0.07/0.15	% Problem    : theBenchmark.p : TPTP v0.0.0. Released v0.0.0.
0.07/0.16	% Command    : twee %s --tstp --casc --quiet --explain-encoding --conditional-encoding if --smaller --drop-non-horn
0.12/0.37	% Computer   : n013.cluster.edu
0.12/0.37	% Model      : x86_64 x86_64
0.12/0.37	% CPU        : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
0.12/0.37	% Memory     : 8042.1875MB
0.12/0.37	% OS         : Linux 3.10.0-693.el7.x86_64
0.12/0.37	% CPULimit   : 180
0.12/0.37	% DateTime   : Thu Aug 29 14:12:46 EDT 2019
0.12/0.37	% CPUTime    : 
131.05/131.33	% SZS status Unsatisfiable
131.05/131.33	
131.05/131.34	% SZS output start Proof
131.05/131.34	Take the following subset of the input axioms:
131.16/131.47	  fof(cn_1, axiom, ![X, Y, Z]: is_a_theorem(implies(implies(X, Y), implies(implies(Y, Z), implies(X, Z))))=true).
131.16/131.47	  fof(cn_2, axiom, ![X]: is_a_theorem(implies(implies(not(X), X), X))=true).
131.16/131.47	  fof(cn_3, axiom, ![X, Y]: is_a_theorem(implies(X, implies(not(X), Y)))=true).
131.16/131.47	  fof(condensed_detachment, axiom, ![X, Y]: ifeq(is_a_theorem(implies(X, Y)), true, ifeq(is_a_theorem(X), true, is_a_theorem(Y), true), true)=true).
131.16/131.47	  fof(ifeq_axiom, axiom, ![B, A, C]: B=ifeq(A, A, B, C)).
131.16/131.47	  fof(prove_cn_54, negated_conjecture, is_a_theorem(implies(implies(a, b), implies(implies(not(a), b), b)))!=true).
131.16/131.47	
131.16/131.47	Now clausify the problem and encode Horn clauses using encoding 3 of
131.16/131.47	http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
131.16/131.47	We repeatedly replace C & s=t => u=v by the two clauses:
131.16/131.47	  fresh(y, y, x1...xn) = u
131.16/131.47	  C => fresh(s, t, x1...xn) = v
131.16/131.47	where fresh is a fresh function symbol and x1..xn are the free
131.16/131.47	variables of u and v.
131.16/131.47	A predicate p(X) is encoded as p(X)=true (this is sound, because the
131.16/131.47	input problem has no model of domain size 1).
131.16/131.47	
131.16/131.47	The encoding turns the above axioms into the following unit equations and goals:
131.16/131.47	
131.16/131.47	Axiom 1 (cn_1): is_a_theorem(implies(implies(X, Y), implies(implies(Y, Z), implies(X, Z)))) = true.
131.16/131.47	Axiom 2 (cn_3): is_a_theorem(implies(X, implies(not(X), Y))) = true.
131.16/131.47	Axiom 3 (condensed_detachment): ifeq(is_a_theorem(implies(X, Y)), true, ifeq(is_a_theorem(X), true, is_a_theorem(Y), true), true) = true.
131.16/131.47	Axiom 4 (cn_2): is_a_theorem(implies(implies(not(X), X), X)) = true.
131.16/131.48	Axiom 5 (ifeq_axiom): X = ifeq(Y, Y, X, Z).
131.16/131.48	
131.16/131.48	Lemma 6: ifeq(is_a_theorem(implies(X, Y)), true, is_a_theorem(implies(implies(Y, Z), implies(X, Z))), true) = true.
131.16/131.48	Proof:
131.16/131.48	  ifeq(is_a_theorem(implies(X, Y)), true, is_a_theorem(implies(implies(Y, Z), implies(X, Z))), true)
131.16/131.48	= { by axiom 5 (ifeq_axiom) }
131.16/131.48	  ifeq(true, true, ifeq(is_a_theorem(implies(X, Y)), true, is_a_theorem(implies(implies(Y, Z), implies(X, Z))), true), true)
131.16/131.48	= { by axiom 1 (cn_1) }
131.16/131.48	  ifeq(is_a_theorem(implies(implies(X, Y), implies(implies(Y, Z), implies(X, Z)))), true, ifeq(is_a_theorem(implies(X, Y)), true, is_a_theorem(implies(implies(Y, Z), implies(X, Z))), true), true)
131.16/131.48	= { by axiom 3 (condensed_detachment) }
131.16/131.48	  true
131.16/131.48	
131.16/131.48	Lemma 7: is_a_theorem(implies(implies(implies(implies(not(X), X), Y), Z), implies(implies(X, Y), Z))) = true.
131.16/131.48	Proof:
131.16/131.48	  is_a_theorem(implies(implies(implies(implies(not(X), X), Y), Z), implies(implies(X, Y), Z)))
131.16/131.48	= { by axiom 5 (ifeq_axiom) }
131.16/131.48	  ifeq(true, true, is_a_theorem(implies(implies(implies(implies(not(X), X), Y), Z), implies(implies(X, Y), Z))), true)
131.16/131.48	= { by axiom 3 (condensed_detachment) }
131.16/131.48	  ifeq(ifeq(is_a_theorem(implies(implies(implies(not(X), X), X), implies(implies(X, Y), implies(implies(not(X), X), Y)))), true, ifeq(is_a_theorem(implies(implies(not(X), X), X)), true, is_a_theorem(implies(implies(X, Y), implies(implies(not(X), X), Y))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(X), X), Y), Z), implies(implies(X, Y), Z))), true)
131.16/131.48	= { by axiom 1 (cn_1) }
131.16/131.48	  ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(not(X), X), X)), true, is_a_theorem(implies(implies(X, Y), implies(implies(not(X), X), Y))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(X), X), Y), Z), implies(implies(X, Y), Z))), true)
131.16/131.48	= { by axiom 4 (cn_2) }
131.16/131.48	  ifeq(ifeq(true, true, ifeq(true, true, is_a_theorem(implies(implies(X, Y), implies(implies(not(X), X), Y))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(X), X), Y), Z), implies(implies(X, Y), Z))), true)
131.16/131.48	= { by axiom 5 (ifeq_axiom) }
131.16/131.48	  ifeq(ifeq(true, true, is_a_theorem(implies(implies(X, Y), implies(implies(not(X), X), Y))), true), true, is_a_theorem(implies(implies(implies(implies(not(X), X), Y), Z), implies(implies(X, Y), Z))), true)
131.16/131.48	= { by axiom 5 (ifeq_axiom) }
131.16/131.48	  ifeq(is_a_theorem(implies(implies(X, Y), implies(implies(not(X), X), Y))), true, is_a_theorem(implies(implies(implies(implies(not(X), X), Y), Z), implies(implies(X, Y), Z))), true)
131.16/131.48	= { by lemma 6 }
131.16/131.48	  true
131.16/131.48	
131.16/131.48	Lemma 8: ifeq(is_a_theorem(implies(implies(implies(X, Y), implies(Z, Y)), W)), true, is_a_theorem(implies(implies(Z, X), W)), true) = true.
131.16/131.48	Proof:
131.16/131.48	  ifeq(is_a_theorem(implies(implies(implies(X, Y), implies(Z, Y)), W)), true, is_a_theorem(implies(implies(Z, X), W)), true)
131.16/131.48	= { by axiom 5 (ifeq_axiom) }
131.16/131.48	  ifeq(true, true, ifeq(is_a_theorem(implies(implies(implies(X, Y), implies(Z, Y)), W)), true, is_a_theorem(implies(implies(Z, X), W)), true), true)
131.16/131.48	= { by lemma 6 }
131.16/131.48	  ifeq(ifeq(is_a_theorem(implies(implies(Z, X), implies(implies(X, Y), implies(Z, Y)))), true, is_a_theorem(implies(implies(implies(implies(X, Y), implies(Z, Y)), W), implies(implies(Z, X), W))), true), true, ifeq(is_a_theorem(implies(implies(implies(X, Y), implies(Z, Y)), W)), true, is_a_theorem(implies(implies(Z, X), W)), true), true)
131.16/131.48	= { by axiom 1 (cn_1) }
131.16/131.48	  ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(implies(X, Y), implies(Z, Y)), W), implies(implies(Z, X), W))), true), true, ifeq(is_a_theorem(implies(implies(implies(X, Y), implies(Z, Y)), W)), true, is_a_theorem(implies(implies(Z, X), W)), true), true)
131.16/131.48	= { by axiom 5 (ifeq_axiom) }
131.16/131.48	  ifeq(is_a_theorem(implies(implies(implies(implies(X, Y), implies(Z, Y)), W), implies(implies(Z, X), W))), true, ifeq(is_a_theorem(implies(implies(implies(X, Y), implies(Z, Y)), W)), true, is_a_theorem(implies(implies(Z, X), W)), true), true)
131.16/131.48	= { by axiom 3 (condensed_detachment) }
131.16/131.48	  true
131.16/131.48	
131.16/131.48	Lemma 9: is_a_theorem(implies(implies(X, implies(not(Y), Y)), implies(implies(Y, Z), implies(X, Z)))) = true.
