(q))), or(not(q), p))), true, theorem(or(not(q), or(not(or(p, not(q))), p))), true), true), true, theorem(or(not(or(p, q)), or(p, implies(or(p, not(q)), p)))), true), true, theorem(or(p, or(not(or(p, q)), implies(or(p, not(q)), p)))), true), true, theorem(or(implies(or(p, q), implies(or(p, not(q)), p)), p)), true), true, theorem(or(not(or(p, not(q))), or(implies(or(p, q), implies(or(p, not(q)), p)), p))), true), true, theorem(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), true), true, theorem(implies(or(p, q), implies(or(p, not(q)), p))), true)
% 12.82/2.14 = { by axiom 5 (rule_1) R->L }
% 12.82/2.14 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(axiom(or(not(or(not(or(p, not(q))), or(not(q), p))), or(not(q), or(not(or(p, not(q))), p)))), true, theorem(or(not(or(not(or(p, not(q))), or(not(q), p))), or(not(q), or(not(or(p, not(q))), p)))), true), true, ifeq(theorem(or(not(or(p, not(q))), or(not(q), p))), true, theorem(or(not(q), or(not(or(p, not(q))), p))), true), true), true, theorem(or(not(or(p, q)), or(p, implies(or(p, not(q)), p)))), true), true, theorem(or(p, or(not(or(p, q)), implies(or(p, not(q)), p)))), true), true, theorem(or(implies(or(p, q), implies(or(p, not(q)), p)), p)), true), true, theorem(or(not(or(p, not(q))), or(implies(or(p, q), implies(or(p, not(q)), p)), p))), true), true, theorem(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), true), true, theorem(implies(or(p, q), implies(or(p, not(q)), p))), true)
% 12.82/2.14 = { by axiom 1 (implies_definition) R->L }
% 12.82/2.14 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(axiom(implies(or(not(or(p, not(q))), or(not(q), p)), or(not(q), or(not(or(p, not(q))), p)))), true, theorem(or(not(or(not(or(p, not(q))), or(not(q), p))), or(not(q), or(not(or(p, not(q))), p)))), true), true, ifeq(theorem(or(not(or(p, not(q))), or(not(q), p))), true, theorem(or(not(q), or(not(or(p, not(q))), p))), true), true), true, theorem(or(not(or(p, q)), or(p, implies(or(p, not(q)), p)))), true), true, theorem(or(p, or(not(or(p, q)), implies(or(p, not(q)), p)))), true), true, theorem(or(implies(or(p, q), implies(or(p, not(q)), p)), p)), true), true, theorem(or(not(or(p, not(q))), or(implies(or(p, q), implies(or(p, not(q)), p)), p))), true), true, theorem(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), true), true, theorem(implies(or(p, q), implies(or(p, not(q)), p))), true)
% 12.82/2.14 = { by axiom 8 (axiom_1_5) }
% 12.82/2.14 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, theorem(or(not(or(not(or(p, not(q))), or(not(q), p))), or(not(q), or(not(or(p, not(q))), p)))), true), true, ifeq(theorem(or(not(or(p, not(q))), or(not(q), p))), true, theorem(or(not(q), or(not(or(p, not(q))), p))), true), true), true, theorem(or(not(or(p, q)), or(p, implies(or(p, not(q)), p)))), true), true, theorem(or(p, or(not(or(p, q)), implies(or(p, not(q)), p)))), true), true, theorem(or(implies(or(p, q), implies(or(p, not(q)), p)), p)), true), true, theorem(or(not(or(p, not(q))), or(implies(or(p, q), implies(or(p, not(q)), p)), p))), true), true, theorem(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), true), true, theorem(implies(or(p, q), implies(or(p, not(q)), p))), true)
% 12.82/2.15 = { by axiom 2 (ifeq_axiom) }
% 12.82/2.15 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(theorem(or(not(or(not(or(p, not(q))), or(not(q), p))), or(not(q), or(not(or(p, not(q))), p)))), true, ifeq(theorem(or(not(or(p, not(q))), or(not(q), p))), true, theorem(or(not(q), or(not(or(p, not(q))), p))), true), true), true, theorem(or(not(or(p, q)), or(p, implies(or(p, not(q)), p)))), true), true, theorem(or(p, or(not(or(p, q)), implies(or(p, not(q)), p)))), true), true, theorem(or(implies(or(p, q), implies(or(p, not(q)), p)), p)), true), true, theorem(or(not(or(p, not(q))), or(implies(or(p, q), implies(or(p, not(q)), p)), p))), true), true, theorem(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), true), true, theorem(implies(or(p, q), implies(or(p, not(q)), p))), true)
% 12.82/2.15 = { by axiom 1 (implies_definition) R->L }
% 12.82/2.15 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(theorem(implies(or(not(or(p, not(q))), or(not(q), p)), or(not(q), or(not(or(p, not(q))), p)))), true, ifeq(theorem(or(not(or(p, not(q))), or(not(q), p))), true, theorem(or(not(q), or(not(or(p, not(q))), p))), true), true), true, theorem(or(not(or(p, q)), or(p, implies(or(p, not(q)), p)))), true), true, theorem(or(p, or(not(or(p, q)), implies(or(p, not(q)), p)))), true), true, theorem(or(implies(or(p, q), implies(or(p, not(q)), p)), p)), true), true, theorem(or(not(or(p, not(q))), or(implies(or(p, q), implies(or(p, not(q)), p)), p))), true), true, theorem(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), true), true, theorem(implies(or(p, q), implies(or(p, not(q)), p))), true)
% 12.82/2.15 = { by axiom 9 (rule_2) }
% 12.82/2.15 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, theorem(or(not(or(p, q)), or(p, implies(or(p, not(q)), p)))), true), true, theorem(or(p, or(not(or(p, q)), implies(or(p, not(q)), p)))), true), true, theorem(or(implies(or(p, q), implies(or(p, not(q)), p)), p)), true), true, theorem(or(not(or(p, not(q))), or(implies(or(p, q), implies(or(p, not(q)), p)), p))), true), true, theorem(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), true), true, theorem(implies(or(p, q), implies(or(p, not(q)), p))), true)
% 12.82/2.15 = { by axiom 2 (ifeq_axiom) }
% 12.82/2.15 ifeq(ifeq(ifeq(ifeq(ifeq(theorem(or(not(or(p, q)), or(p, implies(or(p, not(q)), p)))), true, theorem(or(p, or(not(or(p, q)), implies(or(p, not(q)), p)))), true), true, theorem(or(implies(or(p, q), implies(or(p, not(q)), p)), p)), true), true, theorem(or(not(or(p, not(q))), or(implies(or(p, q), implies(or(p, not(q)), p)), p))), true), true, theorem(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), true), true, theorem(implies(or(p, q), implies(or(p, not(q)), p))), true)
% 12.82/2.15 = { by axiom 2 (ifeq_axiom) R->L }
