%------------------------------------------------------------------------------
% File : Twee---2.7
% Problem : LCL039-1 : TPTP v9.3.1. Released v1.0.0.
% Transfm : none
% Format : tptp:raw
% Command : run_twee /export/starexec/sandbox/benchmark/theBenchmark.p
% Computer : n001.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 11:48:53 AM UTC 2026
% Result : Unsatisfiable 44.17s 6.15s
% Output : Proof 46.55s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.05 % Problem : LCL039-1 : TPTP v9.3.1. Released v1.0.0.
% 0.00/0.07 % Command : run_twee /export/starexec/sandbox/benchmark/theBenchmark.p
% 0.18/0.45 % Computer : n001.cluster.edu
% 0.18/0.45 % Model : x86_64 x86_64
% 0.18/0.45 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.18/0.45 % Memory : 8046.5625MB
% 0.18/0.45 % OS : Linux 6.8.0-71-generic
% 0.18/0.45 % CPULimit : 300
% 0.18/0.45 % WCLimit : 300
% 0.18/0.45 % DateTime : Sun Sep 27 15:24:15 UTC 2026
% 0.18/0.46 % CPUTime :
% 0.18/0.46 Running run_twee /export/starexec/sandbox/benchmark/theBenchmark.p
% 44.17/6.15 Command-line arguments: --no-flatten-goal
% 44.17/6.15
% 44.17/6.15 % SZS status Unsatisfiable
% 44.17/6.15
% 45.76/6.38 % SZS output start Proof
% 45.76/6.38 Axiom 1 (necessitation2): is_a_theorem(implies(necessary(X), X)) = true.
% 45.76/6.38 Axiom 2 (ifeq_axiom): ifeq(X, X, Y, Z) = Y.
% 45.76/6.38 Axiom 3 (cn_18): is_a_theorem(implies(X, implies(Y, X))) = true.
% 45.76/6.38 Axiom 4 (axiom_of_necessitation): ifeq(is_a_theorem(X), true, is_a_theorem(necessary(X)), true) = true.
% 45.76/6.38 Axiom 5 (cn_49): is_a_theorem(implies(implies(not(X), not(Y)), implies(Y, X))) = true.
% 45.76/6.38 Axiom 6 (necessitation1): is_a_theorem(implies(necessary(implies(X, Y)), implies(necessary(X), necessary(Y)))) = true.
% 45.76/6.38 Axiom 7 (condensed_detachment): ifeq(is_a_theorem(implies(X, Y)), true, ifeq(is_a_theorem(X), true, is_a_theorem(Y), true), true) = true.
% 45.76/6.38 Axiom 8 (cn_35): is_a_theorem(implies(implies(X, implies(Y, Z)), implies(implies(X, Y), implies(X, Z)))) = true.
% 45.76/6.38
% 45.76/6.38 Goal 1 (prove_this): is_a_theorem(implies(necessary(a), not(necessary(not(a))))) = true.
% 45.76/6.38 Proof:
% 45.76/6.38 is_a_theorem(implies(necessary(a), not(necessary(not(a)))))
% 45.76/6.38 = { by axiom 2 (ifeq_axiom) R->L }
% 45.76/6.38 ifeq(true, true, is_a_theorem(implies(necessary(a), not(necessary(not(a))))), true)
% 45.76/6.38 = { by axiom 7 (condensed_detachment) R->L }
% 45.76/6.38 ifeq(ifeq(is_a_theorem(implies(necessary(implies(a, not(necessary(not(a))))), implies(necessary(a), necessary(not(necessary(not(a))))))), true, ifeq(is_a_theorem(necessary(implies(a, not(necessary(not(a)))))), true, is_a_theorem(implies(necessary(a), necessary(not(necessary(not(a)))))), true), true), true, is_a_theorem(implies(necessary(a), not(necessary(not(a))))), true)
% 45.76/6.38 = { by axiom 6 (necessitation1) }
% 45.76/6.38 ifeq(ifeq(true, true, ifeq(is_a_theorem(necessary(implies(a, not(necessary(not(a)))))), true, is_a_theorem(implies(necessary(a), necessary(not(necessary(not(a)))))), true), true), true, is_a_theorem(implies(necessary(a), not(necessary(not(a))))), true)
% 45.76/6.38 = { by axiom 2 (ifeq_axiom) }
% 45.76/6.38 ifeq(ifeq(is_a_theorem(necessary(implies(a, not(necessary(not(a)))))), true, is_a_theorem(implies(necessary(a), necessary(not(necessary(not(a)))))), true), true, is_a_theorem(implies(necessary(a), not(necessary(not(a))))), true)
% 45.76/6.38 = { by axiom 2 (ifeq_axiom) R->L }
% 45.76/6.38 ifeq(ifeq(ifeq(true, true, is_a_theorem(necessary(implies(a, not(necessary(not(a)))))), true), true, is_a_theorem(implies(necessary(a), necessary(not(necessary(not(a)))))), true), true, is_a_theorem(implies(necessary(a), not(necessary(not(a))))), true)
% 45.76/6.38 = { by axiom 7 (condensed_detachment) R->L }
% 45.76/6.39 ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(not(necessary(not(a)))), not(a)), implies(a, not(necessary(not(a)))))), true, ifeq(is_a_theorem(implies(not(not(necessary(not(a)))), not(a))), true, is_a_theorem(implies(a, not(necessary(not(a))))), true), true), true, is_a_theorem(necessary(implies(a, not(necessary(not(a)))))), true), true, is_a_theorem(implies(necessary(a), necessary(not(necessary(not(a)))))), true), true, is_a_theorem(implies(necessary(a), not(necessary(not(a))))), true)
% 45.76/6.39 = { by axiom 5 (cn_49) }
% 45.76/6.39 ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(not(not(necessary(not(a)))), not(a))), true, is_a_theorem(implies(a, not(necessary(not(a))))), true), true), true, is_a_theorem(necessary(implies(a, not(necessary(not(a)))))), true), true, is_a_theorem(implies(necessary(a), necessary(not(necessary(not(a)))))), true), true, is_a_theorem(implies(necessary(a), not(necessary(not(a))))), true)
% 45.76/6.39 = { by axiom 2 (ifeq_axiom) }
% 45.76/6.39 ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(not(not(necessary(not(a)))), not(a))), true, is_a_theorem(implies(a, not(necessary(not(a))))), true), true, is_a_theorem(necessary(implies(a, not(necessary(not(a)))))), true), true, is_a_theorem(implies(necessary(a), necessary(not(necessary(not(a)))))), true), true, is_a_theorem(implies(necessary(a), not(necessary(not(a))))), true)
% 45.76/6.39 = { by axiom 2 (ifeq_axiom) R->L }
% 45.76/6.39 ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(not(not(necessary(not(a)))), not(a))), true), true, is_a_theorem(implies(a, not(necessary(not(a))))), true), true, is_a_theorem(necessary(implies(a, not(necessary(not(a)))))), true), true, is_a_theorem(implies(necessary(a), necessary(not(necessary(not(a)))))), true), true, is_a_theorem(implies(necessary(a), not(necessary(not(a))))), true)
% 45.76/6.39 = { by axiom 7 (condensed_detachment) R->L }
% 45.76/6.39 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(not(necessary(not(a)))), implies(X, not(not(necessary(not(a)))))), implies(not(not(necessary(not(a)))), necessary(not(a))))), true, ifeq(is_a_theorem(implies(not(not(necessary(not(a)))), implies(X, not(not(necessary(not(a))))))), true, is_a_theorem(implies(not(not(necessary(not(a)))), necessary(not(a)))), true), true), true, is_a_theorem(implies(not(not(necessary(not(a)))), not(a))), true), true, is_a_theorem(implies(a, not(necessary(not(a))))), true), true, is_a_theorem(necessary(implies(a, not(necessary(not(a)))))), true), true, is_a_theorem(implies(necessary(a), necessary(not(necessary(not(a)))))), true), true, is_a_theorem(implies(necessary(a), not(necessary(not(a))))), true)
% 45.76/6.39 = { by axiom 3 (cn_18) }
% 45.76/6.39 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(not(necessary(not(a)))), implies(X, not(not(necessary(not(a)))))), implies(not(not(necessary(not(a)))), necessary(not(a))))), true, ifeq(true, true, is_a_theorem(implies(not(not(necessary(not(a)))), necessary(not(a)))), true), true), true, is_a_theorem(implies(not(not(necessary(not(a)))), not(a))), true), true, is_a_theorem(implies(a, not(necessary(not(a))))), true), true, is_a_theorem(necessary(implies(a, not(necessary(not(a)))))), true), true, is_a_theorem(implies(necessary(a), necessary(not(necessary(not(a)))))), true), true, is_a_theorem(implies(necessary(a), not(necessary(not(a))))), true)
% 45.76/6.39 = { by axiom 2 (ifeq_axiom) }
% 45.76/6.39 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(not(necessary(not(a)))), implies(X, not(not(necessary(not(a)))))), implies(not(not(necessary(not(a)))), necessary(not(a))))), true, is_a_theorem(implies(not(not(necessary(not(a)))), necessary(not(a)))), true), true, is_a_theorem(implies(not(not(necessary(not(a)))), not(a))), true), true, is_a_theorem(implies(a, not(necessary(not(a))))), true), true, is_a_theorem(necessary(implies(a, not(necessary(not(a)))))), true), true, is_a_theorem(implies(necessary(a), necessary(not(necessary(not(a)))))), true), true, is_a_theorem(implies(necessary(a), not(necessary(not(a))))), true)
% 45.76/6.39 = { by axiom 2 (ifeq_axiom) R->L }
% 45.76/6.39 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(not(not(necessary(not(a)))), implies(X, not(not(necessary(not(a)))))), implies(not(not(necessary(not(a)))), necessary(not(a))))), true), true, is_a_theorem(implies(not(not(necessary(not(a)))), necessary(not(a)))), true), true, is_a_theorem(implies(not(not(necessary(not(a)))), not(a))), true), true, is_a_theorem(implies(a, not(necessary(not(a))))), true), true, is_a_theorem(necessary(implies(a, not(necessary(not(a)))))), true), true, is_a_theorem(implies(necessary(a), necessary(not(necessary(not(a)))))), true), true, is_a_theorem(implies(necessary(a), not(necessary(not(a))))), true)
