0.03/0.13	% Problem    : theBenchmark.p : TPTP v0.0.0. Released v0.0.0.
0.03/0.14	% Command    : twee %s --tstp --casc --quiet --explain-encoding --conditional-encoding if --smaller --drop-non-horn
0.14/0.36	% Computer   : n008.cluster.edu
0.14/0.36	% Model      : x86_64 x86_64
0.14/0.36	% CPU        : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
0.14/0.36	% Memory     : 8042.1875MB
0.14/0.36	% OS         : Linux 3.10.0-693.el7.x86_64
0.14/0.36	% CPULimit   : 180
0.14/0.36	% DateTime   : Thu Aug 29 16:59:53 EDT 2019
0.14/0.36	% CPUTime    : 
50.57/50.79	% SZS status Theorem
50.57/50.79	
50.57/50.79	% SZS output start Proof
50.57/50.79	Take the following subset of the input axioms:
50.98/51.17	  fof(and_1, axiom, ![X, Y]: is_a_theorem(implies(and(X, Y), X)) <=> and_1).
50.98/51.17	  fof(and_2, axiom, and_2 <=> ![X, Y]: is_a_theorem(implies(and(X, Y), Y))).
50.98/51.17	  fof(and_3, axiom, ![X, Y]: is_a_theorem(implies(X, implies(Y, and(X, Y)))) <=> and_3).
50.98/51.17	  fof(axiom_5, axiom, ![X]: is_a_theorem(implies(possibly(X), necessarily(possibly(X)))) <=> axiom_5).
50.98/51.17	  fof(axiom_K, axiom, ![X, Y]: is_a_theorem(implies(necessarily(implies(X, Y)), implies(necessarily(X), necessarily(Y)))) <=> axiom_K).
50.98/51.17	  fof(axiom_M, axiom, axiom_M <=> ![X]: is_a_theorem(implies(necessarily(X), X))).
50.98/51.17	  fof(axiom_s3, axiom, axiom_s3 <=> ![X, Y]: is_a_theorem(strict_implies(strict_implies(X, Y), strict_implies(not(possibly(Y)), not(possibly(X)))))).
50.98/51.17	  fof(hilbert_and_1, axiom, and_1).
50.98/51.17	  fof(hilbert_and_2, axiom, and_2).
50.98/51.17	  fof(hilbert_and_3, axiom, and_3).
50.98/51.17	  fof(hilbert_implies_2, axiom, implies_2).
50.98/51.17	  fof(hilbert_modus_ponens, axiom, modus_ponens).
50.98/51.17	  fof(hilbert_modus_tollens, axiom, modus_tollens).
50.98/51.17	  fof(hilbert_op_equiv, axiom, op_equiv).
50.98/51.17	  fof(hilbert_op_implies_and, axiom, op_implies_and).
50.98/51.17	  fof(hilbert_op_or, axiom, op_or).
50.98/51.17	  fof(hilbert_or_1, axiom, or_1).
50.98/51.17	  fof(hilbert_or_3, axiom, or_3).
50.98/51.17	  fof(implies_2, axiom, implies_2 <=> ![X, Y]: is_a_theorem(implies(implies(X, implies(X, Y)), implies(X, Y)))).
50.98/51.17	  fof(km5_axiom_5, axiom, axiom_5).
50.98/51.17	  fof(km5_axiom_K, axiom, axiom_K).
50.98/51.17	  fof(km5_axiom_M, axiom, axiom_M).
50.98/51.17	  fof(km5_necessitation, axiom, necessitation).
50.98/51.17	  fof(km5_op_possibly, axiom, op_possibly).
50.98/51.17	  fof(kn1, axiom, ![P]: is_a_theorem(implies(P, and(P, P))) <=> kn1).
50.98/51.17	  fof(modus_ponens, axiom, modus_ponens <=> ![X, Y]: ((is_a_theorem(X) & is_a_theorem(implies(X, Y))) => is_a_theorem(Y))).
50.98/51.17	  fof(modus_tollens, axiom, ![X, Y]: is_a_theorem(implies(implies(not(Y), not(X)), implies(X, Y))) <=> modus_tollens).