% 12.82/2.15 ifeq(ifeq(ifeq(ifeq(ifeq(true, true, ifeq(theorem(or(not(or(p, q)), or(p, implies(or(p, not(q)), p)))), true, theorem(or(p, or(not(or(p, q)), implies(or(p, not(q)), p)))), true), true), true, theorem(or(implies(or(p, q), implies(or(p, not(q)), p)), p)), true), true, theorem(or(not(or(p, not(q))), or(implies(or(p, q), implies(or(p, not(q)), p)), p))), true), true, theorem(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), true), true, theorem(implies(or(p, q), implies(or(p, not(q)), p))), true)
% 12.82/2.15 = { by axiom 5 (rule_1) R->L }
% 12.82/2.15 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(axiom(or(not(or(not(or(p, q)), or(p, implies(or(p, not(q)), p)))), or(p, or(not(or(p, q)), implies(or(p, not(q)), p))))), true, theorem(or(not(or(not(or(p, q)), or(p, implies(or(p, not(q)), p)))), or(p, or(not(or(p, q)), implies(or(p, not(q)), p))))), true), true, ifeq(theorem(or(not(or(p, q)), or(p, implies(or(p, not(q)), p)))), true, theorem(or(p, or(not(or(p, q)), implies(or(p, not(q)), p)))), true), true), true, theorem(or(implies(or(p, q), implies(or(p, not(q)), p)), p)), true), true, theorem(or(not(or(p, not(q))), or(implies(or(p, q), implies(or(p, not(q)), p)), p))), true), true, theorem(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), true), true, theorem(implies(or(p, q), implies(or(p, not(q)), p))), true)
% 12.82/2.16 = { by axiom 1 (implies_definition) R->L }
% 12.82/2.16 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(axiom(implies(or(not(or(p, q)), or(p, implies(or(p, not(q)), p))), or(p, or(not(or(p, q)), implies(or(p, not(q)), p))))), true, theorem(or(not(or(not(or(p, q)), or(p, implies(or(p, not(q)), p)))), or(p, or(not(or(p, q)), implies(or(p, not(q)), p))))), true), true, ifeq(theorem(or(not(or(p, q)), or(p, implies(or(p, not(q)), p)))), true, theorem(or(p, or(not(or(p, q)), implies(or(p, not(q)), p)))), true), true), true, theorem(or(implies(or(p, q), implies(or(p, not(q)), p)), p)), true), true, theorem(or(not(or(p, not(q))), or(implies(or(p, q), implies(or(p, not(q)), p)), p))), true), true, theorem(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), true), true, theorem(implies(or(p, q), implies(or(p, not(q)), p))), true)
% 12.82/2.16 = { by axiom 8 (axiom_1_5) }
% 12.82/2.16 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, theorem(or(not(or(not(or(p, q)), or(p, implies(or(p, not(q)), p)))), or(p, or(not(or(p, q)), implies(or(p, not(q)), p))))), true), true, ifeq(theorem(or(not(or(p, q)), or(p, implies(or(p, not(q)), p)))), true, theorem(or(p, or(not(or(p, q)), implies(or(p, not(q)), p)))), true), true), true, theorem(or(implies(or(p, q), implies(or(p, not(q)), p)), p)), true), true, theorem(or(not(or(p, not(q))), or(implies(or(p, q), implies(or(p, not(q)), p)), p))), true), true, theorem(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), true), true, theorem(implies(or(p, q), implies(or(p, not(q)), p))), true)
% 12.82/2.16 = { by axiom 2 (ifeq_axiom) }
% 12.82/2.16 ifeq(ifeq(ifeq(ifeq(ifeq(theorem(or(not(or(not(or(p, q)), or(p, implies(or(p, not(q)), p)))), or(p, or(not(or(p, q)), implies(or(p, not(q)), p))))), true, ifeq(theorem(or(not(or(p, q)), or(p, implies(or(p, not(q)), p)))), true, theorem(or(p, or(not(or(p, q)), implies(or(p, not(q)), p)))), true), true), true, theorem(or(implies(or(p, q), implies(or(p, not(q)), p)), p)), true), true, theorem(or(not(or(p, not(q))), or(implies(or(p, q), implies(or(p, not(q)), p)), p))), true), true, theorem(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), true), true, theorem(implies(or(p, q), implies(or(p, not(q)), p))), true)
% 12.82/2.16 = { by axiom 1 (implies_definition) R->L }
% 12.82/2.16 ifeq(ifeq(ifeq(ifeq(ifeq(theorem(implies(or(not(or(p, q)), or(p, implies(or(p, not(q)), p))), or(p, or(not(or(p, q)), implies(or(p, not(q)), p))))), true, ifeq(theorem(or(not(or(p, q)), or(p, implies(or(p, not(q)), p)))), true, theorem(or(p, or(not(or(p, q)), implies(or(p, not(q)), p)))), true), true), true, theorem(or(implies(or(p, q), implies(or(p, not(q)), p)), p)), true), true, theorem(or(not(or(p, not(q))), or(implies(or(p, q), implies(or(p, not(q)), p)), p))), true), true, theorem(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), true), true, theorem(implies(or(p, q), implies(or(p, not(q)), p))), true)
% 12.82/2.16 = { by axiom 9 (rule_2) }
% 12.82/2.16 ifeq(ifeq(ifeq(ifeq(true, true, theorem(or(implies(or(p, q), implies(or(p, not(q)), p)), p)), true), true, theorem(or(not(or(p, not(q))), or(implies(or(p, q), implies(or(p, not(q)), p)), p))), true), true, theorem(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), true), true, theorem(implies(or(p, q), implies(or(p, not(q)), p))), true)
% 12.82/2.16 = { by axiom 2 (ifeq_axiom) }
% 12.82/2.16 ifeq(ifeq(ifeq(theorem(or(implies(or(p, q), implies(or(p, not(q)), p)), p)), true, theorem(or(not(or(p, not(q))), or(implies(or(p, q), implies(or(p, not(q)), p)), p))), true), true, theorem(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), true), true, theorem(implies(or(p, q), implies(or(p, not(q)), p))), true)
% 12.82/2.16 = { by axiom 2 (ifeq_axiom) R->L }
% 12.82/2.16 ifeq(ifeq(ifeq(true, true, ifeq(theorem(or(implies(or(p, q), implies(or(p, not(q)), p)), p)), true, theorem(or(not(or(p, not(q))), or(implies(or(p, q), implies(or(p, not(q)), p)), p))), true), true), true, theorem(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), true), true, theorem(implies(or(p, q), implies(or(p, not(q)), p))), true)
% 12.82/2.17 = { by axiom 5 (rule_1) R->L }
% 12.82/2.17 ifeq(ifeq(ifeq(ifeq(axiom(or(not(or(implies(or(p, q), implies(or(p, not(q)), p)), p)), or(not(or(p, not(q))), or(implies(or(p, q), implies(or(p, not(q)), p)), p)))), true, theorem(or(not(or(implies(or(p, q), implies(or(p, not(q)), p)), p)), or(not(or(p, not(q))), or(implies(or(p, q), implies(or(p, not(q)), p)), p)))), true), true, ifeq(theorem(or(implies(or(p, q), implies(or(p, not(q)), p)), p)), true, theorem(or(not(or(p, not(q))), or(implies(or(p, q), implies(or(p, not(q)), p)), p))), true), true), true, theorem(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), true), true, theorem(implies(or(p, q), implies(or(p, not(q)), p))), true)