% 45.76/6.39 = { by axiom 7 (condensed_detachment) R->L }
% 45.76/6.40 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(not(necessary(not(a)))), implies(not(necessary(not(a))), not(implies(X, not(not(necessary(not(a)))))))), implies(not(not(necessary(not(a)))), implies(implies(X, not(not(necessary(not(a))))), necessary(not(a)))))), true, ifeq(is_a_theorem(implies(not(not(necessary(not(a)))), implies(not(necessary(not(a))), not(implies(X, not(not(necessary(not(a))))))))), true, is_a_theorem(implies(not(not(necessary(not(a)))), implies(implies(X, not(not(necessary(not(a))))), necessary(not(a))))), true), true), true, is_a_theorem(implies(implies(not(not(necessary(not(a)))), implies(X, not(not(necessary(not(a)))))), implies(not(not(necessary(not(a)))), necessary(not(a))))), true), true, is_a_theorem(implies(not(not(necessary(not(a)))), necessary(not(a)))), true), true, is_a_theorem(implies(not(not(necessary(not(a)))), not(a))), true), true, is_a_theorem(implies(a, not(necessary(not(a))))), true), true, is_a_theorem(necessary(implies(a, not(necessary(not(a)))))), true), true, is_a_theorem(implies(necessary(a), necessary(not(necessary(not(a)))))), true), true, is_a_theorem(implies(necessary(a), not(necessary(not(a))))), true)
% 45.76/6.40 = { by axiom 2 (ifeq_axiom) R->L }
% 45.76/6.40 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(not(necessary(not(a)))), implies(not(necessary(not(a))), not(implies(X, not(not(necessary(not(a)))))))), implies(not(not(necessary(not(a)))), implies(implies(X, not(not(necessary(not(a))))), necessary(not(a)))))), true, ifeq(ifeq(true, true, is_a_theorem(implies(not(not(necessary(not(a)))), implies(not(necessary(not(a))), not(implies(X, not(not(necessary(not(a))))))))), true), true, is_a_theorem(implies(not(not(necessary(not(a)))), implies(implies(X, not(not(necessary(not(a))))), necessary(not(a))))), true), true), true, is_a_theorem(implies(implies(not(not(necessary(not(a)))), implies(X, not(not(necessary(not(a)))))), implies(not(not(necessary(not(a)))), necessary(not(a))))), true), true, is_a_theorem(implies(not(not(necessary(not(a)))), necessary(not(a)))), true), true, is_a_theorem(implies(not(not(necessary(not(a)))), not(a))), true), true, is_a_theorem(implies(a, not(necessary(not(a))))), true), true, is_a_theorem(necessary(implies(a, not(necessary(not(a)))))), true), true, is_a_theorem(implies(necessary(a), necessary(not(necessary(not(a)))))), true), true, is_a_theorem(implies(necessary(a), not(necessary(not(a))))), true)
% 45.76/6.40 = { by axiom 7 (condensed_detachment) R->L }
% 45.76/6.40 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(not(necessary(not(a)))), implies(not(necessary(not(a))), not(implies(X, not(not(necessary(not(a)))))))), implies(not(not(necessary(not(a)))), implies(implies(X, not(not(necessary(not(a))))), necessary(not(a)))))), true, ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(not(necessary(not(a)))), implies(implies(not(not(implies(X, not(not(necessary(not(a))))))), not(not(necessary(not(a))))), implies(not(necessary(not(a))), not(implies(X, not(not(necessary(not(a))))))))), implies(implies(not(not(necessary(not(a)))), implies(not(not(implies(X, not(not(necessary(not(a))))))), not(not(necessary(not(a)))))), implies(not(not(necessary(not(a)))), implies(not(necessary(not(a))), not(implies(X, not(not(necessary(not(a))))))))))), true, ifeq(is_a_theorem(implies(not(not(necessary(not(a)))), implies(implies(not(not(implies(X, not(not(necessary(not(a))))))), not(not(necessary(not(a))))), implies(not(necessary(not(a))), not(implies(X, not(not(necessary(not(a)))))))))), true, is_a_theorem(implies(implies(not(not(necessary(not(a)))), implies(not(not(implies(X, not(not(necessary(not(a))))))), not(not(necessary(not(a)))))), implies(not(not(necessary(not(a)))), implies(not(necessary(not(a))), not(implies(X, not(not(necessary(not(a)))))))))), true), true), true, is_a_theorem(implies(not(not(necessary(not(a)))), implies(not(necessary(not(a))), not(implies(X, not(not(necessary(not(a))))))))), true), true, is_a_theorem(implies(not(not(necessary(not(a)))), implies(implies(X, not(not(necessary(not(a))))), necessary(not(a))))), true), true), true, is_a_theorem(implies(implies(not(not(necessary(not(a)))), implies(X, not(not(necessary(not(a)))))), implies(not(not(necessary(not(a)))), necessary(not(a))))), true), true, is_a_theorem(implies(not(not(necessary(not(a)))), necessary(not(a)))), true), true, is_a_theorem(implies(not(not(necessary(not(a)))), not(a))), true), true, is_a_theorem(implies(a, not(necessary(not(a))))), true), true, is_a_theorem(necessary(implies(a, not(necessary(not(a)))))), true), true, is_a_theorem(implies(necessary(a), necessary(not(necessary(not(a)))))), true), true, is_a_theorem(implies(necessary(a), not(necessary(not(a))))), true)
% 46.55/6.40 = { by axiom 8 (cn_35) }
% 46.55/6.41 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(not(necessary(not(a)))), implies(not(necessary(not(a))), not(implies(X, not(not(necessary(not(a)))))))), implies(not(not(necessary(not(a)))), implies(implies(X, not(not(necessary(not(a))))), necessary(not(a)))))), true, ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(not(not(necessary(not(a)))), implies(implies(not(not(implies(X, not(not(necessary(not(a))))))), not(not(necessary(not(a))))), implies(not(necessary(not(a))), not(implies(X, not(not(necessary(not(a)))))))))), true, is_a_theorem(implies(implies(not(not(necessary(not(a)))), implies(not(not(implies(X, not(not(necessary(not(a))))))), not(not(necessary(not(a)))))), implies(not(not(necessary(not(a)))), implies(not(necessary(not(a))), not(implies(X, not(not(necessary(not(a)))))))))), true), true), true, is_a_theorem(implies(not(not(necessary(not(a)))), implies(not(necessary(not(a))), not(implies(X, not(not(necessary(not(a))))))))), true), true, is_a_theorem(implies(not(not(necessary(not(a)))), implies(implies(X, not(not(necessary(not(a))))), necessary(not(a))))), true), true), true, is_a_theorem(implies(implies(not(not(necessary(not(a)))), implies(X, not(not(necessary(not(a)))))), implies(not(not(necessary(not(a)))), necessary(not(a))))), true), true, is_a_theorem(implies(not(not(necessary(not(a)))), necessary(not(a)))), true), true, is_a_theorem(implies(not(not(necessary(not(a)))), not(a))), true), true, is_a_theorem(implies(a, not(necessary(not(a))))), true), true, is_a_theorem(necessary(implies(a, not(necessary(not(a)))))), true), true, is_a_theorem(implies(necessary(a), necessary(not(necessary(not(a)))))), true), true, is_a_theorem(implies(necessary(a), not(necessary(not(a))))), true)
% 46.55/6.41 = { by axiom 2 (ifeq_axiom) }
% 46.55/6.41 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(not(necessary(not(a)))), implies(not(necessary(not(a))), not(implies(X, not(not(necessary(not(a)))))))), implies(not(not(necessary(not(a)))), implies(implies(X, not(not(necessary(not(a))))), necessary(not(a)))))), true, ifeq(ifeq(ifeq(is_a_theorem(implies(not(not(necessary(not(a)))), implies(implies(not(not(implies(X, not(not(necessary(not(a))))))), not(not(necessary(not(a))))), implies(not(necessary(not(a))), not(implies(X, not(not(necessary(not(a)))))))))), true, is_a_theorem(implies(implies(not(not(necessary(not(a)))), implies(not(not(implies(X, not(not(necessary(not(a))))))), not(not(necessary(not(a)))))), implies(not(not(necessary(not(a)))), implies(not(necessary(not(a))), not(implies(X, not(not(necessary(not(a)))))))))), true), true, is_a_theorem(implies(not(not(necessary(not(a)))), implies(not(necessary(not(a))), not(implies(X, not(not(necessary(not(a))))))))), true), true, is_a_theorem(implies(not(not(necessary(not(a)))), implies(implies(X, not(not(necessary(not(a))))), necessary(not(a))))), true), true), true, is_a_theorem(implies(implies(not(not(necessary(not(a)))), implies(X, not(not(necessary(not(a)))))), implies(not(not(necessary(not(a)))), necessary(not(a))))), true), true, is_a_theorem(implies(not(not(necessary(not(a)))), necessary(not(a)))), true), true, is_a_theorem(implies(not(not(necessary(not(a)))), not(a))), true), true, is_a_theorem(implies(a, not(necessary(not(a))))), true), true, is_a_theorem(necessary(implies(a, not(necessary(not(a)))))), true), true, is_a_theorem(implies(necessary(a), necessary(not(necessary(not(a)))))), true), true, is_a_theorem(implies(necessary(a), not(necessary(not(a))))), true)
