0.07/0.12	% Problem    : theBenchmark.p : TPTP v0.0.0. Released v0.0.0.
0.07/0.12	% Command    : twee %s --tstp --casc --quiet --explain-encoding --conditional-encoding if --smaller --drop-non-horn
0.13/0.34	% Computer   : n003.cluster.edu
0.13/0.34	% Model      : x86_64 x86_64
0.13/0.34	% CPU        : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
0.13/0.34	% Memory     : 8042.1875MB
0.13/0.34	% OS         : Linux 3.10.0-693.el7.x86_64
0.13/0.34	% CPULimit   : 180
0.13/0.34	% DateTime   : Thu Aug 29 14:53:17 EDT 2019
0.13/0.34	% CPUTime    : 
45.01/45.21	% SZS status Theorem
45.01/45.21	
45.01/45.21	% SZS output start Proof
45.01/45.21	Take the following subset of the input axioms:
45.18/45.38	  fof(and_1, axiom, ![X, Y]: is_a_theorem(implies(and(X, Y), X)) <=> and_1).
45.18/45.38	  fof(and_2, axiom, and_2 <=> ![X, Y]: is_a_theorem(implies(and(X, Y), Y))).
45.18/45.38	  fof(and_3, axiom, ![X, Y]: is_a_theorem(implies(X, implies(Y, and(X, Y)))) <=> and_3).
45.18/45.38	  fof(axiom_M, axiom, axiom_M <=> ![X]: is_a_theorem(implies(necessarily(X), X))).
45.18/45.38	  fof(axiom_m1, axiom, axiom_m1 <=> ![X, Y]: is_a_theorem(strict_implies(and(X, Y), and(Y, X)))).
45.18/45.38	  fof(hilbert_and_1, axiom, and_1).
45.18/45.38	  fof(hilbert_and_2, axiom, and_2).
45.18/45.38	  fof(hilbert_and_3, axiom, and_3).
45.18/45.38	  fof(hilbert_implies_1, axiom, implies_1).
45.18/45.38	  fof(hilbert_implies_2, axiom, implies_2).
45.18/45.38	  fof(hilbert_modus_ponens, axiom, modus_ponens).
45.18/45.38	  fof(hilbert_modus_tollens, axiom, modus_tollens).
45.18/45.38	  fof(hilbert_op_equiv, axiom, op_equiv).
45.18/45.38	  fof(hilbert_op_implies_and, axiom, op_implies_and).
45.18/45.38	  fof(hilbert_op_or, axiom, op_or).
45.18/45.38	  fof(hilbert_or_1, axiom, or_1).
45.18/45.38	  fof(hilbert_or_2, axiom, or_2).
45.18/45.38	  fof(hilbert_or_3, axiom, or_3).
45.18/45.38	  fof(implies_1, axiom, implies_1 <=> ![X, Y]: is_a_theorem(implies(X, implies(Y, X)))).
45.18/45.38	  fof(implies_2, axiom, implies_2 <=> ![X, Y]: is_a_theorem(implies(implies(X, implies(X, Y)), implies(X, Y)))).
45.18/45.38	  fof(km5_axiom_M, axiom, axiom_M).
45.18/45.38	  fof(km5_necessitation, axiom, necessitation).
45.18/45.38	  fof(km5_op_possibly, axiom, op_possibly).
45.18/45.38	  fof(kn1, axiom, ![P]: is_a_theorem(implies(P, and(P, P))) <=> kn1).
45.18/45.38	  fof(modus_ponens, axiom, modus_ponens <=> ![X, Y]: ((is_a_theorem(X) & is_a_theorem(implies(X, Y))) => is_a_theorem(Y))).
45.18/45.38	  fof(modus_tollens, axiom, ![X, Y]: is_a_theorem(implies(implies(not(Y), not(X)), implies(X, Y))) <=> modus_tollens).
45.18/45.38	  fof(necessitation, axiom, necessitation <=> ![X]: (is_a_theorem(necessarily(X)) <= is_a_theorem(X))).
45.18/45.38	  fof(op_and, axiom, ![X, Y]: not(or(not(X), not(Y)))=and(X, Y) <= op_and).
45.18/45.38	  fof(op_equiv, axiom, op_equiv => ![X, Y]: equiv(X, Y)=and(implies(X, Y), implies(Y, X))).
45.18/45.38	  fof(op_implies_and, axiom, op_implies_and => ![X, Y]: implies(X, Y)=not(and(X, not(Y)))).
45.18/45.38	  fof(op_or, axiom, op_or => ![X, Y]: or(X, Y)=not(and(not(X), not(Y)))).
45.18/45.38	  fof(op_possibly, axiom, ![X]: not(necessarily(not(X)))=possibly(X) <= op_possibly).
45.18/45.38	  fof(op_strict_implies, axiom, ![X, Y]: necessarily(implies(X, Y))=strict_implies(X, Y) <= op_strict_implies).
45.18/45.38	  fof(or_1, axiom, or_1 <=> ![X, Y]: is_a_theorem(implies(X, or(X, Y)))).
45.18/45.38	  fof(or_2, axiom, or_2 <=> ![X, Y]: is_a_theorem(implies(Y, or(X, Y)))).
45.18/45.38	  fof(or_3, axiom, ![X, Y, Z]: is_a_theorem(implies(implies(X, Z), implies(implies(Y, Z), implies(or(X, Y), Z)))) <=> or_3).
45.18/45.38	  fof(s1_0_axiom_m1, conjecture, axiom_m1).
45.18/45.38	  fof(s1_0_op_strict_implies, axiom, op_strict_implies).
45.18/45.38	  fof(substitution_of_equivalents, axiom, substitution_of_equivalents <=> ![X, Y]: (is_a_theorem(equiv(X, Y)) => X=Y)).
45.18/45.38	  fof(substitution_of_equivalents, axiom, substitution_of_equivalents).
45.18/45.38	
45.18/45.38	Now clausify the problem and encode Horn clauses using encoding 3 of
45.18/45.38	http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
45.18/45.38	We repeatedly replace C & s=t => u=v by the two clauses:
45.18/45.38	  fresh(y, y, x1...xn) = u
45.18/45.38	  C => fresh(s, t, x1...xn) = v
45.18/45.38	where fresh is a fresh function symbol and x1..xn are the free
45.18/45.38	variables of u and v.
45.18/45.38	A predicate p(X) is encoded as p(X)=true (this is sound, because the
45.18/45.38	input problem has no model of domain size 1).
45.18/45.38	
45.18/45.38	The encoding turns the above axioms into the following unit equations and goals:
45.18/45.38	
45.18/45.38	Axiom 1 (and_1_1): fresh107(X, X, Y, Z) = true.
45.18/45.38	Axiom 2 (and_2_1): fresh105(X, X, Y, Z) = true.
45.18/45.38	Axiom 3 (and_3_1): fresh103(X, X, Y, Z) = true.
45.18/45.38	Axiom 4 (axiom_M_1): fresh93(X, X, Y) = true.
45.18/45.38	Axiom 5 (axiom_m1): fresh92(X, X) = true.
45.18/45.38	Axiom 6 (implies_1_1): fresh51(X, X, Y, Z) = true.
45.18/45.38	Axiom 7 (implies_2_1): fresh49(X, X, Y, Z) = true.
45.18/45.38	Axiom 8 (modus_ponens_2): fresh40(X, X, Y, Z) = is_a_theorem(Z).
45.18/45.38	Axiom 9 (modus_ponens_2): fresh116(X, X, Y) = true.
45.18/45.38	Axiom 10 (modus_ponens_2): fresh115(X, X, Y, Z) = fresh116(is_a_theorem(Y), true, Z).
45.18/45.38	Axiom 11 (modus_tollens_1): fresh35(X, X, Y, Z) = true.
45.18/45.38	Axiom 12 (necessitation_1): fresh34(X, X, Y) = is_a_theorem(necessarily(Y)).
45.18/45.38	Axiom 13 (necessitation_1): fresh33(X, X, Y) = true.
45.18/45.38	Axiom 14 (op_equiv): fresh30(X, X, Y, Z) = equiv(Y, Z).
45.18/45.38	Axiom 15 (op_implies_and): fresh29(X, X, Y, Z) = implies(Y, Z).
45.18/45.38	Axiom 16 (op_or): fresh26(X, X, Y, Z) = or(Y, Z).
45.18/45.38	Axiom 17 (op_possibly): fresh25(X, X, Y) = possibly(Y).
45.18/45.38	Axiom 18 (op_strict_implies): fresh23(X, X, Y, Z) = strict_implies(Y, Z).
45.18/45.38	Axiom 19 (or_1_1): fresh21(X, X, Y, Z) = true.
45.18/45.38	Axiom 20 (or_2_1): fresh19(X, X, Y, Z) = true.
45.18/45.38	Axiom 21 (or_3_1): fresh17(X, X, Y, Z, W) = true.
45.18/45.38	Axiom 22 (substitution_of_equivalents_2): fresh4(X, X, Y, Z) = Y.
45.18/45.38	Axiom 23 (substitution_of_equivalents_2): fresh3(X, X, Y, Z) = Z.
45.18/45.38	Axiom 24 (and_3_1): fresh103(and_3, true, X, Y) = is_a_theorem(implies(X, implies(Y, and(X, Y)))).
45.18/45.38	Axiom 25 (substitution_of_equivalents_2): fresh4(substitution_of_equivalents, true, X, Y) = fresh3(is_a_theorem(equiv(X, Y)), true, X, Y).
45.18/45.38	Axiom 26 (modus_ponens_2): fresh115(modus_ponens, true, X, Y) = fresh40(is_a_theorem(implies(X, Y)), true, X, Y).
45.18/45.38	Axiom 27 (kn1_1): fresh45(kn1, true, X) = is_a_theorem(implies(X, and(X, X))).
45.18/45.38	Axiom 28 (and_1_1): fresh107(and_1, true, X, Y) = is_a_theorem(implies(and(X, Y), X)).
45.18/45.38	Axiom 29 (and_2_1): fresh105(and_2, true, X, Y) = is_a_theorem(implies(and(X, Y), Y)).
45.18/45.38	Axiom 30 (or_2_1): fresh19(or_2, true, X, Y) = is_a_theorem(implies(Y, or(X, Y))).
45.18/45.38	Axiom 31 (implies_2_1): fresh49(implies_2, true, X, Y) = is_a_theorem(implies(implies(X, implies(X, Y)), implies(X, Y))).
45.18/45.38	Axiom 32 (or_3_1): fresh17(or_3, true, X, Y, Z) = is_a_theorem(implies(implies(X, Z), implies(implies(Y, Z), implies(or(X, Y), Z)))).
45.18/45.38	Axiom 33 (modus_tollens_1): fresh35(modus_tollens, true, X, Y) = is_a_theorem(implies(implies(not(Y), not(X)), implies(X, Y))).
45.18/45.38	Axiom 34 (implies_1_1): fresh51(implies_1, true, X, Y) = is_a_theorem(implies(X, implies(Y, X))).
45.18/45.38	Axiom 35 (or_1_1): fresh21(or_1, true, X, Y) = is_a_theorem(implies(X, or(X, Y))).