% 12.82/2.17 = { by axiom 1 (implies_definition) R->L }
% 12.82/2.17 ifeq(ifeq(ifeq(ifeq(axiom(implies(or(implies(or(p, q), implies(or(p, not(q)), p)), p), or(not(or(p, not(q))), or(implies(or(p, q), implies(or(p, not(q)), p)), p)))), true, theorem(or(not(or(implies(or(p, q), implies(or(p, not(q)), p)), p)), or(not(or(p, not(q))), or(implies(or(p, q), implies(or(p, not(q)), p)), p)))), true), true, ifeq(theorem(or(implies(or(p, q), implies(or(p, not(q)), p)), p)), true, theorem(or(not(or(p, not(q))), or(implies(or(p, q), implies(or(p, not(q)), p)), p))), true), true), true, theorem(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), true), true, theorem(implies(or(p, q), implies(or(p, not(q)), p))), true)
% 12.82/2.17 = { by axiom 3 (axiom_1_3) }
% 12.82/2.17 ifeq(ifeq(ifeq(ifeq(true, true, theorem(or(not(or(implies(or(p, q), implies(or(p, not(q)), p)), p)), or(not(or(p, not(q))), or(implies(or(p, q), implies(or(p, not(q)), p)), p)))), true), true, ifeq(theorem(or(implies(or(p, q), implies(or(p, not(q)), p)), p)), true, theorem(or(not(or(p, not(q))), or(implies(or(p, q), implies(or(p, not(q)), p)), p))), true), true), true, theorem(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), true), true, theorem(implies(or(p, q), implies(or(p, not(q)), p))), true)
% 12.82/2.17 = { by axiom 2 (ifeq_axiom) }
% 12.82/2.17 ifeq(ifeq(ifeq(theorem(or(not(or(implies(or(p, q), implies(or(p, not(q)), p)), p)), or(not(or(p, not(q))), or(implies(or(p, q), implies(or(p, not(q)), p)), p)))), true, ifeq(theorem(or(implies(or(p, q), implies(or(p, not(q)), p)), p)), true, theorem(or(not(or(p, not(q))), or(implies(or(p, q), implies(or(p, not(q)), p)), p))), true), true), true, theorem(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), true), true, theorem(implies(or(p, q), implies(or(p, not(q)), p))), true)
% 12.82/2.17 = { by axiom 1 (implies_definition) R->L }
% 13.60/2.17 ifeq(ifeq(ifeq(theorem(implies(or(implies(or(p, q), implies(or(p, not(q)), p)), p), or(not(or(p, not(q))), or(implies(or(p, q), implies(or(p, not(q)), p)), p)))), true, ifeq(theorem(or(implies(or(p, q), implies(or(p, not(q)), p)), p)), true, theorem(or(not(or(p, not(q))), or(implies(or(p, q), implies(or(p, not(q)), p)), p))), true), true), true, theorem(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), true), true, theorem(implies(or(p, q), implies(or(p, not(q)), p))), true)
% 13.60/2.17 = { by axiom 9 (rule_2) }
% 13.60/2.17 ifeq(ifeq(true, true, theorem(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), true), true, theorem(implies(or(p, q), implies(or(p, not(q)), p))), true)
% 13.60/2.17 = { by axiom 2 (ifeq_axiom) }
% 13.60/2.17 ifeq(theorem(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), true, theorem(implies(or(p, q), implies(or(p, not(q)), p))), true)
% 13.60/2.17 = { by axiom 2 (ifeq_axiom) R->L }
% 13.60/2.17 ifeq(true, true, ifeq(theorem(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), true, theorem(implies(or(p, q), implies(or(p, not(q)), p))), true), true)
% 13.60/2.17 = { by axiom 9 (rule_2) R->L }
% 13.60/2.18 ifeq(ifeq(theorem(implies(or(not(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, q), implies(or(p, not(q)), p)))), or(not(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), implies(or(p, q), implies(or(p, not(q)), p))))), true, ifeq(theorem(or(not(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, q), implies(or(p, not(q)), p))))), true, theorem(or(not(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), implies(or(p, q), implies(or(p, not(q)), p)))), true), true), true, ifeq(theorem(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), true, theorem(implies(or(p, q), implies(or(p, not(q)), p))), true), true)
% 13.60/2.18 = { by axiom 1 (implies_definition) }
% 13.60/2.18 ifeq(ifeq(theorem(or(not(or(not(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, q), implies(or(p, not(q)), p))))), or(not(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), implies(or(p, q), implies(or(p, not(q)), p))))), true, ifeq(theorem(or(not(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, q), implies(or(p, not(q)), p))))), true, theorem(or(not(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), implies(or(p, q), implies(or(p, not(q)), p)))), true), true), true, ifeq(theorem(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), true, theorem(implies(or(p, q), implies(or(p, not(q)), p))), true), true)
% 13.60/2.18 = { by axiom 2 (ifeq_axiom) R->L }
% 13.60/2.18 ifeq(ifeq(ifeq(true, true, theorem(or(not(or(not(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, q), implies(or(p, not(q)), p))))), or(not(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), implies(or(p, q), implies(or(p, not(q)), p))))), true), true, ifeq(theorem(or(not(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, q), implies(or(p, not(q)), p))))), true, theorem(or(not(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), implies(or(p, q), implies(or(p, not(q)), p)))), true), true), true, ifeq(theorem(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), true, theorem(implies(or(p, q), implies(or(p, not(q)), p))), true), true)
% 13.60/2.19 = { by axiom 5 (rule_1) R->L }