% 46.55/6.41 = { by axiom 2 (ifeq_axiom) R->L }
% 46.55/6.41 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(not(necessary(not(a)))), implies(not(necessary(not(a))), not(implies(X, not(not(necessary(not(a)))))))), implies(not(not(necessary(not(a)))), implies(implies(X, not(not(necessary(not(a))))), necessary(not(a)))))), true, ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(not(not(necessary(not(a)))), implies(implies(not(not(implies(X, not(not(necessary(not(a))))))), not(not(necessary(not(a))))), implies(not(necessary(not(a))), not(implies(X, not(not(necessary(not(a)))))))))), true), true, is_a_theorem(implies(implies(not(not(necessary(not(a)))), implies(not(not(implies(X, not(not(necessary(not(a))))))), not(not(necessary(not(a)))))), implies(not(not(necessary(not(a)))), implies(not(necessary(not(a))), not(implies(X, not(not(necessary(not(a)))))))))), true), true, is_a_theorem(implies(not(not(necessary(not(a)))), implies(not(necessary(not(a))), not(implies(X, not(not(necessary(not(a))))))))), true), true, is_a_theorem(implies(not(not(necessary(not(a)))), implies(implies(X, not(not(necessary(not(a))))), necessary(not(a))))), true), true), true, is_a_theorem(implies(implies(not(not(necessary(not(a)))), implies(X, not(not(necessary(not(a)))))), implies(not(not(necessary(not(a)))), necessary(not(a))))), true), true, is_a_theorem(implies(not(not(necessary(not(a)))), necessary(not(a)))), true), true, is_a_theorem(implies(not(not(necessary(not(a)))), not(a))), true), true, is_a_theorem(implies(a, not(necessary(not(a))))), true), true, is_a_theorem(necessary(implies(a, not(necessary(not(a)))))), true), true, is_a_theorem(implies(necessary(a), necessary(not(necessary(not(a)))))), true), true, is_a_theorem(implies(necessary(a), not(necessary(not(a))))), true)
% 46.55/6.42 = { by axiom 5 (cn_49) R->L }
% 46.55/6.42 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(not(necessary(not(a)))), implies(not(necessary(not(a))), not(implies(X, not(not(necessary(not(a)))))))), implies(not(not(necessary(not(a)))), implies(implies(X, not(not(necessary(not(a))))), necessary(not(a)))))), true, ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(not(implies(X, not(not(necessary(not(a))))))), not(not(necessary(not(a))))), implies(not(necessary(not(a))), not(implies(X, not(not(necessary(not(a))))))))), true, is_a_theorem(implies(not(not(necessary(not(a)))), implies(implies(not(not(implies(X, not(not(necessary(not(a))))))), not(not(necessary(not(a))))), implies(not(necessary(not(a))), not(implies(X, not(not(necessary(not(a)))))))))), true), true, is_a_theorem(implies(implies(not(not(necessary(not(a)))), implies(not(not(implies(X, not(not(necessary(not(a))))))), not(not(necessary(not(a)))))), implies(not(not(necessary(not(a)))), implies(not(necessary(not(a))), not(implies(X, not(not(necessary(not(a)))))))))), true), true, is_a_theorem(implies(not(not(necessary(not(a)))), implies(not(necessary(not(a))), not(implies(X, not(not(necessary(not(a))))))))), true), true, is_a_theorem(implies(not(not(necessary(not(a)))), implies(implies(X, not(not(necessary(not(a))))), necessary(not(a))))), true), true), true, is_a_theorem(implies(implies(not(not(necessary(not(a)))), implies(X, not(not(necessary(not(a)))))), implies(not(not(necessary(not(a)))), necessary(not(a))))), true), true, is_a_theorem(implies(not(not(necessary(not(a)))), necessary(not(a)))), true), true, is_a_theorem(implies(not(not(necessary(not(a)))), not(a))), true), true, is_a_theorem(implies(a, not(necessary(not(a))))), true), true, is_a_theorem(necessary(implies(a, not(necessary(not(a)))))), true), true, is_a_theorem(implies(necessary(a), necessary(not(necessary(not(a)))))), true), true, is_a_theorem(implies(necessary(a), not(necessary(not(a))))), true)
% 46.55/6.42 = { by axiom 2 (ifeq_axiom) R->L }
% 46.55/6.42 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(not(necessary(not(a)))), implies(not(necessary(not(a))), not(implies(X, not(not(necessary(not(a)))))))), implies(not(not(necessary(not(a)))), implies(implies(X, not(not(necessary(not(a))))), necessary(not(a)))))), true, ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(not(not(implies(X, not(not(necessary(not(a))))))), not(not(necessary(not(a))))), implies(not(necessary(not(a))), not(implies(X, not(not(necessary(not(a))))))))), true, is_a_theorem(implies(not(not(necessary(not(a)))), implies(implies(not(not(implies(X, not(not(necessary(not(a))))))), not(not(necessary(not(a))))), implies(not(necessary(not(a))), not(implies(X, not(not(necessary(not(a)))))))))), true), true), true, is_a_theorem(implies(implies(not(not(necessary(not(a)))), implies(not(not(implies(X, not(not(necessary(not(a))))))), not(not(necessary(not(a)))))), implies(not(not(necessary(not(a)))), implies(not(necessary(not(a))), not(implies(X, not(not(necessary(not(a)))))))))), true), true, is_a_theorem(implies(not(not(necessary(not(a)))), implies(not(necessary(not(a))), not(implies(X, not(not(necessary(not(a))))))))), true), true, is_a_theorem(implies(not(not(necessary(not(a)))), implies(implies(X, not(not(necessary(not(a))))), necessary(not(a))))), true), true), true, is_a_theorem(implies(implies(not(not(necessary(not(a)))), implies(X, not(not(necessary(not(a)))))), implies(not(not(necessary(not(a)))), necessary(not(a))))), true), true, is_a_theorem(implies(not(not(necessary(not(a)))), necessary(not(a)))), true), true, is_a_theorem(implies(not(not(necessary(not(a)))), not(a))), true), true, is_a_theorem(implies(a, not(necessary(not(a))))), true), true, is_a_theorem(necessary(implies(a, not(necessary(not(a)))))), true), true, is_a_theorem(implies(necessary(a), necessary(not(necessary(not(a)))))), true), true, is_a_theorem(implies(necessary(a), not(necessary(not(a))))), true)
% 46.55/6.42 = { by axiom 3 (cn_18) R->L }
% 46.55/6.43 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(not(necessary(not(a)))), implies(not(necessary(not(a))), not(implies(X, not(not(necessary(not(a)))))))), implies(not(not(necessary(not(a)))), implies(implies(X, not(not(necessary(not(a))))), necessary(not(a)))))), true, ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(not(not(implies(X, not(not(necessary(not(a))))))), not(not(necessary(not(a))))), implies(not(necessary(not(a))), not(implies(X, not(not(necessary(not(a)))))))), implies(not(not(necessary(not(a)))), implies(implies(not(not(implies(X, not(not(necessary(not(a))))))), not(not(necessary(not(a))))), implies(not(necessary(not(a))), not(implies(X, not(not(necessary(not(a))))))))))), true, ifeq(is_a_theorem(implies(implies(not(not(implies(X, not(not(necessary(not(a))))))), not(not(necessary(not(a))))), implies(not(necessary(not(a))), not(implies(X, not(not(necessary(not(a))))))))), true, is_a_theorem(implies(not(not(necessary(not(a)))), implies(implies(not(not(implies(X, not(not(necessary(not(a))))))), not(not(necessary(not(a))))), implies(not(necessary(not(a))), not(implies(X, not(not(necessary(not(a)))))))))), true), true), true, is_a_theorem(implies(implies(not(not(necessary(not(a)))), implies(not(not(implies(X, not(not(necessary(not(a))))))), not(not(necessary(not(a)))))), implies(not(not(necessary(not(a)))), implies(not(necessary(not(a))), not(implies(X, not(not(necessary(not(a)))))))))), true), true, is_a_theorem(implies(not(not(necessary(not(a)))), implies(not(necessary(not(a))), not(implies(X, not(not(necessary(not(a))))))))), true), true, is_a_theorem(implies(not(not(necessary(not(a)))), implies(implies(X, not(not(necessary(not(a))))), necessary(not(a))))), true), true), true, is_a_theorem(implies(implies(not(not(necessary(not(a)))), implies(X, not(not(necessary(not(a)))))), implies(not(not(necessary(not(a)))), necessary(not(a))))), true), true, is_a_theorem(implies(not(not(necessary(not(a)))), necessary(not(a)))), true), true, is_a_theorem(implies(not(not(necessary(not(a)))), not(a))), true), true, is_a_theorem(implies(a, not(necessary(not(a))))), true), true, is_a_theorem(necessary(implies(a, not(necessary(not(a)))))), true), true, is_a_theorem(implies(necessary(a), necessary(not(necessary(not(a)))))), true), true, is_a_theorem(implies(necessary(a), not(necessary(not(a))))), true)