45.18/45.38	Axiom 36 (op_and): fresh31(op_and, true, X, Y) = not(or(not(X), not(Y))).
45.18/45.38	Axiom 37 (op_or): fresh26(op_or, true, X, Y) = not(and(not(X), not(Y))).
45.18/45.38	Axiom 38 (op_equiv): fresh30(op_equiv, true, X, Y) = and(implies(X, Y), implies(Y, X)).
45.18/45.38	Axiom 39 (op_implies_and): fresh29(op_implies_and, true, X, Y) = not(and(X, not(Y))).
45.18/45.38	Axiom 40 (hilbert_modus_tollens): modus_tollens = true.
45.18/45.38	Axiom 41 (hilbert_and_2): and_2 = true.
45.18/45.38	Axiom 42 (hilbert_and_1): and_1 = true.
45.18/45.38	Axiom 43 (substitution_of_equivalents): substitution_of_equivalents = true.
45.18/45.38	Axiom 44 (hilbert_and_3): and_3 = true.
45.18/45.38	Axiom 45 (hilbert_implies_1): implies_1 = true.
45.18/45.38	Axiom 46 (hilbert_or_1): or_1 = true.
45.18/45.38	Axiom 47 (hilbert_or_2): or_2 = true.
45.18/45.38	Axiom 48 (hilbert_modus_ponens): modus_ponens = true.
45.18/45.38	Axiom 49 (hilbert_or_3): or_3 = true.
45.18/45.38	Axiom 50 (hilbert_implies_2): implies_2 = true.
45.18/45.38	Axiom 51 (hilbert_op_equiv): op_equiv = true.
45.18/45.38	Axiom 52 (hilbert_op_implies_and): op_implies_and = true.
45.18/45.38	Axiom 53 (hilbert_op_or): op_or = true.
45.18/45.38	Axiom 54 (axiom_m1_1): fresh89(axiom_m1, true, X, Y) = is_a_theorem(strict_implies(and(X, Y), and(Y, X))).
45.18/45.38	Axiom 55 (axiom_m1): fresh92(is_a_theorem(strict_implies(and(sK28_axiom_m1_X, sK27_axiom_m1_Y), and(sK27_axiom_m1_Y, sK28_axiom_m1_X))), true) = axiom_m1.
45.18/45.38	Axiom 56 (necessitation_1): fresh34(necessitation, true, X) = fresh33(is_a_theorem(X), true, X).
45.18/45.38	Axiom 57 (axiom_M_1): fresh93(axiom_M, true, X) = is_a_theorem(implies(necessarily(X), X)).
45.18/45.38	Axiom 58 (op_strict_implies): fresh23(op_strict_implies, true, X, Y) = necessarily(implies(X, Y)).
45.18/45.38	Axiom 59 (op_possibly): fresh25(op_possibly, true, X) = not(necessarily(not(X))).
45.18/45.38	Axiom 60 (km5_axiom_M): axiom_M = true.
45.18/45.38	Axiom 61 (km5_necessitation): necessitation = true.
45.18/45.38	Axiom 62 (km5_op_possibly): op_possibly = true.
45.18/45.41	Axiom 63 (s1_0_op_strict_implies): op_strict_implies = true.
45.18/45.41	
45.18/45.41	Lemma 64: not(and(X, not(Y))) = implies(X, Y).
45.18/45.41	Proof:
45.18/45.41	  not(and(X, not(Y)))
45.18/45.41	= { by axiom 39 (op_implies_and) }
45.18/45.41	  fresh29(op_implies_and, true, X, Y)
45.18/45.41	= { by axiom 52 (hilbert_op_implies_and) }
45.18/45.41	  fresh29(true, true, X, Y)
45.18/45.41	= { by axiom 15 (op_implies_and) }
45.18/45.41	  implies(X, Y)
45.18/45.41	
45.18/45.41	Lemma 65: fresh3(is_a_theorem(equiv(X, Y)), true, X, Y) = X.
45.18/45.41	Proof:
45.18/45.41	  fresh3(is_a_theorem(equiv(X, Y)), true, X, Y)
45.18/45.41	= { by axiom 25 (substitution_of_equivalents_2) }
45.18/45.41	  fresh4(substitution_of_equivalents, true, X, Y)
45.18/45.41	= { by axiom 43 (substitution_of_equivalents) }
45.18/45.41	  fresh4(true, true, X, Y)
45.18/45.41	= { by axiom 22 (substitution_of_equivalents_2) }
45.18/45.41	  X
45.18/45.41	
45.18/45.41	Lemma 66: and(implies(X, Y), implies(Y, X)) = equiv(X, Y).
45.18/45.41	Proof:
45.18/45.41	  and(implies(X, Y), implies(Y, X))
45.18/45.41	= { by axiom 38 (op_equiv) }
45.18/45.41	  fresh30(op_equiv, true, X, Y)
45.18/45.41	= { by axiom 51 (hilbert_op_equiv) }
45.18/45.41	  fresh30(true, true, X, Y)
45.18/45.41	= { by axiom 14 (op_equiv) }
45.18/45.41	  equiv(X, Y)
45.18/45.41	
45.18/45.41	Lemma 67: fresh40(is_a_theorem(implies(X, Y)), true, X, Y) = fresh116(is_a_theorem(X), true, Y).
45.18/45.41	Proof:
45.18/45.41	  fresh40(is_a_theorem(implies(X, Y)), true, X, Y)
45.18/45.41	= { by axiom 26 (modus_ponens_2) }
45.18/45.41	  fresh115(modus_ponens, true, X, Y)
45.18/45.41	= { by axiom 48 (hilbert_modus_ponens) }
45.18/45.41	  fresh115(true, true, X, Y)
45.18/45.41	= { by axiom 10 (modus_ponens_2) }
45.18/45.41	  fresh116(is_a_theorem(X), true, Y)
45.18/45.41	
45.18/45.41	Lemma 68: is_a_theorem(implies(X, implies(Y, and(X, Y)))) = true.
45.18/45.41	Proof:
45.18/45.41	  is_a_theorem(implies(X, implies(Y, and(X, Y))))
45.18/45.41	= { by axiom 24 (and_3_1) }
45.18/45.41	  fresh103(and_3, true, X, Y)
45.18/45.41	= { by axiom 44 (hilbert_and_3) }
45.18/45.41	  fresh103(true, true, X, Y)
45.18/45.41	= { by axiom 3 (and_3_1) }
45.18/45.41	  true
45.18/45.41	
45.18/45.41	Lemma 69: fresh116(is_a_theorem(X), true, implies(Y, and(X, Y))) = is_a_theorem(implies(Y, and(X, Y))).
45.18/45.41	Proof:
45.18/45.41	  fresh116(is_a_theorem(X), true, implies(Y, and(X, Y)))
45.18/45.41	= { by lemma 67 }
45.18/45.41	  fresh40(is_a_theorem(implies(X, implies(Y, and(X, Y)))), true, X, implies(Y, and(X, Y)))
45.18/45.41	= { by lemma 68 }
45.18/45.41	  fresh40(true, true, X, implies(Y, and(X, Y)))
45.18/45.41	= { by axiom 8 (modus_ponens_2) }
45.18/45.41	  is_a_theorem(implies(Y, and(X, Y)))
45.18/45.41	
45.18/45.41	Lemma 70: fresh116(is_a_theorem(implies(X, implies(X, Y))), true, implies(X, Y)) = is_a_theorem(implies(X, Y)).
45.18/45.41	Proof:
45.18/45.41	  fresh116(is_a_theorem(implies(X, implies(X, Y))), true, implies(X, Y))
45.18/45.41	= { by lemma 67 }
45.18/45.41	  fresh40(is_a_theorem(implies(implies(X, implies(X, Y)), implies(X, Y))), true, implies(X, implies(X, Y)), implies(X, Y))
45.18/45.41	= { by axiom 31 (implies_2_1) }
45.18/45.41	  fresh40(fresh49(implies_2, true, X, Y), true, implies(X, implies(X, Y)), implies(X, Y))
45.18/45.41	= { by axiom 50 (hilbert_implies_2) }
45.18/45.41	  fresh40(fresh49(true, true, X, Y), true, implies(X, implies(X, Y)), implies(X, Y))
45.18/45.41	= { by axiom 7 (implies_2_1) }
45.18/45.41	  fresh40(true, true, implies(X, implies(X, Y)), implies(X, Y))
45.18/45.41	= { by axiom 8 (modus_ponens_2) }
45.18/45.41	  is_a_theorem(implies(X, Y))
45.18/45.41	
45.18/45.41	Lemma 71: is_a_theorem(implies(X, X)) = true.
45.18/45.41	Proof:
45.18/45.41	  is_a_theorem(implies(X, X))
45.18/45.41	= { by lemma 70 }
45.18/45.41	  fresh116(is_a_theorem(implies(X, implies(X, X))), true, implies(X, X))
45.18/45.41	= { by axiom 34 (implies_1_1) }
45.18/45.41	  fresh116(fresh51(implies_1, true, X, X), true, implies(X, X))
45.18/45.41	= { by axiom 45 (hilbert_implies_1) }
45.18/45.41	  fresh116(fresh51(true, true, X, X), true, implies(X, X))
45.18/45.41	= { by axiom 6 (implies_1_1) }
45.18/45.41	  fresh116(true, true, implies(X, X))
45.18/45.41	= { by axiom 9 (modus_ponens_2) }
45.18/45.42	  true
45.18/45.42	
45.18/45.42	Lemma 72: or(Y, implies(X, X)) = implies(X, X).