% 13.60/2.19 ifeq(ifeq(ifeq(ifeq(axiom(or(not(or(not(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, q), implies(or(p, not(q)), p)))), implies(or(p, q), implies(or(p, not(q)), p)))), or(not(or(not(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, q), implies(or(p, not(q)), p))))), or(not(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), implies(or(p, q), implies(or(p, not(q)), p)))))), true, theorem(or(not(or(not(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, q), implies(or(p, not(q)), p)))), implies(or(p, q), implies(or(p, not(q)), p)))), or(not(or(not(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, q), implies(or(p, not(q)), p))))), or(not(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), implies(or(p, q), implies(or(p, not(q)), p)))))), true), true, theorem(or(not(or(not(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, q), implies(or(p, not(q)), p))))), or(not(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), implies(or(p, q), implies(or(p, not(q)), p))))), true), true, ifeq(theorem(or(not(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, q), implies(or(p, not(q)), p))))), true, theorem(or(not(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), implies(or(p, q), implies(or(p, not(q)), p)))), true), true), true, ifeq(theorem(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), true, theorem(implies(or(p, q), implies(or(p, not(q)), p))), true), true)
% 13.60/2.19 = { by axiom 1 (implies_definition) R->L }
% 13.60/2.20 ifeq(ifeq(ifeq(ifeq(axiom(or(not(or(not(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, q), implies(or(p, not(q)), p)))), implies(or(p, q), implies(or(p, not(q)), p)))), implies(or(not(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, q), implies(or(p, not(q)), p)))), or(not(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), implies(or(p, q), implies(or(p, not(q)), p)))))), true, theorem(or(not(or(not(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, q), implies(or(p, not(q)), p)))), implies(or(p, q), implies(or(p, not(q)), p)))), or(not(or(not(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, q), implies(or(p, not(q)), p))))), or(not(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), implies(or(p, q), implies(or(p, not(q)), p)))))), true), true, theorem(or(not(or(not(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, q), implies(or(p, not(q)), p))))), or(not(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), implies(or(p, q), implies(or(p, not(q)), p))))), true), true, ifeq(theorem(or(not(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, q), implies(or(p, not(q)), p))))), true, theorem(or(not(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), implies(or(p, q), implies(or(p, not(q)), p)))), true), true), true, ifeq(theorem(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), true, theorem(implies(or(p, q), implies(or(p, not(q)), p))), true), true)
% 13.60/2.20 = { by axiom 1 (implies_definition) R->L }
% 13.84/2.20 ifeq(ifeq(ifeq(ifeq(axiom(or(not(implies(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, q), implies(or(p, not(q)), p))), implies(or(p, q), implies(or(p, not(q)), p)))), implies(or(not(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, q), implies(or(p, not(q)), p)))), or(not(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), implies(or(p, q), implies(or(p, not(q)), p)))))), true, theorem(or(not(or(not(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, q), implies(or(p, not(q)), p)))), implies(or(p, q), implies(or(p, not(q)), p)))), or(not(or(not(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, q), implies(or(p, not(q)), p))))), or(not(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), implies(or(p, q), implies(or(p, not(q)), p)))))), true), true, theorem(or(not(or(not(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, q), implies(or(p, not(q)), p))))), or(not(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), implies(or(p, q), implies(or(p, not(q)), p))))), true), true, ifeq(theorem(or(not(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, q), implies(or(p, not(q)), p))))), true, theorem(or(not(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), implies(or(p, q), implies(or(p, not(q)), p)))), true), true), true, ifeq(theorem(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), true, theorem(implies(or(p, q), implies(or(p, not(q)), p))), true), true)
% 13.84/2.21 = { by axiom 1 (implies_definition) R->L }
% 13.84/2.21 ifeq(ifeq(ifeq(ifeq(axiom(implies(implies(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, q), implies(or(p, not(q)), p))), implies(or(p, q), implies(or(p, not(q)), p))), implies(or(not(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, q), implies(or(p, not(q)), p)))), or(not(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), implies(or(p, q), implies(or(p, not(q)), p)))))), true, theorem(or(not(or(not(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, q), implies(or(p, not(q)), p)))), implies(or(p, q), implies(or(p, not(q)), p)))), or(not(or(not(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, q), implies(or(p, not(q)), p))))), or(not(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), implies(or(p, q), implies(or(p, not(q)), p)))))), true), true, theorem(or(not(or(not(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, q), implies(or(p, not(q)), p))))), or(not(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), implies(or(p, q), implies(or(p, not(q)), p))))), true), true, ifeq(theorem(or(not(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, q), implies(or(p, not(q)), p))))), true, theorem(or(not(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), implies(or(p, q), implies(or(p, not(q)), p)))), true), true), true, ifeq(theorem(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), true, theorem(implies(or(p, q), implies(or(p, not(q)), p))), true), true)
% 13.84/2.21 = { by axiom 7 (axiom_1_6) }
% 13.84/2.22 ifeq(ifeq(ifeq(ifeq(true, true, theorem(or(not(or(not(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, q), implies(or(p, not(q)), p)))), implies(or(p, q), implies(or(p, not(q)), p)))), or(not(or(not(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, q), implies(or(p, not(q)), p))))), or(not(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), implies(or(p, q), implies(or(p, not(q)), p)))))), true), true, theorem(or(not(or(not(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, q), implies(or(p, not(q)), p))))), or(not(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), implies(or(p, q), implies(or(p, not(q)), p))))), true), true, ifeq(theorem(or(not(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, q), implies(or(p, not(q)), p))))), true, theorem(or(not(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), implies(or(p, q), implies(or(p, not(q)), p)))), true), true), true, ifeq(theorem(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), true, theorem(implies(or(p, q), implies(or(p, not(q)), p))), true), true)