50.98/51.17	  fof(necessitation, axiom, necessitation <=> ![X]: (is_a_theorem(necessarily(X)) <= is_a_theorem(X))).
50.98/51.17	  fof(op_equiv, axiom, op_equiv => ![X, Y]: equiv(X, Y)=and(implies(X, Y), implies(Y, X))).
50.98/51.17	  fof(op_implies_and, axiom, op_implies_and => ![X, Y]: implies(X, Y)=not(and(X, not(Y)))).
50.98/51.17	  fof(op_or, axiom, op_or => ![X, Y]: or(X, Y)=not(and(not(X), not(Y)))).
50.98/51.17	  fof(op_possibly, axiom, ![X]: not(necessarily(not(X)))=possibly(X) <= op_possibly).
50.98/51.17	  fof(op_strict_implies, axiom, ![X, Y]: necessarily(implies(X, Y))=strict_implies(X, Y) <= op_strict_implies).
50.98/51.17	  fof(or_1, axiom, or_1 <=> ![X, Y]: is_a_theorem(implies(X, or(X, Y)))).
50.98/51.17	  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).
50.98/51.17	  fof(r3, axiom, r3 <=> ![P, Q]: is_a_theorem(implies(or(P, Q), or(Q, P)))).
50.98/51.17	  fof(s1_0_m6s3m9b_axiom_s3, conjecture, axiom_s3).
50.98/51.17	  fof(s1_0_op_strict_implies, axiom, op_strict_implies).
50.98/51.17	  fof(substitution_of_equivalents, axiom, substitution_of_equivalents <=> ![X, Y]: (is_a_theorem(equiv(X, Y)) => X=Y)).
50.98/51.17	  fof(substitution_of_equivalents, axiom, substitution_of_equivalents).
50.98/51.17	
50.98/51.17	Now clausify the problem and encode Horn clauses using encoding 3 of
50.98/51.17	http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
50.98/51.17	We repeatedly replace C & s=t => u=v by the two clauses:
50.98/51.17	  fresh(y, y, x1...xn) = u
50.98/51.17	  C => fresh(s, t, x1...xn) = v
50.98/51.17	where fresh is a fresh function symbol and x1..xn are the free
50.98/51.17	variables of u and v.
50.98/51.17	A predicate p(X) is encoded as p(X)=true (this is sound, because the
50.98/51.17	input problem has no model of domain size 1).
50.98/51.17	
50.98/51.17	The encoding turns the above axioms into the following unit equations and goals:
50.98/51.17	
50.98/51.17	Axiom 1 (and_1_1): fresh107(X, X, Y, Z) = true.
50.98/51.17	Axiom 2 (and_2_1): fresh105(X, X, Y, Z) = true.
50.98/51.17	Axiom 3 (and_3_1): fresh103(X, X, Y, Z) = true.
50.98/51.17	Axiom 4 (axiom_5_1): fresh99(X, X, Y) = true.
50.98/51.17	Axiom 5 (axiom_K_1): fresh95(X, X, Y, Z) = true.
50.98/51.17	Axiom 6 (axiom_M_1): fresh93(X, X, Y) = true.
50.98/51.17	Axiom 7 (axiom_s3): fresh68(X, X) = true.
50.98/51.17	Axiom 8 (implies_2_1): fresh49(X, X, Y, Z) = true.
50.98/51.17	Axiom 9 (modus_ponens_2): fresh40(X, X, Y, Z) = is_a_theorem(Z).
50.98/51.17	Axiom 10 (modus_ponens_2): fresh116(X, X, Y) = true.
50.98/51.17	Axiom 11 (modus_ponens_2): fresh115(X, X, Y, Z) = fresh116(is_a_theorem(Y), true, Z).
50.98/51.17	Axiom 12 (modus_tollens_1): fresh35(X, X, Y, Z) = true.
50.98/51.17	Axiom 13 (necessitation_1): fresh34(X, X, Y) = is_a_theorem(necessarily(Y)).
50.98/51.17	Axiom 14 (necessitation_1): fresh33(X, X, Y) = true.
50.98/51.17	Axiom 15 (op_equiv): fresh30(X, X, Y, Z) = equiv(Y, Z).