% 46.55/6.43 = { by axiom 7 (condensed_detachment) }
% 46.55/6.43 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(not(necessary(not(a)))), implies(not(necessary(not(a))), not(implies(X, not(not(necessary(not(a)))))))), implies(not(not(necessary(not(a)))), implies(implies(X, not(not(necessary(not(a))))), necessary(not(a)))))), true, ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(not(not(necessary(not(a)))), implies(not(not(implies(X, not(not(necessary(not(a))))))), not(not(necessary(not(a)))))), implies(not(not(necessary(not(a)))), implies(not(necessary(not(a))), not(implies(X, not(not(necessary(not(a)))))))))), true), true, is_a_theorem(implies(not(not(necessary(not(a)))), implies(not(necessary(not(a))), not(implies(X, not(not(necessary(not(a))))))))), true), true, is_a_theorem(implies(not(not(necessary(not(a)))), implies(implies(X, not(not(necessary(not(a))))), necessary(not(a))))), true), true), true, is_a_theorem(implies(implies(not(not(necessary(not(a)))), implies(X, not(not(necessary(not(a)))))), implies(not(not(necessary(not(a)))), necessary(not(a))))), true), true, is_a_theorem(implies(not(not(necessary(not(a)))), necessary(not(a)))), true), true, is_a_theorem(implies(not(not(necessary(not(a)))), not(a))), true), true, is_a_theorem(implies(a, not(necessary(not(a))))), true), true, is_a_theorem(necessary(implies(a, not(necessary(not(a)))))), true), true, is_a_theorem(implies(necessary(a), necessary(not(necessary(not(a)))))), true), true, is_a_theorem(implies(necessary(a), not(necessary(not(a))))), true)
% 46.55/6.43 = { by axiom 2 (ifeq_axiom) }
% 46.55/6.43 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(not(necessary(not(a)))), implies(not(necessary(not(a))), not(implies(X, not(not(necessary(not(a)))))))), implies(not(not(necessary(not(a)))), implies(implies(X, not(not(necessary(not(a))))), necessary(not(a)))))), true, ifeq(ifeq(is_a_theorem(implies(implies(not(not(necessary(not(a)))), implies(not(not(implies(X, not(not(necessary(not(a))))))), not(not(necessary(not(a)))))), implies(not(not(necessary(not(a)))), implies(not(necessary(not(a))), not(implies(X, not(not(necessary(not(a)))))))))), true, is_a_theorem(implies(not(not(necessary(not(a)))), implies(not(necessary(not(a))), not(implies(X, not(not(necessary(not(a))))))))), true), true, is_a_theorem(implies(not(not(necessary(not(a)))), implies(implies(X, not(not(necessary(not(a))))), necessary(not(a))))), true), true), true, is_a_theorem(implies(implies(not(not(necessary(not(a)))), implies(X, not(not(necessary(not(a)))))), implies(not(not(necessary(not(a)))), necessary(not(a))))), true), true, is_a_theorem(implies(not(not(necessary(not(a)))), necessary(not(a)))), true), true, is_a_theorem(implies(not(not(necessary(not(a)))), not(a))), true), true, is_a_theorem(implies(a, not(necessary(not(a))))), true), true, is_a_theorem(necessary(implies(a, not(necessary(not(a)))))), true), true, is_a_theorem(implies(necessary(a), necessary(not(necessary(not(a)))))), true), true, is_a_theorem(implies(necessary(a), not(necessary(not(a))))), true)
% 46.55/6.44 = { by axiom 2 (ifeq_axiom) R->L }
% 46.55/6.44 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(not(necessary(not(a)))), implies(not(necessary(not(a))), not(implies(X, not(not(necessary(not(a)))))))), implies(not(not(necessary(not(a)))), implies(implies(X, not(not(necessary(not(a))))), necessary(not(a)))))), true, ifeq(ifeq(is_a_theorem(implies(implies(not(not(necessary(not(a)))), implies(not(not(implies(X, not(not(necessary(not(a))))))), not(not(necessary(not(a)))))), implies(not(not(necessary(not(a)))), implies(not(necessary(not(a))), not(implies(X, not(not(necessary(not(a)))))))))), true, ifeq(true, true, is_a_theorem(implies(not(not(necessary(not(a)))), implies(not(necessary(not(a))), not(implies(X, not(not(necessary(not(a))))))))), true), true), true, is_a_theorem(implies(not(not(necessary(not(a)))), implies(implies(X, not(not(necessary(not(a))))), necessary(not(a))))), true), true), true, is_a_theorem(implies(implies(not(not(necessary(not(a)))), implies(X, not(not(necessary(not(a)))))), implies(not(not(necessary(not(a)))), necessary(not(a))))), true), true, is_a_theorem(implies(not(not(necessary(not(a)))), necessary(not(a)))), true), true, is_a_theorem(implies(not(not(necessary(not(a)))), not(a))), true), true, is_a_theorem(implies(a, not(necessary(not(a))))), true), true, is_a_theorem(necessary(implies(a, not(necessary(not(a)))))), true), true, is_a_theorem(implies(necessary(a), necessary(not(necessary(not(a)))))), true), true, is_a_theorem(implies(necessary(a), not(necessary(not(a))))), true)
% 46.55/6.44 = { by axiom 3 (cn_18) R->L }
% 46.55/6.44 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(not(necessary(not(a)))), implies(not(necessary(not(a))), not(implies(X, not(not(necessary(not(a)))))))), implies(not(not(necessary(not(a)))), implies(implies(X, not(not(necessary(not(a))))), necessary(not(a)))))), true, ifeq(ifeq(is_a_theorem(implies(implies(not(not(necessary(not(a)))), implies(not(not(implies(X, not(not(necessary(not(a))))))), not(not(necessary(not(a)))))), implies(not(not(necessary(not(a)))), implies(not(necessary(not(a))), not(implies(X, not(not(necessary(not(a)))))))))), true, ifeq(is_a_theorem(implies(not(not(necessary(not(a)))), implies(not(not(implies(X, not(not(necessary(not(a))))))), not(not(necessary(not(a))))))), true, is_a_theorem(implies(not(not(necessary(not(a)))), implies(not(necessary(not(a))), not(implies(X, not(not(necessary(not(a))))))))), true), true), true, is_a_theorem(implies(not(not(necessary(not(a)))), implies(implies(X, not(not(necessary(not(a))))), necessary(not(a))))), true), true), true, is_a_theorem(implies(implies(not(not(necessary(not(a)))), implies(X, not(not(necessary(not(a)))))), implies(not(not(necessary(not(a)))), necessary(not(a))))), true), true, is_a_theorem(implies(not(not(necessary(not(a)))), necessary(not(a)))), true), true, is_a_theorem(implies(not(not(necessary(not(a)))), not(a))), true), true, is_a_theorem(implies(a, not(necessary(not(a))))), true), true, is_a_theorem(necessary(implies(a, not(necessary(not(a)))))), true), true, is_a_theorem(implies(necessary(a), necessary(not(necessary(not(a)))))), true), true, is_a_theorem(implies(necessary(a), not(necessary(not(a))))), true)
% 46.55/6.44 = { by axiom 7 (condensed_detachment) }
% 46.55/6.44 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(not(necessary(not(a)))), implies(not(necessary(not(a))), not(implies(X, not(not(necessary(not(a)))))))), implies(not(not(necessary(not(a)))), implies(implies(X, not(not(necessary(not(a))))), necessary(not(a)))))), true, ifeq(true, true, is_a_theorem(implies(not(not(necessary(not(a)))), implies(implies(X, not(not(necessary(not(a))))), necessary(not(a))))), true), true), true, is_a_theorem(implies(implies(not(not(necessary(not(a)))), implies(X, not(not(necessary(not(a)))))), implies(not(not(necessary(not(a)))), necessary(not(a))))), true), true, is_a_theorem(implies(not(not(necessary(not(a)))), necessary(not(a)))), true), true, is_a_theorem(implies(not(not(necessary(not(a)))), not(a))), true), true, is_a_theorem(implies(a, not(necessary(not(a))))), true), true, is_a_theorem(necessary(implies(a, not(necessary(not(a)))))), true), true, is_a_theorem(implies(necessary(a), necessary(not(necessary(not(a)))))), true), true, is_a_theorem(implies(necessary(a), not(necessary(not(a))))), true)
% 46.55/6.44 = { by axiom 2 (ifeq_axiom) }
% 46.55/6.44 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(not(necessary(not(a)))), implies(not(necessary(not(a))), not(implies(X, not(not(necessary(not(a)))))))), implies(not(not(necessary(not(a)))), implies(implies(X, not(not(necessary(not(a))))), necessary(not(a)))))), true, is_a_theorem(implies(not(not(necessary(not(a)))), implies(implies(X, not(not(necessary(not(a))))), necessary(not(a))))), true), true, is_a_theorem(implies(implies(not(not(necessary(not(a)))), implies(X, not(not(necessary(not(a)))))), implies(not(not(necessary(not(a)))), necessary(not(a))))), true), true, is_a_theorem(implies(not(not(necessary(not(a)))), necessary(not(a)))), true), true, is_a_theorem(implies(not(not(necessary(not(a)))), not(a))), true), true, is_a_theorem(implies(a, not(necessary(not(a))))), true), true, is_a_theorem(necessary(implies(a, not(necessary(not(a)))))), true), true, is_a_theorem(implies(necessary(a), necessary(not(necessary(not(a)))))), true), true, is_a_theorem(implies(necessary(a), not(necessary(not(a))))), true)