% 13.84/2.22 = { by axiom 2 (ifeq_axiom) }
% 13.84/2.22 ifeq(ifeq(ifeq(theorem(or(not(or(not(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, q), implies(or(p, not(q)), p)))), implies(or(p, q), implies(or(p, not(q)), p)))), or(not(or(not(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, q), implies(or(p, not(q)), p))))), or(not(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), implies(or(p, q), implies(or(p, not(q)), p)))))), true, theorem(or(not(or(not(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, q), implies(or(p, not(q)), p))))), or(not(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), implies(or(p, q), implies(or(p, not(q)), p))))), true), true, ifeq(theorem(or(not(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, q), implies(or(p, not(q)), p))))), true, theorem(or(not(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), implies(or(p, q), implies(or(p, not(q)), p)))), true), true), true, ifeq(theorem(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), true, theorem(implies(or(p, q), implies(or(p, not(q)), p))), true), true)
% 13.84/2.22 = { by axiom 2 (ifeq_axiom) R->L }
% 13.84/2.23 ifeq(ifeq(ifeq(theorem(or(not(or(not(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, q), implies(or(p, not(q)), p)))), implies(or(p, q), implies(or(p, not(q)), p)))), or(not(or(not(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, q), implies(or(p, not(q)), p))))), or(not(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), implies(or(p, q), implies(or(p, not(q)), p)))))), true, ifeq(true, true, theorem(or(not(or(not(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, q), implies(or(p, not(q)), p))))), or(not(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), implies(or(p, q), implies(or(p, not(q)), p))))), true), true), true, ifeq(theorem(or(not(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, q), implies(or(p, not(q)), p))))), true, theorem(or(not(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), implies(or(p, q), implies(or(p, not(q)), p)))), true), true), true, ifeq(theorem(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), true, theorem(implies(or(p, q), implies(or(p, not(q)), p))), true), true)
% 13.84/2.23 = { by axiom 5 (rule_1) R->L }
% 13.84/2.23 ifeq(ifeq(ifeq(theorem(or(not(or(not(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, q), implies(or(p, not(q)), p)))), implies(or(p, q), implies(or(p, not(q)), p)))), or(not(or(not(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, q), implies(or(p, not(q)), p))))), or(not(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), implies(or(p, q), implies(or(p, not(q)), p)))))), true, ifeq(ifeq(axiom(or(not(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, q), implies(or(p, not(q)), p)))), implies(or(p, q), implies(or(p, not(q)), p)))), true, theorem(or(not(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, q), implies(or(p, not(q)), p)))), implies(or(p, q), implies(or(p, not(q)), p)))), true), true, theorem(or(not(or(not(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, q), implies(or(p, not(q)), p))))), or(not(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), implies(or(p, q), implies(or(p, not(q)), p))))), true), true), true, ifeq(theorem(or(not(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, q), implies(or(p, not(q)), p))))), true, theorem(or(not(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), implies(or(p, q), implies(or(p, not(q)), p)))), true), true), true, ifeq(theorem(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), true, theorem(implies(or(p, q), implies(or(p, not(q)), p))), true), true)
% 13.84/2.24 = { by axiom 1 (implies_definition) R->L }
% 13.84/2.24 ifeq(ifeq(ifeq(theorem(or(not(or(not(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, q), implies(or(p, not(q)), p)))), implies(or(p, q), implies(or(p, not(q)), p)))), or(not(or(not(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, q), implies(or(p, not(q)), p))))), or(not(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), implies(or(p, q), implies(or(p, not(q)), p)))))), true, ifeq(ifeq(axiom(implies(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, q), implies(or(p, not(q)), p))), implies(or(p, q), implies(or(p, not(q)), p)))), true, theorem(or(not(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, q), implies(or(p, not(q)), p)))), implies(or(p, q), implies(or(p, not(q)), p)))), true), true, theorem(or(not(or(not(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, q), implies(or(p, not(q)), p))))), or(not(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), implies(or(p, q), implies(or(p, not(q)), p))))), true), true), true, ifeq(theorem(or(not(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, q), implies(or(p, not(q)), p))))), true, theorem(or(not(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), implies(or(p, q), implies(or(p, not(q)), p)))), true), true), true, ifeq(theorem(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), true, theorem(implies(or(p, q), implies(or(p, not(q)), p))), true), true)
% 13.84/2.24 = { by axiom 4 (axiom_1_2) }