45.18/45.42	Proof:
45.18/45.42	  or(Y, implies(X, X))
45.18/45.42	= { by axiom 23 (substitution_of_equivalents_2) }
45.18/45.42	  fresh3(true, true, implies(X, X), or(Y, implies(X, X)))
45.18/45.42	= { by axiom 9 (modus_ponens_2) }
45.18/45.42	  fresh3(fresh116(true, true, and(implies(implies(X, X), or(Y, implies(X, X))), implies(or(Y, implies(X, X)), implies(X, X)))), true, implies(X, X), or(Y, implies(X, X)))
45.18/45.42	= { by axiom 9 (modus_ponens_2) }
45.18/45.42	  fresh3(fresh116(fresh116(true, true, implies(or(Y, implies(X, X)), implies(X, X))), true, and(implies(implies(X, X), or(Y, implies(X, X))), implies(or(Y, implies(X, X)), implies(X, X)))), true, implies(X, X), or(Y, implies(X, X)))
45.18/45.42	= { by lemma 71 }
45.18/45.42	  fresh3(fresh116(fresh116(is_a_theorem(implies(X, X)), true, implies(or(Y, implies(X, X)), implies(X, X))), true, and(implies(implies(X, X), or(Y, implies(X, X))), implies(or(Y, implies(X, X)), implies(X, X)))), true, implies(X, X), or(Y, implies(X, X)))
45.18/45.42	= { by axiom 10 (modus_ponens_2) }
45.18/45.42	  fresh3(fresh116(fresh115(true, true, implies(X, X), implies(or(Y, implies(X, X)), implies(X, X))), true, and(implies(implies(X, X), or(Y, implies(X, X))), implies(or(Y, implies(X, X)), implies(X, X)))), true, implies(X, X), or(Y, implies(X, X)))
45.18/45.42	= { by axiom 48 (hilbert_modus_ponens) }
45.18/45.42	  fresh3(fresh116(fresh115(modus_ponens, true, implies(X, X), implies(or(Y, implies(X, X)), implies(X, X))), true, and(implies(implies(X, X), or(Y, implies(X, X))), implies(or(Y, implies(X, X)), implies(X, X)))), true, implies(X, X), or(Y, implies(X, X)))
45.18/45.42	= { by axiom 26 (modus_ponens_2) }
45.18/45.42	  fresh3(fresh116(fresh40(is_a_theorem(implies(implies(X, X), implies(or(Y, implies(X, X)), implies(X, X)))), true, implies(X, X), implies(or(Y, implies(X, X)), implies(X, X))), true, and(implies(implies(X, X), or(Y, implies(X, X))), implies(or(Y, implies(X, X)), implies(X, X)))), true, implies(X, X), or(Y, implies(X, X)))
45.18/45.42	= { by axiom 34 (implies_1_1) }
45.18/45.42	  fresh3(fresh116(fresh40(fresh51(implies_1, true, implies(X, X), or(Y, implies(X, X))), true, implies(X, X), implies(or(Y, implies(X, X)), implies(X, X))), true, and(implies(implies(X, X), or(Y, implies(X, X))), implies(or(Y, implies(X, X)), implies(X, X)))), true, implies(X, X), or(Y, implies(X, X)))
45.18/45.42	= { by axiom 45 (hilbert_implies_1) }
45.18/45.42	  fresh3(fresh116(fresh40(fresh51(true, true, implies(X, X), or(Y, implies(X, X))), true, implies(X, X), implies(or(Y, implies(X, X)), implies(X, X))), true, and(implies(implies(X, X), or(Y, implies(X, X))), implies(or(Y, implies(X, X)), implies(X, X)))), true, implies(X, X), or(Y, implies(X, X)))
45.18/45.42	= { by axiom 6 (implies_1_1) }
45.18/45.42	  fresh3(fresh116(fresh40(true, true, implies(X, X), implies(or(Y, implies(X, X)), implies(X, X))), true, and(implies(implies(X, X), or(Y, implies(X, X))), implies(or(Y, implies(X, X)), implies(X, X)))), true, implies(X, X), or(Y, implies(X, X)))
45.18/45.42	= { by axiom 8 (modus_ponens_2) }
45.18/45.42	  fresh3(fresh116(is_a_theorem(implies(or(Y, implies(X, X)), implies(X, X))), true, and(implies(implies(X, X), or(Y, implies(X, X))), implies(or(Y, implies(X, X)), implies(X, X)))), true, implies(X, X), or(Y, implies(X, X)))
45.18/45.42	= { by lemma 67 }
45.25/45.42	  fresh3(fresh40(is_a_theorem(implies(implies(or(Y, implies(X, X)), implies(X, X)), and(implies(implies(X, X), or(Y, implies(X, X))), implies(or(Y, implies(X, X)), implies(X, X))))), true, implies(or(Y, implies(X, X)), implies(X, X)), and(implies(implies(X, X), or(Y, implies(X, X))), implies(or(Y, implies(X, X)), implies(X, X)))), true, implies(X, X), or(Y, implies(X, X)))
45.25/45.42	= { by lemma 69 }
45.25/45.42	  fresh3(fresh40(fresh116(is_a_theorem(implies(implies(X, X), or(Y, implies(X, X)))), true, implies(implies(or(Y, implies(X, X)), implies(X, X)), and(implies(implies(X, X), or(Y, implies(X, X))), implies(or(Y, implies(X, X)), implies(X, X))))), true, implies(or(Y, implies(X, X)), implies(X, X)), and(implies(implies(X, X), or(Y, implies(X, X))), implies(or(Y, implies(X, X)), implies(X, X)))), true, implies(X, X), or(Y, implies(X, X)))
45.25/45.42	= { by axiom 30 (or_2_1) }
45.25/45.42	  fresh3(fresh40(fresh116(fresh19(or_2, true, Y, implies(X, X)), true, implies(implies(or(Y, implies(X, X)), implies(X, X)), and(implies(implies(X, X), or(Y, implies(X, X))), implies(or(Y, implies(X, X)), implies(X, X))))), true, implies(or(Y, implies(X, X)), implies(X, X)), and(implies(implies(X, X), or(Y, implies(X, X))), implies(or(Y, implies(X, X)), implies(X, X)))), true, implies(X, X), or(Y, implies(X, X)))
45.25/45.42	= { by axiom 47 (hilbert_or_2) }
45.25/45.42	  fresh3(fresh40(fresh116(fresh19(true, true, Y, implies(X, X)), true, implies(implies(or(Y, implies(X, X)), implies(X, X)), and(implies(implies(X, X), or(Y, implies(X, X))), implies(or(Y, implies(X, X)), implies(X, X))))), true, implies(or(Y, implies(X, X)), implies(X, X)), and(implies(implies(X, X), or(Y, implies(X, X))), implies(or(Y, implies(X, X)), implies(X, X)))), true, implies(X, X), or(Y, implies(X, X)))
45.25/45.42	= { by axiom 20 (or_2_1) }
45.25/45.42	  fresh3(fresh40(fresh116(true, true, implies(implies(or(Y, implies(X, X)), implies(X, X)), and(implies(implies(X, X), or(Y, implies(X, X))), implies(or(Y, implies(X, X)), implies(X, X))))), true, implies(or(Y, implies(X, X)), implies(X, X)), and(implies(implies(X, X), or(Y, implies(X, X))), implies(or(Y, implies(X, X)), implies(X, X)))), true, implies(X, X), or(Y, implies(X, X)))
45.25/45.42	= { by axiom 9 (modus_ponens_2) }
45.25/45.42	  fresh3(fresh40(true, true, implies(or(Y, implies(X, X)), implies(X, X)), and(implies(implies(X, X), or(Y, implies(X, X))), implies(or(Y, implies(X, X)), implies(X, X)))), true, implies(X, X), or(Y, implies(X, X)))
45.25/45.42	= { by axiom 8 (modus_ponens_2) }
45.25/45.42	  fresh3(is_a_theorem(and(implies(implies(X, X), or(Y, implies(X, X))), implies(or(Y, implies(X, X)), implies(X, X)))), true, implies(X, X), or(Y, implies(X, X)))
45.25/45.42	= { by lemma 66 }
45.25/45.42	  fresh3(is_a_theorem(equiv(implies(X, X), or(Y, implies(X, X)))), true, implies(X, X), or(Y, implies(X, X)))
45.25/45.42	= { by lemma 65 }
45.25/45.42	  implies(X, X)
45.25/45.42	
45.25/45.42	Lemma 73: implies(not(X), Y) = or(X, Y).
45.25/45.42	Proof:
45.25/45.42	  implies(not(X), Y)
45.25/45.42	= { by lemma 64 }
45.25/45.42	  not(and(not(X), not(Y)))
45.25/45.42	= { by axiom 37 (op_or) }
45.25/45.42	  fresh26(op_or, true, X, Y)
45.25/45.42	= { by axiom 53 (hilbert_op_or) }
45.25/45.42	  fresh26(true, true, X, Y)
45.25/45.42	= { by axiom 16 (op_or) }
45.25/45.42	  or(X, Y)
45.25/45.42	
45.25/45.42	Lemma 74: and(implies(X, not(Y)), or(Y, X)) = equiv(X, not(Y)).
45.25/45.42	Proof:
45.25/45.42	  and(implies(X, not(Y)), or(Y, X))
45.25/45.42	= { by lemma 73 }
45.25/45.42	  and(implies(X, not(Y)), implies(not(Y), X))
45.25/45.42	= { by axiom 38 (op_equiv) }
45.25/45.42	  fresh30(op_equiv, true, X, not(Y))
45.25/45.42	= { by axiom 51 (hilbert_op_equiv) }
45.25/45.42	  fresh30(true, true, X, not(Y))
45.25/45.42	= { by axiom 14 (op_equiv) }
45.25/45.42	  equiv(X, not(Y))
45.25/45.42	
45.25/45.42	Lemma 75: is_a_theorem(implies(and(X, Y), X)) = true.
45.25/45.42	Proof:
45.25/45.42	  is_a_theorem(implies(and(X, Y), X))
45.25/45.42	= { by axiom 28 (and_1_1) }
45.25/45.42	  fresh107(and_1, true, X, Y)
45.25/45.42	= { by axiom 42 (hilbert_and_1) }
45.25/45.42	  fresh107(true, true, X, Y)
45.25/45.42	= { by axiom 1 (and_1_1) }
45.25/45.43	  true
45.25/45.43	
45.25/45.43	Lemma 76: is_a_theorem(or(implies(X, X), Y)) = true.
45.25/45.43	Proof:
45.25/45.43	  is_a_theorem(or(implies(X, X), Y))
45.25/45.43	= { by axiom 8 (modus_ponens_2) }
45.25/45.43	  fresh40(true, true, implies(X, X), or(implies(X, X), Y))
45.25/45.43	= { by axiom 19 (or_1_1) }
45.25/45.43	  fresh40(fresh21(true, true, implies(X, X), Y), true, implies(X, X), or(implies(X, X), Y))
45.25/45.43	= { by axiom 46 (hilbert_or_1) }
45.25/45.43	  fresh40(fresh21(or_1, true, implies(X, X), Y), true, implies(X, X), or(implies(X, X), Y))
45.25/45.43	= { by axiom 35 (or_1_1) }
45.25/45.43	  fresh40(is_a_theorem(implies(implies(X, X), or(implies(X, X), Y))), true, implies(X, X), or(implies(X, X), Y))
45.25/45.43	= { by axiom 26 (modus_ponens_2) }
45.25/45.43	  fresh115(modus_ponens, true, implies(X, X), or(implies(X, X), Y))
45.25/45.43	= { by axiom 48 (hilbert_modus_ponens) }
45.25/45.43	  fresh115(true, true, implies(X, X), or(implies(X, X), Y))
45.25/45.43	= { by axiom 10 (modus_ponens_2) }
45.25/45.43	  fresh116(is_a_theorem(implies(X, X)), true, or(implies(X, X), Y))
45.25/45.43	= { by lemma 71 }
45.25/45.43	  fresh116(true, true, or(implies(X, X), Y))
45.25/45.43	= { by axiom 9 (modus_ponens_2) }
45.25/45.43	  true
45.25/45.43	
45.25/45.43	Lemma 77: not(not(implies(X, X))) = possibly(implies(X, X)).