% 46.55/6.44 = { by axiom 2 (ifeq_axiom) R->L }
% 46.55/6.45 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(not(not(necessary(not(a)))), implies(not(necessary(not(a))), not(implies(X, not(not(necessary(not(a)))))))), implies(not(not(necessary(not(a)))), implies(implies(X, not(not(necessary(not(a))))), necessary(not(a)))))), true), true, is_a_theorem(implies(not(not(necessary(not(a)))), implies(implies(X, not(not(necessary(not(a))))), necessary(not(a))))), true), true, is_a_theorem(implies(implies(not(not(necessary(not(a)))), implies(X, not(not(necessary(not(a)))))), implies(not(not(necessary(not(a)))), necessary(not(a))))), true), true, is_a_theorem(implies(not(not(necessary(not(a)))), necessary(not(a)))), true), true, is_a_theorem(implies(not(not(necessary(not(a)))), not(a))), true), true, is_a_theorem(implies(a, not(necessary(not(a))))), true), true, is_a_theorem(necessary(implies(a, not(necessary(not(a)))))), true), true, is_a_theorem(implies(necessary(a), necessary(not(necessary(not(a)))))), true), true, is_a_theorem(implies(necessary(a), not(necessary(not(a))))), true)
% 46.55/6.45 = { by axiom 7 (condensed_detachment) R->L }
% 46.55/6.45 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(not(necessary(not(a))), not(implies(X, not(not(necessary(not(a))))))), implies(implies(X, not(not(necessary(not(a))))), necessary(not(a)))), implies(not(not(necessary(not(a)))), implies(implies(not(necessary(not(a))), not(implies(X, not(not(necessary(not(a))))))), implies(implies(X, not(not(necessary(not(a))))), necessary(not(a))))))), true, ifeq(is_a_theorem(implies(implies(not(necessary(not(a))), not(implies(X, not(not(necessary(not(a))))))), implies(implies(X, not(not(necessary(not(a))))), necessary(not(a))))), true, is_a_theorem(implies(not(not(necessary(not(a)))), implies(implies(not(necessary(not(a))), not(implies(X, not(not(necessary(not(a))))))), implies(implies(X, not(not(necessary(not(a))))), necessary(not(a)))))), true), true), true, is_a_theorem(implies(implies(not(not(necessary(not(a)))), implies(not(necessary(not(a))), not(implies(X, not(not(necessary(not(a)))))))), implies(not(not(necessary(not(a)))), implies(implies(X, not(not(necessary(not(a))))), necessary(not(a)))))), true), true, is_a_theorem(implies(not(not(necessary(not(a)))), implies(implies(X, not(not(necessary(not(a))))), necessary(not(a))))), true), true, is_a_theorem(implies(implies(not(not(necessary(not(a)))), implies(X, not(not(necessary(not(a)))))), implies(not(not(necessary(not(a)))), necessary(not(a))))), true), true, is_a_theorem(implies(not(not(necessary(not(a)))), necessary(not(a)))), true), true, is_a_theorem(implies(not(not(necessary(not(a)))), not(a))), true), true, is_a_theorem(implies(a, not(necessary(not(a))))), true), true, is_a_theorem(necessary(implies(a, not(necessary(not(a)))))), true), true, is_a_theorem(implies(necessary(a), necessary(not(necessary(not(a)))))), true), true, is_a_theorem(implies(necessary(a), not(necessary(not(a))))), true)
% 46.55/6.45 = { by axiom 3 (cn_18) }
% 46.55/6.45 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(not(necessary(not(a))), not(implies(X, not(not(necessary(not(a))))))), implies(implies(X, not(not(necessary(not(a))))), necessary(not(a))))), true, is_a_theorem(implies(not(not(necessary(not(a)))), implies(implies(not(necessary(not(a))), not(implies(X, not(not(necessary(not(a))))))), implies(implies(X, not(not(necessary(not(a))))), necessary(not(a)))))), true), true), true, is_a_theorem(implies(implies(not(not(necessary(not(a)))), implies(not(necessary(not(a))), not(implies(X, not(not(necessary(not(a)))))))), implies(not(not(necessary(not(a)))), implies(implies(X, not(not(necessary(not(a))))), necessary(not(a)))))), true), true, is_a_theorem(implies(not(not(necessary(not(a)))), implies(implies(X, not(not(necessary(not(a))))), necessary(not(a))))), true), true, is_a_theorem(implies(implies(not(not(necessary(not(a)))), implies(X, not(not(necessary(not(a)))))), implies(not(not(necessary(not(a)))), necessary(not(a))))), true), true, is_a_theorem(implies(not(not(necessary(not(a)))), necessary(not(a)))), true), true, is_a_theorem(implies(not(not(necessary(not(a)))), not(a))), true), true, is_a_theorem(implies(a, not(necessary(not(a))))), true), true, is_a_theorem(necessary(implies(a, not(necessary(not(a)))))), true), true, is_a_theorem(implies(necessary(a), necessary(not(necessary(not(a)))))), true), true, is_a_theorem(implies(necessary(a), not(necessary(not(a))))), true)
% 46.55/6.45 = { by axiom 2 (ifeq_axiom) }
% 46.55/6.45 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(necessary(not(a))), not(implies(X, not(not(necessary(not(a))))))), implies(implies(X, not(not(necessary(not(a))))), necessary(not(a))))), true, is_a_theorem(implies(not(not(necessary(not(a)))), implies(implies(not(necessary(not(a))), not(implies(X, not(not(necessary(not(a))))))), implies(implies(X, not(not(necessary(not(a))))), necessary(not(a)))))), true), true, is_a_theorem(implies(implies(not(not(necessary(not(a)))), implies(not(necessary(not(a))), not(implies(X, not(not(necessary(not(a)))))))), implies(not(not(necessary(not(a)))), implies(implies(X, not(not(necessary(not(a))))), necessary(not(a)))))), true), true, is_a_theorem(implies(not(not(necessary(not(a)))), implies(implies(X, not(not(necessary(not(a))))), necessary(not(a))))), true), true, is_a_theorem(implies(implies(not(not(necessary(not(a)))), implies(X, not(not(necessary(not(a)))))), implies(not(not(necessary(not(a)))), necessary(not(a))))), true), true, is_a_theorem(implies(not(not(necessary(not(a)))), necessary(not(a)))), true), true, is_a_theorem(implies(not(not(necessary(not(a)))), not(a))), true), true, is_a_theorem(implies(a, not(necessary(not(a))))), true), true, is_a_theorem(necessary(implies(a, not(necessary(not(a)))))), true), true, is_a_theorem(implies(necessary(a), necessary(not(necessary(not(a)))))), true), true, is_a_theorem(implies(necessary(a), not(necessary(not(a))))), true)
% 46.55/6.45 = { by axiom 5 (cn_49) }
% 46.55/6.45 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(not(not(necessary(not(a)))), implies(implies(not(necessary(not(a))), not(implies(X, not(not(necessary(not(a))))))), implies(implies(X, not(not(necessary(not(a))))), necessary(not(a)))))), true), true, is_a_theorem(implies(implies(not(not(necessary(not(a)))), implies(not(necessary(not(a))), not(implies(X, not(not(necessary(not(a)))))))), implies(not(not(necessary(not(a)))), implies(implies(X, not(not(necessary(not(a))))), necessary(not(a)))))), true), true, is_a_theorem(implies(not(not(necessary(not(a)))), implies(implies(X, not(not(necessary(not(a))))), necessary(not(a))))), true), true, is_a_theorem(implies(implies(not(not(necessary(not(a)))), implies(X, not(not(necessary(not(a)))))), implies(not(not(necessary(not(a)))), necessary(not(a))))), true), true, is_a_theorem(implies(not(not(necessary(not(a)))), necessary(not(a)))), true), true, is_a_theorem(implies(not(not(necessary(not(a)))), not(a))), true), true, is_a_theorem(implies(a, not(necessary(not(a))))), true), true, is_a_theorem(necessary(implies(a, not(necessary(not(a)))))), true), true, is_a_theorem(implies(necessary(a), necessary(not(necessary(not(a)))))), true), true, is_a_theorem(implies(necessary(a), not(necessary(not(a))))), true)
% 46.55/6.46 = { by axiom 2 (ifeq_axiom) }
% 46.55/6.46 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(not(not(necessary(not(a)))), implies(implies(not(necessary(not(a))), not(implies(X, not(not(necessary(not(a))))))), implies(implies(X, not(not(necessary(not(a))))), necessary(not(a)))))), true, is_a_theorem(implies(implies(not(not(necessary(not(a)))), implies(not(necessary(not(a))), not(implies(X, not(not(necessary(not(a)))))))), implies(not(not(necessary(not(a)))), implies(implies(X, not(not(necessary(not(a))))), necessary(not(a)))))), true), true, is_a_theorem(implies(not(not(necessary(not(a)))), implies(implies(X, not(not(necessary(not(a))))), necessary(not(a))))), true), true, is_a_theorem(implies(implies(not(not(necessary(not(a)))), implies(X, not(not(necessary(not(a)))))), implies(not(not(necessary(not(a)))), necessary(not(a))))), true), true, is_a_theorem(implies(not(not(necessary(not(a)))), necessary(not(a)))), true), true, is_a_theorem(implies(not(not(necessary(not(a)))), not(a))), true), true, is_a_theorem(implies(a, not(necessary(not(a))))), true), true, is_a_theorem(necessary(implies(a, not(necessary(not(a)))))), true), true, is_a_theorem(implies(necessary(a), necessary(not(necessary(not(a)))))), true), true, is_a_theorem(implies(necessary(a), not(necessary(not(a))))), true)