% 13.84/2.24 ifeq(ifeq(ifeq(theorem(or(not(or(not(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, q), implies(or(p, not(q)), p)))), implies(or(p, q), implies(or(p, not(q)), p)))), or(not(or(not(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, q), implies(or(p, not(q)), p))))), or(not(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), implies(or(p, q), implies(or(p, not(q)), p)))))), true, ifeq(ifeq(true, true, theorem(or(not(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, q), implies(or(p, not(q)), p)))), implies(or(p, q), implies(or(p, not(q)), p)))), true), true, theorem(or(not(or(not(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, q), implies(or(p, not(q)), p))))), or(not(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), implies(or(p, q), implies(or(p, not(q)), p))))), true), true), true, ifeq(theorem(or(not(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, q), implies(or(p, not(q)), p))))), true, theorem(or(not(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), implies(or(p, q), implies(or(p, not(q)), p)))), true), true), true, ifeq(theorem(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), true, theorem(implies(or(p, q), implies(or(p, not(q)), p))), true), true)
% 13.84/2.25 = { by axiom 2 (ifeq_axiom) }
% 13.84/2.25 ifeq(ifeq(ifeq(theorem(or(not(or(not(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, q), implies(or(p, not(q)), p)))), implies(or(p, q), implies(or(p, not(q)), p)))), or(not(or(not(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, q), implies(or(p, not(q)), p))))), or(not(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), implies(or(p, q), implies(or(p, not(q)), p)))))), true, ifeq(theorem(or(not(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, q), implies(or(p, not(q)), p)))), implies(or(p, q), implies(or(p, not(q)), p)))), true, theorem(or(not(or(not(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, q), implies(or(p, not(q)), p))))), or(not(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), implies(or(p, q), implies(or(p, not(q)), p))))), true), true), true, ifeq(theorem(or(not(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, q), implies(or(p, not(q)), p))))), true, theorem(or(not(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), implies(or(p, q), implies(or(p, not(q)), p)))), true), true), true, ifeq(theorem(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), true, theorem(implies(or(p, q), implies(or(p, not(q)), p))), true), true)
% 13.84/2.25 = { by axiom 1 (implies_definition) R->L }
% 13.84/2.25 ifeq(ifeq(ifeq(theorem(implies(or(not(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, q), implies(or(p, not(q)), p)))), implies(or(p, q), implies(or(p, not(q)), p))), or(not(or(not(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, q), implies(or(p, not(q)), p))))), or(not(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), implies(or(p, q), implies(or(p, not(q)), p)))))), true, ifeq(theorem(or(not(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, q), implies(or(p, not(q)), p)))), implies(or(p, q), implies(or(p, not(q)), p)))), true, theorem(or(not(or(not(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, q), implies(or(p, not(q)), p))))), or(not(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), implies(or(p, q), implies(or(p, not(q)), p))))), true), true), true, ifeq(theorem(or(not(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, q), implies(or(p, not(q)), p))))), true, theorem(or(not(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), implies(or(p, q), implies(or(p, not(q)), p)))), true), true), true, ifeq(theorem(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), true, theorem(implies(or(p, q), implies(or(p, not(q)), p))), true), true)
% 13.84/2.26 = { by axiom 9 (rule_2) }
% 13.84/2.26 ifeq(ifeq(true, true, ifeq(theorem(or(not(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, q), implies(or(p, not(q)), p))))), true, theorem(or(not(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), implies(or(p, q), implies(or(p, not(q)), p)))), true), true), true, ifeq(theorem(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), true, theorem(implies(or(p, q), implies(or(p, not(q)), p))), true), true)
% 13.84/2.26 = { by axiom 2 (ifeq_axiom) }
% 13.84/2.26 ifeq(ifeq(theorem(or(not(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, q), implies(or(p, not(q)), p))))), true, theorem(or(not(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), implies(or(p, q), implies(or(p, not(q)), p)))), true), true, ifeq(theorem(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), true, theorem(implies(or(p, q), implies(or(p, not(q)), p))), true), true)
% 13.84/2.26 = { by axiom 1 (implies_definition) }
% 13.84/2.26 ifeq(ifeq(theorem(or(not(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), or(implies(or(p, q), implies(or(p, not(q)), p)), or(not(or(p, q)), implies(or(p, not(q)), p))))), true, theorem(or(not(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), implies(or(p, q), implies(or(p, not(q)), p)))), true), true, ifeq(theorem(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), true, theorem(implies(or(p, q), implies(or(p, not(q)), p))), true), true)
% 13.84/2.26 = { by axiom 2 (ifeq_axiom) R->L }
% 13.84/2.26 ifeq(ifeq(ifeq(true, true, theorem(or(not(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), or(implies(or(p, q), implies(or(p, not(q)), p)), or(not(or(p, q)), implies(or(p, not(q)), p))))), true), true, theorem(or(not(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), implies(or(p, q), implies(or(p, not(q)), p)))), true), true, ifeq(theorem(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), true, theorem(implies(or(p, q), implies(or(p, not(q)), p))), true), true)
% 13.84/2.26 = { by axiom 5 (rule_1) R->L }
% 13.84/2.26 ifeq(ifeq(ifeq(ifeq(axiom(or(not(or(not(implies(or(p, not(q)), p)), or(not(or(p, q)), implies(or(p, not(q)), p)))), or(not(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), or(implies(or(p, q), implies(or(p, not(q)), p)), or(not(or(p, q)), implies(or(p, not(q)), p)))))), true, theorem(or(not(or(not(implies(or(p, not(q)), p)), or(not(or(p, q)), implies(or(p, not(q)), p)))), or(not(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), or(implies(or(p, q), implies(or(p, not(q)), p)), or(not(or(p, q)), implies(or(p, not(q)), p)))))), true), true, theorem(or(not(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), or(implies(or(p, q), implies(or(p, not(q)), p)), or(not(or(p, q)), implies(or(p, not(q)), p))))), true), true, theorem(or(not(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), implies(or(p, q), implies(or(p, not(q)), p)))), true), true, ifeq(theorem(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), true, theorem(implies(or(p, q), implies(or(p, not(q)), p))), true), true)