45.25/45.43	Proof:
45.25/45.43	  not(not(implies(X, X)))
45.25/45.43	= { by axiom 23 (substitution_of_equivalents_2) }
45.25/45.43	  not(fresh3(true, true, necessarily(not(implies(X, X))), not(implies(X, X))))
45.25/45.43	= { by axiom 9 (modus_ponens_2) }
45.25/45.43	  not(fresh3(fresh116(true, true, and(implies(necessarily(not(implies(X, X))), not(implies(X, X))), or(implies(X, X), necessarily(not(implies(X, X)))))), true, necessarily(not(implies(X, X))), not(implies(X, X))))
45.25/45.43	= { by lemma 76 }
45.25/45.43	  not(fresh3(fresh116(is_a_theorem(or(implies(X, X), necessarily(not(implies(X, X))))), true, and(implies(necessarily(not(implies(X, X))), not(implies(X, X))), or(implies(X, X), necessarily(not(implies(X, X)))))), true, necessarily(not(implies(X, X))), not(implies(X, X))))
45.25/45.43	= { by lemma 67 }
45.25/45.43	  not(fresh3(fresh40(is_a_theorem(implies(or(implies(X, X), necessarily(not(implies(X, X)))), and(implies(necessarily(not(implies(X, X))), not(implies(X, X))), or(implies(X, X), necessarily(not(implies(X, X))))))), true, or(implies(X, X), necessarily(not(implies(X, X)))), and(implies(necessarily(not(implies(X, X))), not(implies(X, X))), or(implies(X, X), necessarily(not(implies(X, X)))))), true, necessarily(not(implies(X, X))), not(implies(X, X))))
45.25/45.43	= { by lemma 69 }
45.25/45.43	  not(fresh3(fresh40(fresh116(is_a_theorem(implies(necessarily(not(implies(X, X))), not(implies(X, X)))), true, implies(or(implies(X, X), necessarily(not(implies(X, X)))), and(implies(necessarily(not(implies(X, X))), not(implies(X, X))), or(implies(X, X), necessarily(not(implies(X, X))))))), true, or(implies(X, X), necessarily(not(implies(X, X)))), and(implies(necessarily(not(implies(X, X))), not(implies(X, X))), or(implies(X, X), necessarily(not(implies(X, X)))))), true, necessarily(not(implies(X, X))), not(implies(X, X))))
45.25/45.43	= { by axiom 57 (axiom_M_1) }
45.25/45.43	  not(fresh3(fresh40(fresh116(fresh93(axiom_M, true, not(implies(X, X))), true, implies(or(implies(X, X), necessarily(not(implies(X, X)))), and(implies(necessarily(not(implies(X, X))), not(implies(X, X))), or(implies(X, X), necessarily(not(implies(X, X))))))), true, or(implies(X, X), necessarily(not(implies(X, X)))), and(implies(necessarily(not(implies(X, X))), not(implies(X, X))), or(implies(X, X), necessarily(not(implies(X, X)))))), true, necessarily(not(implies(X, X))), not(implies(X, X))))
45.25/45.43	= { by axiom 60 (km5_axiom_M) }
45.25/45.43	  not(fresh3(fresh40(fresh116(fresh93(true, true, not(implies(X, X))), true, implies(or(implies(X, X), necessarily(not(implies(X, X)))), and(implies(necessarily(not(implies(X, X))), not(implies(X, X))), or(implies(X, X), necessarily(not(implies(X, X))))))), true, or(implies(X, X), necessarily(not(implies(X, X)))), and(implies(necessarily(not(implies(X, X))), not(implies(X, X))), or(implies(X, X), necessarily(not(implies(X, X)))))), true, necessarily(not(implies(X, X))), not(implies(X, X))))
45.25/45.43	= { by axiom 4 (axiom_M_1) }
45.25/45.43	  not(fresh3(fresh40(fresh116(true, true, implies(or(implies(X, X), necessarily(not(implies(X, X)))), and(implies(necessarily(not(implies(X, X))), not(implies(X, X))), or(implies(X, X), necessarily(not(implies(X, X))))))), true, or(implies(X, X), necessarily(not(implies(X, X)))), and(implies(necessarily(not(implies(X, X))), not(implies(X, X))), or(implies(X, X), necessarily(not(implies(X, X)))))), true, necessarily(not(implies(X, X))), not(implies(X, X))))
45.25/45.43	= { by axiom 9 (modus_ponens_2) }
45.25/45.43	  not(fresh3(fresh40(true, true, or(implies(X, X), necessarily(not(implies(X, X)))), and(implies(necessarily(not(implies(X, X))), not(implies(X, X))), or(implies(X, X), necessarily(not(implies(X, X)))))), true, necessarily(not(implies(X, X))), not(implies(X, X))))
45.25/45.43	= { by axiom 8 (modus_ponens_2) }
45.25/45.43	  not(fresh3(is_a_theorem(and(implies(necessarily(not(implies(X, X))), not(implies(X, X))), or(implies(X, X), necessarily(not(implies(X, X)))))), true, necessarily(not(implies(X, X))), not(implies(X, X))))
45.25/45.43	= { by lemma 74 }
45.25/45.43	  not(fresh3(is_a_theorem(equiv(necessarily(not(implies(X, X))), not(implies(X, X)))), true, necessarily(not(implies(X, X))), not(implies(X, X))))
45.25/45.43	= { by lemma 65 }
45.25/45.43	  not(necessarily(not(implies(X, X))))
45.25/45.43	= { by axiom 59 (op_possibly) }
45.25/45.43	  fresh25(op_possibly, true, implies(X, X))
45.25/45.43	= { by axiom 62 (km5_op_possibly) }
45.25/45.43	  fresh25(true, true, implies(X, X))
45.25/45.43	= { by axiom 17 (op_possibly) }
45.25/45.43	  possibly(implies(X, X))
45.25/45.43	
45.25/45.43	Lemma 78: or(implies(X, X), Y) = possibly(implies(X, X)).
45.25/45.43	Proof:
45.25/45.43	  or(implies(X, X), Y)
45.25/45.43	= { by lemma 73 }
45.25/45.43	  implies(not(implies(X, X)), Y)
45.25/45.43	= { by lemma 64 }
45.25/45.43	  not(and(not(implies(X, X)), not(Y)))
45.25/45.43	= { by lemma 65 }
45.25/45.43	  not(fresh3(is_a_theorem(equiv(and(not(implies(X, X)), not(Y)), not(implies(X, X)))), true, and(not(implies(X, X)), not(Y)), not(implies(X, X))))
45.25/45.43	= { by lemma 74 }
45.25/45.43	  not(fresh3(is_a_theorem(and(implies(and(not(implies(X, X)), not(Y)), not(implies(X, X))), or(implies(X, X), and(not(implies(X, X)), not(Y))))), true, and(not(implies(X, X)), not(Y)), not(implies(X, X))))
45.25/45.43	= { by axiom 8 (modus_ponens_2) }
45.25/45.43	  not(fresh3(fresh40(true, true, or(implies(X, X), and(not(implies(X, X)), not(Y))), and(implies(and(not(implies(X, X)), not(Y)), not(implies(X, X))), or(implies(X, X), and(not(implies(X, X)), not(Y))))), true, and(not(implies(X, X)), not(Y)), not(implies(X, X))))
45.25/45.43	= { by axiom 9 (modus_ponens_2) }
45.25/45.43	  not(fresh3(fresh40(fresh116(true, true, implies(or(implies(X, X), and(not(implies(X, X)), not(Y))), and(implies(and(not(implies(X, X)), not(Y)), not(implies(X, X))), or(implies(X, X), and(not(implies(X, X)), not(Y)))))), true, or(implies(X, X), and(not(implies(X, X)), not(Y))), and(implies(and(not(implies(X, X)), not(Y)), not(implies(X, X))), or(implies(X, X), and(not(implies(X, X)), not(Y))))), true, and(not(implies(X, X)), not(Y)), not(implies(X, X))))
45.25/45.43	= { by lemma 75 }
45.25/45.43	  not(fresh3(fresh40(fresh116(is_a_theorem(implies(and(not(implies(X, X)), not(Y)), not(implies(X, X)))), true, implies(or(implies(X, X), and(not(implies(X, X)), not(Y))), and(implies(and(not(implies(X, X)), not(Y)), not(implies(X, X))), or(implies(X, X), and(not(implies(X, X)), not(Y)))))), true, or(implies(X, X), and(not(implies(X, X)), not(Y))), and(implies(and(not(implies(X, X)), not(Y)), not(implies(X, X))), or(implies(X, X), and(not(implies(X, X)), not(Y))))), true, and(not(implies(X, X)), not(Y)), not(implies(X, X))))
45.25/45.43	= { by lemma 69 }
45.25/45.43	  not(fresh3(fresh40(is_a_theorem(implies(or(implies(X, X), and(not(implies(X, X)), not(Y))), and(implies(and(not(implies(X, X)), not(Y)), not(implies(X, X))), or(implies(X, X), and(not(implies(X, X)), not(Y)))))), true, or(implies(X, X), and(not(implies(X, X)), not(Y))), and(implies(and(not(implies(X, X)), not(Y)), not(implies(X, X))), or(implies(X, X), and(not(implies(X, X)), not(Y))))), true, and(not(implies(X, X)), not(Y)), not(implies(X, X))))
45.25/45.43	= { by lemma 67 }
45.25/45.43	  not(fresh3(fresh116(is_a_theorem(or(implies(X, X), and(not(implies(X, X)), not(Y)))), true, and(implies(and(not(implies(X, X)), not(Y)), not(implies(X, X))), or(implies(X, X), and(not(implies(X, X)), not(Y))))), true, and(not(implies(X, X)), not(Y)), not(implies(X, X))))
45.25/45.43	= { by lemma 76 }
45.25/45.43	  not(fresh3(fresh116(true, true, and(implies(and(not(implies(X, X)), not(Y)), not(implies(X, X))), or(implies(X, X), and(not(implies(X, X)), not(Y))))), true, and(not(implies(X, X)), not(Y)), not(implies(X, X))))
45.25/45.43	= { by axiom 9 (modus_ponens_2) }
45.25/45.43	  not(fresh3(true, true, and(not(implies(X, X)), not(Y)), not(implies(X, X))))
45.25/45.43	= { by axiom 23 (substitution_of_equivalents_2) }
45.25/45.43	  not(not(implies(X, X)))
45.25/45.43	= { by lemma 77 }
45.25/45.43	  possibly(implies(X, X))
45.25/45.43	
45.25/45.43	Lemma 79: is_a_theorem(implies(and(X, Y), Y)) = true.
45.25/45.43	Proof:
45.25/45.43	  is_a_theorem(implies(and(X, Y), Y))
45.25/45.43	= { by axiom 29 (and_2_1) }
45.25/45.43	  fresh105(and_2, true, X, Y)
45.25/45.43	= { by axiom 41 (hilbert_and_2) }
45.25/45.43	  fresh105(true, true, X, Y)
45.25/45.43	= { by axiom 2 (and_2_1) }
45.25/45.44	  true
45.25/45.44	
45.25/45.44	Lemma 80: and(X, X) = X.