50.98/51.17	Axiom 16 (op_implies_and): fresh29(X, X, Y, Z) = implies(Y, Z).
50.98/51.17	Axiom 17 (op_or): fresh26(X, X, Y, Z) = or(Y, Z).
50.98/51.17	Axiom 18 (op_possibly): fresh25(X, X, Y) = possibly(Y).
50.98/51.17	Axiom 19 (op_strict_implies): fresh23(X, X, Y, Z) = strict_implies(Y, Z).
50.98/51.17	Axiom 20 (or_1_1): fresh21(X, X, Y, Z) = true.
50.98/51.17	Axiom 21 (or_3_1): fresh17(X, X, Y, Z, W) = true.
50.98/51.17	Axiom 22 (substitution_of_equivalents_2): fresh4(X, X, Y, Z) = Y.
50.98/51.17	Axiom 23 (substitution_of_equivalents_2): fresh3(X, X, Y, Z) = Z.
50.98/51.17	Axiom 24 (and_3_1): fresh103(and_3, true, X, Y) = is_a_theorem(implies(X, implies(Y, and(X, Y)))).
50.98/51.17	Axiom 25 (substitution_of_equivalents_2): fresh4(substitution_of_equivalents, true, X, Y) = fresh3(is_a_theorem(equiv(X, Y)), true, X, Y).
50.98/51.17	Axiom 26 (modus_ponens_2): fresh115(modus_ponens, true, X, Y) = fresh40(is_a_theorem(implies(X, Y)), true, X, Y).
50.98/51.17	Axiom 27 (kn1_1): fresh45(kn1, true, X) = is_a_theorem(implies(X, and(X, X))).
50.98/51.17	Axiom 28 (and_1_1): fresh107(and_1, true, X, Y) = is_a_theorem(implies(and(X, Y), X)).
50.98/51.17	Axiom 29 (and_2_1): fresh105(and_2, true, X, Y) = is_a_theorem(implies(and(X, Y), Y)).
50.98/51.17	Axiom 30 (r3_1): fresh11(r3, true, X, Y) = is_a_theorem(implies(or(X, Y), or(Y, X))).
50.98/51.17	Axiom 31 (implies_2_1): fresh49(implies_2, true, X, Y) = is_a_theorem(implies(implies(X, implies(X, Y)), implies(X, Y))).
50.98/51.17	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)))).
50.98/51.17	Axiom 33 (modus_tollens_1): fresh35(modus_tollens, true, X, Y) = is_a_theorem(implies(implies(not(Y), not(X)), implies(X, Y))).
50.98/51.17	Axiom 34 (or_1_1): fresh21(or_1, true, X, Y) = is_a_theorem(implies(X, or(X, Y))).
50.98/51.17	Axiom 35 (op_or): fresh26(op_or, true, X, Y) = not(and(not(X), not(Y))).
50.98/51.17	Axiom 36 (op_equiv): fresh30(op_equiv, true, X, Y) = and(implies(X, Y), implies(Y, X)).
50.98/51.17	Axiom 37 (op_implies_and): fresh29(op_implies_and, true, X, Y) = not(and(X, not(Y))).
50.98/51.17	Axiom 38 (hilbert_modus_tollens): modus_tollens = true.
50.98/51.17	Axiom 39 (hilbert_and_2): and_2 = true.
50.98/51.17	Axiom 40 (hilbert_and_1): and_1 = true.
50.98/51.17	Axiom 41 (substitution_of_equivalents): substitution_of_equivalents = true.
50.98/51.17	Axiom 42 (hilbert_and_3): and_3 = true.
50.98/51.17	Axiom 43 (hilbert_or_1): or_1 = true.
50.98/51.17	Axiom 44 (hilbert_modus_ponens): modus_ponens = true.
50.98/51.17	Axiom 45 (hilbert_or_3): or_3 = true.
50.98/51.17	Axiom 46 (hilbert_implies_2): implies_2 = true.
50.98/51.17	Axiom 47 (hilbert_op_equiv): op_equiv = true.
50.98/51.17	Axiom 48 (hilbert_op_implies_and): op_implies_and = true.
