0.02/0.10	% Problem    : theBenchmark.p : TPTP v0.0.0. Released v0.0.0.
0.02/0.11	% Command    : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
0.10/0.31	% Computer   : n007.cluster.edu
0.10/0.31	% Model      : x86_64 x86_64
0.10/0.31	% CPU        : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
0.10/0.31	% Memory     : 8042.1875MB
0.10/0.31	% OS         : Linux 3.10.0-693.el7.x86_64
0.10/0.31	% CPULimit   : 1200
0.10/0.31	% WCLimit    : 120
0.10/0.31	% DateTime   : Tue Jul 13 16:26:49 EDT 2021
0.10/0.31	% CPUTime    : 
50.35/6.71	% SZS status Theorem
50.35/6.71	
51.46/6.82	% SZS output start Proof
51.46/6.82	Take the following subset of the input axioms:
51.46/6.82	  fof(and_1, axiom, ![X, Y]: is_a_theorem(implies(and(X, Y), X)) <=> and_1).
51.46/6.82	  fof(and_3, axiom, and_3 <=> ![X, Y]: is_a_theorem(implies(X, implies(Y, and(X, Y))))).
51.46/6.82	  fof(axiom_5, axiom, ![X]: is_a_theorem(implies(possibly(X), necessarily(possibly(X)))) <=> axiom_5).
51.46/6.82	  fof(axiom_M, axiom, axiom_M <=> ![X]: is_a_theorem(implies(necessarily(X), X))).
51.46/6.82	  fof(axiom_m9, axiom, ![X]: is_a_theorem(strict_implies(possibly(possibly(X)), possibly(X))) <=> axiom_m9).
51.46/6.82	  fof(hilbert_and_1, axiom, and_1).
51.46/6.82	  fof(hilbert_and_3, axiom, and_3).
51.46/6.82	  fof(hilbert_implies_1, axiom, implies_1).
51.46/6.82	  fof(hilbert_implies_2, axiom, implies_2).
51.46/6.82	  fof(hilbert_modus_ponens, axiom, modus_ponens).
51.46/6.82	  fof(hilbert_modus_tollens, axiom, modus_tollens).
51.46/6.82	  fof(hilbert_op_equiv, axiom, op_equiv).
51.46/6.82	  fof(hilbert_op_implies_and, axiom, op_implies_and).
51.46/6.82	  fof(hilbert_op_or, axiom, op_or).
51.46/6.82	  fof(implies_1, axiom, implies_1 <=> ![X, Y]: is_a_theorem(implies(X, implies(Y, X)))).
51.46/6.82	  fof(implies_2, axiom, ![X, Y]: is_a_theorem(implies(implies(X, implies(X, Y)), implies(X, Y))) <=> implies_2).
51.46/6.82	  fof(km5_axiom_5, axiom, axiom_5).
51.46/6.82	  fof(km5_axiom_M, axiom, axiom_M).
51.46/6.82	  fof(km5_necessitation, axiom, necessitation).
51.46/6.82	  fof(km5_op_possibly, axiom, op_possibly).
51.46/6.82	  fof(modus_ponens, axiom, ![X, Y]: (is_a_theorem(Y) <= (is_a_theorem(implies(X, Y)) & is_a_theorem(X))) <=> modus_ponens).
51.46/6.82	  fof(modus_tollens, axiom, ![X, Y]: is_a_theorem(implies(implies(not(Y), not(X)), implies(X, Y))) <=> modus_tollens).
51.46/6.82	  fof(necessitation, axiom, ![X]: (is_a_theorem(necessarily(X)) <= is_a_theorem(X)) <=> necessitation).
51.46/6.82	  fof(op_equiv, axiom, op_equiv => ![X, Y]: and(implies(X, Y), implies(Y, X))=equiv(X, Y)).
51.46/6.82	  fof(op_implies_and, axiom, op_implies_and => ![X, Y]: implies(X, Y)=not(and(X, not(Y)))).
51.46/6.82	  fof(op_or, axiom, op_or => ![X, Y]: or(X, Y)=not(and(not(X), not(Y)))).
51.46/6.82	  fof(op_possibly, axiom, ![X]: possibly(X)=not(necessarily(not(X))) <= op_possibly).
51.46/6.82	  fof(op_strict_implies, axiom, op_strict_implies => ![X, Y]: necessarily(implies(X, Y))=strict_implies(X, Y)).
51.46/6.82	  fof(s1_0_m6s3m9b_axiom_m9, conjecture, axiom_m9).
51.54/6.82	  fof(s1_0_op_implies, axiom, op_implies).
51.54/6.82	  fof(s1_0_op_strict_implies, axiom, op_strict_implies).
51.54/6.82	  fof(substitution_of_equivalents, axiom, ![X, Y]: (is_a_theorem(equiv(X, Y)) => X=Y) <=> substitution_of_equivalents).
51.54/6.82	  fof(substitution_of_equivalents, axiom, substitution_of_equivalents).
51.54/6.82	
51.54/6.82	Now clausify the problem and encode Horn clauses using encoding 3 of
51.54/6.82	http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
51.54/6.82	We repeatedly replace C & s=t => u=v by the two clauses:
51.54/6.82	  fresh(y, y, x1...xn) = u
51.54/6.82	  C => fresh(s, t, x1...xn) = v
51.54/6.82	where fresh is a fresh function symbol and x1..xn are the free
51.54/6.82	variables of u and v.
51.54/6.82	A predicate p(X) is encoded as p(X)=true (this is sound, because the
51.54/6.82	input problem has no model of domain size 1).
51.54/6.83	
51.54/6.83	The encoding turns the above axioms into the following unit equations and goals:
51.54/6.83	
51.54/6.83	Axiom 1 (s1_0_op_implies): op_implies = true.
51.54/6.83	Axiom 2 (hilbert_op_implies_and): op_implies_and = true.
51.54/6.83	Axiom 3 (s1_0_op_strict_implies): op_strict_implies = true.
51.54/6.83	Axiom 4 (hilbert_and_1): and_1 = true.
51.54/6.83	Axiom 5 (hilbert_implies_2): implies_2 = true.
51.54/6.83	Axiom 6 (hilbert_modus_ponens): modus_ponens = true.
51.54/6.83	Axiom 7 (hilbert_and_3): and_3 = true.
51.54/6.83	Axiom 8 (hilbert_modus_tollens): modus_tollens = true.
51.54/6.83	Axiom 9 (hilbert_implies_1): implies_1 = true.
51.54/6.83	Axiom 10 (substitution_of_equivalents): substitution_of_equivalents = true.
51.54/6.83	Axiom 11 (hilbert_op_or): op_or = true.
51.54/6.83	Axiom 12 (hilbert_op_equiv): op_equiv = true.
51.54/6.83	Axiom 13 (km5_axiom_5): axiom_5 = true.
51.54/6.83	Axiom 14 (km5_necessitation): necessitation = true.
51.54/6.83	Axiom 15 (km5_axiom_M): axiom_M = true.
51.54/6.83	Axiom 16 (km5_op_possibly): op_possibly = true.
51.54/6.83	Axiom 17 (axiom_m9): fresh74(X, X) = true.
51.54/6.83	Axiom 18 (modus_ponens_2): fresh116(X, X, Y) = true.
51.54/6.83	Axiom 19 (axiom_5_1): fresh99(X, X, Y) = true.
51.54/6.83	Axiom 20 (axiom_M_1): fresh93(X, X, Y) = true.
51.54/6.83	Axiom 21 (necessitation_1): fresh34(X, X, Y) = is_a_theorem(necessarily(Y)).
51.54/6.83	Axiom 22 (necessitation_1): fresh33(X, X, Y) = true.
51.54/6.83	Axiom 23 (op_possibly): fresh25(X, X, Y) = possibly(Y).
51.54/6.83	Axiom 24 (op_possibly): fresh25(op_possibly, true, X) = not(necessarily(not(X))).
51.54/6.83	Axiom 25 (modus_ponens_2): fresh115(X, X, Y, Z) = fresh116(is_a_theorem(Y), true, Z).
51.54/6.83	Axiom 26 (and_1_1): fresh107(X, X, Y, Z) = true.