45.25/45.44	Proof:
45.25/45.44	  and(X, X)
45.25/45.44	= { by axiom 23 (substitution_of_equivalents_2) }
45.25/45.44	  fresh3(true, true, X, and(X, X))
45.25/45.44	= { by axiom 9 (modus_ponens_2) }
45.25/45.44	  fresh3(fresh116(true, true, equiv(X, and(X, X))), true, X, and(X, X))
45.25/45.44	= { by lemma 79 }
45.25/45.44	  fresh3(fresh116(is_a_theorem(implies(and(X, X), X)), true, equiv(X, and(X, X))), true, X, and(X, X))
45.25/45.44	= { by lemma 67 }
45.25/45.44	  fresh3(fresh40(is_a_theorem(implies(implies(and(X, X), X), equiv(X, and(X, X)))), true, implies(and(X, X), X), equiv(X, and(X, X))), true, X, and(X, X))
45.25/45.44	= { by lemma 66 }
45.25/45.44	  fresh3(fresh40(is_a_theorem(implies(implies(and(X, X), X), and(implies(X, and(X, X)), implies(and(X, X), X)))), true, implies(and(X, X), X), equiv(X, and(X, X))), true, X, and(X, X))
45.25/45.44	= { by lemma 69 }
45.25/45.44	  fresh3(fresh40(fresh116(is_a_theorem(implies(X, and(X, X))), true, implies(implies(and(X, X), X), and(implies(X, and(X, X)), implies(and(X, X), X)))), true, implies(and(X, X), X), equiv(X, and(X, X))), true, X, and(X, X))
45.25/45.44	= { by lemma 70 }
45.25/45.44	  fresh3(fresh40(fresh116(fresh116(is_a_theorem(implies(X, implies(X, and(X, X)))), true, implies(X, and(X, X))), true, implies(implies(and(X, X), X), and(implies(X, and(X, X)), implies(and(X, X), X)))), true, implies(and(X, X), X), equiv(X, and(X, X))), true, X, and(X, X))
45.25/45.44	= { by lemma 68 }
45.25/45.44	  fresh3(fresh40(fresh116(fresh116(true, true, implies(X, and(X, X))), true, implies(implies(and(X, X), X), and(implies(X, and(X, X)), implies(and(X, X), X)))), true, implies(and(X, X), X), equiv(X, and(X, X))), true, X, and(X, X))
45.25/45.44	= { by axiom 9 (modus_ponens_2) }
45.25/45.44	  fresh3(fresh40(fresh116(true, true, implies(implies(and(X, X), X), and(implies(X, and(X, X)), implies(and(X, X), X)))), true, implies(and(X, X), X), equiv(X, and(X, X))), true, X, and(X, X))
45.25/45.44	= { by axiom 9 (modus_ponens_2) }
45.25/45.44	  fresh3(fresh40(true, true, implies(and(X, X), X), equiv(X, and(X, X))), true, X, and(X, X))
45.25/45.44	= { by axiom 8 (modus_ponens_2) }
45.25/45.44	  fresh3(is_a_theorem(equiv(X, and(X, X))), true, X, and(X, X))
45.25/45.44	= { by lemma 65 }
45.25/45.44	  X
45.25/45.44	
45.25/45.44	Lemma 81: or(X, X) = not(not(X)).
45.25/45.44	Proof:
45.25/45.44	  or(X, X)
45.25/45.44	= { by lemma 73 }
45.25/45.44	  implies(not(X), X)
45.25/45.44	= { by lemma 64 }
45.25/45.44	  not(and(not(X), not(X)))
45.25/45.44	= { by lemma 80 }
45.25/45.44	  not(not(X))
45.25/45.44	
45.25/45.44	Lemma 82: possibly(implies(X, X)) = implies(X, X).
45.25/45.44	Proof:
45.25/45.44	  possibly(implies(X, X))
45.25/45.44	= { by lemma 77 }
45.25/45.44	  not(not(implies(X, X)))
45.25/45.44	= { by lemma 81 }
45.25/45.44	  or(implies(X, X), implies(X, X))
45.25/45.44	= { by lemma 72 }
45.25/45.44	  implies(X, X)
45.25/45.44	
45.25/45.44	Lemma 83: and(implies(?, ?), X) = X.
45.25/45.44	Proof:
45.25/45.44	  and(implies(?, ?), X)
45.25/45.44	= { by lemma 65 }
45.25/45.44	  fresh3(is_a_theorem(equiv(and(implies(?, ?), X), X)), true, and(implies(?, ?), X), X)
45.25/45.44	= { by axiom 8 (modus_ponens_2) }
45.25/45.44	  fresh3(fresh40(true, true, implies(X, and(implies(?, ?), X)), equiv(and(implies(?, ?), X), X)), true, and(implies(?, ?), X), X)
45.25/45.44	= { by axiom 9 (modus_ponens_2) }
45.25/45.44	  fresh3(fresh40(fresh116(true, true, implies(implies(X, and(implies(?, ?), X)), and(implies(and(implies(?, ?), X), X), implies(X, and(implies(?, ?), X))))), true, implies(X, and(implies(?, ?), X)), equiv(and(implies(?, ?), X), X)), true, and(implies(?, ?), X), X)
45.25/45.44	= { by lemma 79 }
45.25/45.44	  fresh3(fresh40(fresh116(is_a_theorem(implies(and(implies(?, ?), X), X)), true, implies(implies(X, and(implies(?, ?), X)), and(implies(and(implies(?, ?), X), X), implies(X, and(implies(?, ?), X))))), true, implies(X, and(implies(?, ?), X)), equiv(and(implies(?, ?), X), X)), true, and(implies(?, ?), X), X)
45.25/45.44	= { by lemma 69 }
45.25/45.44	  fresh3(fresh40(is_a_theorem(implies(implies(X, and(implies(?, ?), X)), and(implies(and(implies(?, ?), X), X), implies(X, and(implies(?, ?), X))))), true, implies(X, and(implies(?, ?), X)), equiv(and(implies(?, ?), X), X)), true, and(implies(?, ?), X), X)
45.25/45.44	= { by lemma 66 }
45.25/45.44	  fresh3(fresh40(is_a_theorem(implies(implies(X, and(implies(?, ?), X)), equiv(and(implies(?, ?), X), X))), true, implies(X, and(implies(?, ?), X)), equiv(and(implies(?, ?), X), X)), true, and(implies(?, ?), X), X)
45.25/45.44	= { by lemma 67 }
45.25/45.44	  fresh3(fresh116(is_a_theorem(implies(X, and(implies(?, ?), X))), true, equiv(and(implies(?, ?), X), X)), true, and(implies(?, ?), X), X)
45.25/45.44	= { by lemma 69 }
45.25/45.44	  fresh3(fresh116(fresh116(is_a_theorem(implies(?, ?)), true, implies(X, and(implies(?, ?), X))), true, equiv(and(implies(?, ?), X), X)), true, and(implies(?, ?), X), X)
45.25/45.44	= { by lemma 71 }
45.25/45.44	  fresh3(fresh116(fresh116(true, true, implies(X, and(implies(?, ?), X))), true, equiv(and(implies(?, ?), X), X)), true, and(implies(?, ?), X), X)
45.25/45.44	= { by axiom 9 (modus_ponens_2) }
45.25/45.44	  fresh3(fresh116(true, true, equiv(and(implies(?, ?), X), X)), true, and(implies(?, ?), X), X)
45.25/45.44	= { by axiom 9 (modus_ponens_2) }
45.25/45.44	  fresh3(true, true, and(implies(?, ?), X), X)
45.25/45.44	= { by axiom 23 (substitution_of_equivalents_2) }
45.25/45.44	  X
45.25/45.44	
45.25/45.44	Lemma 84: or(and(X, not(Y)), Z) = implies(implies(X, Y), Z).
45.25/45.44	Proof:
45.25/45.44	  or(and(X, not(Y)), Z)
45.25/45.44	= { by axiom 16 (op_or) }
45.25/45.44	  fresh26(true, true, and(X, not(Y)), Z)
45.25/45.44	= { by axiom 53 (hilbert_op_or) }
45.25/45.44	  fresh26(op_or, true, and(X, not(Y)), Z)
45.25/45.44	= { by axiom 37 (op_or) }
45.25/45.44	  not(and(not(and(X, not(Y))), not(Z)))
45.25/45.44	= { by lemma 64 }
45.25/45.44	  not(and(implies(X, Y), not(Z)))
45.25/45.44	= { by lemma 64 }
45.25/45.44	  implies(implies(X, Y), Z)
45.25/45.44	
45.25/45.44	Lemma 85: implies(or(X, X), Y) = or(not(X), Y).
45.25/45.44	Proof:
45.25/45.44	  implies(or(X, X), Y)
45.25/45.44	= { by lemma 73 }
45.25/45.44	  implies(implies(not(X), X), Y)
45.25/45.44	= { by lemma 84 }
45.25/45.44	  or(and(not(X), not(X)), Y)
45.25/45.44	= { by lemma 80 }
45.25/45.44	  or(not(X), Y)
45.25/45.44	
45.25/45.44	Lemma 86: fresh116(is_a_theorem(implies(X, Y)), true, or(not(X), Y)) = is_a_theorem(or(not(X), Y)).
45.25/45.44	Proof:
45.25/45.44	  fresh116(is_a_theorem(implies(X, Y)), true, or(not(X), Y))
45.25/45.44	= { by lemma 67 }
45.25/45.44	  fresh40(is_a_theorem(implies(implies(X, Y), or(not(X), Y))), true, implies(X, Y), or(not(X), Y))
45.25/45.44	= { by lemma 85 }
45.25/45.44	  fresh40(is_a_theorem(implies(implies(X, Y), implies(or(X, X), Y))), true, implies(X, Y), or(not(X), Y))
45.25/45.44	= { by lemma 70 }
45.25/45.44	  fresh40(fresh116(is_a_theorem(implies(implies(X, Y), implies(implies(X, Y), implies(or(X, X), Y)))), true, implies(implies(X, Y), implies(or(X, X), Y))), true, implies(X, Y), or(not(X), Y))
45.25/45.44	= { by axiom 32 (or_3_1) }
45.25/45.44	  fresh40(fresh116(fresh17(or_3, true, X, X, Y), true, implies(implies(X, Y), implies(or(X, X), Y))), true, implies(X, Y), or(not(X), Y))
45.25/45.44	= { by axiom 49 (hilbert_or_3) }
45.25/45.44	  fresh40(fresh116(fresh17(true, true, X, X, Y), true, implies(implies(X, Y), implies(or(X, X), Y))), true, implies(X, Y), or(not(X), Y))
45.25/45.44	= { by axiom 21 (or_3_1) }
45.25/45.44	  fresh40(fresh116(true, true, implies(implies(X, Y), implies(or(X, X), Y))), true, implies(X, Y), or(not(X), Y))
45.25/45.44	= { by axiom 9 (modus_ponens_2) }
45.25/45.44	  fresh40(true, true, implies(X, Y), or(not(X), Y))
45.25/45.44	= { by axiom 8 (modus_ponens_2) }
45.25/45.46	  is_a_theorem(or(not(X), Y))
45.25/45.46	
45.25/45.46	Lemma 87: or(X, X) = X.