50.98/51.17	Axiom 49 (hilbert_op_or): op_or = true.
50.98/51.17	Axiom 50 (necessitation_1): fresh34(necessitation, true, X) = fresh33(is_a_theorem(X), true, X).
50.98/51.17	Axiom 51 (axiom_K_1): fresh95(axiom_K, true, X, Y) = is_a_theorem(implies(necessarily(implies(X, Y)), implies(necessarily(X), necessarily(Y)))).
50.98/51.17	Axiom 52 (axiom_5_1): fresh99(axiom_5, true, X) = is_a_theorem(implies(possibly(X), necessarily(possibly(X)))).
50.98/51.17	Axiom 53 (axiom_M_1): fresh93(axiom_M, true, X) = is_a_theorem(implies(necessarily(X), X)).
50.98/51.17	Axiom 54 (axiom_s3_1): fresh67(axiom_s3, true, X, Y) = is_a_theorem(strict_implies(strict_implies(X, Y), strict_implies(not(possibly(Y)), not(possibly(X))))).
50.98/51.17	Axiom 55 (axiom_s3): fresh68(is_a_theorem(strict_implies(strict_implies(sK5_axiom_s3_X, sK4_axiom_s3_Y), strict_implies(not(possibly(sK4_axiom_s3_Y)), not(possibly(sK5_axiom_s3_X))))), true) = axiom_s3.
50.98/51.17	Axiom 56 (op_strict_implies): fresh23(op_strict_implies, true, X, Y) = necessarily(implies(X, Y)).
50.98/51.17	Axiom 57 (op_possibly): fresh25(op_possibly, true, X) = not(necessarily(not(X))).
50.98/51.17	Axiom 58 (km5_axiom_K): axiom_K = true.
50.98/51.17	Axiom 59 (km5_axiom_5): axiom_5 = true.
50.98/51.17	Axiom 60 (km5_axiom_M): axiom_M = true.
50.98/51.17	Axiom 61 (km5_necessitation): necessitation = true.
50.98/51.17	Axiom 62 (km5_op_possibly): op_possibly = true.
51.03/51.22	Axiom 63 (s1_0_op_strict_implies): op_strict_implies = true.
51.03/51.22	
51.03/51.22	Lemma 64: not(necessarily(not(X))) = possibly(X).
51.03/51.22	Proof:
51.03/51.22	  not(necessarily(not(X)))
51.03/51.22	= { by axiom 57 (op_possibly) }
51.03/51.22	  fresh25(op_possibly, true, X)
51.03/51.22	= { by axiom 62 (km5_op_possibly) }
51.03/51.22	  fresh25(true, true, X)
51.03/51.22	= { by axiom 18 (op_possibly) }
51.03/51.22	  possibly(X)
51.03/51.22	
51.03/51.22	Lemma 65: not(and(X, not(Y))) = implies(X, Y).
51.03/51.22	Proof:
51.03/51.22	  not(and(X, not(Y)))
51.03/51.22	= { by axiom 37 (op_implies_and) }
51.03/51.22	  fresh29(op_implies_and, true, X, Y)
51.03/51.22	= { by axiom 48 (hilbert_op_implies_and) }
51.03/51.22	  fresh29(true, true, X, Y)
51.03/51.22	= { by axiom 16 (op_implies_and) }
51.03/51.22	  implies(X, Y)
51.03/51.22	
51.03/51.22	Lemma 66: and(implies(X, Y), implies(Y, X)) = equiv(X, Y).
51.03/51.22	Proof:
51.03/51.22	  and(implies(X, Y), implies(Y, X))
51.03/51.22	= { by axiom 36 (op_equiv) }
51.03/51.22	  fresh30(op_equiv, true, X, Y)
51.03/51.22	= { by axiom 47 (hilbert_op_equiv) }
51.03/51.22	  fresh30(true, true, X, Y)
51.03/51.22	= { by axiom 15 (op_equiv) }
51.03/51.22	  equiv(X, Y)
51.03/51.22	
51.03/51.22	Lemma 67: fresh40(is_a_theorem(implies(X, Y)), true, X, Y) = fresh116(is_a_theorem(X), true, Y).