% 46.55/6.46 = { by axiom 2 (ifeq_axiom) R->L }
% 46.55/6.46 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(not(not(necessary(not(a)))), implies(implies(not(necessary(not(a))), not(implies(X, not(not(necessary(not(a))))))), implies(implies(X, not(not(necessary(not(a))))), necessary(not(a)))))), true, is_a_theorem(implies(implies(not(not(necessary(not(a)))), implies(not(necessary(not(a))), not(implies(X, not(not(necessary(not(a)))))))), implies(not(not(necessary(not(a)))), implies(implies(X, not(not(necessary(not(a))))), necessary(not(a)))))), true), true), true, is_a_theorem(implies(not(not(necessary(not(a)))), implies(implies(X, not(not(necessary(not(a))))), necessary(not(a))))), true), true, is_a_theorem(implies(implies(not(not(necessary(not(a)))), implies(X, not(not(necessary(not(a)))))), implies(not(not(necessary(not(a)))), necessary(not(a))))), true), true, is_a_theorem(implies(not(not(necessary(not(a)))), necessary(not(a)))), true), true, is_a_theorem(implies(not(not(necessary(not(a)))), not(a))), true), true, is_a_theorem(implies(a, not(necessary(not(a))))), true), true, is_a_theorem(necessary(implies(a, not(necessary(not(a)))))), true), true, is_a_theorem(implies(necessary(a), necessary(not(necessary(not(a)))))), true), true, is_a_theorem(implies(necessary(a), not(necessary(not(a))))), true)
% 46.55/6.46 = { by axiom 8 (cn_35) R->L }
% 46.55/6.46 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(not(necessary(not(a)))), implies(implies(not(necessary(not(a))), not(implies(X, not(not(necessary(not(a))))))), implies(implies(X, not(not(necessary(not(a))))), necessary(not(a))))), implies(implies(not(not(necessary(not(a)))), implies(not(necessary(not(a))), not(implies(X, not(not(necessary(not(a)))))))), implies(not(not(necessary(not(a)))), implies(implies(X, not(not(necessary(not(a))))), necessary(not(a))))))), true, ifeq(is_a_theorem(implies(not(not(necessary(not(a)))), implies(implies(not(necessary(not(a))), not(implies(X, not(not(necessary(not(a))))))), implies(implies(X, not(not(necessary(not(a))))), necessary(not(a)))))), true, is_a_theorem(implies(implies(not(not(necessary(not(a)))), implies(not(necessary(not(a))), not(implies(X, not(not(necessary(not(a)))))))), implies(not(not(necessary(not(a)))), implies(implies(X, not(not(necessary(not(a))))), necessary(not(a)))))), true), true), true, is_a_theorem(implies(not(not(necessary(not(a)))), implies(implies(X, not(not(necessary(not(a))))), necessary(not(a))))), true), true, is_a_theorem(implies(implies(not(not(necessary(not(a)))), implies(X, not(not(necessary(not(a)))))), implies(not(not(necessary(not(a)))), necessary(not(a))))), true), true, is_a_theorem(implies(not(not(necessary(not(a)))), necessary(not(a)))), true), true, is_a_theorem(implies(not(not(necessary(not(a)))), not(a))), true), true, is_a_theorem(implies(a, not(necessary(not(a))))), true), true, is_a_theorem(necessary(implies(a, not(necessary(not(a)))))), true), true, is_a_theorem(implies(necessary(a), necessary(not(necessary(not(a)))))), true), true, is_a_theorem(implies(necessary(a), not(necessary(not(a))))), true)
% 46.55/6.46 = { by axiom 7 (condensed_detachment) }
% 46.55/6.46 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(not(not(necessary(not(a)))), implies(implies(X, not(not(necessary(not(a))))), necessary(not(a))))), true), true, is_a_theorem(implies(implies(not(not(necessary(not(a)))), implies(X, not(not(necessary(not(a)))))), implies(not(not(necessary(not(a)))), necessary(not(a))))), true), true, is_a_theorem(implies(not(not(necessary(not(a)))), necessary(not(a)))), true), true, is_a_theorem(implies(not(not(necessary(not(a)))), not(a))), true), true, is_a_theorem(implies(a, not(necessary(not(a))))), true), true, is_a_theorem(necessary(implies(a, not(necessary(not(a)))))), true), true, is_a_theorem(implies(necessary(a), necessary(not(necessary(not(a)))))), true), true, is_a_theorem(implies(necessary(a), not(necessary(not(a))))), true)
% 46.55/6.46 = { by axiom 2 (ifeq_axiom) }
% 46.55/6.46 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(not(not(necessary(not(a)))), implies(implies(X, not(not(necessary(not(a))))), necessary(not(a))))), true, is_a_theorem(implies(implies(not(not(necessary(not(a)))), implies(X, not(not(necessary(not(a)))))), implies(not(not(necessary(not(a)))), necessary(not(a))))), true), true, is_a_theorem(implies(not(not(necessary(not(a)))), necessary(not(a)))), true), true, is_a_theorem(implies(not(not(necessary(not(a)))), not(a))), true), true, is_a_theorem(implies(a, not(necessary(not(a))))), true), true, is_a_theorem(necessary(implies(a, not(necessary(not(a)))))), true), true, is_a_theorem(implies(necessary(a), necessary(not(necessary(not(a)))))), true), true, is_a_theorem(implies(necessary(a), not(necessary(not(a))))), true)
% 46.55/6.46 = { by axiom 2 (ifeq_axiom) R->L }
% 46.55/6.46 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(not(not(necessary(not(a)))), implies(implies(X, not(not(necessary(not(a))))), necessary(not(a))))), true, is_a_theorem(implies(implies(not(not(necessary(not(a)))), implies(X, not(not(necessary(not(a)))))), implies(not(not(necessary(not(a)))), necessary(not(a))))), true), true), true, is_a_theorem(implies(not(not(necessary(not(a)))), necessary(not(a)))), true), true, is_a_theorem(implies(not(not(necessary(not(a)))), not(a))), true), true, is_a_theorem(implies(a, not(necessary(not(a))))), true), true, is_a_theorem(necessary(implies(a, not(necessary(not(a)))))), true), true, is_a_theorem(implies(necessary(a), necessary(not(necessary(not(a)))))), true), true, is_a_theorem(implies(necessary(a), not(necessary(not(a))))), true)
% 46.55/6.46 = { by axiom 8 (cn_35) R->L }
% 46.55/6.46 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(not(necessary(not(a)))), implies(implies(X, not(not(necessary(not(a))))), necessary(not(a)))), implies(implies(not(not(necessary(not(a)))), implies(X, not(not(necessary(not(a)))))), implies(not(not(necessary(not(a)))), necessary(not(a)))))), true, ifeq(is_a_theorem(implies(not(not(necessary(not(a)))), implies(implies(X, not(not(necessary(not(a))))), necessary(not(a))))), true, is_a_theorem(implies(implies(not(not(necessary(not(a)))), implies(X, not(not(necessary(not(a)))))), implies(not(not(necessary(not(a)))), necessary(not(a))))), true), true), true, is_a_theorem(implies(not(not(necessary(not(a)))), necessary(not(a)))), true), true, is_a_theorem(implies(not(not(necessary(not(a)))), not(a))), true), true, is_a_theorem(implies(a, not(necessary(not(a))))), true), true, is_a_theorem(necessary(implies(a, not(necessary(not(a)))))), true), true, is_a_theorem(implies(necessary(a), necessary(not(necessary(not(a)))))), true), true, is_a_theorem(implies(necessary(a), not(necessary(not(a))))), true)
% 46.55/6.46 = { by axiom 7 (condensed_detachment) }
% 46.55/6.46 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(not(not(necessary(not(a)))), necessary(not(a)))), true), true, is_a_theorem(implies(not(not(necessary(not(a)))), not(a))), true), true, is_a_theorem(implies(a, not(necessary(not(a))))), true), true, is_a_theorem(necessary(implies(a, not(necessary(not(a)))))), true), true, is_a_theorem(implies(necessary(a), necessary(not(necessary(not(a)))))), true), true, is_a_theorem(implies(necessary(a), not(necessary(not(a))))), true)
% 46.55/6.46 = { by axiom 2 (ifeq_axiom) }
% 46.55/6.47 ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(not(not(necessary(not(a)))), necessary(not(a)))), true, is_a_theorem(implies(not(not(necessary(not(a)))), not(a))), true), true, is_a_theorem(implies(a, not(necessary(not(a))))), true), true, is_a_theorem(necessary(implies(a, not(necessary(not(a)))))), true), true, is_a_theorem(implies(necessary(a), necessary(not(necessary(not(a)))))), true), true, is_a_theorem(implies(necessary(a), not(necessary(not(a))))), true)
% 46.55/6.47 = { by axiom 2 (ifeq_axiom) R->L }