51.54/6.83	Axiom 27 (and_3_1): fresh103(X, X, Y, Z) = true.
51.54/6.83	Axiom 28 (implies_1_1): fresh51(X, X, Y, Z) = true.
51.54/6.83	Axiom 29 (implies_2_1): fresh49(X, X, Y, Z) = true.
51.54/6.83	Axiom 30 (modus_ponens_2): fresh40(X, X, Y, Z) = is_a_theorem(Z).
51.54/6.83	Axiom 31 (modus_tollens_1): fresh35(X, X, Y, Z) = true.
51.54/6.83	Axiom 32 (necessitation_1): fresh34(necessitation, true, X) = fresh33(is_a_theorem(X), true, X).
51.54/6.83	Axiom 33 (op_equiv): fresh30(X, X, Y, Z) = equiv(Y, Z).
51.54/6.83	Axiom 34 (op_implies_and): fresh29(X, X, Y, Z) = implies(Y, Z).
51.54/6.83	Axiom 35 (op_or): fresh26(X, X, Y, Z) = or(Y, Z).
51.54/6.83	Axiom 36 (op_strict_implies): fresh23(X, X, Y, Z) = strict_implies(Y, Z).
51.54/6.83	Axiom 37 (op_strict_implies): fresh23(op_strict_implies, true, X, Y) = necessarily(implies(X, Y)).
51.54/6.83	Axiom 38 (substitution_of_equivalents_2): fresh4(X, X, Y, Z) = Y.
51.54/6.83	Axiom 39 (substitution_of_equivalents_2): fresh3(X, X, Y, Z) = Z.
51.54/6.83	Axiom 40 (op_implies_and): fresh29(op_implies_and, true, X, Y) = not(and(X, not(Y))).
51.54/6.83	Axiom 41 (axiom_M_1): fresh93(axiom_M, true, X) = is_a_theorem(implies(necessarily(X), X)).
51.54/6.83	Axiom 42 (op_or): fresh26(op_or, true, X, Y) = not(and(not(X), not(Y))).
51.54/6.83	Axiom 43 (implies_1_1): fresh51(implies_1, true, X, Y) = is_a_theorem(implies(X, implies(Y, X))).
51.54/6.83	Axiom 44 (and_1_1): fresh107(and_1, true, X, Y) = is_a_theorem(implies(and(X, Y), X)).
51.54/6.83	Axiom 45 (op_equiv): fresh30(op_equiv, true, X, Y) = and(implies(X, Y), implies(Y, X)).
51.54/6.83	Axiom 46 (axiom_m9_1): fresh73(axiom_m9, true, X) = is_a_theorem(strict_implies(possibly(possibly(X)), possibly(X))).
51.54/6.83	Axiom 47 (axiom_5_1): fresh99(axiom_5, true, X) = is_a_theorem(implies(possibly(X), necessarily(possibly(X)))).
51.54/6.83	Axiom 48 (modus_ponens_2): fresh115(modus_ponens, true, X, Y) = fresh40(is_a_theorem(implies(X, Y)), true, X, Y).
51.54/6.83	Axiom 49 (substitution_of_equivalents_2): fresh4(substitution_of_equivalents, true, X, Y) = fresh3(is_a_theorem(equiv(X, Y)), true, X, Y).
51.54/6.83	Axiom 50 (and_3_1): fresh103(and_3, true, X, Y) = is_a_theorem(implies(X, implies(Y, and(X, Y)))).
51.54/6.83	Axiom 51 (axiom_m9): fresh74(is_a_theorem(strict_implies(possibly(possibly(x17)), possibly(x17))), true) = axiom_m9.
51.54/6.83	Axiom 52 (implies_2_1): fresh49(implies_2, true, X, Y) = is_a_theorem(implies(implies(X, implies(X, Y)), implies(X, Y))).
51.54/6.83	Axiom 53 (modus_tollens_1): fresh35(modus_tollens, true, X, Y) = is_a_theorem(implies(implies(not(Y), not(X)), implies(X, Y))).
51.54/6.83	
51.54/6.83	Lemma 54: fresh116(X, X, Y) = op_implies.
51.54/6.83	Proof:
51.54/6.83	  fresh116(X, X, Y)
51.54/6.83	= { by axiom 18 (modus_ponens_2) }
51.54/6.83	  true
51.54/6.83	= { by axiom 1 (s1_0_op_implies) R->L }
51.54/6.83	  op_implies
51.54/6.83	
51.54/6.83	Lemma 55: fresh40(is_a_theorem(implies(X, Y)), op_implies, X, Y) = fresh116(is_a_theorem(X), op_implies, Y).
51.54/6.83	Proof:
51.54/6.83	  fresh40(is_a_theorem(implies(X, Y)), op_implies, X, Y)
51.54/6.83	= { by axiom 1 (s1_0_op_implies) }
51.54/6.83	  fresh40(is_a_theorem(implies(X, Y)), true, X, Y)
51.54/6.83	= { by axiom 48 (modus_ponens_2) R->L }
51.54/6.83	  fresh115(modus_ponens, true, X, Y)
51.54/6.83	= { by axiom 6 (hilbert_modus_ponens) }
51.54/6.83	  fresh115(true, true, X, Y)
51.54/6.83	= { by axiom 1 (s1_0_op_implies) R->L }
51.54/6.83	  fresh115(op_implies, true, X, Y)
51.54/6.83	= { by axiom 1 (s1_0_op_implies) R->L }
51.54/6.83	  fresh115(op_implies, op_implies, X, Y)
51.54/6.83	= { by axiom 25 (modus_ponens_2) }
51.54/6.83	  fresh116(is_a_theorem(X), true, Y)
51.54/6.83	= { by axiom 1 (s1_0_op_implies) R->L }
51.54/6.83	  fresh116(is_a_theorem(X), op_implies, Y)
51.54/6.83	
51.54/6.83	Lemma 56: and(implies(X, Y), implies(Y, X)) = equiv(X, Y).
51.54/6.83	Proof:
51.54/6.83	  and(implies(X, Y), implies(Y, X))
51.54/6.83	= { by axiom 45 (op_equiv) R->L }
51.54/6.83	  fresh30(op_equiv, true, X, Y)
51.54/6.83	= { by axiom 12 (hilbert_op_equiv) }
51.54/6.83	  fresh30(true, true, X, Y)
51.54/6.83	= { by axiom 1 (s1_0_op_implies) R->L }
51.54/6.83	  fresh30(op_implies, true, X, Y)
51.54/6.83	= { by axiom 1 (s1_0_op_implies) R->L }
51.54/6.83	  fresh30(op_implies, op_implies, X, Y)
51.54/6.83	= { by axiom 33 (op_equiv) }
51.54/6.83	  equiv(X, Y)
51.54/6.83	
51.54/6.83	Lemma 57: is_a_theorem(implies(X, implies(Y, and(X, Y)))) = op_implies.
51.54/6.83	Proof:
51.54/6.83	  is_a_theorem(implies(X, implies(Y, and(X, Y))))
51.54/6.83	= { by axiom 50 (and_3_1) R->L }
51.54/6.83	  fresh103(and_3, true, X, Y)
51.54/6.83	= { by axiom 7 (hilbert_and_3) }
51.54/6.83	  fresh103(true, true, X, Y)
51.54/6.83	= { by axiom 1 (s1_0_op_implies) R->L }
51.54/6.83	  fresh103(op_implies, true, X, Y)
51.54/6.83	= { by axiom 1 (s1_0_op_implies) R->L }
51.54/6.83	  fresh103(op_implies, op_implies, X, Y)
51.54/6.83	= { by axiom 27 (and_3_1) }
51.54/6.83	  true
51.54/6.83	= { by axiom 1 (s1_0_op_implies) R->L }
51.54/6.83	  op_implies
51.54/6.83	
51.54/6.83	Lemma 58: fresh116(is_a_theorem(X), op_implies, implies(Y, and(X, Y))) = is_a_theorem(implies(Y, and(X, Y))).