51.03/51.22	Proof:
51.03/51.22	  fresh40(is_a_theorem(implies(X, Y)), true, X, Y)
51.03/51.22	= { by axiom 26 (modus_ponens_2) }
51.03/51.22	  fresh115(modus_ponens, true, X, Y)
51.03/51.22	= { by axiom 44 (hilbert_modus_ponens) }
51.03/51.22	  fresh115(true, true, X, Y)
51.03/51.22	= { by axiom 11 (modus_ponens_2) }
51.03/51.22	  fresh116(is_a_theorem(X), true, Y)
51.03/51.22	
51.03/51.22	Lemma 68: is_a_theorem(implies(X, implies(Y, and(X, Y)))) = true.
51.03/51.22	Proof:
51.03/51.22	  is_a_theorem(implies(X, implies(Y, and(X, Y))))
51.03/51.22	= { by axiom 24 (and_3_1) }
51.03/51.22	  fresh103(and_3, true, X, Y)
51.03/51.22	= { by axiom 42 (hilbert_and_3) }
51.03/51.22	  fresh103(true, true, X, Y)
51.03/51.22	= { by axiom 3 (and_3_1) }
51.03/51.22	  true
51.03/51.22	
51.03/51.22	Lemma 69: fresh116(is_a_theorem(X), true, implies(Y, and(X, Y))) = is_a_theorem(implies(Y, and(X, Y))).
51.03/51.22	Proof:
51.03/51.22	  fresh116(is_a_theorem(X), true, implies(Y, and(X, Y)))
51.03/51.22	= { by lemma 67 }
51.03/51.22	  fresh40(is_a_theorem(implies(X, implies(Y, and(X, Y)))), true, X, implies(Y, and(X, Y)))
51.03/51.22	= { by lemma 68 }
51.03/51.22	  fresh40(true, true, X, implies(Y, and(X, Y)))
51.03/51.22	= { by axiom 9 (modus_ponens_2) }
51.03/51.22	  is_a_theorem(implies(Y, and(X, Y)))
51.03/51.22	
51.03/51.22	Lemma 70: fresh116(is_a_theorem(implies(X, implies(X, Y))), true, implies(X, Y)) = is_a_theorem(implies(X, Y)).
51.03/51.22	Proof:
51.03/51.22	  fresh116(is_a_theorem(implies(X, implies(X, Y))), true, implies(X, Y))
51.03/51.22	= { by lemma 67 }
51.03/51.22	  fresh40(is_a_theorem(implies(implies(X, implies(X, Y)), implies(X, Y))), true, implies(X, implies(X, Y)), implies(X, Y))
51.03/51.22	= { by axiom 31 (implies_2_1) }
51.03/51.22	  fresh40(fresh49(implies_2, true, X, Y), true, implies(X, implies(X, Y)), implies(X, Y))
51.03/51.22	= { by axiom 46 (hilbert_implies_2) }
51.03/51.22	  fresh40(fresh49(true, true, X, Y), true, implies(X, implies(X, Y)), implies(X, Y))
51.03/51.22	= { by axiom 8 (implies_2_1) }
51.03/51.22	  fresh40(true, true, implies(X, implies(X, Y)), implies(X, Y))
51.03/51.22	= { by axiom 9 (modus_ponens_2) }
51.03/51.22	  is_a_theorem(implies(X, Y))
51.03/51.22	
51.03/51.22	Lemma 71: fresh3(is_a_theorem(equiv(X, Y)), true, X, Y) = X.
51.03/51.22	Proof:
51.03/51.22	  fresh3(is_a_theorem(equiv(X, Y)), true, X, Y)
51.03/51.22	= { by axiom 25 (substitution_of_equivalents_2) }
51.03/51.22	  fresh4(substitution_of_equivalents, true, X, Y)
51.03/51.22	= { by axiom 41 (substitution_of_equivalents) }
51.03/51.22	  fresh4(true, true, X, Y)
51.03/51.22	= { by axiom 22 (substitution_of_equivalents_2) }
51.03/51.22	  X
51.03/51.22	
51.03/51.22	Lemma 72: and(X, X) = X.