% 46.55/6.47 ifeq(ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(not(not(necessary(not(a)))), necessary(not(a)))), true, is_a_theorem(implies(not(not(necessary(not(a)))), not(a))), true), true), true, is_a_theorem(implies(a, not(necessary(not(a))))), true), true, is_a_theorem(necessary(implies(a, not(necessary(not(a)))))), true), true, is_a_theorem(implies(necessary(a), necessary(not(necessary(not(a)))))), true), true, is_a_theorem(implies(necessary(a), not(necessary(not(a))))), true)
% 46.55/6.47 = { by axiom 7 (condensed_detachment) R->L }
% 46.55/6.47 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(not(necessary(not(a)))), implies(necessary(not(a)), not(a))), implies(implies(not(not(necessary(not(a)))), necessary(not(a))), implies(not(not(necessary(not(a)))), not(a))))), true, ifeq(is_a_theorem(implies(not(not(necessary(not(a)))), implies(necessary(not(a)), not(a)))), true, is_a_theorem(implies(implies(not(not(necessary(not(a)))), necessary(not(a))), implies(not(not(necessary(not(a)))), not(a)))), true), true), true, ifeq(is_a_theorem(implies(not(not(necessary(not(a)))), necessary(not(a)))), true, is_a_theorem(implies(not(not(necessary(not(a)))), not(a))), true), true), true, is_a_theorem(implies(a, not(necessary(not(a))))), true), true, is_a_theorem(necessary(implies(a, not(necessary(not(a)))))), true), true, is_a_theorem(implies(necessary(a), necessary(not(necessary(not(a)))))), true), true, is_a_theorem(implies(necessary(a), not(necessary(not(a))))), true)
% 46.55/6.47 = { by axiom 2 (ifeq_axiom) R->L }
% 46.55/6.47 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(not(necessary(not(a)))), implies(necessary(not(a)), not(a))), implies(implies(not(not(necessary(not(a)))), necessary(not(a))), implies(not(not(necessary(not(a)))), not(a))))), true, ifeq(ifeq(true, true, is_a_theorem(implies(not(not(necessary(not(a)))), implies(necessary(not(a)), not(a)))), true), true, is_a_theorem(implies(implies(not(not(necessary(not(a)))), necessary(not(a))), implies(not(not(necessary(not(a)))), not(a)))), true), true), true, ifeq(is_a_theorem(implies(not(not(necessary(not(a)))), necessary(not(a)))), true, is_a_theorem(implies(not(not(necessary(not(a)))), not(a))), true), true), true, is_a_theorem(implies(a, not(necessary(not(a))))), true), true, is_a_theorem(necessary(implies(a, not(necessary(not(a)))))), true), true, is_a_theorem(implies(necessary(a), necessary(not(necessary(not(a)))))), true), true, is_a_theorem(implies(necessary(a), not(necessary(not(a))))), true)
% 46.55/6.47 = { by axiom 1 (necessitation2) R->L }
% 46.55/6.47 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(not(necessary(not(a)))), implies(necessary(not(a)), not(a))), implies(implies(not(not(necessary(not(a)))), necessary(not(a))), implies(not(not(necessary(not(a)))), not(a))))), true, ifeq(ifeq(is_a_theorem(implies(necessary(not(a)), not(a))), true, is_a_theorem(implies(not(not(necessary(not(a)))), implies(necessary(not(a)), not(a)))), true), true, is_a_theorem(implies(implies(not(not(necessary(not(a)))), necessary(not(a))), implies(not(not(necessary(not(a)))), not(a)))), true), true), true, ifeq(is_a_theorem(implies(not(not(necessary(not(a)))), necessary(not(a)))), true, is_a_theorem(implies(not(not(necessary(not(a)))), not(a))), true), true), true, is_a_theorem(implies(a, not(necessary(not(a))))), true), true, is_a_theorem(necessary(implies(a, not(necessary(not(a)))))), true), true, is_a_theorem(implies(necessary(a), necessary(not(necessary(not(a)))))), true), true, is_a_theorem(implies(necessary(a), not(necessary(not(a))))), true)
% 46.55/6.47 = { by axiom 2 (ifeq_axiom) R->L }
% 46.55/6.47 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(not(necessary(not(a)))), implies(necessary(not(a)), not(a))), implies(implies(not(not(necessary(not(a)))), necessary(not(a))), implies(not(not(necessary(not(a)))), not(a))))), true, ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(necessary(not(a)), not(a))), true, is_a_theorem(implies(not(not(necessary(not(a)))), implies(necessary(not(a)), not(a)))), true), true), true, is_a_theorem(implies(implies(not(not(necessary(not(a)))), necessary(not(a))), implies(not(not(necessary(not(a)))), not(a)))), true), true), true, ifeq(is_a_theorem(implies(not(not(necessary(not(a)))), necessary(not(a)))), true, is_a_theorem(implies(not(not(necessary(not(a)))), not(a))), true), true), true, is_a_theorem(implies(a, not(necessary(not(a))))), true), true, is_a_theorem(necessary(implies(a, not(necessary(not(a)))))), true), true, is_a_theorem(implies(necessary(a), necessary(not(necessary(not(a)))))), true), true, is_a_theorem(implies(necessary(a), not(necessary(not(a))))), true)
% 46.55/6.47 = { by axiom 3 (cn_18) R->L }
% 46.55/6.47 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(not(necessary(not(a)))), implies(necessary(not(a)), not(a))), implies(implies(not(not(necessary(not(a)))), necessary(not(a))), implies(not(not(necessary(not(a)))), not(a))))), true, ifeq(ifeq(is_a_theorem(implies(implies(necessary(not(a)), not(a)), implies(not(not(necessary(not(a)))), implies(necessary(not(a)), not(a))))), true, ifeq(is_a_theorem(implies(necessary(not(a)), not(a))), true, is_a_theorem(implies(not(not(necessary(not(a)))), implies(necessary(not(a)), not(a)))), true), true), true, is_a_theorem(implies(implies(not(not(necessary(not(a)))), necessary(not(a))), implies(not(not(necessary(not(a)))), not(a)))), true), true), true, ifeq(is_a_theorem(implies(not(not(necessary(not(a)))), necessary(not(a)))), true, is_a_theorem(implies(not(not(necessary(not(a)))), not(a))), true), true), true, is_a_theorem(implies(a, not(necessary(not(a))))), true), true, is_a_theorem(necessary(implies(a, not(necessary(not(a)))))), true), true, is_a_theorem(implies(necessary(a), necessary(not(necessary(not(a)))))), true), true, is_a_theorem(implies(necessary(a), not(necessary(not(a))))), true)
% 46.55/6.47 = { by axiom 7 (condensed_detachment) }
% 46.55/6.47 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(not(necessary(not(a)))), implies(necessary(not(a)), not(a))), implies(implies(not(not(necessary(not(a)))), necessary(not(a))), implies(not(not(necessary(not(a)))), not(a))))), true, ifeq(true, true, is_a_theorem(implies(implies(not(not(necessary(not(a)))), necessary(not(a))), implies(not(not(necessary(not(a)))), not(a)))), true), true), true, ifeq(is_a_theorem(implies(not(not(necessary(not(a)))), necessary(not(a)))), true, is_a_theorem(implies(not(not(necessary(not(a)))), not(a))), true), true), true, is_a_theorem(implies(a, not(necessary(not(a))))), true), true, is_a_theorem(necessary(implies(a, not(necessary(not(a)))))), true), true, is_a_theorem(implies(necessary(a), necessary(not(necessary(not(a)))))), true), true, is_a_theorem(implies(necessary(a), not(necessary(not(a))))), true)
% 46.55/6.47 = { by axiom 2 (ifeq_axiom) }
% 46.55/6.47 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(not(necessary(not(a)))), implies(necessary(not(a)), not(a))), implies(implies(not(not(necessary(not(a)))), necessary(not(a))), implies(not(not(necessary(not(a)))), not(a))))), true, is_a_theorem(implies(implies(not(not(necessary(not(a)))), necessary(not(a))), implies(not(not(necessary(not(a)))), not(a)))), true), true, ifeq(is_a_theorem(implies(not(not(necessary(not(a)))), necessary(not(a)))), true, is_a_theorem(implies(not(not(necessary(not(a)))), not(a))), true), true), true, is_a_theorem(implies(a, not(necessary(not(a))))), true), true, is_a_theorem(necessary(implies(a, not(necessary(not(a)))))), true), true, is_a_theorem(implies(necessary(a), necessary(not(necessary(not(a)))))), true), true, is_a_theorem(implies(necessary(a), not(necessary(not(a))))), true)
% 46.55/6.47 = { by axiom 8 (cn_35) }
% 46.55/6.47 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(not(not(necessary(not(a)))), necessary(not(a))), implies(not(not(necessary(not(a)))), not(a)))), true), true, ifeq(is_a_theorem(implies(not(not(necessary(not(a)))), necessary(not(a)))), true, is_a_theorem(implies(not(not(necessary(not(a)))), not(a))), true), true), true, is_a_theorem(implies(a, not(necessary(not(a))))), true), true, is_a_theorem(necessary(implies(a, not(necessary(not(a)))))), true), true, is_a_theorem(implies(necessary(a), necessary(not(necessary(not(a)))))), true), true, is_a_theorem(implies(necessary(a), not(necessary(not(a))))), true)
% 46.55/6.47 = { by axiom 2 (ifeq_axiom) }