51.54/6.83	Proof:
51.54/6.83	  fresh116(is_a_theorem(X), op_implies, implies(Y, and(X, Y)))
51.54/6.83	= { by lemma 55 R->L }
51.54/6.83	  fresh40(is_a_theorem(implies(X, implies(Y, and(X, Y)))), op_implies, X, implies(Y, and(X, Y)))
51.54/6.83	= { by lemma 57 }
51.54/6.83	  fresh40(op_implies, op_implies, X, implies(Y, and(X, Y)))
51.54/6.83	= { by axiom 30 (modus_ponens_2) }
51.54/6.83	  is_a_theorem(implies(Y, and(X, Y)))
51.54/6.83	
51.54/6.83	Lemma 59: fresh3(is_a_theorem(equiv(X, Y)), op_implies, X, Y) = X.
51.54/6.83	Proof:
51.54/6.83	  fresh3(is_a_theorem(equiv(X, Y)), op_implies, X, Y)
51.54/6.83	= { by axiom 1 (s1_0_op_implies) }
51.54/6.83	  fresh3(is_a_theorem(equiv(X, Y)), true, X, Y)
51.54/6.83	= { by axiom 49 (substitution_of_equivalents_2) R->L }
51.54/6.83	  fresh4(substitution_of_equivalents, true, X, Y)
51.54/6.83	= { by axiom 10 (substitution_of_equivalents) }
51.54/6.83	  fresh4(true, true, X, Y)
51.54/6.83	= { by axiom 1 (s1_0_op_implies) R->L }
51.54/6.83	  fresh4(op_implies, true, X, Y)
51.54/6.83	= { by axiom 1 (s1_0_op_implies) R->L }
51.54/6.83	  fresh4(op_implies, op_implies, X, Y)
51.54/6.83	= { by axiom 38 (substitution_of_equivalents_2) }
51.54/6.83	  X
51.54/6.83	
51.54/6.83	Lemma 60: necessarily(possibly(X)) = possibly(X).
51.54/6.83	Proof:
51.54/6.83	  necessarily(possibly(X))
51.54/6.83	= { by axiom 39 (substitution_of_equivalents_2) R->L }
51.54/6.83	  fresh3(op_implies, op_implies, possibly(X), necessarily(possibly(X)))
51.54/6.83	= { by lemma 54 R->L }
51.54/6.83	  fresh3(fresh116(op_implies, op_implies, equiv(possibly(X), necessarily(possibly(X)))), op_implies, possibly(X), necessarily(possibly(X)))
51.54/6.83	= { by axiom 1 (s1_0_op_implies) }
51.54/6.83	  fresh3(fresh116(true, op_implies, equiv(possibly(X), necessarily(possibly(X)))), op_implies, possibly(X), necessarily(possibly(X)))
51.54/6.83	= { by axiom 20 (axiom_M_1) R->L }
51.54/6.83	  fresh3(fresh116(fresh93(op_implies, op_implies, possibly(X)), op_implies, equiv(possibly(X), necessarily(possibly(X)))), op_implies, possibly(X), necessarily(possibly(X)))
51.54/6.83	= { by axiom 1 (s1_0_op_implies) }
51.54/6.83	  fresh3(fresh116(fresh93(op_implies, true, possibly(X)), op_implies, equiv(possibly(X), necessarily(possibly(X)))), op_implies, possibly(X), necessarily(possibly(X)))
51.54/6.83	= { by axiom 1 (s1_0_op_implies) }
51.54/6.83	  fresh3(fresh116(fresh93(true, true, possibly(X)), op_implies, equiv(possibly(X), necessarily(possibly(X)))), op_implies, possibly(X), necessarily(possibly(X)))
51.54/6.83	= { by axiom 15 (km5_axiom_M) R->L }
51.54/6.83	  fresh3(fresh116(fresh93(axiom_M, true, possibly(X)), op_implies, equiv(possibly(X), necessarily(possibly(X)))), op_implies, possibly(X), necessarily(possibly(X)))
51.54/6.83	= { by axiom 41 (axiom_M_1) }
51.54/6.83	  fresh3(fresh116(is_a_theorem(implies(necessarily(possibly(X)), possibly(X))), op_implies, equiv(possibly(X), necessarily(possibly(X)))), op_implies, possibly(X), necessarily(possibly(X)))
51.54/6.83	= { by lemma 55 R->L }
51.54/6.83	  fresh3(fresh40(is_a_theorem(implies(implies(necessarily(possibly(X)), possibly(X)), equiv(possibly(X), necessarily(possibly(X))))), op_implies, implies(necessarily(possibly(X)), possibly(X)), equiv(possibly(X), necessarily(possibly(X)))), op_implies, possibly(X), necessarily(possibly(X)))
51.54/6.83	= { by lemma 56 R->L }
51.54/6.83	  fresh3(fresh40(is_a_theorem(implies(implies(necessarily(possibly(X)), possibly(X)), and(implies(possibly(X), necessarily(possibly(X))), implies(necessarily(possibly(X)), possibly(X))))), op_implies, implies(necessarily(possibly(X)), possibly(X)), equiv(possibly(X), necessarily(possibly(X)))), op_implies, possibly(X), necessarily(possibly(X)))
51.54/6.83	= { by lemma 58 R->L }
51.54/6.83	  fresh3(fresh40(fresh116(is_a_theorem(implies(possibly(X), necessarily(possibly(X)))), op_implies, implies(implies(necessarily(possibly(X)), possibly(X)), and(implies(possibly(X), necessarily(possibly(X))), implies(necessarily(possibly(X)), possibly(X))))), op_implies, implies(necessarily(possibly(X)), possibly(X)), equiv(possibly(X), necessarily(possibly(X)))), op_implies, possibly(X), necessarily(possibly(X)))
51.54/6.83	= { by axiom 47 (axiom_5_1) R->L }
51.54/6.83	  fresh3(fresh40(fresh116(fresh99(axiom_5, true, X), op_implies, implies(implies(necessarily(possibly(X)), possibly(X)), and(implies(possibly(X), necessarily(possibly(X))), implies(necessarily(possibly(X)), possibly(X))))), op_implies, implies(necessarily(possibly(X)), possibly(X)), equiv(possibly(X), necessarily(possibly(X)))), op_implies, possibly(X), necessarily(possibly(X)))
51.54/6.83	= { by axiom 13 (km5_axiom_5) }
51.54/6.83	  fresh3(fresh40(fresh116(fresh99(true, true, X), op_implies, implies(implies(necessarily(possibly(X)), possibly(X)), and(implies(possibly(X), necessarily(possibly(X))), implies(necessarily(possibly(X)), possibly(X))))), op_implies, implies(necessarily(possibly(X)), possibly(X)), equiv(possibly(X), necessarily(possibly(X)))), op_implies, possibly(X), necessarily(possibly(X)))
51.54/6.83	= { by axiom 1 (s1_0_op_implies) R->L }
51.54/6.83	  fresh3(fresh40(fresh116(fresh99(op_implies, true, X), op_implies, implies(implies(necessarily(possibly(X)), possibly(X)), and(implies(possibly(X), necessarily(possibly(X))), implies(necessarily(possibly(X)), possibly(X))))), op_implies, implies(necessarily(possibly(X)), possibly(X)), equiv(possibly(X), necessarily(possibly(X)))), op_implies, possibly(X), necessarily(possibly(X)))
51.54/6.83	= { by axiom 1 (s1_0_op_implies) R->L }
51.54/6.83	  fresh3(fresh40(fresh116(fresh99(op_implies, op_implies, X), op_implies, implies(implies(necessarily(possibly(X)), possibly(X)), and(implies(possibly(X), necessarily(possibly(X))), implies(necessarily(possibly(X)), possibly(X))))), op_implies, implies(necessarily(possibly(X)), possibly(X)), equiv(possibly(X), necessarily(possibly(X)))), op_implies, possibly(X), necessarily(possibly(X)))
51.54/6.83	= { by axiom 19 (axiom_5_1) }
51.54/6.83	  fresh3(fresh40(fresh116(true, op_implies, implies(implies(necessarily(possibly(X)), possibly(X)), and(implies(possibly(X), necessarily(possibly(X))), implies(necessarily(possibly(X)), possibly(X))))), op_implies, implies(necessarily(possibly(X)), possibly(X)), equiv(possibly(X), necessarily(possibly(X)))), op_implies, possibly(X), necessarily(possibly(X)))
51.54/6.83	= { by axiom 1 (s1_0_op_implies) R->L }
51.54/6.83	  fresh3(fresh40(fresh116(op_implies, op_implies, implies(implies(necessarily(possibly(X)), possibly(X)), and(implies(possibly(X), necessarily(possibly(X))), implies(necessarily(possibly(X)), possibly(X))))), op_implies, implies(necessarily(possibly(X)), possibly(X)), equiv(possibly(X), necessarily(possibly(X)))), op_implies, possibly(X), necessarily(possibly(X)))
51.54/6.83	= { by lemma 54 }
51.54/6.84	  fresh3(fresh40(op_implies, op_implies, implies(necessarily(possibly(X)), possibly(X)), equiv(possibly(X), necessarily(possibly(X)))), op_implies, possibly(X), necessarily(possibly(X)))
51.54/6.84	= { by axiom 30 (modus_ponens_2) }
51.54/6.84	  fresh3(is_a_theorem(equiv(possibly(X), necessarily(possibly(X)))), op_implies, possibly(X), necessarily(possibly(X)))
51.54/6.84	= { by lemma 59 }
51.54/6.84	  possibly(X)
51.54/6.84	
51.54/6.84	Lemma 61: not(and(X, not(Y))) = implies(X, Y).