% 13.84/2.27 = { by axiom 1 (implies_definition) R->L }
% 13.84/2.27 ifeq(ifeq(ifeq(ifeq(axiom(or(not(or(not(implies(or(p, not(q)), p)), or(not(or(p, q)), implies(or(p, not(q)), p)))), implies(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p)), or(implies(or(p, q), implies(or(p, not(q)), p)), or(not(or(p, q)), implies(or(p, not(q)), p)))))), true, theorem(or(not(or(not(implies(or(p, not(q)), p)), or(not(or(p, q)), implies(or(p, not(q)), p)))), or(not(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), or(implies(or(p, q), implies(or(p, not(q)), p)), or(not(or(p, q)), implies(or(p, not(q)), p)))))), true), true, theorem(or(not(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), or(implies(or(p, q), implies(or(p, not(q)), p)), or(not(or(p, q)), implies(or(p, not(q)), p))))), true), true, theorem(or(not(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), implies(or(p, q), implies(or(p, not(q)), p)))), true), true, ifeq(theorem(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), true, theorem(implies(or(p, q), implies(or(p, not(q)), p))), true), true)
% 13.84/2.27 = { by axiom 1 (implies_definition) R->L }
% 13.84/2.27 ifeq(ifeq(ifeq(ifeq(axiom(or(not(implies(implies(or(p, not(q)), p), or(not(or(p, q)), implies(or(p, not(q)), p)))), implies(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p)), or(implies(or(p, q), implies(or(p, not(q)), p)), or(not(or(p, q)), implies(or(p, not(q)), p)))))), true, theorem(or(not(or(not(implies(or(p, not(q)), p)), or(not(or(p, q)), implies(or(p, not(q)), p)))), or(not(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), or(implies(or(p, q), implies(or(p, not(q)), p)), or(not(or(p, q)), implies(or(p, not(q)), p)))))), true), true, theorem(or(not(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), or(implies(or(p, q), implies(or(p, not(q)), p)), or(not(or(p, q)), implies(or(p, not(q)), p))))), true), true, theorem(or(not(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), implies(or(p, q), implies(or(p, not(q)), p)))), true), true, ifeq(theorem(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), true, theorem(implies(or(p, q), implies(or(p, not(q)), p))), true), true)
% 13.84/2.27 = { by axiom 1 (implies_definition) R->L }
% 13.84/2.27 ifeq(ifeq(ifeq(ifeq(axiom(implies(implies(implies(or(p, not(q)), p), or(not(or(p, q)), implies(or(p, not(q)), p))), implies(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p)), or(implies(or(p, q), implies(or(p, not(q)), p)), or(not(or(p, q)), implies(or(p, not(q)), p)))))), true, theorem(or(not(or(not(implies(or(p, not(q)), p)), or(not(or(p, q)), implies(or(p, not(q)), p)))), or(not(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), or(implies(or(p, q), implies(or(p, not(q)), p)), or(not(or(p, q)), implies(or(p, not(q)), p)))))), true), true, theorem(or(not(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), or(implies(or(p, q), implies(or(p, not(q)), p)), or(not(or(p, q)), implies(or(p, not(q)), p))))), true), true, theorem(or(not(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), implies(or(p, q), implies(or(p, not(q)), p)))), true), true, ifeq(theorem(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), true, theorem(implies(or(p, q), implies(or(p, not(q)), p))), true), true)
% 13.84/2.27 = { by axiom 7 (axiom_1_6) }
% 13.84/2.28 ifeq(ifeq(ifeq(ifeq(true, true, theorem(or(not(or(not(implies(or(p, not(q)), p)), or(not(or(p, q)), implies(or(p, not(q)), p)))), or(not(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), or(implies(or(p, q), implies(or(p, not(q)), p)), or(not(or(p, q)), implies(or(p, not(q)), p)))))), true), true, theorem(or(not(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), or(implies(or(p, q), implies(or(p, not(q)), p)), or(not(or(p, q)), implies(or(p, not(q)), p))))), true), true, theorem(or(not(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), implies(or(p, q), implies(or(p, not(q)), p)))), true), true, ifeq(theorem(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), true, theorem(implies(or(p, q), implies(or(p, not(q)), p))), true), true)
% 13.84/2.28 = { by axiom 2 (ifeq_axiom) }
% 13.84/2.28 ifeq(ifeq(ifeq(theorem(or(not(or(not(implies(or(p, not(q)), p)), or(not(or(p, q)), implies(or(p, not(q)), p)))), or(not(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), or(implies(or(p, q), implies(or(p, not(q)), p)), or(not(or(p, q)), implies(or(p, not(q)), p)))))), true, theorem(or(not(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), or(implies(or(p, q), implies(or(p, not(q)), p)), or(not(or(p, q)), implies(or(p, not(q)), p))))), true), true, theorem(or(not(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), implies(or(p, q), implies(or(p, not(q)), p)))), true), true, ifeq(theorem(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), true, theorem(implies(or(p, q), implies(or(p, not(q)), p))), true), true)
% 13.84/2.28 = { by axiom 2 (ifeq_axiom) R->L }
% 13.84/2.28 ifeq(ifeq(ifeq(theorem(or(not(or(not(implies(or(p, not(q)), p)), or(not(or(p, q)), implies(or(p, not(q)), p)))), or(not(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), or(implies(or(p, q), implies(or(p, not(q)), p)), or(not(or(p, q)), implies(or(p, not(q)), p)))))), true, ifeq(true, true, theorem(or(not(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), or(implies(or(p, q), implies(or(p, not(q)), p)), or(not(or(p, q)), implies(or(p, not(q)), p))))), true), true), true, theorem(or(not(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), implies(or(p, q), implies(or(p, not(q)), p)))), true), true, ifeq(theorem(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), true, theorem(implies(or(p, q), implies(or(p, not(q)), p))), true), true)
% 13.84/2.28 = { by axiom 5 (rule_1) R->L }