% 46.55/6.47 ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(not(necessary(not(a)))), necessary(not(a))), implies(not(not(necessary(not(a)))), not(a)))), true, ifeq(is_a_theorem(implies(not(not(necessary(not(a)))), necessary(not(a)))), true, is_a_theorem(implies(not(not(necessary(not(a)))), not(a))), true), true), true, is_a_theorem(implies(a, not(necessary(not(a))))), true), true, is_a_theorem(necessary(implies(a, not(necessary(not(a)))))), true), true, is_a_theorem(implies(necessary(a), necessary(not(necessary(not(a)))))), true), true, is_a_theorem(implies(necessary(a), not(necessary(not(a))))), true)
% 46.55/6.48 = { by axiom 7 (condensed_detachment) }
% 46.55/6.48 ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(a, not(necessary(not(a))))), true), true, is_a_theorem(necessary(implies(a, not(necessary(not(a)))))), true), true, is_a_theorem(implies(necessary(a), necessary(not(necessary(not(a)))))), true), true, is_a_theorem(implies(necessary(a), not(necessary(not(a))))), true)
% 46.55/6.48 = { by axiom 2 (ifeq_axiom) }
% 46.55/6.48 ifeq(ifeq(ifeq(is_a_theorem(implies(a, not(necessary(not(a))))), true, is_a_theorem(necessary(implies(a, not(necessary(not(a)))))), true), true, is_a_theorem(implies(necessary(a), necessary(not(necessary(not(a)))))), true), true, is_a_theorem(implies(necessary(a), not(necessary(not(a))))), true)
% 46.55/6.48 = { by axiom 4 (axiom_of_necessitation) }
% 46.55/6.48 ifeq(ifeq(true, true, is_a_theorem(implies(necessary(a), necessary(not(necessary(not(a)))))), true), true, is_a_theorem(implies(necessary(a), not(necessary(not(a))))), true)
% 46.55/6.48 = { by axiom 2 (ifeq_axiom) }
% 46.55/6.48 ifeq(is_a_theorem(implies(necessary(a), necessary(not(necessary(not(a)))))), true, is_a_theorem(implies(necessary(a), not(necessary(not(a))))), true)
% 46.55/6.48 = { by axiom 2 (ifeq_axiom) R->L }
% 46.55/6.48 ifeq(true, true, ifeq(is_a_theorem(implies(necessary(a), necessary(not(necessary(not(a)))))), true, is_a_theorem(implies(necessary(a), not(necessary(not(a))))), true), true)
% 46.55/6.48 = { by axiom 7 (condensed_detachment) R->L }
% 46.55/6.48 ifeq(ifeq(is_a_theorem(implies(implies(necessary(a), implies(necessary(not(necessary(not(a)))), not(necessary(not(a))))), implies(implies(necessary(a), necessary(not(necessary(not(a))))), implies(necessary(a), not(necessary(not(a))))))), true, ifeq(is_a_theorem(implies(necessary(a), implies(necessary(not(necessary(not(a)))), not(necessary(not(a)))))), true, is_a_theorem(implies(implies(necessary(a), necessary(not(necessary(not(a))))), implies(necessary(a), not(necessary(not(a)))))), true), true), true, ifeq(is_a_theorem(implies(necessary(a), necessary(not(necessary(not(a)))))), true, is_a_theorem(implies(necessary(a), not(necessary(not(a))))), true), true)
% 46.55/6.48 = { by axiom 2 (ifeq_axiom) R->L }
% 46.55/6.48 ifeq(ifeq(is_a_theorem(implies(implies(necessary(a), implies(necessary(not(necessary(not(a)))), not(necessary(not(a))))), implies(implies(necessary(a), necessary(not(necessary(not(a))))), implies(necessary(a), not(necessary(not(a))))))), true, ifeq(ifeq(true, true, is_a_theorem(implies(necessary(a), implies(necessary(not(necessary(not(a)))), not(necessary(not(a)))))), true), true, is_a_theorem(implies(implies(necessary(a), necessary(not(necessary(not(a))))), implies(necessary(a), not(necessary(not(a)))))), true), true), true, ifeq(is_a_theorem(implies(necessary(a), necessary(not(necessary(not(a)))))), true, is_a_theorem(implies(necessary(a), not(necessary(not(a))))), true), true)
% 46.55/6.48 = { by axiom 1 (necessitation2) R->L }
% 46.55/6.48 ifeq(ifeq(is_a_theorem(implies(implies(necessary(a), implies(necessary(not(necessary(not(a)))), not(necessary(not(a))))), implies(implies(necessary(a), necessary(not(necessary(not(a))))), implies(necessary(a), not(necessary(not(a))))))), true, ifeq(ifeq(is_a_theorem(implies(necessary(not(necessary(not(a)))), not(necessary(not(a))))), true, is_a_theorem(implies(necessary(a), implies(necessary(not(necessary(not(a)))), not(necessary(not(a)))))), true), true, is_a_theorem(implies(implies(necessary(a), necessary(not(necessary(not(a))))), implies(necessary(a), not(necessary(not(a)))))), true), true), true, ifeq(is_a_theorem(implies(necessary(a), necessary(not(necessary(not(a)))))), true, is_a_theorem(implies(necessary(a), not(necessary(not(a))))), true), true)
% 46.55/6.48 = { by axiom 2 (ifeq_axiom) R->L }
% 46.55/6.48 ifeq(ifeq(is_a_theorem(implies(implies(necessary(a), implies(necessary(not(necessary(not(a)))), not(necessary(not(a))))), implies(implies(necessary(a), necessary(not(necessary(not(a))))), implies(necessary(a), not(necessary(not(a))))))), true, ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(necessary(not(necessary(not(a)))), not(necessary(not(a))))), true, is_a_theorem(implies(necessary(a), implies(necessary(not(necessary(not(a)))), not(necessary(not(a)))))), true), true), true, is_a_theorem(implies(implies(necessary(a), necessary(not(necessary(not(a))))), implies(necessary(a), not(necessary(not(a)))))), true), true), true, ifeq(is_a_theorem(implies(necessary(a), necessary(not(necessary(not(a)))))), true, is_a_theorem(implies(necessary(a), not(necessary(not(a))))), true), true)
% 46.55/6.48 = { by axiom 3 (cn_18) R->L }
% 46.55/6.48 ifeq(ifeq(is_a_theorem(implies(implies(necessary(a), implies(necessary(not(necessary(not(a)))), not(necessary(not(a))))), implies(implies(necessary(a), necessary(not(necessary(not(a))))), implies(necessary(a), not(necessary(not(a))))))), true, ifeq(ifeq(is_a_theorem(implies(implies(necessary(not(necessary(not(a)))), not(necessary(not(a)))), implies(necessary(a), implies(necessary(not(necessary(not(a)))), not(necessary(not(a))))))), true, ifeq(is_a_theorem(implies(necessary(not(necessary(not(a)))), not(necessary(not(a))))), true, is_a_theorem(implies(necessary(a), implies(necessary(not(necessary(not(a)))), not(necessary(not(a)))))), true), true), true, is_a_theorem(implies(implies(necessary(a), necessary(not(necessary(not(a))))), implies(necessary(a), not(necessary(not(a)))))), true), true), true, ifeq(is_a_theorem(implies(necessary(a), necessary(not(necessary(not(a)))))), true, is_a_theorem(implies(necessary(a), not(necessary(not(a))))), true), true)
% 46.55/6.48 = { by axiom 7 (condensed_detachment) }
% 46.55/6.48 ifeq(ifeq(is_a_theorem(implies(implies(necessary(a), implies(necessary(not(necessary(not(a)))), not(necessary(not(a))))), implies(implies(necessary(a), necessary(not(necessary(not(a))))), implies(necessary(a), not(necessary(not(a))))))), true, ifeq(true, true, is_a_theorem(implies(implies(necessary(a), necessary(not(necessary(not(a))))), implies(necessary(a), not(necessary(not(a)))))), true), true), true, ifeq(is_a_theorem(implies(necessary(a), necessary(not(necessary(not(a)))))), true, is_a_theorem(implies(necessary(a), not(necessary(not(a))))), true), true)
% 46.55/6.48 = { by axiom 2 (ifeq_axiom) }
% 46.55/6.48 ifeq(ifeq(is_a_theorem(implies(implies(necessary(a), implies(necessary(not(necessary(not(a)))), not(necessary(not(a))))), implies(implies(necessary(a), necessary(not(necessary(not(a))))), implies(necessary(a), not(necessary(not(a))))))), true, is_a_theorem(implies(implies(necessary(a), necessary(not(necessary(not(a))))), implies(necessary(a), not(necessary(not(a)))))), true), true, ifeq(is_a_theorem(implies(necessary(a), necessary(not(necessary(not(a)))))), true, is_a_theorem(implies(necessary(a), not(necessary(not(a))))), true), true)
% 46.55/6.48 = { by axiom 8 (cn_35) }
% 46.55/6.48 ifeq(ifeq(true, true, is_a_theorem(implies(implies(necessary(a), necessary(not(necessary(not(a))))), implies(necessary(a), not(necessary(not(a)))))), true), true, ifeq(is_a_theorem(implies(necessary(a), necessary(not(necessary(not(a)))))), true, is_a_theorem(implies(necessary(a), not(necessary(not(a))))), true), true)
% 46.55/6.48 = { by axiom 2 (ifeq_axiom) }
% 46.55/6.48 ifeq(is_a_theorem(implies(implies(necessary(a), necessary(not(necessary(not(a))))), implies(necessary(a), not(necessary(not(a)))))), true, ifeq(is_a_theorem(implies(necessary(a), necessary(not(necessary(not(a)))))), true, is_a_theorem(implies(necessary(a), not(necessary(not(a))))), true), true)
% 46.55/6.48 = { by axiom 7 (condensed_detachment) }
% 46.55/6.48 true
% 46.55/6.48 % SZS output end Proof
% 46.55/6.48
% 46.55/6.48 RESULT: Unsatisfiable (the axioms are contradictory).
%------------------------------------------------------------------------------