131.16/131.48	Proof:
131.16/131.48	  is_a_theorem(implies(implies(X, implies(not(Y), Y)), implies(implies(Y, Z), implies(X, Z))))
131.16/131.48	= { by axiom 5 (ifeq_axiom) }
131.16/131.48	  ifeq(true, true, is_a_theorem(implies(implies(X, implies(not(Y), Y)), implies(implies(Y, Z), implies(X, Z)))), true)
131.16/131.48	= { by lemma 7 }
131.16/131.48	  ifeq(is_a_theorem(implies(implies(implies(implies(not(Y), Y), Z), implies(X, Z)), implies(implies(Y, Z), implies(X, Z)))), true, is_a_theorem(implies(implies(X, implies(not(Y), Y)), implies(implies(Y, Z), implies(X, Z)))), true)
131.16/131.48	= { by lemma 8 }
131.16/131.48	  true
131.16/131.48	
131.16/131.48	Lemma 10: is_a_theorem(implies(implies(implies(not(X), Y), Z), implies(X, Z))) = true.
131.16/131.48	Proof:
131.16/131.48	  is_a_theorem(implies(implies(implies(not(X), Y), Z), implies(X, Z)))
131.16/131.48	= { by axiom 5 (ifeq_axiom) }
131.16/131.48	  ifeq(true, true, is_a_theorem(implies(implies(implies(not(X), Y), Z), implies(X, Z))), true)
131.16/131.48	= { by axiom 1 (cn_1) }
131.16/131.48	  ifeq(is_a_theorem(implies(implies(X, implies(not(X), Y)), implies(implies(implies(not(X), Y), Z), implies(X, Z)))), true, is_a_theorem(implies(implies(implies(not(X), Y), Z), implies(X, Z))), true)
131.16/131.48	= { by axiom 5 (ifeq_axiom) }
131.16/131.48	  ifeq(is_a_theorem(implies(implies(X, implies(not(X), Y)), implies(implies(implies(not(X), Y), Z), implies(X, Z)))), true, ifeq(true, true, is_a_theorem(implies(implies(implies(not(X), Y), Z), implies(X, Z))), true), true)
131.16/131.48	= { by axiom 2 (cn_3) }
131.16/131.48	  ifeq(is_a_theorem(implies(implies(X, implies(not(X), Y)), implies(implies(implies(not(X), Y), Z), implies(X, Z)))), true, ifeq(is_a_theorem(implies(X, implies(not(X), Y))), true, is_a_theorem(implies(implies(implies(not(X), Y), Z), implies(X, Z))), true), true)
131.16/131.48	= { by axiom 3 (condensed_detachment) }
131.16/131.48	  true
131.16/131.48	
131.16/131.48	Lemma 11: ifeq(is_a_theorem(implies(implies(X, Y), Z)), true, is_a_theorem(implies(implies(implies(not(X), W), Y), Z)), true) = true.
131.16/131.48	Proof:
131.16/131.48	  ifeq(is_a_theorem(implies(implies(X, Y), Z)), true, is_a_theorem(implies(implies(implies(not(X), W), Y), Z)), true)
131.16/131.48	= { by axiom 5 (ifeq_axiom) }
131.16/131.48	  ifeq(true, true, ifeq(is_a_theorem(implies(implies(X, Y), Z)), true, is_a_theorem(implies(implies(implies(not(X), W), Y), Z)), true), true)
131.16/131.48	= { by lemma 6 }
131.16/131.48	  ifeq(ifeq(is_a_theorem(implies(implies(implies(not(X), W), Y), implies(X, Y))), true, is_a_theorem(implies(implies(implies(X, Y), Z), implies(implies(implies(not(X), W), Y), Z))), true), true, ifeq(is_a_theorem(implies(implies(X, Y), Z)), true, is_a_theorem(implies(implies(implies(not(X), W), Y), Z)), true), true)
131.16/131.48	= { by lemma 10 }
131.16/131.48	  ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(X, Y), Z), implies(implies(implies(not(X), W), Y), Z))), true), true, ifeq(is_a_theorem(implies(implies(X, Y), Z)), true, is_a_theorem(implies(implies(implies(not(X), W), Y), Z)), true), true)
131.16/131.48	= { by axiom 5 (ifeq_axiom) }
131.16/131.48	  ifeq(is_a_theorem(implies(implies(implies(X, Y), Z), implies(implies(implies(not(X), W), Y), Z))), true, ifeq(is_a_theorem(implies(implies(X, Y), Z)), true, is_a_theorem(implies(implies(implies(not(X), W), Y), Z)), true), true)
131.16/131.48	= { by axiom 3 (condensed_detachment) }
131.16/131.48	  true
131.16/131.48	
131.16/131.48	Lemma 12: ifeq(is_a_theorem(implies(implies(not(X), Y), Z)), true, is_a_theorem(implies(X, Z)), true) = true.
131.16/131.48	Proof:
131.16/131.48	  ifeq(is_a_theorem(implies(implies(not(X), Y), Z)), true, is_a_theorem(implies(X, Z)), true)
131.16/131.48	= { by axiom 5 (ifeq_axiom) }
131.16/131.48	  ifeq(true, true, ifeq(is_a_theorem(implies(implies(not(X), Y), Z)), true, is_a_theorem(implies(X, Z)), true), true)
131.16/131.48	= { by lemma 10 }
131.16/131.48	  ifeq(is_a_theorem(implies(implies(implies(not(X), Y), Z), implies(X, Z))), true, ifeq(is_a_theorem(implies(implies(not(X), Y), Z)), true, is_a_theorem(implies(X, Z)), true), true)
131.16/131.48	= { by axiom 3 (condensed_detachment) }
131.16/131.48	  true
131.16/131.48	
131.16/131.48	Lemma 13: is_a_theorem(implies(X, implies(implies(X, Y), implies(Z, Y)))) = true.
131.16/131.48	Proof:
131.16/131.48	  is_a_theorem(implies(X, implies(implies(X, Y), implies(Z, Y))))
131.16/131.48	= { by axiom 5 (ifeq_axiom) }
131.16/131.48	  ifeq(true, true, is_a_theorem(implies(X, implies(implies(X, Y), implies(Z, Y)))), true)
131.16/131.48	= { by lemma 8 }
131.16/131.48	  ifeq(ifeq(is_a_theorem(implies(implies(implies(not(Z), X), implies(not(X), X)), implies(implies(X, Y), implies(Z, Y)))), true, is_a_theorem(implies(implies(not(X), not(Z)), implies(implies(X, Y), implies(Z, Y)))), true), true, is_a_theorem(implies(X, implies(implies(X, Y), implies(Z, Y)))), true)
131.16/131.48	= { by axiom 5 (ifeq_axiom) }
131.16/131.48	  ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(not(Z), X), implies(not(X), X)), implies(implies(X, Y), implies(Z, Y)))), true), true, is_a_theorem(implies(implies(not(X), not(Z)), implies(implies(X, Y), implies(Z, Y)))), true), true, is_a_theorem(implies(X, implies(implies(X, Y), implies(Z, Y)))), true)
131.16/131.48	= { by lemma 9 }
131.16/131.48	  ifeq(ifeq(ifeq(is_a_theorem(implies(implies(Z, implies(not(X), X)), implies(implies(X, Y), implies(Z, Y)))), true, is_a_theorem(implies(implies(implies(not(Z), X), implies(not(X), X)), implies(implies(X, Y), implies(Z, Y)))), true), true, is_a_theorem(implies(implies(not(X), not(Z)), implies(implies(X, Y), implies(Z, Y)))), true), true, is_a_theorem(implies(X, implies(implies(X, Y), implies(Z, Y)))), true)
131.16/131.48	= { by lemma 11 }
131.16/131.48	  ifeq(ifeq(true, true, is_a_theorem(implies(implies(not(X), not(Z)), implies(implies(X, Y), implies(Z, Y)))), true), true, is_a_theorem(implies(X, implies(implies(X, Y), implies(Z, Y)))), true)
131.16/131.48	= { by axiom 5 (ifeq_axiom) }
131.16/131.48	  ifeq(is_a_theorem(implies(implies(not(X), not(Z)), implies(implies(X, Y), implies(Z, Y)))), true, is_a_theorem(implies(X, implies(implies(X, Y), implies(Z, Y)))), true)
131.16/131.48	= { by lemma 12 }
131.16/131.48	  true
131.16/131.48	
131.16/131.48	Lemma 14: ifeq(is_a_theorem(implies(implies(X, X), Y)), true, is_a_theorem(Y), true) = true.