51.54/6.84	Proof:
51.54/6.84	  not(and(X, not(Y)))
51.54/6.84	= { by axiom 40 (op_implies_and) R->L }
51.54/6.84	  fresh29(op_implies_and, true, X, Y)
51.54/6.84	= { by axiom 2 (hilbert_op_implies_and) }
51.54/6.84	  fresh29(true, true, X, Y)
51.54/6.84	= { by axiom 1 (s1_0_op_implies) R->L }
51.54/6.84	  fresh29(op_implies, true, X, Y)
51.54/6.84	= { by axiom 1 (s1_0_op_implies) R->L }
51.54/6.84	  fresh29(op_implies, op_implies, X, Y)
51.54/6.84	= { by axiom 34 (op_implies_and) }
51.54/6.84	  implies(X, Y)
51.54/6.84	
51.54/6.84	Lemma 62: implies(not(X), Y) = or(X, Y).
51.54/6.84	Proof:
51.54/6.84	  implies(not(X), Y)
51.54/6.84	= { by lemma 61 R->L }
51.54/6.84	  not(and(not(X), not(Y)))
51.54/6.84	= { by axiom 42 (op_or) R->L }
51.54/6.84	  fresh26(op_or, true, X, Y)
51.54/6.84	= { by axiom 11 (hilbert_op_or) }
51.54/6.84	  fresh26(true, true, X, Y)
51.54/6.84	= { by axiom 1 (s1_0_op_implies) R->L }
51.54/6.84	  fresh26(op_implies, true, X, Y)
51.54/6.84	= { by axiom 1 (s1_0_op_implies) R->L }
51.54/6.84	  fresh26(op_implies, op_implies, X, Y)
51.54/6.84	= { by axiom 35 (op_or) }
51.54/6.84	  or(X, Y)
51.54/6.84	
51.54/6.84	Lemma 63: not(necessarily(not(X))) = possibly(X).
51.54/6.84	Proof:
51.54/6.84	  not(necessarily(not(X)))
51.54/6.84	= { by axiom 24 (op_possibly) R->L }
51.54/6.84	  fresh25(op_possibly, true, X)
51.54/6.84	= { by axiom 16 (km5_op_possibly) }
51.54/6.84	  fresh25(true, true, X)
51.54/6.84	= { by axiom 1 (s1_0_op_implies) R->L }
51.54/6.84	  fresh25(op_implies, true, X)
51.54/6.84	= { by axiom 1 (s1_0_op_implies) R->L }
51.54/6.84	  fresh25(op_implies, op_implies, X)
51.54/6.84	= { by axiom 23 (op_possibly) }
51.54/6.84	  possibly(X)
51.54/6.84	
51.54/6.84	Lemma 64: possibly(necessarily(not(X))) = not(necessarily(possibly(X))).
51.54/6.84	Proof:
51.54/6.84	  possibly(necessarily(not(X)))
51.54/6.84	= { by lemma 63 R->L }
51.54/6.84	  not(necessarily(not(necessarily(not(X)))))
51.54/6.84	= { by lemma 63 }
51.54/6.84	  not(necessarily(possibly(X)))
51.54/6.84	
51.54/6.84	Lemma 65: is_a_theorem(strict_implies(possibly(possibly(X)), possibly(X))) = fresh73(axiom_m9, op_implies, X).
51.54/6.84	Proof:
51.54/6.84	  is_a_theorem(strict_implies(possibly(possibly(X)), possibly(X)))
51.54/6.84	= { by axiom 46 (axiom_m9_1) R->L }
51.54/6.84	  fresh73(axiom_m9, true, X)
51.54/6.84	= { by axiom 1 (s1_0_op_implies) R->L }
51.54/6.84	  fresh73(axiom_m9, op_implies, X)
51.54/6.84	
51.54/6.84	Goal 1 (s1_0_m6s3m9b_axiom_m9): axiom_m9 = true.
51.54/6.84	Proof:
51.54/6.84	  axiom_m9
51.54/6.84	= { by axiom 51 (axiom_m9) R->L }
51.54/6.84	  fresh74(is_a_theorem(strict_implies(possibly(possibly(x17)), possibly(x17))), true)
51.54/6.84	= { by lemma 65 }
51.54/6.84	  fresh74(fresh73(axiom_m9, op_implies, x17), true)
51.54/6.84	= { by axiom 1 (s1_0_op_implies) R->L }
51.54/6.84	  fresh74(fresh73(axiom_m9, op_implies, x17), op_implies)
51.54/6.84	= { by lemma 65 R->L }
51.54/6.84	  fresh74(is_a_theorem(strict_implies(possibly(possibly(x17)), possibly(x17))), op_implies)
51.54/6.84	= { by lemma 63 R->L }
51.54/6.84	  fresh74(is_a_theorem(strict_implies(not(necessarily(not(possibly(x17)))), possibly(x17))), op_implies)
51.54/6.84	= { by lemma 60 R->L }
51.54/6.84	  fresh74(is_a_theorem(strict_implies(not(necessarily(not(necessarily(possibly(x17))))), possibly(x17))), op_implies)
51.54/6.84	= { by lemma 64 R->L }
51.54/6.84	  fresh74(is_a_theorem(strict_implies(not(necessarily(possibly(necessarily(not(x17))))), possibly(x17))), op_implies)
51.54/6.84	= { by lemma 60 }
51.54/6.84	  fresh74(is_a_theorem(strict_implies(not(possibly(necessarily(not(x17)))), possibly(x17))), op_implies)
51.54/6.84	= { by lemma 64 }
51.54/6.84	  fresh74(is_a_theorem(strict_implies(not(not(necessarily(possibly(x17)))), possibly(x17))), op_implies)
51.54/6.84	= { by lemma 60 }
51.54/6.84	  fresh74(is_a_theorem(strict_implies(not(not(possibly(x17))), possibly(x17))), op_implies)
51.54/6.84	= { by axiom 36 (op_strict_implies) R->L }
51.54/6.84	  fresh74(is_a_theorem(fresh23(op_implies, op_implies, not(not(possibly(x17))), possibly(x17))), op_implies)
51.54/6.84	= { by axiom 1 (s1_0_op_implies) }
51.54/6.84	  fresh74(is_a_theorem(fresh23(op_implies, true, not(not(possibly(x17))), possibly(x17))), op_implies)
51.54/6.84	= { by axiom 1 (s1_0_op_implies) }
51.54/6.84	  fresh74(is_a_theorem(fresh23(true, true, not(not(possibly(x17))), possibly(x17))), op_implies)
51.54/6.84	= { by axiom 3 (s1_0_op_strict_implies) R->L }
51.54/6.84	  fresh74(is_a_theorem(fresh23(op_strict_implies, true, not(not(possibly(x17))), possibly(x17))), op_implies)
51.54/6.84	= { by axiom 37 (op_strict_implies) }
51.54/6.84	  fresh74(is_a_theorem(necessarily(implies(not(not(possibly(x17))), possibly(x17)))), op_implies)
51.54/6.84	= { by lemma 62 }
51.54/6.84	  fresh74(is_a_theorem(necessarily(or(not(possibly(x17)), possibly(x17)))), op_implies)
51.54/6.84	= { by axiom 21 (necessitation_1) R->L }
51.54/6.84	  fresh74(fresh34(op_implies, op_implies, or(not(possibly(x17)), possibly(x17))), op_implies)
51.54/6.84	= { by axiom 1 (s1_0_op_implies) }
51.54/6.84	  fresh74(fresh34(op_implies, true, or(not(possibly(x17)), possibly(x17))), op_implies)
51.54/6.84	= { by axiom 1 (s1_0_op_implies) }
51.54/6.84	  fresh74(fresh34(true, true, or(not(possibly(x17)), possibly(x17))), op_implies)
51.54/6.84	= { by axiom 14 (km5_necessitation) R->L }
51.54/6.84	  fresh74(fresh34(necessitation, true, or(not(possibly(x17)), possibly(x17))), op_implies)
51.54/6.84	= { by axiom 32 (necessitation_1) }
51.54/6.84	  fresh74(fresh33(is_a_theorem(or(not(possibly(x17)), possibly(x17))), true, or(not(possibly(x17)), possibly(x17))), op_implies)
51.54/6.84	= { by axiom 1 (s1_0_op_implies) R->L }
51.54/6.84	  fresh74(fresh33(is_a_theorem(or(not(possibly(x17)), possibly(x17))), op_implies, or(not(possibly(x17)), possibly(x17))), op_implies)
51.54/6.84	= { by lemma 62 R->L }
51.54/6.84	  fresh74(fresh33(is_a_theorem(implies(not(not(possibly(x17))), possibly(x17))), op_implies, or(not(possibly(x17)), possibly(x17))), op_implies)
51.54/6.84	= { by axiom 30 (modus_ponens_2) R->L }