45.25/45.46	Proof:
45.25/45.46	  or(X, X)
45.25/45.46	= { by lemma 65 }
45.25/45.46	  fresh3(is_a_theorem(equiv(or(X, X), X)), true, or(X, X), X)
45.25/45.46	= { by axiom 8 (modus_ponens_2) }
45.25/45.46	  fresh3(fresh40(true, true, implies(X, or(X, X)), equiv(or(X, X), X)), true, or(X, X), X)
45.25/45.46	= { by axiom 9 (modus_ponens_2) }
45.25/45.46	  fresh3(fresh40(fresh116(true, true, implies(implies(X, not(not(X))), and(or(not(and(X, X)), X), implies(X, not(not(X)))))), true, implies(X, or(X, X)), equiv(or(X, X), X)), true, or(X, X), X)
45.25/45.46	= { by axiom 9 (modus_ponens_2) }
45.25/45.46	  fresh3(fresh40(fresh116(fresh116(true, true, or(not(and(X, X)), X)), true, implies(implies(X, not(not(X))), and(or(not(and(X, X)), X), implies(X, not(not(X)))))), true, implies(X, or(X, X)), equiv(or(X, X), X)), true, or(X, X), X)
45.25/45.46	= { by lemma 75 }
45.25/45.46	  fresh3(fresh40(fresh116(fresh116(is_a_theorem(implies(and(X, X), X)), true, or(not(and(X, X)), X)), true, implies(implies(X, not(not(X))), and(or(not(and(X, X)), X), implies(X, not(not(X)))))), true, implies(X, or(X, X)), equiv(or(X, X), X)), true, or(X, X), X)
45.25/45.46	= { by lemma 86 }
45.25/45.46	  fresh3(fresh40(fresh116(is_a_theorem(or(not(and(X, X)), X)), true, implies(implies(X, not(not(X))), and(or(not(and(X, X)), X), implies(X, not(not(X)))))), true, implies(X, or(X, X)), equiv(or(X, X), X)), true, or(X, X), X)
45.25/45.46	= { by lemma 69 }
45.25/45.46	  fresh3(fresh40(is_a_theorem(implies(implies(X, not(not(X))), and(or(not(and(X, X)), X), implies(X, not(not(X)))))), true, implies(X, or(X, X)), equiv(or(X, X), X)), true, or(X, X), X)
45.25/45.46	= { by lemma 80 }
45.25/45.46	  fresh3(fresh40(is_a_theorem(implies(implies(X, not(and(not(X), not(X)))), and(or(not(and(X, X)), X), implies(X, not(not(X)))))), true, implies(X, or(X, X)), equiv(or(X, X), X)), true, or(X, X), X)
45.25/45.46	= { by lemma 64 }
45.25/45.46	  fresh3(fresh40(is_a_theorem(implies(implies(X, implies(not(X), X)), and(or(not(and(X, X)), X), implies(X, not(not(X)))))), true, implies(X, or(X, X)), equiv(or(X, X), X)), true, or(X, X), X)
45.25/45.46	= { by lemma 73 }
45.25/45.46	  fresh3(fresh40(is_a_theorem(implies(implies(X, or(X, X)), and(or(not(and(X, X)), X), implies(X, not(not(X)))))), true, implies(X, or(X, X)), equiv(or(X, X), X)), true, or(X, X), X)
45.25/45.46	= { by lemma 80 }
45.25/45.46	  fresh3(fresh40(is_a_theorem(implies(implies(X, or(X, X)), and(or(not(X), X), implies(X, not(not(X)))))), true, implies(X, or(X, X)), equiv(or(X, X), X)), true, or(X, X), X)
45.25/45.46	= { by lemma 73 }
45.25/45.46	  fresh3(fresh40(is_a_theorem(implies(implies(X, or(X, X)), and(implies(not(not(X)), X), implies(X, not(not(X)))))), true, implies(X, or(X, X)), equiv(or(X, X), X)), true, or(X, X), X)
45.25/45.46	= { by axiom 38 (op_equiv) }
45.25/45.46	  fresh3(fresh40(is_a_theorem(implies(implies(X, or(X, X)), fresh30(op_equiv, true, not(not(X)), X))), true, implies(X, or(X, X)), equiv(or(X, X), X)), true, or(X, X), X)
45.25/45.46	= { by axiom 51 (hilbert_op_equiv) }
45.25/45.46	  fresh3(fresh40(is_a_theorem(implies(implies(X, or(X, X)), fresh30(true, true, not(not(X)), X))), true, implies(X, or(X, X)), equiv(or(X, X), X)), true, or(X, X), X)
45.25/45.46	= { by axiom 14 (op_equiv) }
45.25/45.46	  fresh3(fresh40(is_a_theorem(implies(implies(X, or(X, X)), equiv(not(not(X)), X))), true, implies(X, or(X, X)), equiv(or(X, X), X)), true, or(X, X), X)
45.25/45.46	= { by lemma 80 }
45.25/45.46	  fresh3(fresh40(is_a_theorem(implies(implies(X, or(X, X)), equiv(not(and(not(X), not(X))), X))), true, implies(X, or(X, X)), equiv(or(X, X), X)), true, or(X, X), X)
45.25/45.46	= { by lemma 64 }
45.25/45.46	  fresh3(fresh40(is_a_theorem(implies(implies(X, or(X, X)), equiv(implies(not(X), X), X))), true, implies(X, or(X, X)), equiv(or(X, X), X)), true, or(X, X), X)
45.25/45.46	= { by lemma 73 }
45.25/45.46	  fresh3(fresh40(is_a_theorem(implies(implies(X, or(X, X)), equiv(or(X, X), X))), true, implies(X, or(X, X)), equiv(or(X, X), X)), true, or(X, X), X)
45.25/45.46	= { by lemma 67 }
45.25/45.46	  fresh3(fresh116(is_a_theorem(implies(X, or(X, X))), true, equiv(or(X, X), X)), true, or(X, X), X)
45.25/45.46	= { by axiom 35 (or_1_1) }
45.25/45.46	  fresh3(fresh116(fresh21(or_1, true, X, X), true, equiv(or(X, X), X)), true, or(X, X), X)
45.25/45.46	= { by axiom 46 (hilbert_or_1) }
45.25/45.46	  fresh3(fresh116(fresh21(true, true, X, X), true, equiv(or(X, X), X)), true, or(X, X), X)
45.25/45.46	= { by axiom 19 (or_1_1) }
45.25/45.46	  fresh3(fresh116(true, true, equiv(or(X, X), X)), true, or(X, X), X)
45.25/45.46	= { by axiom 9 (modus_ponens_2) }
45.25/45.46	  fresh3(true, true, or(X, X), X)
45.25/45.46	= { by axiom 23 (substitution_of_equivalents_2) }
45.25/45.46	  X
45.25/45.46	
45.25/45.46	Lemma 88: not(not(X)) = X.
45.25/45.46	Proof:
45.25/45.46	  not(not(X))
45.25/45.46	= { by lemma 81 }
45.25/45.46	  or(X, X)
45.25/45.46	= { by lemma 87 }
45.39/45.62	  X
45.39/45.62	
45.39/45.62	Goal 1 (s1_0_axiom_m1): axiom_m1 = true.
45.39/45.62	Proof:
45.39/45.62	  axiom_m1
45.39/45.62	= { by axiom 55 (axiom_m1) }
45.39/45.62	  fresh92(is_a_theorem(strict_implies(and(sK28_axiom_m1_X, sK27_axiom_m1_Y), and(sK27_axiom_m1_Y, sK28_axiom_m1_X))), true)
45.39/45.62	= { by axiom 18 (op_strict_implies) }
45.39/45.62	  fresh92(is_a_theorem(fresh23(true, true, and(sK28_axiom_m1_X, sK27_axiom_m1_Y), and(sK27_axiom_m1_Y, sK28_axiom_m1_X))), true)
45.39/45.62	= { by axiom 63 (s1_0_op_strict_implies) }
45.39/45.62	  fresh92(is_a_theorem(fresh23(op_strict_implies, true, and(sK28_axiom_m1_X, sK27_axiom_m1_Y), and(sK27_axiom_m1_Y, sK28_axiom_m1_X))), true)
45.39/45.62	= { by axiom 58 (op_strict_implies) }
45.39/45.62	  fresh92(is_a_theorem(necessarily(implies(and(sK28_axiom_m1_X, sK27_axiom_m1_Y), and(sK27_axiom_m1_Y, sK28_axiom_m1_X)))), true)
45.39/45.62	= { by axiom 12 (necessitation_1) }
45.39/45.62	  fresh92(fresh34(true, true, implies(and(sK28_axiom_m1_X, sK27_axiom_m1_Y), and(sK27_axiom_m1_Y, sK28_axiom_m1_X))), true)
45.39/45.62	= { by axiom 61 (km5_necessitation) }
45.39/45.62	  fresh92(fresh34(necessitation, true, implies(and(sK28_axiom_m1_X, sK27_axiom_m1_Y), and(sK27_axiom_m1_Y, sK28_axiom_m1_X))), true)
45.39/45.62	= { by axiom 56 (necessitation_1) }
45.39/45.62	  fresh92(fresh33(is_a_theorem(implies(and(sK28_axiom_m1_X, sK27_axiom_m1_Y), and(sK27_axiom_m1_Y, sK28_axiom_m1_X))), true, implies(and(sK28_axiom_m1_X, sK27_axiom_m1_Y), and(sK27_axiom_m1_Y, sK28_axiom_m1_X))), true)
45.39/45.62	= { by lemma 87 }
45.39/45.62	  fresh92(fresh33(is_a_theorem(implies(and(sK28_axiom_m1_X, or(sK27_axiom_m1_Y, sK27_axiom_m1_Y)), and(sK27_axiom_m1_Y, sK28_axiom_m1_X))), true, implies(and(sK28_axiom_m1_X, sK27_axiom_m1_Y), and(sK27_axiom_m1_Y, sK28_axiom_m1_X))), true)
45.39/45.62	= { by lemma 73 }
45.39/45.62	  fresh92(fresh33(is_a_theorem(implies(and(sK28_axiom_m1_X, implies(not(sK27_axiom_m1_Y), sK27_axiom_m1_Y)), and(sK27_axiom_m1_Y, sK28_axiom_m1_X))), true, implies(and(sK28_axiom_m1_X, sK27_axiom_m1_Y), and(sK27_axiom_m1_Y, sK28_axiom_m1_X))), true)