131.16/131.48	Proof:
131.16/131.48	  ifeq(is_a_theorem(implies(implies(X, X), Y)), true, is_a_theorem(Y), true)
131.16/131.48	= { by axiom 5 (ifeq_axiom) }
131.16/131.48	  ifeq(is_a_theorem(implies(implies(X, X), Y)), true, ifeq(true, true, is_a_theorem(Y), true), true)
131.16/131.48	= { by lemma 12 }
131.16/131.48	  ifeq(is_a_theorem(implies(implies(X, X), Y)), true, ifeq(ifeq(is_a_theorem(implies(implies(not(X), X), X)), true, is_a_theorem(implies(X, X)), true), true, is_a_theorem(Y), true), true)
131.16/131.48	= { by axiom 4 (cn_2) }
131.16/131.48	  ifeq(is_a_theorem(implies(implies(X, X), Y)), true, ifeq(ifeq(true, true, is_a_theorem(implies(X, X)), true), true, is_a_theorem(Y), true), true)
131.16/131.48	= { by axiom 5 (ifeq_axiom) }
131.16/131.48	  ifeq(is_a_theorem(implies(implies(X, X), Y)), true, ifeq(is_a_theorem(implies(X, X)), true, is_a_theorem(Y), true), true)
131.16/131.48	= { by axiom 3 (condensed_detachment) }
131.16/131.49	  true
131.16/131.49	
131.16/131.49	Lemma 15: is_a_theorem(implies(implies(implies(implies(not(X), X), X), Y), implies(Z, Y))) = true.
131.16/131.49	Proof:
131.16/131.49	  is_a_theorem(implies(implies(implies(implies(not(X), X), X), Y), implies(Z, Y)))
131.16/131.49	= { by axiom 5 (ifeq_axiom) }
131.16/131.49	  ifeq(true, true, is_a_theorem(implies(implies(implies(implies(not(X), X), X), Y), implies(Z, Y))), true)
131.16/131.49	= { by axiom 3 (condensed_detachment) }
131.16/131.49	  ifeq(ifeq(is_a_theorem(implies(implies(not(implies(implies(not(X), X), X)), not(Z)), implies(implies(implies(not(implies(implies(not(X), X), X)), implies(implies(not(X), X), X)), Y), implies(Z, Y)))), true, ifeq(is_a_theorem(implies(not(implies(implies(not(X), X), X)), not(Z))), true, is_a_theorem(implies(implies(implies(not(implies(implies(not(X), X), X)), implies(implies(not(X), X), X)), Y), implies(Z, Y))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(X), X), X), Y), implies(Z, Y))), true)
131.16/131.49	= { by axiom 5 (ifeq_axiom) }
131.16/131.49	  ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(not(implies(implies(not(X), X), X)), not(Z)), implies(implies(implies(not(implies(implies(not(X), X), X)), implies(implies(not(X), X), X)), Y), implies(Z, Y)))), true), true, ifeq(is_a_theorem(implies(not(implies(implies(not(X), X), X)), not(Z))), true, is_a_theorem(implies(implies(implies(not(implies(implies(not(X), X), X)), implies(implies(not(X), X), X)), Y), implies(Z, Y))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(X), X), X), Y), implies(Z, Y))), true)
131.16/131.49	= { by lemma 11 }
131.16/131.49	  ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(Z, implies(not(implies(implies(not(X), X), X)), implies(implies(not(X), X), X))), implies(implies(implies(not(implies(implies(not(X), X), X)), implies(implies(not(X), X), X)), Y), implies(Z, Y)))), true, is_a_theorem(implies(implies(implies(not(Z), implies(implies(not(X), X), X)), implies(not(implies(implies(not(X), X), X)), implies(implies(not(X), X), X))), implies(implies(implies(not(implies(implies(not(X), X), X)), implies(implies(not(X), X), X)), Y), implies(Z, Y)))), true), true, is_a_theorem(implies(implies(not(implies(implies(not(X), X), X)), not(Z)), implies(implies(implies(not(implies(implies(not(X), X), X)), implies(implies(not(X), X), X)), Y), implies(Z, Y)))), true), true, ifeq(is_a_theorem(implies(not(implies(implies(not(X), X), X)), not(Z))), true, is_a_theorem(implies(implies(implies(not(implies(implies(not(X), X), X)), implies(implies(not(X), X), X)), Y), implies(Z, Y))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(X), X), X), Y), implies(Z, Y))), true)
131.16/131.49	= { by axiom 1 (cn_1) }
131.16/131.49	  ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(not(Z), implies(implies(not(X), X), X)), implies(not(implies(implies(not(X), X), X)), implies(implies(not(X), X), X))), implies(implies(implies(not(implies(implies(not(X), X), X)), implies(implies(not(X), X), X)), Y), implies(Z, Y)))), true), true, is_a_theorem(implies(implies(not(implies(implies(not(X), X), X)), not(Z)), implies(implies(implies(not(implies(implies(not(X), X), X)), implies(implies(not(X), X), X)), Y), implies(Z, Y)))), true), true, ifeq(is_a_theorem(implies(not(implies(implies(not(X), X), X)), not(Z))), true, is_a_theorem(implies(implies(implies(not(implies(implies(not(X), X), X)), implies(implies(not(X), X), X)), Y), implies(Z, Y))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(X), X), X), Y), implies(Z, Y))), true)
131.16/131.49	= { by axiom 5 (ifeq_axiom) }
131.16/131.49	  ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(not(Z), implies(implies(not(X), X), X)), implies(not(implies(implies(not(X), X), X)), implies(implies(not(X), X), X))), implies(implies(implies(not(implies(implies(not(X), X), X)), implies(implies(not(X), X), X)), Y), implies(Z, Y)))), true, is_a_theorem(implies(implies(not(implies(implies(not(X), X), X)), not(Z)), implies(implies(implies(not(implies(implies(not(X), X), X)), implies(implies(not(X), X), X)), Y), implies(Z, Y)))), true), true, ifeq(is_a_theorem(implies(not(implies(implies(not(X), X), X)), not(Z))), true, is_a_theorem(implies(implies(implies(not(implies(implies(not(X), X), X)), implies(implies(not(X), X), X)), Y), implies(Z, Y))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(X), X), X), Y), implies(Z, Y))), true)
131.16/131.49	= { by lemma 8 }
131.16/131.49	  ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(not(implies(implies(not(X), X), X)), not(Z))), true, is_a_theorem(implies(implies(implies(not(implies(implies(not(X), X), X)), implies(implies(not(X), X), X)), Y), implies(Z, Y))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(X), X), X), Y), implies(Z, Y))), true)
131.16/131.49	= { by axiom 5 (ifeq_axiom) }
131.16/131.49	  ifeq(ifeq(true, true, ifeq(ifeq(true, true, is_a_theorem(implies(not(implies(implies(not(X), X), X)), not(Z))), true), true, is_a_theorem(implies(implies(implies(not(implies(implies(not(X), X), X)), implies(implies(not(X), X), X)), Y), implies(Z, Y))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(X), X), X), Y), implies(Z, Y))), true)
131.16/131.49	= { by axiom 4 (cn_2) }
131.16/131.49	  ifeq(ifeq(true, true, ifeq(ifeq(is_a_theorem(implies(implies(not(X), X), X)), true, is_a_theorem(implies(not(implies(implies(not(X), X), X)), not(Z))), true), true, is_a_theorem(implies(implies(implies(not(implies(implies(not(X), X), X)), implies(implies(not(X), X), X)), Y), implies(Z, Y))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(X), X), X), Y), implies(Z, Y))), true)
131.16/131.49	= { by axiom 5 (ifeq_axiom) }
131.16/131.49	  ifeq(ifeq(true, true, ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(not(X), X), X)), true, is_a_theorem(implies(not(implies(implies(not(X), X), X)), not(Z))), true), true), true, is_a_theorem(implies(implies(implies(not(implies(implies(not(X), X), X)), implies(implies(not(X), X), X)), Y), implies(Z, Y))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(X), X), X), Y), implies(Z, Y))), true)
131.16/131.49	= { by axiom 2 (cn_3) }
131.16/131.49	  ifeq(ifeq(true, true, ifeq(ifeq(is_a_theorem(implies(implies(implies(not(X), X), X), implies(not(implies(implies(not(X), X), X)), not(Z)))), true, ifeq(is_a_theorem(implies(implies(not(X), X), X)), true, is_a_theorem(implies(not(implies(implies(not(X), X), X)), not(Z))), true), true), true, is_a_theorem(implies(implies(implies(not(implies(implies(not(X), X), X)), implies(implies(not(X), X), X)), Y), implies(Z, Y))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(X), X), X), Y), implies(Z, Y))), true)
131.16/131.49	= { by axiom 3 (condensed_detachment) }
131.16/131.49	  ifeq(ifeq(true, true, ifeq(true, true, is_a_theorem(implies(implies(implies(not(implies(implies(not(X), X), X)), implies(implies(not(X), X), X)), Y), implies(Z, Y))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(X), X), X), Y), implies(Z, Y))), true)
131.16/131.49	= { by axiom 5 (ifeq_axiom) }
131.16/131.49	  ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(not(implies(implies(not(X), X), X)), implies(implies(not(X), X), X)), Y), implies(Z, Y))), true), true, is_a_theorem(implies(implies(implies(implies(not(X), X), X), Y), implies(Z, Y))), true)
131.16/131.49	= { by axiom 5 (ifeq_axiom) }
131.16/131.49	  ifeq(is_a_theorem(implies(implies(implies(not(implies(implies(not(X), X), X)), implies(implies(not(X), X), X)), Y), implies(Z, Y))), true, is_a_theorem(implies(implies(implies(implies(not(X), X), X), Y), implies(Z, Y))), true)
131.16/131.49	= { by axiom 5 (ifeq_axiom) }
131.16/131.49	  ifeq(true, true, ifeq(is_a_theorem(implies(implies(implies(not(implies(implies(not(X), X), X)), implies(implies(not(X), X), X)), Y), implies(Z, Y))), true, is_a_theorem(implies(implies(implies(implies(not(X), X), X), Y), implies(Z, Y))), true), true)
131.16/131.49	= { by lemma 7 }
131.16/131.49	  ifeq(is_a_theorem(implies(implies(implies(implies(not(implies(implies(not(X), X), X)), implies(implies(not(X), X), X)), Y), implies(Z, Y)), implies(implies(implies(implies(not(X), X), X), Y), implies(Z, Y)))), true, ifeq(is_a_theorem(implies(implies(implies(not(implies(implies(not(X), X), X)), implies(implies(not(X), X), X)), Y), implies(Z, Y))), true, is_a_theorem(implies(implies(implies(implies(not(X), X), X), Y), implies(Z, Y))), true), true)
131.16/131.49	= { by axiom 3 (condensed_detachment) }
131.16/131.49	  true
131.16/131.49	
131.16/131.49	Lemma 16: is_a_theorem(implies(X, implies(Y, implies(implies(X, Z), Z)))) = true.