51.54/6.84	  fresh74(fresh33(fresh40(op_implies, op_implies, or(possibly(x17), not(not(not(possibly(x17))))), implies(not(not(possibly(x17))), possibly(x17))), op_implies, or(not(possibly(x17)), possibly(x17))), op_implies)
51.54/6.84	= { by axiom 1 (s1_0_op_implies) }
51.54/6.84	  fresh74(fresh33(fresh40(true, op_implies, or(possibly(x17), not(not(not(possibly(x17))))), implies(not(not(possibly(x17))), possibly(x17))), op_implies, or(not(possibly(x17)), possibly(x17))), op_implies)
51.54/6.85	= { by axiom 31 (modus_tollens_1) R->L }
51.54/6.85	  fresh74(fresh33(fresh40(fresh35(op_implies, op_implies, not(not(possibly(x17))), possibly(x17)), op_implies, or(possibly(x17), not(not(not(possibly(x17))))), implies(not(not(possibly(x17))), possibly(x17))), op_implies, or(not(possibly(x17)), possibly(x17))), op_implies)
51.54/6.85	= { by axiom 1 (s1_0_op_implies) }
51.54/6.85	  fresh74(fresh33(fresh40(fresh35(op_implies, true, not(not(possibly(x17))), possibly(x17)), op_implies, or(possibly(x17), not(not(not(possibly(x17))))), implies(not(not(possibly(x17))), possibly(x17))), op_implies, or(not(possibly(x17)), possibly(x17))), op_implies)
51.54/6.85	= { by axiom 1 (s1_0_op_implies) }
51.54/6.85	  fresh74(fresh33(fresh40(fresh35(true, true, not(not(possibly(x17))), possibly(x17)), op_implies, or(possibly(x17), not(not(not(possibly(x17))))), implies(not(not(possibly(x17))), possibly(x17))), op_implies, or(not(possibly(x17)), possibly(x17))), op_implies)
51.54/6.85	= { by axiom 8 (hilbert_modus_tollens) R->L }
51.54/6.85	  fresh74(fresh33(fresh40(fresh35(modus_tollens, true, not(not(possibly(x17))), possibly(x17)), op_implies, or(possibly(x17), not(not(not(possibly(x17))))), implies(not(not(possibly(x17))), possibly(x17))), op_implies, or(not(possibly(x17)), possibly(x17))), op_implies)
51.54/6.85	= { by axiom 53 (modus_tollens_1) }
51.54/6.85	  fresh74(fresh33(fresh40(is_a_theorem(implies(implies(not(possibly(x17)), not(not(not(possibly(x17))))), implies(not(not(possibly(x17))), possibly(x17)))), op_implies, or(possibly(x17), not(not(not(possibly(x17))))), implies(not(not(possibly(x17))), possibly(x17))), op_implies, or(not(possibly(x17)), possibly(x17))), op_implies)
51.54/6.85	= { by lemma 62 }
51.54/6.85	  fresh74(fresh33(fresh40(is_a_theorem(implies(or(possibly(x17), not(not(not(possibly(x17))))), implies(not(not(possibly(x17))), possibly(x17)))), op_implies, or(possibly(x17), not(not(not(possibly(x17))))), implies(not(not(possibly(x17))), possibly(x17))), op_implies, or(not(possibly(x17)), possibly(x17))), op_implies)
51.54/6.85	= { by lemma 55 }
51.54/6.85	  fresh74(fresh33(fresh116(is_a_theorem(or(possibly(x17), not(not(not(possibly(x17)))))), op_implies, implies(not(not(possibly(x17))), possibly(x17))), op_implies, or(not(possibly(x17)), possibly(x17))), op_implies)
51.54/6.86	= { by lemma 59 R->L }
51.54/6.86	  fresh74(fresh33(fresh116(is_a_theorem(or(possibly(x17), not(fresh3(is_a_theorem(equiv(not(not(possibly(x17))), and(not(not(possibly(x17))), not(not(possibly(x17)))))), op_implies, not(not(possibly(x17))), and(not(not(possibly(x17))), not(not(possibly(x17)))))))), op_implies, implies(not(not(possibly(x17))), possibly(x17))), op_implies, or(not(possibly(x17)), possibly(x17))), op_implies)
51.54/6.86	= { by axiom 30 (modus_ponens_2) R->L }
51.54/6.86	  fresh74(fresh33(fresh116(is_a_theorem(or(possibly(x17), not(fresh3(fresh40(op_implies, op_implies, implies(and(not(not(possibly(x17))), not(not(possibly(x17)))), not(not(possibly(x17)))), equiv(not(not(possibly(x17))), and(not(not(possibly(x17))), not(not(possibly(x17)))))), op_implies, not(not(possibly(x17))), and(not(not(possibly(x17))), not(not(possibly(x17)))))))), op_implies, implies(not(not(possibly(x17))), possibly(x17))), op_implies, or(not(possibly(x17)), possibly(x17))), op_implies)
51.54/6.86	= { by lemma 54 R->L }
51.54/6.86	  fresh74(fresh33(fresh116(is_a_theorem(or(possibly(x17), not(fresh3(fresh40(fresh116(op_implies, op_implies, implies(implies(and(not(not(possibly(x17))), not(not(possibly(x17)))), not(not(possibly(x17)))), and(implies(not(not(possibly(x17))), and(not(not(possibly(x17))), not(not(possibly(x17))))), implies(and(not(not(possibly(x17))), not(not(possibly(x17)))), not(not(possibly(x17))))))), op_implies, implies(and(not(not(possibly(x17))), not(not(possibly(x17)))), not(not(possibly(x17)))), equiv(not(not(possibly(x17))), and(not(not(possibly(x17))), not(not(possibly(x17)))))), op_implies, not(not(possibly(x17))), and(not(not(possibly(x17))), not(not(possibly(x17)))))))), op_implies, implies(not(not(possibly(x17))), possibly(x17))), op_implies, or(not(possibly(x17)), possibly(x17))), op_implies)
51.54/6.86	= { by lemma 54 R->L }
51.54/6.86	  fresh74(fresh33(fresh116(is_a_theorem(or(possibly(x17), not(fresh3(fresh40(fresh116(fresh116(op_implies, op_implies, implies(not(not(possibly(x17))), and(not(not(possibly(x17))), not(not(possibly(x17)))))), op_implies, implies(implies(and(not(not(possibly(x17))), not(not(possibly(x17)))), not(not(possibly(x17)))), and(implies(not(not(possibly(x17))), and(not(not(possibly(x17))), not(not(possibly(x17))))), implies(and(not(not(possibly(x17))), not(not(possibly(x17)))), not(not(possibly(x17))))))), op_implies, implies(and(not(not(possibly(x17))), not(not(possibly(x17)))), not(not(possibly(x17)))), equiv(not(not(possibly(x17))), and(not(not(possibly(x17))), not(not(possibly(x17)))))), op_implies, not(not(possibly(x17))), and(not(not(possibly(x17))), not(not(possibly(x17)))))))), op_implies, implies(not(not(possibly(x17))), possibly(x17))), op_implies, or(not(possibly(x17)), possibly(x17))), op_implies)
51.54/6.86	= { by lemma 57 R->L }