45.39/45.62	= { by lemma 64 }
45.39/45.62	  fresh92(fresh33(is_a_theorem(implies(and(sK28_axiom_m1_X, not(and(not(sK27_axiom_m1_Y), not(sK27_axiom_m1_Y)))), and(sK27_axiom_m1_Y, sK28_axiom_m1_X))), true, implies(and(sK28_axiom_m1_X, sK27_axiom_m1_Y), and(sK27_axiom_m1_Y, sK28_axiom_m1_X))), true)
45.39/45.62	= { by lemma 80 }
45.39/45.62	  fresh92(fresh33(is_a_theorem(implies(and(sK28_axiom_m1_X, not(not(sK27_axiom_m1_Y))), and(sK27_axiom_m1_Y, sK28_axiom_m1_X))), true, implies(and(sK28_axiom_m1_X, sK27_axiom_m1_Y), and(sK27_axiom_m1_Y, sK28_axiom_m1_X))), true)
45.39/45.62	= { by lemma 88 }
45.39/45.62	  fresh92(fresh33(is_a_theorem(implies(not(not(and(sK28_axiom_m1_X, not(not(sK27_axiom_m1_Y))))), and(sK27_axiom_m1_Y, sK28_axiom_m1_X))), true, implies(and(sK28_axiom_m1_X, sK27_axiom_m1_Y), and(sK27_axiom_m1_Y, sK28_axiom_m1_X))), true)
45.39/45.62	= { by lemma 83 }
45.39/45.62	  fresh92(fresh33(is_a_theorem(implies(not(and(implies(?, ?), not(and(sK28_axiom_m1_X, not(not(sK27_axiom_m1_Y)))))), and(sK27_axiom_m1_Y, sK28_axiom_m1_X))), true, implies(and(sK28_axiom_m1_X, sK27_axiom_m1_Y), and(sK27_axiom_m1_Y, sK28_axiom_m1_X))), true)
45.39/45.62	= { by lemma 64 }
45.39/45.62	  fresh92(fresh33(is_a_theorem(implies(implies(implies(?, ?), and(sK28_axiom_m1_X, not(not(sK27_axiom_m1_Y)))), and(sK27_axiom_m1_Y, sK28_axiom_m1_X))), true, implies(and(sK28_axiom_m1_X, sK27_axiom_m1_Y), and(sK27_axiom_m1_Y, sK28_axiom_m1_X))), true)
45.39/45.62	= { by lemma 84 }
45.39/45.62	  fresh92(fresh33(is_a_theorem(or(and(implies(?, ?), not(and(sK28_axiom_m1_X, not(not(sK27_axiom_m1_Y))))), and(sK27_axiom_m1_Y, sK28_axiom_m1_X))), true, implies(and(sK28_axiom_m1_X, sK27_axiom_m1_Y), and(sK27_axiom_m1_Y, sK28_axiom_m1_X))), true)
45.39/45.62	= { by lemma 64 }
45.39/45.62	  fresh92(fresh33(is_a_theorem(or(and(implies(?, ?), implies(sK28_axiom_m1_X, not(sK27_axiom_m1_Y))), and(sK27_axiom_m1_Y, sK28_axiom_m1_X))), true, implies(and(sK28_axiom_m1_X, sK27_axiom_m1_Y), and(sK27_axiom_m1_Y, sK28_axiom_m1_X))), true)
45.39/45.62	= { by lemma 83 }
45.39/45.62	  fresh92(fresh33(is_a_theorem(or(implies(sK28_axiom_m1_X, not(sK27_axiom_m1_Y)), and(sK27_axiom_m1_Y, sK28_axiom_m1_X))), true, implies(and(sK28_axiom_m1_X, sK27_axiom_m1_Y), and(sK27_axiom_m1_Y, sK28_axiom_m1_X))), true)
45.39/45.62	= { by axiom 8 (modus_ponens_2) }
45.39/45.62	  fresh92(fresh33(fresh40(true, true, or(and(sK27_axiom_m1_Y, sK28_axiom_m1_X), not(not(implies(sK28_axiom_m1_X, not(sK27_axiom_m1_Y))))), or(implies(sK28_axiom_m1_X, not(sK27_axiom_m1_Y)), and(sK27_axiom_m1_Y, sK28_axiom_m1_X))), true, implies(and(sK28_axiom_m1_X, sK27_axiom_m1_Y), and(sK27_axiom_m1_Y, sK28_axiom_m1_X))), true)
45.39/45.62	= { by axiom 11 (modus_tollens_1) }
45.39/45.62	  fresh92(fresh33(fresh40(fresh35(true, true, not(implies(sK28_axiom_m1_X, not(sK27_axiom_m1_Y))), and(sK27_axiom_m1_Y, sK28_axiom_m1_X)), true, or(and(sK27_axiom_m1_Y, sK28_axiom_m1_X), not(not(implies(sK28_axiom_m1_X, not(sK27_axiom_m1_Y))))), or(implies(sK28_axiom_m1_X, not(sK27_axiom_m1_Y)), and(sK27_axiom_m1_Y, sK28_axiom_m1_X))), true, implies(and(sK28_axiom_m1_X, sK27_axiom_m1_Y), and(sK27_axiom_m1_Y, sK28_axiom_m1_X))), true)
45.39/45.62	= { by axiom 40 (hilbert_modus_tollens) }
45.39/45.62	  fresh92(fresh33(fresh40(fresh35(modus_tollens, true, not(implies(sK28_axiom_m1_X, not(sK27_axiom_m1_Y))), and(sK27_axiom_m1_Y, sK28_axiom_m1_X)), true, or(and(sK27_axiom_m1_Y, sK28_axiom_m1_X), not(not(implies(sK28_axiom_m1_X, not(sK27_axiom_m1_Y))))), or(implies(sK28_axiom_m1_X, not(sK27_axiom_m1_Y)), and(sK27_axiom_m1_Y, sK28_axiom_m1_X))), true, implies(and(sK28_axiom_m1_X, sK27_axiom_m1_Y), and(sK27_axiom_m1_Y, sK28_axiom_m1_X))), true)
45.39/45.62	= { by axiom 33 (modus_tollens_1) }
45.39/45.62	  fresh92(fresh33(fresh40(is_a_theorem(implies(implies(not(and(sK27_axiom_m1_Y, sK28_axiom_m1_X)), not(not(implies(sK28_axiom_m1_X, not(sK27_axiom_m1_Y))))), implies(not(implies(sK28_axiom_m1_X, not(sK27_axiom_m1_Y))), and(sK27_axiom_m1_Y, sK28_axiom_m1_X)))), true, or(and(sK27_axiom_m1_Y, sK28_axiom_m1_X), not(not(implies(sK28_axiom_m1_X, not(sK27_axiom_m1_Y))))), or(implies(sK28_axiom_m1_X, not(sK27_axiom_m1_Y)), and(sK27_axiom_m1_Y, sK28_axiom_m1_X))), true, implies(and(sK28_axiom_m1_X, sK27_axiom_m1_Y), and(sK27_axiom_m1_Y, sK28_axiom_m1_X))), true)
45.39/45.62	= { by lemma 73 }
45.39/45.62	  fresh92(fresh33(fresh40(is_a_theorem(implies(or(and(sK27_axiom_m1_Y, sK28_axiom_m1_X), not(not(implies(sK28_axiom_m1_X, not(sK27_axiom_m1_Y))))), implies(not(implies(sK28_axiom_m1_X, not(sK27_axiom_m1_Y))), and(sK27_axiom_m1_Y, sK28_axiom_m1_X)))), true, or(and(sK27_axiom_m1_Y, sK28_axiom_m1_X), not(not(implies(sK28_axiom_m1_X, not(sK27_axiom_m1_Y))))), or(implies(sK28_axiom_m1_X, not(sK27_axiom_m1_Y)), and(sK27_axiom_m1_Y, sK28_axiom_m1_X))), true, implies(and(sK28_axiom_m1_X, sK27_axiom_m1_Y), and(sK27_axiom_m1_Y, sK28_axiom_m1_X))), true)
45.39/45.62	= { by lemma 73 }
45.39/45.62	  fresh92(fresh33(fresh40(is_a_theorem(implies(or(and(sK27_axiom_m1_Y, sK28_axiom_m1_X), not(not(implies(sK28_axiom_m1_X, not(sK27_axiom_m1_Y))))), or(implies(sK28_axiom_m1_X, not(sK27_axiom_m1_Y)), and(sK27_axiom_m1_Y, sK28_axiom_m1_X)))), true, or(and(sK27_axiom_m1_Y, sK28_axiom_m1_X), not(not(implies(sK28_axiom_m1_X, not(sK27_axiom_m1_Y))))), or(implies(sK28_axiom_m1_X, not(sK27_axiom_m1_Y)), and(sK27_axiom_m1_Y, sK28_axiom_m1_X))), true, implies(and(sK28_axiom_m1_X, sK27_axiom_m1_Y), and(sK27_axiom_m1_Y, sK28_axiom_m1_X))), true)
45.39/45.62	= { by lemma 67 }
45.39/45.62	  fresh92(fresh33(fresh116(is_a_theorem(or(and(sK27_axiom_m1_Y, sK28_axiom_m1_X), not(not(implies(sK28_axiom_m1_X, not(sK27_axiom_m1_Y)))))), true, or(implies(sK28_axiom_m1_X, not(sK27_axiom_m1_Y)), and(sK27_axiom_m1_Y, sK28_axiom_m1_X))), true, implies(and(sK28_axiom_m1_X, sK27_axiom_m1_Y), and(sK27_axiom_m1_Y, sK28_axiom_m1_X))), true)
45.39/45.62	= { by lemma 88 }
45.39/45.62	  fresh92(fresh33(fresh116(is_a_theorem(or(and(sK27_axiom_m1_Y, not(not(sK28_axiom_m1_X))), not(not(implies(sK28_axiom_m1_X, not(sK27_axiom_m1_Y)))))), true, or(implies(sK28_axiom_m1_X, not(sK27_axiom_m1_Y)), and(sK27_axiom_m1_Y, sK28_axiom_m1_X))), true, implies(and(sK28_axiom_m1_X, sK27_axiom_m1_Y), and(sK27_axiom_m1_Y, sK28_axiom_m1_X))), true)
45.39/45.62	= { by lemma 87 }
45.39/45.62	  fresh92(fresh33(fresh116(is_a_theorem(or(or(and(sK27_axiom_m1_Y, not(not(sK28_axiom_m1_X))), and(sK27_axiom_m1_Y, not(not(sK28_axiom_m1_X)))), not(not(implies(sK28_axiom_m1_X, not(sK27_axiom_m1_Y)))))), true, or(implies(sK28_axiom_m1_X, not(sK27_axiom_m1_Y)), and(sK27_axiom_m1_Y, sK28_axiom_m1_X))), true, implies(and(sK28_axiom_m1_X, sK27_axiom_m1_Y), and(sK27_axiom_m1_Y, sK28_axiom_m1_X))), true)
45.39/45.62	= { by lemma 84 }
45.39/45.62	  fresh92(fresh33(fresh116(is_a_theorem(or(implies(implies(sK27_axiom_m1_Y, not(sK28_axiom_m1_X)), and(sK27_axiom_m1_Y, not(not(sK28_axiom_m1_X)))), not(not(implies(sK28_axiom_m1_X, not(sK27_axiom_m1_Y)))))), true, or(implies(sK28_axiom_m1_X, not(sK27_axiom_m1_Y)), and(sK27_axiom_m1_Y, sK28_axiom_m1_X))), true, implies(and(sK28_axiom_m1_X, sK27_axiom_m1_Y), and(sK27_axiom_m1_Y, sK28_axiom_m1_X))), true)