131.16/131.49	Proof:
131.16/131.49	  is_a_theorem(implies(X, implies(Y, implies(implies(X, Z), Z))))
131.16/131.49	= { by axiom 5 (ifeq_axiom) }
131.16/131.49	  ifeq(true, true, is_a_theorem(implies(X, implies(Y, implies(implies(X, Z), Z)))), true)
131.16/131.49	= { by lemma 8 }
131.16/131.49	  ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(not(Z), Z), Z), implies(implies(X, Z), Z)), implies(Y, implies(implies(X, Z), Z)))), true, is_a_theorem(implies(implies(implies(X, Z), implies(not(Z), Z)), implies(Y, implies(implies(X, Z), Z)))), true), true, is_a_theorem(implies(X, implies(Y, implies(implies(X, Z), Z)))), true)
131.16/131.49	= { by lemma 15 }
131.16/131.49	  ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(X, Z), implies(not(Z), Z)), implies(Y, implies(implies(X, Z), Z)))), true), true, is_a_theorem(implies(X, implies(Y, implies(implies(X, Z), Z)))), true)
131.16/131.49	= { by axiom 5 (ifeq_axiom) }
131.16/131.49	  ifeq(is_a_theorem(implies(implies(implies(X, Z), implies(not(Z), Z)), implies(Y, implies(implies(X, Z), Z)))), true, is_a_theorem(implies(X, implies(Y, implies(implies(X, Z), Z)))), true)
131.16/131.49	= { by axiom 5 (ifeq_axiom) }
131.16/131.49	  ifeq(true, true, ifeq(is_a_theorem(implies(implies(implies(X, Z), implies(not(Z), Z)), implies(Y, implies(implies(X, Z), Z)))), true, is_a_theorem(implies(X, implies(Y, implies(implies(X, Z), Z)))), true), true)
131.16/131.49	= { by lemma 6 }
131.16/131.49	  ifeq(ifeq(is_a_theorem(implies(X, implies(implies(X, Z), implies(not(Z), Z)))), true, is_a_theorem(implies(implies(implies(implies(X, Z), implies(not(Z), Z)), implies(Y, implies(implies(X, Z), Z))), implies(X, implies(Y, implies(implies(X, Z), Z))))), true), true, ifeq(is_a_theorem(implies(implies(implies(X, Z), implies(not(Z), Z)), implies(Y, implies(implies(X, Z), Z)))), true, is_a_theorem(implies(X, implies(Y, implies(implies(X, Z), Z)))), true), true)
131.16/131.49	= { by lemma 13 }
131.16/131.49	  ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(implies(X, Z), implies(not(Z), Z)), implies(Y, implies(implies(X, Z), Z))), implies(X, implies(Y, implies(implies(X, Z), Z))))), true), true, ifeq(is_a_theorem(implies(implies(implies(X, Z), implies(not(Z), Z)), implies(Y, implies(implies(X, Z), Z)))), true, is_a_theorem(implies(X, implies(Y, implies(implies(X, Z), Z)))), true), true)
131.16/131.49	= { by axiom 5 (ifeq_axiom) }
131.16/131.49	  ifeq(is_a_theorem(implies(implies(implies(implies(X, Z), implies(not(Z), Z)), implies(Y, implies(implies(X, Z), Z))), implies(X, implies(Y, implies(implies(X, Z), Z))))), true, ifeq(is_a_theorem(implies(implies(implies(X, Z), implies(not(Z), Z)), implies(Y, implies(implies(X, Z), Z)))), true, is_a_theorem(implies(X, implies(Y, implies(implies(X, Z), Z)))), true), true)
131.16/131.49	= { by axiom 3 (condensed_detachment) }
131.16/131.49	  true
131.16/131.49	
131.16/131.49	Lemma 17: ifeq(is_a_theorem(implies(X, implies(not(Y), Y))), true, is_a_theorem(implies(implies(Y, Z), implies(X, Z))), true) = true.
131.16/131.49	Proof:
131.16/131.49	  ifeq(is_a_theorem(implies(X, implies(not(Y), Y))), true, is_a_theorem(implies(implies(Y, Z), implies(X, Z))), true)
131.16/131.49	= { by axiom 5 (ifeq_axiom) }
131.16/131.49	  ifeq(true, true, ifeq(is_a_theorem(implies(X, implies(not(Y), Y))), true, is_a_theorem(implies(implies(Y, Z), implies(X, Z))), true), true)
131.16/131.49	= { by lemma 9 }
131.16/131.49	  ifeq(is_a_theorem(implies(implies(X, implies(not(Y), Y)), implies(implies(Y, Z), implies(X, Z)))), true, ifeq(is_a_theorem(implies(X, implies(not(Y), Y))), true, is_a_theorem(implies(implies(Y, Z), implies(X, Z))), true), true)
131.16/131.49	= { by axiom 3 (condensed_detachment) }
131.16/131.49	  true
131.16/131.49	
131.16/131.49	Lemma 18: is_a_theorem(implies(implies(implies(implies(X, Y), Y), Z), implies(X, Z))) = true.
131.16/131.49	Proof:
131.16/131.49	  is_a_theorem(implies(implies(implies(implies(X, Y), Y), Z), implies(X, Z)))
131.16/131.49	= { by axiom 5 (ifeq_axiom) }
131.16/131.49	  ifeq(true, true, is_a_theorem(implies(implies(implies(implies(X, Y), Y), Z), implies(X, Z))), true)
131.16/131.49	= { by lemma 16 }
131.16/131.49	  ifeq(is_a_theorem(implies(X, implies(not(implies(implies(X, Y), Y)), implies(implies(X, Y), Y)))), true, is_a_theorem(implies(implies(implies(implies(X, Y), Y), Z), implies(X, Z))), true)
131.16/131.49	= { by lemma 17 }
131.35/131.64	  true
131.35/131.64	
131.35/131.64	Goal 1 (prove_cn_54): is_a_theorem(implies(implies(a, b), implies(implies(not(a), b), b))) = true.