51.54/6.86	  fresh74(fresh33(fresh116(is_a_theorem(or(possibly(x17), not(fresh3(fresh40(fresh116(fresh116(is_a_theorem(implies(not(not(possibly(x17))), implies(not(not(possibly(x17))), and(not(not(possibly(x17))), not(not(possibly(x17))))))), op_implies, implies(not(not(possibly(x17))), and(not(not(possibly(x17))), not(not(possibly(x17)))))), op_implies, implies(implies(and(not(not(possibly(x17))), not(not(possibly(x17)))), not(not(possibly(x17)))), and(implies(not(not(possibly(x17))), and(not(not(possibly(x17))), not(not(possibly(x17))))), implies(and(not(not(possibly(x17))), not(not(possibly(x17)))), not(not(possibly(x17))))))), op_implies, implies(and(not(not(possibly(x17))), not(not(possibly(x17)))), not(not(possibly(x17)))), equiv(not(not(possibly(x17))), and(not(not(possibly(x17))), not(not(possibly(x17)))))), op_implies, not(not(possibly(x17))), and(not(not(possibly(x17))), not(not(possibly(x17)))))))), op_implies, implies(not(not(possibly(x17))), possibly(x17))), op_implies, or(not(possibly(x17)), possibly(x17))), op_implies)
51.54/6.86	= { by lemma 55 R->L }
51.54/6.86	  fresh74(fresh33(fresh116(is_a_theorem(or(possibly(x17), not(fresh3(fresh40(fresh116(fresh40(is_a_theorem(implies(implies(not(not(possibly(x17))), implies(not(not(possibly(x17))), and(not(not(possibly(x17))), not(not(possibly(x17)))))), implies(not(not(possibly(x17))), and(not(not(possibly(x17))), not(not(possibly(x17))))))), op_implies, implies(not(not(possibly(x17))), implies(not(not(possibly(x17))), and(not(not(possibly(x17))), not(not(possibly(x17)))))), implies(not(not(possibly(x17))), and(not(not(possibly(x17))), not(not(possibly(x17)))))), op_implies, implies(implies(and(not(not(possibly(x17))), not(not(possibly(x17)))), not(not(possibly(x17)))), and(implies(not(not(possibly(x17))), and(not(not(possibly(x17))), not(not(possibly(x17))))), implies(and(not(not(possibly(x17))), not(not(possibly(x17)))), not(not(possibly(x17))))))), op_implies, implies(and(not(not(possibly(x17))), not(not(possibly(x17)))), not(not(possibly(x17)))), equiv(not(not(possibly(x17))), and(not(not(possibly(x17))), not(not(possibly(x17)))))), op_implies, not(not(possibly(x17))), and(not(not(possibly(x17))), not(not(possibly(x17)))))))), op_implies, implies(not(not(possibly(x17))), possibly(x17))), op_implies, or(not(possibly(x17)), possibly(x17))), op_implies)
51.54/6.87	= { by axiom 52 (implies_2_1) R->L }
51.54/6.87	  fresh74(fresh33(fresh116(is_a_theorem(or(possibly(x17), not(fresh3(fresh40(fresh116(fresh40(fresh49(implies_2, true, not(not(possibly(x17))), and(not(not(possibly(x17))), not(not(possibly(x17))))), op_implies, implies(not(not(possibly(x17))), implies(not(not(possibly(x17))), and(not(not(possibly(x17))), not(not(possibly(x17)))))), implies(not(not(possibly(x17))), and(not(not(possibly(x17))), not(not(possibly(x17)))))), op_implies, implies(implies(and(not(not(possibly(x17))), not(not(possibly(x17)))), not(not(possibly(x17)))), and(implies(not(not(possibly(x17))), and(not(not(possibly(x17))), not(not(possibly(x17))))), implies(and(not(not(possibly(x17))), not(not(possibly(x17)))), not(not(possibly(x17))))))), op_implies, implies(and(not(not(possibly(x17))), not(not(possibly(x17)))), not(not(possibly(x17)))), equiv(not(not(possibly(x17))), and(not(not(possibly(x17))), not(not(possibly(x17)))))), op_implies, not(not(possibly(x17))), and(not(not(possibly(x17))), not(not(possibly(x17)))))))), op_implies, implies(not(not(possibly(x17))), possibly(x17))), op_implies, or(not(possibly(x17)), possibly(x17))), op_implies)
51.54/6.87	= { by axiom 5 (hilbert_implies_2) }
51.54/6.87	  fresh74(fresh33(fresh116(is_a_theorem(or(possibly(x17), not(fresh3(fresh40(fresh116(fresh40(fresh49(true, true, not(not(possibly(x17))), and(not(not(possibly(x17))), not(not(possibly(x17))))), op_implies, implies(not(not(possibly(x17))), implies(not(not(possibly(x17))), and(not(not(possibly(x17))), not(not(possibly(x17)))))), implies(not(not(possibly(x17))), and(not(not(possibly(x17))), not(not(possibly(x17)))))), op_implies, implies(implies(and(not(not(possibly(x17))), not(not(possibly(x17)))), not(not(possibly(x17)))), and(implies(not(not(possibly(x17))), and(not(not(possibly(x17))), not(not(possibly(x17))))), implies(and(not(not(possibly(x17))), not(not(possibly(x17)))), not(not(possibly(x17))))))), op_implies, implies(and(not(not(possibly(x17))), not(not(possibly(x17)))), not(not(possibly(x17)))), equiv(not(not(possibly(x17))), and(not(not(possibly(x17))), not(not(possibly(x17)))))), op_implies, not(not(possibly(x17))), and(not(not(possibly(x17))), not(not(possibly(x17)))))))), op_implies, implies(not(not(possibly(x17))), possibly(x17))), op_implies, or(not(possibly(x17)), possibly(x17))), op_implies)
51.54/6.87	= { by axiom 1 (s1_0_op_implies) R->L }
51.54/6.87	  fresh74(fresh33(fresh116(is_a_theorem(or(possibly(x17), not(fresh3(fresh40(fresh116(fresh40(fresh49(op_implies, true, not(not(possibly(x17))), and(not(not(possibly(x17))), not(not(possibly(x17))))), op_implies, implies(not(not(possibly(x17))), implies(not(not(possibly(x17))), and(not(not(possibly(x17))), not(not(possibly(x17)))))), implies(not(not(possibly(x17))), and(not(not(possibly(x17))), not(not(possibly(x17)))))), op_implies, implies(implies(and(not(not(possibly(x17))), not(not(possibly(x17)))), not(not(possibly(x17)))), and(implies(not(not(possibly(x17))), and(not(not(possibly(x17))), not(not(possibly(x17))))), implies(and(not(not(possibly(x17))), not(not(possibly(x17)))), not(not(possibly(x17))))))), op_implies, implies(and(not(not(possibly(x17))), not(not(possibly(x17)))), not(not(possibly(x17)))), equiv(not(not(possibly(x17))), and(not(not(possibly(x17))), not(not(possibly(x17)))))), op_implies, not(not(possibly(x17))), and(not(not(possibly(x17))), not(not(possibly(x17)))))))), op_implies, implies(not(not(possibly(x17))), possibly(x17))), op_implies, or(not(possibly(x17)), possibly(x17))), op_implies)
51.54/6.87	= { by axiom 1 (s1_0_op_implies) R->L }