45.39/45.62	= { by lemma 64 }
45.39/45.62	  fresh92(fresh33(fresh116(is_a_theorem(or(not(and(implies(sK27_axiom_m1_Y, not(sK28_axiom_m1_X)), not(and(sK27_axiom_m1_Y, not(not(sK28_axiom_m1_X)))))), not(not(implies(sK28_axiom_m1_X, not(sK27_axiom_m1_Y)))))), true, or(implies(sK28_axiom_m1_X, not(sK27_axiom_m1_Y)), and(sK27_axiom_m1_Y, sK28_axiom_m1_X))), true, implies(and(sK28_axiom_m1_X, sK27_axiom_m1_Y), and(sK27_axiom_m1_Y, sK28_axiom_m1_X))), true)
45.39/45.62	= { by lemma 64 }
45.39/45.62	  fresh92(fresh33(fresh116(is_a_theorem(or(not(and(implies(sK27_axiom_m1_Y, not(sK28_axiom_m1_X)), implies(sK27_axiom_m1_Y, not(sK28_axiom_m1_X)))), not(not(implies(sK28_axiom_m1_X, not(sK27_axiom_m1_Y)))))), true, or(implies(sK28_axiom_m1_X, not(sK27_axiom_m1_Y)), and(sK27_axiom_m1_Y, sK28_axiom_m1_X))), true, implies(and(sK28_axiom_m1_X, sK27_axiom_m1_Y), and(sK27_axiom_m1_Y, sK28_axiom_m1_X))), true)
45.39/45.62	= { by lemma 80 }
45.39/45.62	  fresh92(fresh33(fresh116(is_a_theorem(or(not(implies(sK27_axiom_m1_Y, not(sK28_axiom_m1_X))), not(not(implies(sK28_axiom_m1_X, not(sK27_axiom_m1_Y)))))), true, or(implies(sK28_axiom_m1_X, not(sK27_axiom_m1_Y)), and(sK27_axiom_m1_Y, sK28_axiom_m1_X))), true, implies(and(sK28_axiom_m1_X, sK27_axiom_m1_Y), and(sK27_axiom_m1_Y, sK28_axiom_m1_X))), true)
45.39/45.62	= { by lemma 87 }
45.39/45.62	  fresh92(fresh33(fresh116(is_a_theorem(or(not(implies(or(sK27_axiom_m1_Y, sK27_axiom_m1_Y), not(sK28_axiom_m1_X))), not(not(implies(sK28_axiom_m1_X, not(sK27_axiom_m1_Y)))))), true, or(implies(sK28_axiom_m1_X, not(sK27_axiom_m1_Y)), and(sK27_axiom_m1_Y, sK28_axiom_m1_X))), true, implies(and(sK28_axiom_m1_X, sK27_axiom_m1_Y), and(sK27_axiom_m1_Y, sK28_axiom_m1_X))), true)
45.39/45.62	= { by lemma 85 }
45.39/45.62	  fresh92(fresh33(fresh116(is_a_theorem(or(not(or(not(sK27_axiom_m1_Y), not(sK28_axiom_m1_X))), not(not(implies(sK28_axiom_m1_X, not(sK27_axiom_m1_Y)))))), true, or(implies(sK28_axiom_m1_X, not(sK27_axiom_m1_Y)), and(sK27_axiom_m1_Y, sK28_axiom_m1_X))), true, implies(and(sK28_axiom_m1_X, sK27_axiom_m1_Y), and(sK27_axiom_m1_Y, sK28_axiom_m1_X))), true)
45.39/45.62	= { by lemma 80 }
45.39/45.62	  fresh92(fresh33(fresh116(is_a_theorem(or(not(or(not(sK27_axiom_m1_Y), not(sK28_axiom_m1_X))), not(and(not(implies(sK28_axiom_m1_X, not(sK27_axiom_m1_Y))), not(implies(sK28_axiom_m1_X, not(sK27_axiom_m1_Y))))))), true, or(implies(sK28_axiom_m1_X, not(sK27_axiom_m1_Y)), and(sK27_axiom_m1_Y, sK28_axiom_m1_X))), true, implies(and(sK28_axiom_m1_X, sK27_axiom_m1_Y), and(sK27_axiom_m1_Y, sK28_axiom_m1_X))), true)
45.39/45.62	= { by lemma 64 }
45.39/45.62	  fresh92(fresh33(fresh116(is_a_theorem(or(not(or(not(sK27_axiom_m1_Y), not(sK28_axiom_m1_X))), implies(not(implies(sK28_axiom_m1_X, not(sK27_axiom_m1_Y))), implies(sK28_axiom_m1_X, not(sK27_axiom_m1_Y))))), true, or(implies(sK28_axiom_m1_X, not(sK27_axiom_m1_Y)), and(sK27_axiom_m1_Y, sK28_axiom_m1_X))), true, implies(and(sK28_axiom_m1_X, sK27_axiom_m1_Y), and(sK27_axiom_m1_Y, sK28_axiom_m1_X))), true)
45.39/45.62	= { by lemma 73 }
45.39/45.62	  fresh92(fresh33(fresh116(is_a_theorem(or(not(or(not(sK27_axiom_m1_Y), not(sK28_axiom_m1_X))), or(implies(sK28_axiom_m1_X, not(sK27_axiom_m1_Y)), implies(sK28_axiom_m1_X, not(sK27_axiom_m1_Y))))), true, or(implies(sK28_axiom_m1_X, not(sK27_axiom_m1_Y)), and(sK27_axiom_m1_Y, sK28_axiom_m1_X))), true, implies(and(sK28_axiom_m1_X, sK27_axiom_m1_Y), and(sK27_axiom_m1_Y, sK28_axiom_m1_X))), true)
45.39/45.62	= { by lemma 87 }
45.39/45.62	  fresh92(fresh33(fresh116(is_a_theorem(or(not(or(not(sK27_axiom_m1_Y), not(sK28_axiom_m1_X))), implies(sK28_axiom_m1_X, not(sK27_axiom_m1_Y)))), true, or(implies(sK28_axiom_m1_X, not(sK27_axiom_m1_Y)), and(sK27_axiom_m1_Y, sK28_axiom_m1_X))), true, implies(and(sK28_axiom_m1_X, sK27_axiom_m1_Y), and(sK27_axiom_m1_Y, sK28_axiom_m1_X))), true)
45.39/45.62	= { by lemma 86 }
45.39/45.62	  fresh92(fresh33(fresh116(fresh116(is_a_theorem(implies(or(not(sK27_axiom_m1_Y), not(sK28_axiom_m1_X)), implies(sK28_axiom_m1_X, not(sK27_axiom_m1_Y)))), true, or(not(or(not(sK27_axiom_m1_Y), not(sK28_axiom_m1_X))), implies(sK28_axiom_m1_X, not(sK27_axiom_m1_Y)))), true, or(implies(sK28_axiom_m1_X, not(sK27_axiom_m1_Y)), and(sK27_axiom_m1_Y, sK28_axiom_m1_X))), true, implies(and(sK28_axiom_m1_X, sK27_axiom_m1_Y), and(sK27_axiom_m1_Y, sK28_axiom_m1_X))), true)
45.39/45.62	= { by lemma 73 }
45.39/45.62	  fresh92(fresh33(fresh116(fresh116(is_a_theorem(implies(implies(not(not(sK27_axiom_m1_Y)), not(sK28_axiom_m1_X)), implies(sK28_axiom_m1_X, not(sK27_axiom_m1_Y)))), true, or(not(or(not(sK27_axiom_m1_Y), not(sK28_axiom_m1_X))), implies(sK28_axiom_m1_X, not(sK27_axiom_m1_Y)))), true, or(implies(sK28_axiom_m1_X, not(sK27_axiom_m1_Y)), and(sK27_axiom_m1_Y, sK28_axiom_m1_X))), true, implies(and(sK28_axiom_m1_X, sK27_axiom_m1_Y), and(sK27_axiom_m1_Y, sK28_axiom_m1_X))), true)
45.39/45.62	= { by axiom 33 (modus_tollens_1) }
45.39/45.62	  fresh92(fresh33(fresh116(fresh116(fresh35(modus_tollens, true, sK28_axiom_m1_X, not(sK27_axiom_m1_Y)), true, or(not(or(not(sK27_axiom_m1_Y), not(sK28_axiom_m1_X))), implies(sK28_axiom_m1_X, not(sK27_axiom_m1_Y)))), true, or(implies(sK28_axiom_m1_X, not(sK27_axiom_m1_Y)), and(sK27_axiom_m1_Y, sK28_axiom_m1_X))), true, implies(and(sK28_axiom_m1_X, sK27_axiom_m1_Y), and(sK27_axiom_m1_Y, sK28_axiom_m1_X))), true)
45.39/45.62	= { by axiom 40 (hilbert_modus_tollens) }
45.39/45.62	  fresh92(fresh33(fresh116(fresh116(fresh35(true, true, sK28_axiom_m1_X, not(sK27_axiom_m1_Y)), true, or(not(or(not(sK27_axiom_m1_Y), not(sK28_axiom_m1_X))), implies(sK28_axiom_m1_X, not(sK27_axiom_m1_Y)))), true, or(implies(sK28_axiom_m1_X, not(sK27_axiom_m1_Y)), and(sK27_axiom_m1_Y, sK28_axiom_m1_X))), true, implies(and(sK28_axiom_m1_X, sK27_axiom_m1_Y), and(sK27_axiom_m1_Y, sK28_axiom_m1_X))), true)
45.39/45.62	= { by axiom 11 (modus_tollens_1) }
45.39/45.62	  fresh92(fresh33(fresh116(fresh116(true, true, or(not(or(not(sK27_axiom_m1_Y), not(sK28_axiom_m1_X))), implies(sK28_axiom_m1_X, not(sK27_axiom_m1_Y)))), true, or(implies(sK28_axiom_m1_X, not(sK27_axiom_m1_Y)), and(sK27_axiom_m1_Y, sK28_axiom_m1_X))), true, implies(and(sK28_axiom_m1_X, sK27_axiom_m1_Y), and(sK27_axiom_m1_Y, sK28_axiom_m1_X))), true)
45.39/45.62	= { by axiom 9 (modus_ponens_2) }
45.39/45.62	  fresh92(fresh33(fresh116(true, true, or(implies(sK28_axiom_m1_X, not(sK27_axiom_m1_Y)), and(sK27_axiom_m1_Y, sK28_axiom_m1_X))), true, implies(and(sK28_axiom_m1_X, sK27_axiom_m1_Y), and(sK27_axiom_m1_Y, sK28_axiom_m1_X))), true)
45.39/45.62	= { by axiom 9 (modus_ponens_2) }
45.39/45.62	  fresh92(fresh33(true, true, implies(and(sK28_axiom_m1_X, sK27_axiom_m1_Y), and(sK27_axiom_m1_Y, sK28_axiom_m1_X))), true)
45.39/45.62	= { by axiom 13 (necessitation_1) }
45.39/45.62	  fresh92(true, true)
45.39/45.62	= { by axiom 5 (axiom_m1) }
45.39/45.62	  true
45.39/45.62	% SZS output end Proof
45.39/45.62	
45.39/45.62	RESULT: Theorem (the conjecture is true).
45.46/45.64	EOF