131.35/131.64	Proof:
131.35/131.64	  is_a_theorem(implies(implies(a, b), implies(implies(not(a), b), b)))
131.35/131.64	= { by axiom 5 (ifeq_axiom) }
131.35/131.64	  ifeq(true, true, is_a_theorem(implies(implies(a, b), implies(implies(not(a), b), b))), true)
131.35/131.64	= { by axiom 3 (condensed_detachment) }
131.35/131.64	  ifeq(ifeq(is_a_theorem(implies(implies(a, implies(implies(not(a), b), implies(implies(a, b), b))), implies(implies(not(a), b), implies(implies(a, b), b)))), true, ifeq(is_a_theorem(implies(a, implies(implies(not(a), b), implies(implies(a, b), b)))), true, is_a_theorem(implies(implies(not(a), b), implies(implies(a, b), b))), true), true), true, is_a_theorem(implies(implies(a, b), implies(implies(not(a), b), b))), true)
131.35/131.64	= { by axiom 5 (ifeq_axiom) }
131.35/131.64	  ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(a, implies(implies(not(a), b), implies(implies(a, b), b))), implies(implies(not(a), b), implies(implies(a, b), b)))), true), true, ifeq(is_a_theorem(implies(a, implies(implies(not(a), b), implies(implies(a, b), b)))), true, is_a_theorem(implies(implies(not(a), b), implies(implies(a, b), b))), true), true), true, is_a_theorem(implies(implies(a, b), implies(implies(not(a), b), b))), true)
131.35/131.64	= { by axiom 3 (condensed_detachment) }
131.35/131.64	  ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(not(implies(implies(not(a), b), implies(implies(a, b), b))), a), a), a), implies(not(implies(implies(not(a), b), implies(implies(a, b), b))), a))), true, ifeq(is_a_theorem(implies(implies(implies(not(implies(implies(not(a), b), implies(implies(a, b), b))), a), a), a)), true, is_a_theorem(implies(not(implies(implies(not(a), b), implies(implies(a, b), b))), a)), true), true), true, is_a_theorem(implies(implies(a, implies(implies(not(a), b), implies(implies(a, b), b))), implies(implies(not(a), b), implies(implies(a, b), b)))), true), true, ifeq(is_a_theorem(implies(a, implies(implies(not(a), b), implies(implies(a, b), b)))), true, is_a_theorem(implies(implies(not(a), b), implies(implies(a, b), b))), true), true), true, is_a_theorem(implies(implies(a, b), implies(implies(not(a), b), b))), true)
131.35/131.64	= { by lemma 18 }
131.35/131.64	  ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(implies(not(implies(implies(not(a), b), implies(implies(a, b), b))), a), a), a)), true, is_a_theorem(implies(not(implies(implies(not(a), b), implies(implies(a, b), b))), a)), true), true), true, is_a_theorem(implies(implies(a, implies(implies(not(a), b), implies(implies(a, b), b))), implies(implies(not(a), b), implies(implies(a, b), b)))), true), true, ifeq(is_a_theorem(implies(a, implies(implies(not(a), b), implies(implies(a, b), b)))), true, is_a_theorem(implies(implies(not(a), b), implies(implies(a, b), b))), true), true), true, is_a_theorem(implies(implies(a, b), implies(implies(not(a), b), b))), true)
131.35/131.64	= { by axiom 5 (ifeq_axiom) }
131.35/131.64	  ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(not(implies(implies(not(a), b), implies(implies(a, b), b))), a), a), a)), true, is_a_theorem(implies(not(implies(implies(not(a), b), implies(implies(a, b), b))), a)), true), true, is_a_theorem(implies(implies(a, implies(implies(not(a), b), implies(implies(a, b), b))), implies(implies(not(a), b), implies(implies(a, b), b)))), true), true, ifeq(is_a_theorem(implies(a, implies(implies(not(a), b), implies(implies(a, b), b)))), true, is_a_theorem(implies(implies(not(a), b), implies(implies(a, b), b))), true), true), true, is_a_theorem(implies(implies(a, b), implies(implies(not(a), b), b))), true)
131.35/131.64	= { by axiom 5 (ifeq_axiom) }
131.35/131.64	  ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(not(implies(implies(not(a), b), implies(implies(a, b), b))), a), a), a)), true), true, is_a_theorem(implies(not(implies(implies(not(a), b), implies(implies(a, b), b))), a)), true), true, is_a_theorem(implies(implies(a, implies(implies(not(a), b), implies(implies(a, b), b))), implies(implies(not(a), b), implies(implies(a, b), b)))), true), true, ifeq(is_a_theorem(implies(a, implies(implies(not(a), b), implies(implies(a, b), b)))), true, is_a_theorem(implies(implies(not(a), b), implies(implies(a, b), b))), true), true), true, is_a_theorem(implies(implies(a, b), implies(implies(not(a), b), b))), true)
131.35/131.64	= { by lemma 14 }
131.35/131.64	  ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(a, a), implies(implies(implies(implies(not(a), b), implies(implies(a, b), b)), a), a))), true, is_a_theorem(implies(implies(implies(implies(not(a), b), implies(implies(a, b), b)), a), a)), true), true, is_a_theorem(implies(implies(implies(not(implies(implies(not(a), b), implies(implies(a, b), b))), a), a), a)), true), true, is_a_theorem(implies(not(implies(implies(not(a), b), implies(implies(a, b), b))), a)), true), true, is_a_theorem(implies(implies(a, implies(implies(not(a), b), implies(implies(a, b), b))), implies(implies(not(a), b), implies(implies(a, b), b)))), true), true, ifeq(is_a_theorem(implies(a, implies(implies(not(a), b), implies(implies(a, b), b)))), true, is_a_theorem(implies(implies(not(a), b), implies(implies(a, b), b))), true), true), true, is_a_theorem(implies(implies(a, b), implies(implies(not(a), b), b))), true)
131.35/131.64	= { by axiom 5 (ifeq_axiom) }
131.35/131.64	  ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(a, a), implies(implies(implies(implies(not(a), b), implies(implies(a, b), b)), a), a))), true), true, is_a_theorem(implies(implies(implies(implies(not(a), b), implies(implies(a, b), b)), a), a)), true), true, is_a_theorem(implies(implies(implies(not(implies(implies(not(a), b), implies(implies(a, b), b))), a), a), a)), true), true, is_a_theorem(implies(not(implies(implies(not(a), b), implies(implies(a, b), b))), a)), true), true, is_a_theorem(implies(implies(a, implies(implies(not(a), b), implies(implies(a, b), b))), implies(implies(not(a), b), implies(implies(a, b), b)))), true), true, ifeq(is_a_theorem(implies(a, implies(implies(not(a), b), implies(implies(a, b), b)))), true, is_a_theorem(implies(implies(not(a), b), implies(implies(a, b), b))), true), true), true, is_a_theorem(implies(implies(a, b), implies(implies(not(a), b), b))), true)
131.35/131.64	= { by lemma 13 }
131.35/131.64	  ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(not(a), implies(implies(not(a), b), implies(implies(a, b), b)))), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(implies(not(a), b), implies(implies(a, b), b)), a), a))), true), true, is_a_theorem(implies(implies(implies(implies(not(a), b), implies(implies(a, b), b)), a), a)), true), true, is_a_theorem(implies(implies(implies(not(implies(implies(not(a), b), implies(implies(a, b), b))), a), a), a)), true), true, is_a_theorem(implies(not(implies(implies(not(a), b), implies(implies(a, b), b))), a)), true), true, is_a_theorem(implies(implies(a, implies(implies(not(a), b), implies(implies(a, b), b))), implies(implies(not(a), b), implies(implies(a, b), b)))), true), true, ifeq(is_a_theorem(implies(a, implies(implies(not(a), b), implies(implies(a, b), b)))), true, is_a_theorem(implies(implies(not(a), b), implies(implies(a, b), b))), true), true), true, is_a_theorem(implies(implies(a, b), implies(implies(not(a), b), b))), true)
131.35/131.64	= { by axiom 5 (ifeq_axiom) }
131.35/131.64	  ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(not(a), implies(implies(not(a), b), implies(implies(a, b), b)))), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(implies(not(a), b), implies(implies(a, b), b)), a), a))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(a), b), implies(implies(a, b), b)), a), a)), true), true, is_a_theorem(implies(implies(implies(not(implies(implies(not(a), b), implies(implies(a, b), b))), a), a), a)), true), true, is_a_theorem(implies(not(implies(implies(not(a), b), implies(implies(a, b), b))), a)), true), true, is_a_theorem(implies(implies(a, implies(implies(not(a), b), implies(implies(a, b), b))), implies(implies(not(a), b), implies(implies(a, b), b)))), true), true, ifeq(is_a_theorem(implies(a, implies(implies(not(a), b), implies(implies(a, b), b)))), true, is_a_theorem(implies(implies(not(a), b), implies(implies(a, b), b))), true), true), true, is_a_theorem(implies(implies(a, b), implies(implies(not(a), b), b))), true)
131.35/131.64	= { by lemma 8 }