51.54/6.87	  fresh74(fresh33(fresh116(is_a_theorem(or(possibly(x17), not(fresh3(fresh40(fresh116(fresh40(fresh49(op_implies, op_implies, not(not(possibly(x17))), and(not(not(possibly(x17))), not(not(possibly(x17))))), op_implies, implies(not(not(possibly(x17))), implies(not(not(possibly(x17))), and(not(not(possibly(x17))), not(not(possibly(x17)))))), implies(not(not(possibly(x17))), and(not(not(possibly(x17))), not(not(possibly(x17)))))), op_implies, implies(implies(and(not(not(possibly(x17))), not(not(possibly(x17)))), not(not(possibly(x17)))), and(implies(not(not(possibly(x17))), and(not(not(possibly(x17))), not(not(possibly(x17))))), implies(and(not(not(possibly(x17))), not(not(possibly(x17)))), not(not(possibly(x17))))))), op_implies, implies(and(not(not(possibly(x17))), not(not(possibly(x17)))), not(not(possibly(x17)))), equiv(not(not(possibly(x17))), and(not(not(possibly(x17))), not(not(possibly(x17)))))), op_implies, not(not(possibly(x17))), and(not(not(possibly(x17))), not(not(possibly(x17)))))))), op_implies, implies(not(not(possibly(x17))), possibly(x17))), op_implies, or(not(possibly(x17)), possibly(x17))), op_implies)
51.54/6.87	= { by axiom 29 (implies_2_1) }
51.54/6.87	  fresh74(fresh33(fresh116(is_a_theorem(or(possibly(x17), not(fresh3(fresh40(fresh116(fresh40(true, op_implies, implies(not(not(possibly(x17))), implies(not(not(possibly(x17))), and(not(not(possibly(x17))), not(not(possibly(x17)))))), implies(not(not(possibly(x17))), and(not(not(possibly(x17))), not(not(possibly(x17)))))), op_implies, implies(implies(and(not(not(possibly(x17))), not(not(possibly(x17)))), not(not(possibly(x17)))), and(implies(not(not(possibly(x17))), and(not(not(possibly(x17))), not(not(possibly(x17))))), implies(and(not(not(possibly(x17))), not(not(possibly(x17)))), not(not(possibly(x17))))))), op_implies, implies(and(not(not(possibly(x17))), not(not(possibly(x17)))), not(not(possibly(x17)))), equiv(not(not(possibly(x17))), and(not(not(possibly(x17))), not(not(possibly(x17)))))), op_implies, not(not(possibly(x17))), and(not(not(possibly(x17))), not(not(possibly(x17)))))))), op_implies, implies(not(not(possibly(x17))), possibly(x17))), op_implies, or(not(possibly(x17)), possibly(x17))), op_implies)
51.54/6.87	= { by axiom 1 (s1_0_op_implies) R->L }
51.54/6.88	  fresh74(fresh33(fresh116(is_a_theorem(or(possibly(x17), not(fresh3(fresh40(fresh116(fresh40(op_implies, op_implies, implies(not(not(possibly(x17))), implies(not(not(possibly(x17))), and(not(not(possibly(x17))), not(not(possibly(x17)))))), implies(not(not(possibly(x17))), and(not(not(possibly(x17))), not(not(possibly(x17)))))), op_implies, implies(implies(and(not(not(possibly(x17))), not(not(possibly(x17)))), not(not(possibly(x17)))), and(implies(not(not(possibly(x17))), and(not(not(possibly(x17))), not(not(possibly(x17))))), implies(and(not(not(possibly(x17))), not(not(possibly(x17)))), not(not(possibly(x17))))))), op_implies, implies(and(not(not(possibly(x17))), not(not(possibly(x17)))), not(not(possibly(x17)))), equiv(not(not(possibly(x17))), and(not(not(possibly(x17))), not(not(possibly(x17)))))), op_implies, not(not(possibly(x17))), and(not(not(possibly(x17))), not(not(possibly(x17)))))))), op_implies, implies(not(not(possibly(x17))), possibly(x17))), op_implies, or(not(possibly(x17)), possibly(x17))), op_implies)
51.54/6.88	= { by axiom 30 (modus_ponens_2) }
51.54/6.88	  fresh74(fresh33(fresh116(is_a_theorem(or(possibly(x17), not(fresh3(fresh40(fresh116(is_a_theorem(implies(not(not(possibly(x17))), and(not(not(possibly(x17))), not(not(possibly(x17)))))), op_implies, implies(implies(and(not(not(possibly(x17))), not(not(possibly(x17)))), not(not(possibly(x17)))), and(implies(not(not(possibly(x17))), and(not(not(possibly(x17))), not(not(possibly(x17))))), implies(and(not(not(possibly(x17))), not(not(possibly(x17)))), not(not(possibly(x17))))))), op_implies, implies(and(not(not(possibly(x17))), not(not(possibly(x17)))), not(not(possibly(x17)))), equiv(not(not(possibly(x17))), and(not(not(possibly(x17))), not(not(possibly(x17)))))), op_implies, not(not(possibly(x17))), and(not(not(possibly(x17))), not(not(possibly(x17)))))))), op_implies, implies(not(not(possibly(x17))), possibly(x17))), op_implies, or(not(possibly(x17)), possibly(x17))), op_implies)
51.54/6.88	= { by lemma 58 }
51.54/6.88	  fresh74(fresh33(fresh116(is_a_theorem(or(possibly(x17), not(fresh3(fresh40(is_a_theorem(implies(implies(and(not(not(possibly(x17))), not(not(possibly(x17)))), not(not(possibly(x17)))), and(implies(not(not(possibly(x17))), and(not(not(possibly(x17))), not(not(possibly(x17))))), implies(and(not(not(possibly(x17))), not(not(possibly(x17)))), not(not(possibly(x17))))))), op_implies, implies(and(not(not(possibly(x17))), not(not(possibly(x17)))), not(not(possibly(x17)))), equiv(not(not(possibly(x17))), and(not(not(possibly(x17))), not(not(possibly(x17)))))), op_implies, not(not(possibly(x17))), and(not(not(possibly(x17))), not(not(possibly(x17)))))))), op_implies, implies(not(not(possibly(x17))), possibly(x17))), op_implies, or(not(possibly(x17)), possibly(x17))), op_implies)
51.54/6.88	= { by lemma 56 }
51.54/6.88	  fresh74(fresh33(fresh116(is_a_theorem(or(possibly(x17), not(fresh3(fresh40(is_a_theorem(implies(implies(and(not(not(possibly(x17))), not(not(possibly(x17)))), not(not(possibly(x17)))), equiv(not(not(possibly(x17))), and(not(not(possibly(x17))), not(not(possibly(x17))))))), op_implies, implies(and(not(not(possibly(x17))), not(not(possibly(x17)))), not(not(possibly(x17)))), equiv(not(not(possibly(x17))), and(not(not(possibly(x17))), not(not(possibly(x17)))))), op_implies, not(not(possibly(x17))), and(not(not(possibly(x17))), not(not(possibly(x17)))))))), op_implies, implies(not(not(possibly(x17))), possibly(x17))), op_implies, or(not(possibly(x17)), possibly(x17))), op_implies)
51.54/6.88	= { by lemma 55 }
51.54/6.88	  fresh74(fresh33(fresh116(is_a_theorem(or(possibly(x17), not(fresh3(fresh116(is_a_theorem(implies(and(not(not(possibly(x17))), not(not(possibly(x17)))), not(not(possibly(x17))))), op_implies, equiv(not(not(possibly(x17))), and(not(not(possibly(x17))), not(not(possibly(x17)))))), op_implies, not(not(possibly(x17))), and(not(not(possibly(x17))), not(not(possibly(x17)))))))), op_implies, implies(not(not(possibly(x17))), possibly(x17))), op_implies, or(not(possibly(x17)), possibly(x17))), op_implies)
51.54/6.88	= { by axiom 44 (and_1_1) R->L }