% 13.84/2.28 ifeq(ifeq(ifeq(theorem(or(not(or(not(implies(or(p, not(q)), p)), or(not(or(p, q)), implies(or(p, not(q)), p)))), or(not(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), or(implies(or(p, q), implies(or(p, not(q)), p)), or(not(or(p, q)), implies(or(p, not(q)), p)))))), true, ifeq(ifeq(axiom(or(not(implies(or(p, not(q)), p)), or(not(or(p, q)), implies(or(p, not(q)), p)))), true, theorem(or(not(implies(or(p, not(q)), p)), or(not(or(p, q)), implies(or(p, not(q)), p)))), true), true, theorem(or(not(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), or(implies(or(p, q), implies(or(p, not(q)), p)), or(not(or(p, q)), implies(or(p, not(q)), p))))), true), true), true, theorem(or(not(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), implies(or(p, q), implies(or(p, not(q)), p)))), true), true, ifeq(theorem(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), true, theorem(implies(or(p, q), implies(or(p, not(q)), p))), true), true)
% 13.84/2.29 = { by axiom 1 (implies_definition) R->L }
% 13.84/2.29 ifeq(ifeq(ifeq(theorem(or(not(or(not(implies(or(p, not(q)), p)), or(not(or(p, q)), implies(or(p, not(q)), p)))), or(not(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), or(implies(or(p, q), implies(or(p, not(q)), p)), or(not(or(p, q)), implies(or(p, not(q)), p)))))), true, ifeq(ifeq(axiom(implies(implies(or(p, not(q)), p), or(not(or(p, q)), implies(or(p, not(q)), p)))), true, theorem(or(not(implies(or(p, not(q)), p)), or(not(or(p, q)), implies(or(p, not(q)), p)))), true), true, theorem(or(not(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), or(implies(or(p, q), implies(or(p, not(q)), p)), or(not(or(p, q)), implies(or(p, not(q)), p))))), true), true), true, theorem(or(not(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), implies(or(p, q), implies(or(p, not(q)), p)))), true), true, ifeq(theorem(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), true, theorem(implies(or(p, q), implies(or(p, not(q)), p))), true), true)
% 13.84/2.29 = { by axiom 3 (axiom_1_3) }
% 13.84/2.29 ifeq(ifeq(ifeq(theorem(or(not(or(not(implies(or(p, not(q)), p)), or(not(or(p, q)), implies(or(p, not(q)), p)))), or(not(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), or(implies(or(p, q), implies(or(p, not(q)), p)), or(not(or(p, q)), implies(or(p, not(q)), p)))))), true, ifeq(ifeq(true, true, theorem(or(not(implies(or(p, not(q)), p)), or(not(or(p, q)), implies(or(p, not(q)), p)))), true), true, theorem(or(not(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), or(implies(or(p, q), implies(or(p, not(q)), p)), or(not(or(p, q)), implies(or(p, not(q)), p))))), true), true), true, theorem(or(not(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), implies(or(p, q), implies(or(p, not(q)), p)))), true), true, ifeq(theorem(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), true, theorem(implies(or(p, q), implies(or(p, not(q)), p))), true), true)
% 13.84/2.29 = { by axiom 2 (ifeq_axiom) }
% 13.84/2.29 ifeq(ifeq(ifeq(theorem(or(not(or(not(implies(or(p, not(q)), p)), or(not(or(p, q)), implies(or(p, not(q)), p)))), or(not(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), or(implies(or(p, q), implies(or(p, not(q)), p)), or(not(or(p, q)), implies(or(p, not(q)), p)))))), true, ifeq(theorem(or(not(implies(or(p, not(q)), p)), or(not(or(p, q)), implies(or(p, not(q)), p)))), true, theorem(or(not(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), or(implies(or(p, q), implies(or(p, not(q)), p)), or(not(or(p, q)), implies(or(p, not(q)), p))))), true), true), true, theorem(or(not(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), implies(or(p, q), implies(or(p, not(q)), p)))), true), true, ifeq(theorem(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), true, theorem(implies(or(p, q), implies(or(p, not(q)), p))), true), true)
% 13.84/2.29 = { by axiom 1 (implies_definition) R->L }
% 13.84/2.29 ifeq(ifeq(ifeq(theorem(implies(or(not(implies(or(p, not(q)), p)), or(not(or(p, q)), implies(or(p, not(q)), p))), or(not(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), or(implies(or(p, q), implies(or(p, not(q)), p)), or(not(or(p, q)), implies(or(p, not(q)), p)))))), true, ifeq(theorem(or(not(implies(or(p, not(q)), p)), or(not(or(p, q)), implies(or(p, not(q)), p)))), true, theorem(or(not(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), or(implies(or(p, q), implies(or(p, not(q)), p)), or(not(or(p, q)), implies(or(p, not(q)), p))))), true), true), true, theorem(or(not(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), implies(or(p, q), implies(or(p, not(q)), p)))), true), true, ifeq(theorem(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), true, theorem(implies(or(p, q), implies(or(p, not(q)), p))), true), true)
% 13.84/2.29 = { by axiom 9 (rule_2) }
% 13.84/2.29 ifeq(ifeq(true, true, theorem(or(not(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), implies(or(p, q), implies(or(p, not(q)), p)))), true), true, ifeq(theorem(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), true, theorem(implies(or(p, q), implies(or(p, not(q)), p))), true), true)
% 13.84/2.29 = { by axiom 2 (ifeq_axiom) }
% 13.84/2.29 ifeq(theorem(or(not(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), implies(or(p, q), implies(or(p, not(q)), p)))), true, ifeq(theorem(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), true, theorem(implies(or(p, q), implies(or(p, not(q)), p))), true), true)
% 13.84/2.29 = { by axiom 1 (implies_definition) R->L }
% 13.84/2.29 ifeq(theorem(implies(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p)), implies(or(p, q), implies(or(p, not(q)), p)))), true, ifeq(theorem(or(implies(or(p, q), implies(or(p, not(q)), p)), implies(or(p, not(q)), p))), true, theorem(implies(or(p, q), implies(or(p, not(q)), p))), true), true)
% 13.84/2.29 = { by axiom 9 (rule_2) }
% 13.84/2.29 true
% 13.84/2.29 % SZS output end Proof
% 13.84/2.29
% 13.84/2.29 RESULT: Unsatisfiable (the axioms are contradictory).
%------------------------------------------------------------------------------