131.35/131.64	  ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(implies(not(a), b), implies(implies(a, b), b)), a), implies(not(a), a)), implies(implies(a, a), implies(implies(implies(implies(not(a), b), implies(implies(a, b), b)), a), a)))), true, is_a_theorem(implies(implies(not(a), implies(implies(not(a), b), implies(implies(a, b), b))), implies(implies(a, a), implies(implies(implies(implies(not(a), b), implies(implies(a, b), b)), a), a)))), true), true, ifeq(is_a_theorem(implies(not(a), implies(implies(not(a), b), implies(implies(a, b), b)))), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(implies(not(a), b), implies(implies(a, b), b)), a), a))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(a), b), implies(implies(a, b), b)), a), a)), true), true, is_a_theorem(implies(implies(implies(not(implies(implies(not(a), b), implies(implies(a, b), b))), a), a), a)), true), true, is_a_theorem(implies(not(implies(implies(not(a), b), implies(implies(a, b), b))), a)), true), true, is_a_theorem(implies(implies(a, implies(implies(not(a), b), implies(implies(a, b), b))), implies(implies(not(a), b), implies(implies(a, b), b)))), true), true, ifeq(is_a_theorem(implies(a, implies(implies(not(a), b), implies(implies(a, b), b)))), true, is_a_theorem(implies(implies(not(a), b), implies(implies(a, b), b))), true), true), true, is_a_theorem(implies(implies(a, b), implies(implies(not(a), b), b))), true)
131.35/131.64	= { by lemma 9 }
131.35/131.64	  ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(not(a), implies(implies(not(a), b), implies(implies(a, b), b))), implies(implies(a, a), implies(implies(implies(implies(not(a), b), implies(implies(a, b), b)), a), a)))), true), true, ifeq(is_a_theorem(implies(not(a), implies(implies(not(a), b), implies(implies(a, b), b)))), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(implies(not(a), b), implies(implies(a, b), b)), a), a))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(a), b), implies(implies(a, b), b)), a), a)), true), true, is_a_theorem(implies(implies(implies(not(implies(implies(not(a), b), implies(implies(a, b), b))), a), a), a)), true), true, is_a_theorem(implies(not(implies(implies(not(a), b), implies(implies(a, b), b))), a)), true), true, is_a_theorem(implies(implies(a, implies(implies(not(a), b), implies(implies(a, b), b))), implies(implies(not(a), b), implies(implies(a, b), b)))), true), true, ifeq(is_a_theorem(implies(a, implies(implies(not(a), b), implies(implies(a, b), b)))), true, is_a_theorem(implies(implies(not(a), b), implies(implies(a, b), b))), true), true), true, is_a_theorem(implies(implies(a, b), implies(implies(not(a), b), b))), true)
131.35/131.64	= { by axiom 5 (ifeq_axiom) }
131.35/131.64	  ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(a), implies(implies(not(a), b), implies(implies(a, b), b))), implies(implies(a, a), implies(implies(implies(implies(not(a), b), implies(implies(a, b), b)), a), a)))), true, ifeq(is_a_theorem(implies(not(a), implies(implies(not(a), b), implies(implies(a, b), b)))), true, is_a_theorem(implies(implies(a, a), implies(implies(implies(implies(not(a), b), implies(implies(a, b), b)), a), a))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(a), b), implies(implies(a, b), b)), a), a)), true), true, is_a_theorem(implies(implies(implies(not(implies(implies(not(a), b), implies(implies(a, b), b))), a), a), a)), true), true, is_a_theorem(implies(not(implies(implies(not(a), b), implies(implies(a, b), b))), a)), true), true, is_a_theorem(implies(implies(a, implies(implies(not(a), b), implies(implies(a, b), b))), implies(implies(not(a), b), implies(implies(a, b), b)))), true), true, ifeq(is_a_theorem(implies(a, implies(implies(not(a), b), implies(implies(a, b), b)))), true, is_a_theorem(implies(implies(not(a), b), implies(implies(a, b), b))), true), true), true, is_a_theorem(implies(implies(a, b), implies(implies(not(a), b), b))), true)
131.35/131.64	= { by axiom 3 (condensed_detachment) }
131.35/131.64	  ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(implies(not(a), b), implies(implies(a, b), b)), a), a)), true), true, is_a_theorem(implies(implies(implies(not(implies(implies(not(a), b), implies(implies(a, b), b))), a), a), a)), true), true, is_a_theorem(implies(not(implies(implies(not(a), b), implies(implies(a, b), b))), a)), true), true, is_a_theorem(implies(implies(a, implies(implies(not(a), b), implies(implies(a, b), b))), implies(implies(not(a), b), implies(implies(a, b), b)))), true), true, ifeq(is_a_theorem(implies(a, implies(implies(not(a), b), implies(implies(a, b), b)))), true, is_a_theorem(implies(implies(not(a), b), implies(implies(a, b), b))), true), true), true, is_a_theorem(implies(implies(a, b), implies(implies(not(a), b), b))), true)
131.35/131.64	= { by axiom 5 (ifeq_axiom) }
131.35/131.64	  ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(not(a), b), implies(implies(a, b), b)), a), a)), true, is_a_theorem(implies(implies(implies(not(implies(implies(not(a), b), implies(implies(a, b), b))), a), a), a)), true), true, is_a_theorem(implies(not(implies(implies(not(a), b), implies(implies(a, b), b))), a)), true), true, is_a_theorem(implies(implies(a, implies(implies(not(a), b), implies(implies(a, b), b))), implies(implies(not(a), b), implies(implies(a, b), b)))), true), true, ifeq(is_a_theorem(implies(a, implies(implies(not(a), b), implies(implies(a, b), b)))), true, is_a_theorem(implies(implies(not(a), b), implies(implies(a, b), b))), true), true), true, is_a_theorem(implies(implies(a, b), implies(implies(not(a), b), b))), true)
131.35/131.64	= { by lemma 11 }
131.35/131.64	  ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(not(implies(implies(not(a), b), implies(implies(a, b), b))), a)), true), true, is_a_theorem(implies(implies(a, implies(implies(not(a), b), implies(implies(a, b), b))), implies(implies(not(a), b), implies(implies(a, b), b)))), true), true, ifeq(is_a_theorem(implies(a, implies(implies(not(a), b), implies(implies(a, b), b)))), true, is_a_theorem(implies(implies(not(a), b), implies(implies(a, b), b))), true), true), true, is_a_theorem(implies(implies(a, b), implies(implies(not(a), b), b))), true)
131.35/131.64	= { by axiom 5 (ifeq_axiom) }
131.35/131.64	  ifeq(ifeq(ifeq(is_a_theorem(implies(not(implies(implies(not(a), b), implies(implies(a, b), b))), a)), true, is_a_theorem(implies(implies(a, implies(implies(not(a), b), implies(implies(a, b), b))), implies(implies(not(a), b), implies(implies(a, b), b)))), true), true, ifeq(is_a_theorem(implies(a, implies(implies(not(a), b), implies(implies(a, b), b)))), true, is_a_theorem(implies(implies(not(a), b), implies(implies(a, b), b))), true), true), true, is_a_theorem(implies(implies(a, b), implies(implies(not(a), b), b))), true)
131.35/131.64	= { by axiom 5 (ifeq_axiom) }
131.35/131.64	  ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(not(implies(implies(not(a), b), implies(implies(a, b), b))), a)), true, is_a_theorem(implies(implies(a, implies(implies(not(a), b), implies(implies(a, b), b))), implies(implies(not(a), b), implies(implies(a, b), b)))), true), true), true, ifeq(is_a_theorem(implies(a, implies(implies(not(a), b), implies(implies(a, b), b)))), true, is_a_theorem(implies(implies(not(a), b), implies(implies(a, b), b))), true), true), true, is_a_theorem(implies(implies(a, b), implies(implies(not(a), b), b))), true)
131.35/131.64	= { by lemma 8 }
131.35/131.64	  ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(a, implies(implies(not(a), b), implies(implies(a, b), b))), implies(not(implies(implies(not(a), b), implies(implies(a, b), b))), implies(implies(not(a), b), implies(implies(a, b), b)))), implies(implies(a, implies(implies(not(a), b), implies(implies(a, b), b))), implies(implies(not(a), b), implies(implies(a, b), b))))), true, is_a_theorem(implies(implies(not(implies(implies(not(a), b), implies(implies(a, b), b))), a), implies(implies(a, implies(implies(not(a), b), implies(implies(a, b), b))), implies(implies(not(a), b), implies(implies(a, b), b))))), true), true, ifeq(is_a_theorem(implies(not(implies(implies(not(a), b), implies(implies(a, b), b))), a)), true, is_a_theorem(implies(implies(a, implies(implies(not(a), b), implies(implies(a, b), b))), implies(implies(not(a), b), implies(implies(a, b), b)))), true), true), true, ifeq(is_a_theorem(implies(a, implies(implies(not(a), b), implies(implies(a, b), b)))), true, is_a_theorem(implies(implies(not(a), b), implies(implies(a, b), b))), true), true), true, is_a_theorem(implies(implies(a, b), implies(implies(not(a), b), b))), true)
131.35/131.64	= { by axiom 5 (ifeq_axiom) }