51.54/6.89	  fresh74(fresh33(fresh116(is_a_theorem(or(possibly(x17), not(fresh3(fresh116(fresh107(and_1, true, not(not(possibly(x17))), not(not(possibly(x17)))), op_implies, equiv(not(not(possibly(x17))), and(not(not(possibly(x17))), not(not(possibly(x17)))))), op_implies, not(not(possibly(x17))), and(not(not(possibly(x17))), not(not(possibly(x17)))))))), op_implies, implies(not(not(possibly(x17))), possibly(x17))), op_implies, or(not(possibly(x17)), possibly(x17))), op_implies)
51.54/6.89	= { by axiom 4 (hilbert_and_1) }
51.54/6.89	  fresh74(fresh33(fresh116(is_a_theorem(or(possibly(x17), not(fresh3(fresh116(fresh107(true, true, not(not(possibly(x17))), not(not(possibly(x17)))), op_implies, equiv(not(not(possibly(x17))), and(not(not(possibly(x17))), not(not(possibly(x17)))))), op_implies, not(not(possibly(x17))), and(not(not(possibly(x17))), not(not(possibly(x17)))))))), op_implies, implies(not(not(possibly(x17))), possibly(x17))), op_implies, or(not(possibly(x17)), possibly(x17))), op_implies)
51.54/6.89	= { by axiom 1 (s1_0_op_implies) R->L }
51.54/6.89	  fresh74(fresh33(fresh116(is_a_theorem(or(possibly(x17), not(fresh3(fresh116(fresh107(op_implies, true, not(not(possibly(x17))), not(not(possibly(x17)))), op_implies, equiv(not(not(possibly(x17))), and(not(not(possibly(x17))), not(not(possibly(x17)))))), op_implies, not(not(possibly(x17))), and(not(not(possibly(x17))), not(not(possibly(x17)))))))), op_implies, implies(not(not(possibly(x17))), possibly(x17))), op_implies, or(not(possibly(x17)), possibly(x17))), op_implies)
51.54/6.89	= { by axiom 1 (s1_0_op_implies) R->L }
51.54/6.89	  fresh74(fresh33(fresh116(is_a_theorem(or(possibly(x17), not(fresh3(fresh116(fresh107(op_implies, op_implies, not(not(possibly(x17))), not(not(possibly(x17)))), op_implies, equiv(not(not(possibly(x17))), and(not(not(possibly(x17))), not(not(possibly(x17)))))), op_implies, not(not(possibly(x17))), and(not(not(possibly(x17))), not(not(possibly(x17)))))))), op_implies, implies(not(not(possibly(x17))), possibly(x17))), op_implies, or(not(possibly(x17)), possibly(x17))), op_implies)
51.54/6.89	= { by axiom 26 (and_1_1) }
51.54/6.89	  fresh74(fresh33(fresh116(is_a_theorem(or(possibly(x17), not(fresh3(fresh116(true, op_implies, equiv(not(not(possibly(x17))), and(not(not(possibly(x17))), not(not(possibly(x17)))))), op_implies, not(not(possibly(x17))), and(not(not(possibly(x17))), not(not(possibly(x17)))))))), op_implies, implies(not(not(possibly(x17))), possibly(x17))), op_implies, or(not(possibly(x17)), possibly(x17))), op_implies)
51.54/6.89	= { by axiom 1 (s1_0_op_implies) R->L }
51.54/6.89	  fresh74(fresh33(fresh116(is_a_theorem(or(possibly(x17), not(fresh3(fresh116(op_implies, op_implies, equiv(not(not(possibly(x17))), and(not(not(possibly(x17))), not(not(possibly(x17)))))), op_implies, not(not(possibly(x17))), and(not(not(possibly(x17))), not(not(possibly(x17)))))))), op_implies, implies(not(not(possibly(x17))), possibly(x17))), op_implies, or(not(possibly(x17)), possibly(x17))), op_implies)
51.54/6.89	= { by lemma 54 }
51.54/6.89	  fresh74(fresh33(fresh116(is_a_theorem(or(possibly(x17), not(fresh3(op_implies, op_implies, not(not(possibly(x17))), and(not(not(possibly(x17))), not(not(possibly(x17)))))))), op_implies, implies(not(not(possibly(x17))), possibly(x17))), op_implies, or(not(possibly(x17)), possibly(x17))), op_implies)
51.54/6.89	= { by axiom 39 (substitution_of_equivalents_2) }
51.54/6.89	  fresh74(fresh33(fresh116(is_a_theorem(or(possibly(x17), not(and(not(not(possibly(x17))), not(not(possibly(x17))))))), op_implies, implies(not(not(possibly(x17))), possibly(x17))), op_implies, or(not(possibly(x17)), possibly(x17))), op_implies)
51.54/6.89	= { by lemma 61 }
51.54/6.89	  fresh74(fresh33(fresh116(is_a_theorem(or(possibly(x17), implies(not(not(possibly(x17))), not(possibly(x17))))), op_implies, implies(not(not(possibly(x17))), possibly(x17))), op_implies, or(not(possibly(x17)), possibly(x17))), op_implies)
51.54/6.89	= { by lemma 62 R->L }
51.54/6.89	  fresh74(fresh33(fresh116(is_a_theorem(implies(not(possibly(x17)), implies(not(not(possibly(x17))), not(possibly(x17))))), op_implies, implies(not(not(possibly(x17))), possibly(x17))), op_implies, or(not(possibly(x17)), possibly(x17))), op_implies)
51.54/6.89	= { by axiom 43 (implies_1_1) R->L }
51.54/6.89	  fresh74(fresh33(fresh116(fresh51(implies_1, true, not(possibly(x17)), not(not(possibly(x17)))), op_implies, implies(not(not(possibly(x17))), possibly(x17))), op_implies, or(not(possibly(x17)), possibly(x17))), op_implies)
51.54/6.89	= { by axiom 9 (hilbert_implies_1) }
51.54/6.89	  fresh74(fresh33(fresh116(fresh51(true, true, not(possibly(x17)), not(not(possibly(x17)))), op_implies, implies(not(not(possibly(x17))), possibly(x17))), op_implies, or(not(possibly(x17)), possibly(x17))), op_implies)
51.54/6.89	= { by axiom 1 (s1_0_op_implies) R->L }
51.54/6.89	  fresh74(fresh33(fresh116(fresh51(op_implies, true, not(possibly(x17)), not(not(possibly(x17)))), op_implies, implies(not(not(possibly(x17))), possibly(x17))), op_implies, or(not(possibly(x17)), possibly(x17))), op_implies)
51.54/6.89	= { by axiom 1 (s1_0_op_implies) R->L }
51.54/6.89	  fresh74(fresh33(fresh116(fresh51(op_implies, op_implies, not(possibly(x17)), not(not(possibly(x17)))), op_implies, implies(not(not(possibly(x17))), possibly(x17))), op_implies, or(not(possibly(x17)), possibly(x17))), op_implies)
51.54/6.89	= { by axiom 28 (implies_1_1) }
51.54/6.89	  fresh74(fresh33(fresh116(true, op_implies, implies(not(not(possibly(x17))), possibly(x17))), op_implies, or(not(possibly(x17)), possibly(x17))), op_implies)
51.54/6.89	= { by axiom 1 (s1_0_op_implies) R->L }
51.54/6.89	  fresh74(fresh33(fresh116(op_implies, op_implies, implies(not(not(possibly(x17))), possibly(x17))), op_implies, or(not(possibly(x17)), possibly(x17))), op_implies)
51.54/6.89	= { by lemma 54 }
51.54/6.89	  fresh74(fresh33(op_implies, op_implies, or(not(possibly(x17)), possibly(x17))), op_implies)
51.54/6.89	= { by axiom 22 (necessitation_1) }
51.54/6.89	  fresh74(true, op_implies)
51.54/6.89	= { by axiom 1 (s1_0_op_implies) R->L }
51.54/6.89	  fresh74(op_implies, op_implies)
51.54/6.89	= { by axiom 17 (axiom_m9) }
51.54/6.89	  true
51.54/6.89	% SZS output end Proof
51.54/6.89	
51.54/6.89	RESULT: Theorem (the conjecture is true).
51.54/6.91	EOF