131.35/131.64	  ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(a, implies(implies(not(a), b), implies(implies(a, b), b))), implies(not(implies(implies(not(a), b), implies(implies(a, b), b))), implies(implies(not(a), b), implies(implies(a, b), b)))), implies(implies(a, implies(implies(not(a), b), implies(implies(a, b), b))), implies(implies(not(a), b), implies(implies(a, b), b))))), true), true, is_a_theorem(implies(implies(not(implies(implies(not(a), b), implies(implies(a, b), b))), a), implies(implies(a, implies(implies(not(a), b), implies(implies(a, b), b))), implies(implies(not(a), b), implies(implies(a, b), b))))), true), true, ifeq(is_a_theorem(implies(not(implies(implies(not(a), b), implies(implies(a, b), b))), a)), true, is_a_theorem(implies(implies(a, implies(implies(not(a), b), implies(implies(a, b), b))), implies(implies(not(a), b), implies(implies(a, b), b)))), true), true), true, ifeq(is_a_theorem(implies(a, implies(implies(not(a), b), implies(implies(a, b), b)))), true, is_a_theorem(implies(implies(not(a), b), implies(implies(a, b), b))), true), true), true, is_a_theorem(implies(implies(a, b), implies(implies(not(a), b), b))), true)
131.35/131.64	= { by lemma 14 }
131.35/131.64	  ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(a, implies(implies(not(a), b), implies(implies(a, b), b))), implies(implies(not(a), b), implies(implies(a, b), b))), implies(implies(a, implies(implies(not(a), b), implies(implies(a, b), b))), implies(implies(not(a), b), implies(implies(a, b), b)))), implies(implies(implies(implies(not(implies(implies(not(a), b), implies(implies(a, b), b))), implies(implies(not(a), b), implies(implies(a, b), b))), implies(implies(not(a), b), implies(implies(a, b), b))), implies(implies(a, implies(implies(not(a), b), implies(implies(a, b), b))), implies(implies(not(a), b), implies(implies(a, b), b)))), implies(implies(a, implies(implies(not(a), b), implies(implies(a, b), b))), implies(implies(not(a), b), implies(implies(a, b), b)))))), true, is_a_theorem(implies(implies(implies(implies(not(implies(implies(not(a), b), implies(implies(a, b), b))), implies(implies(not(a), b), implies(implies(a, b), b))), implies(implies(not(a), b), implies(implies(a, b), b))), implies(implies(a, implies(implies(not(a), b), implies(implies(a, b), b))), implies(implies(not(a), b), implies(implies(a, b), b)))), implies(implies(a, implies(implies(not(a), b), implies(implies(a, b), b))), implies(implies(not(a), b), implies(implies(a, b), b))))), true), true, is_a_theorem(implies(implies(implies(a, implies(implies(not(a), b), implies(implies(a, b), b))), implies(not(implies(implies(not(a), b), implies(implies(a, b), b))), implies(implies(not(a), b), implies(implies(a, b), b)))), implies(implies(a, implies(implies(not(a), b), implies(implies(a, b), b))), implies(implies(not(a), b), implies(implies(a, b), b))))), true), true, is_a_theorem(implies(implies(not(implies(implies(not(a), b), implies(implies(a, b), b))), a), implies(implies(a, implies(implies(not(a), b), implies(implies(a, b), b))), implies(implies(not(a), b), implies(implies(a, b), b))))), true), true, ifeq(is_a_theorem(implies(not(implies(implies(not(a), b), implies(implies(a, b), b))), a)), true, is_a_theorem(implies(implies(a, implies(implies(not(a), b), implies(implies(a, b), b))), implies(implies(not(a), b), implies(implies(a, b), b)))), true), true), true, ifeq(is_a_theorem(implies(a, implies(implies(not(a), b), implies(implies(a, b), b)))), true, is_a_theorem(implies(implies(not(a), b), implies(implies(a, b), b))), true), true), true, is_a_theorem(implies(implies(a, b), implies(implies(not(a), b), b))), true)
131.35/131.64	= { by axiom 5 (ifeq_axiom) }
131.35/131.64	  ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(implies(a, implies(implies(not(a), b), implies(implies(a, b), b))), implies(implies(not(a), b), implies(implies(a, b), b))), implies(implies(a, implies(implies(not(a), b), implies(implies(a, b), b))), implies(implies(not(a), b), implies(implies(a, b), b)))), implies(implies(implies(implies(not(implies(implies(not(a), b), implies(implies(a, b), b))), implies(implies(not(a), b), implies(implies(a, b), b))), implies(implies(not(a), b), implies(implies(a, b), b))), implies(implies(a, implies(implies(not(a), b), implies(implies(a, b), b))), implies(implies(not(a), b), implies(implies(a, b), b)))), implies(implies(a, implies(implies(not(a), b), implies(implies(a, b), b))), implies(implies(not(a), b), implies(implies(a, b), b)))))), true), true, is_a_theorem(implies(implies(implies(implies(not(implies(implies(not(a), b), implies(implies(a, b), b))), implies(implies(not(a), b), implies(implies(a, b), b))), implies(implies(not(a), b), implies(implies(a, b), b))), implies(implies(a, implies(implies(not(a), b), implies(implies(a, b), b))), implies(implies(not(a), b), implies(implies(a, b), b)))), implies(implies(a, implies(implies(not(a), b), implies(implies(a, b), b))), implies(implies(not(a), b), implies(implies(a, b), b))))), true), true, is_a_theorem(implies(implies(implies(a, implies(implies(not(a), b), implies(implies(a, b), b))), implies(not(implies(implies(not(a), b), implies(implies(a, b), b))), implies(implies(not(a), b), implies(implies(a, b), b)))), implies(implies(a, implies(implies(not(a), b), implies(implies(a, b), b))), implies(implies(not(a), b), implies(implies(a, b), b))))), true), true, is_a_theorem(implies(implies(not(implies(implies(not(a), b), implies(implies(a, b), b))), a), implies(implies(a, implies(implies(not(a), b), implies(implies(a, b), b))), implies(implies(not(a), b), implies(implies(a, b), b))))), true), true, ifeq(is_a_theorem(implies(not(implies(implies(not(a), b), implies(implies(a, b), b))), a)), true, is_a_theorem(implies(implies(a, implies(implies(not(a), b), implies(implies(a, b), b))), implies(implies(not(a), b), implies(implies(a, b), b)))), true), true), true, ifeq(is_a_theorem(implies(a, implies(implies(not(a), b), implies(implies(a, b), b)))), true, is_a_theorem(implies(implies(not(a), b), implies(implies(a, b), b))), true), true), true, is_a_theorem(implies(implies(a, b), implies(implies(not(a), b), b))), true)
131.35/131.64	= { by lemma 15 }
131.35/131.64	  ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(not(implies(implies(not(a), b), implies(implies(a, b), b))), implies(implies(not(a), b), implies(implies(a, b), b))), implies(implies(not(a), b), implies(implies(a, b), b))), implies(implies(a, implies(implies(not(a), b), implies(implies(a, b), b))), implies(implies(not(a), b), implies(implies(a, b), b)))), implies(not(implies(implies(a, implies(implies(not(a), b), implies(implies(a, b), b))), implies(implies(not(a), b), implies(implies(a, b), b)))), implies(implies(a, implies(implies(not(a), b), implies(implies(a, b), b))), implies(implies(not(a), b), implies(implies(a, b), b)))))), true, is_a_theorem(implies(implies(implies(implies(a, implies(implies(not(a), b), implies(implies(a, b), b))), implies(implies(not(a), b), implies(implies(a, b), b))), implies(implies(a, implies(implies(not(a), b), implies(implies(a, b), b))), implies(implies(not(a), b), implies(implies(a, b), b)))), implies(implies(implies(implies(not(implies(implies(not(a), b), implies(implies(a, b), b))), implies(implies(not(a), b), implies(implies(a, b), b))), implies(implies(not(a), b), implies(implies(a, b), b))), implies(implies(a, implies(implies(not(a), b), implies(implies(a, b), b))), implies(implies(not(a), b), implies(implies(a, b), b)))), implies(implies(a, implies(implies(not(a), b), implies(implies(a, b), b))), implies(implies(not(a), b), implies(implies(a, b), b)))))), true), true, is_a_theorem(implies(implies(implies(implies(not(implies(implies(not(a), b), implies(implies(a, b), b))), implies(implies(not(a), b), implies(implies(a, b), b))), implies(implies(not(a), b), implies(implies(a, b), b))), implies(implies(a, implies(implies(not(a), b), implies(implies(a, b), b))), implies(implies(not(a), b), implies(implies(a, b), b)))), implies(implies(a, implies(implies(not(a), b), implies(implies(a, b), b))), implies(implies(not(a), b), implies(implies(a, b), b))))), true), true, is_a_theorem(implies(implies(implies(a, implies(implies(not(a), b), implies(implies(a, b), b))), implies(not(implies(implies(not(a), b), implies(implies(a, b), b))), implies(implies(not(a), b), implies(implies(a, b), b)))), implies(implies(a, implies(implies(not(a), b), implies(implies(a, b), b))), implies(implies(not(a), b), implies(implies(a, b), b))))), true), true, is_a_theorem(implies(implies(not(implies(implies(not(a), b), implies(implies(a, b), b))), a), implies(implies(a, implies(implies(not(a), b), implies(implies(a, b), b))), implies(implies(not(a), b), implies(implies(a, b), b))))), true), true, ifeq(is_a_theorem(implies(not(implies(implies(not(a), b), implies(implies(a, b), b))), a)), true, is_a_theorem(implies(implies(a, implies(implies(not(a), b), implies(implies(a, b), b))), implies(implies(not(a), b), implies(implies(a, b), b)))), true), true), true, ifeq(is_a_theorem(implies(a, implies(implies(not(a), b), implies(implies(a, b), b)))), true, is_a_theorem(implies(implies(not(a), b), implies(implies(a, b), b))), true), true), true, is_a_theorem(implies(implies(a, b), implies(implies(not(a), b), b))), true)
131.35/131.64	= { by lemma 17 }
131.35/131.64	  ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(implies(not(implies(implies(not(a), b), implies(implies(a, b), b))), implies(implies(not(a), b), implies(implies(a, b), b))), implies(implies(not(a), b), implies(implies(a, b), b))), implies(implies(a, implies(implies(not(a), b), implies(implies(a, b), b))), implies(implies(not(a), b), implies(implies(a, b), b)))), implies(implies(a, implies(implies(not(a), b), implies(implies(a, b), b))), implies(implies(not(a), b), implies(implies(a, b), b))))), true), true, is_a_theorem(implies(implies(implies(a, implies(implies(not(a), b), implies(implies(a, b), b))), implies(not(implies(implies(not(a), b), implies(implies(a, b), b))), implies(implies(not(a), b), implies(implies(a, b), b)))), implies(implies(a, implies(implies(not(a), b), implies(implies(a, b), b))), implies(implies(not(a), b), implies(implies(a, b), b))))), true), true, is_a_theorem(implies(implies(not(implies(implies(not(a), b), implies(implies(a, b), b))), a), implies(implies(a, implies(implies(not(a), b), implies(implies(a, b), b))), implies(implies(not(a), b), implies(implies(a, b), b))))), true), true, ifeq(is_a_theorem(implies(not(implies(implies(not(a), b), implies(implies(a, b), b))), a)), true, is_a_theorem(implies(implies(a, implies(implies(not(a), b), implies(implies(a, b), b))), implies(implies(not(a), b), implies(implies(a, b), b)))), true), true), true, ifeq(is_a_theorem(implies(a, implies(implies(not(a), b), implies(implies(a, b), b)))), true, is_a_theorem(implies(implies(not(a), b), implies(implies(a, b), b))), true), true), true, is_a_theorem(implies(implies(a, b), implies(implies(not(a), b), b))), true)
131.35/131.64	= { by axiom 5 (ifeq_axiom) }
131.35/131.65	  ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(not(implies(implies(not(a), b), implies(implies(a, b), b))), implies(implies(not(a), b), implies(implies(a, b), b))), implies(implies(not(a), b), implies(implies(a, b), b))), implies(implies(a, implies(implies(not(a), b), implies(implies(a, b), b))), implies(implies(not(a), b), implies(implies(a, b), b)))), implies(implies(a, implies(implies(not(a), b), implies(implies(a, b), b))), implies(implies(not(a), b), implies(implies(a, b), b))))), true, is_a_theorem(implies(implies(implies(a, implies(implies(not(a), b), implies(implies(a, b), b))), implies(not(implies(implies(not(a), b), implies(implies(a, b), b))), implies(implies(not(a), b), implies(implies(a, b), b)))), implies(implies(a, implies(implies(not(a), b), implies(implies(a, b), b))), implies(implies(not(a), b), implies(implies(a, b), b))))), true), true, is_a_theorem(implies(implies(not(implies(implies(not(a), b), implies(implies(a, b), b))), a), implies(implies(a, implies(implies(not(a), b), implies(implies(a, b), b))), implies(implies(not(a), b), implies(implies(a, b), b))))), true), true, ifeq(is_a_theorem(implies(not(implies(implies(not(a), b), implies(implies(a, b), b))), a)), true, is_a_theorem(implies(implies(a, implies(implies(not(a), b), implies(implies(a, b), b))), implies(implies(not(a), b), implies(implies(a, b), b)))), true), true), true, ifeq(is_a_theorem(implies(a, implies(implies(not(a), b), implies(implies(a, b), b)))), true, is_a_theorem(implies(implies(not(a), b), implies(implies(a, b), b))), true), true), true, is_a_theorem(implies(implies(a, b), implies(implies(not(a), b), b))), true)
131.35/131.65	= { by lemma 8 }
131.35/131.65	  ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(not(implies(implies(not(a), b), implies(implies(a, b), b))), a), implies(implies(a, implies(implies(not(a), b), implies(implies(a, b), b))), implies(implies(not(a), b), implies(implies(a, b), b))))), true), true, ifeq(is_a_theorem(implies(not(implies(implies(not(a), b), implies(implies(a, b), b))), a)), true, is_a_theorem(implies(implies(a, implies(implies(not(a), b), implies(implies(a, b), b))), implies(implies(not(a), b), implies(implies(a, b), b)))), true), true), true, ifeq(is_a_theorem(implies(a, implies(implies(not(a), b), implies(implies(a, b), b)))), true, is_a_theorem(implies(implies(not(a), b), implies(implies(a, b), b))), true), true), true, is_a_theorem(implies(implies(a, b), implies(implies(not(a), b), b))), true)
131.35/131.65	= { by axiom 5 (ifeq_axiom) }
131.35/131.65	  ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(implies(implies(not(a), b), implies(implies(a, b), b))), a), implies(implies(a, implies(implies(not(a), b), implies(implies(a, b), b))), implies(implies(not(a), b), implies(implies(a, b), b))))), true, ifeq(is_a_theorem(implies(not(implies(implies(not(a), b), implies(implies(a, b), b))), a)), true, is_a_theorem(implies(implies(a, implies(implies(not(a), b), implies(implies(a, b), b))), implies(implies(not(a), b), implies(implies(a, b), b)))), true), true), true, ifeq(is_a_theorem(implies(a, implies(implies(not(a), b), implies(implies(a, b), b)))), true, is_a_theorem(implies(implies(not(a), b), implies(implies(a, b), b))), true), true), true, is_a_theorem(implies(implies(a, b), implies(implies(not(a), b), b))), true)
131.35/131.65	= { by axiom 3 (condensed_detachment) }
131.35/131.65	  ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(a, implies(implies(not(a), b), implies(implies(a, b), b)))), true, is_a_theorem(implies(implies(not(a), b), implies(implies(a, b), b))), true), true), true, is_a_theorem(implies(implies(a, b), implies(implies(not(a), b), b))), true)
131.35/131.65	= { by lemma 16 }
131.35/131.65	  ifeq(ifeq(true, true, ifeq(true, true, is_a_theorem(implies(implies(not(a), b), implies(implies(a, b), b))), true), true), true, is_a_theorem(implies(implies(a, b), implies(implies(not(a), b), b))), true)
131.35/131.65	= { by axiom 5 (ifeq_axiom) }
131.35/131.65	  ifeq(ifeq(true, true, is_a_theorem(implies(implies(not(a), b), implies(implies(a, b), b))), true), true, is_a_theorem(implies(implies(a, b), implies(implies(not(a), b), b))), true)
131.35/131.65	= { by axiom 5 (ifeq_axiom) }
131.35/131.65	  ifeq(is_a_theorem(implies(implies(not(a), b), implies(implies(a, b), b))), true, is_a_theorem(implies(implies(a, b), implies(implies(not(a), b), b))), true)
131.35/131.65	= { by axiom 5 (ifeq_axiom) }
131.35/131.65	  ifeq(true, true, ifeq(is_a_theorem(implies(implies(not(a), b), implies(implies(a, b), b))), true, is_a_theorem(implies(implies(a, b), implies(implies(not(a), b), b))), true), true)
131.35/131.65	= { by lemma 8 }
131.35/131.65	  ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(implies(a, b), b), b), implies(implies(not(a), b), b)), implies(implies(a, b), implies(implies(not(a), b), b)))), true, is_a_theorem(implies(implies(implies(not(a), b), implies(implies(a, b), b)), implies(implies(a, b), implies(implies(not(a), b), b)))), true), true, ifeq(is_a_theorem(implies(implies(not(a), b), implies(implies(a, b), b))), true, is_a_theorem(implies(implies(a, b), implies(implies(not(a), b), b))), true), true)
131.35/131.65	= { by lemma 18 }
131.35/131.65	  ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(not(a), b), implies(implies(a, b), b)), implies(implies(a, b), implies(implies(not(a), b), b)))), true), true, ifeq(is_a_theorem(implies(implies(not(a), b), implies(implies(a, b), b))), true, is_a_theorem(implies(implies(a, b), implies(implies(not(a), b), b))), true), true)
131.35/131.65	= { by axiom 5 (ifeq_axiom) }
131.35/131.65	  ifeq(is_a_theorem(implies(implies(implies(not(a), b), implies(implies(a, b), b)), implies(implies(a, b), implies(implies(not(a), b), b)))), true, ifeq(is_a_theorem(implies(implies(not(a), b), implies(implies(a, b), b))), true, is_a_theorem(implies(implies(a, b), implies(implies(not(a), b), b))), true), true)
131.35/131.65	= { by axiom 3 (condensed_detachment) }
131.35/131.65	  true
131.35/131.65	% SZS output end Proof
131.35/131.65	
131.35/131.65	RESULT: Unsatisfiable (the axioms are contradictory).
131.35/131.66	EOF