51.03/51.22	Proof:
51.03/51.22	  and(X, X)
51.03/51.22	= { by axiom 23 (substitution_of_equivalents_2) }
51.03/51.22	  fresh3(true, true, X, and(X, X))
51.03/51.22	= { by axiom 10 (modus_ponens_2) }
51.03/51.22	  fresh3(fresh116(true, true, equiv(X, and(X, X))), true, X, and(X, X))
51.03/51.22	= { by axiom 2 (and_2_1) }
51.03/51.22	  fresh3(fresh116(fresh105(true, true, X, X), true, equiv(X, and(X, X))), true, X, and(X, X))
51.03/51.22	= { by axiom 39 (hilbert_and_2) }
51.03/51.22	  fresh3(fresh116(fresh105(and_2, true, X, X), true, equiv(X, and(X, X))), true, X, and(X, X))
51.03/51.22	= { by axiom 29 (and_2_1) }
51.03/51.22	  fresh3(fresh116(is_a_theorem(implies(and(X, X), X)), true, equiv(X, and(X, X))), true, X, and(X, X))
51.03/51.22	= { by lemma 67 }
51.03/51.22	  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))
51.03/51.22	= { by lemma 66 }
51.03/51.22	  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))
51.03/51.22	= { by lemma 69 }
51.03/51.22	  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))
51.03/51.22	= { by lemma 70 }
51.03/51.22	  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))
51.03/51.22	= { by lemma 68 }
51.03/51.22	  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))
51.03/51.22	= { by axiom 10 (modus_ponens_2) }
51.03/51.22	  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))
51.03/51.22	= { by axiom 10 (modus_ponens_2) }
51.03/51.22	  fresh3(fresh40(true, true, implies(and(X, X), X), equiv(X, and(X, X))), true, X, and(X, X))
51.03/51.22	= { by axiom 9 (modus_ponens_2) }
51.03/51.22	  fresh3(is_a_theorem(equiv(X, and(X, X))), true, X, and(X, X))
51.03/51.22	= { by lemma 71 }
51.03/51.22	  X
51.03/51.22	
51.03/51.22	Lemma 73: implies(not(X), Y) = or(X, Y).
51.03/51.22	Proof:
51.03/51.22	  implies(not(X), Y)
51.03/51.22	= { by lemma 65 }
51.03/51.22	  not(and(not(X), not(Y)))
51.03/51.22	= { by axiom 35 (op_or) }
51.03/51.22	  fresh26(op_or, true, X, Y)
51.03/51.22	= { by axiom 49 (hilbert_op_or) }
51.03/51.22	  fresh26(true, true, X, Y)
51.03/51.22	= { by axiom 17 (op_or) }
51.03/51.22	  or(X, Y)
51.03/51.22	
51.03/51.22	Lemma 74: not(not(X)) = or(X, X).
51.03/51.22	Proof:
51.03/51.22	  not(not(X))
51.03/51.22	= { by lemma 72 }
51.03/51.22	  not(and(not(X), not(X)))
51.03/51.22	= { by lemma 65 }
51.03/51.22	  implies(not(X), X)
51.03/51.22	= { by lemma 73 }
51.03/51.22	  or(X, X)
51.03/51.22	
51.03/51.22	Lemma 75: or(and(X, not(Y)), Z) = implies(implies(X, Y), Z).
51.03/51.22	Proof:
51.03/51.22	  or(and(X, not(Y)), Z)
51.03/51.22	= { by axiom 17 (op_or) }
51.03/51.22	  fresh26(true, true, and(X, not(Y)), Z)
51.03/51.22	= { by axiom 49 (hilbert_op_or) }
51.03/51.22	  fresh26(op_or, true, and(X, not(Y)), Z)
51.03/51.22	= { by axiom 35 (op_or) }
51.03/51.22	  not(and(not(and(X, not(Y))), not(Z)))
51.03/51.22	= { by lemma 65 }
51.03/51.22	  not(and(implies(X, Y), not(Z)))
51.03/51.22	= { by lemma 65 }
51.08/51.27	  implies(implies(X, Y), Z)
51.08/51.27	
51.08/51.27	Lemma 76: or(X, X) = X.
51.08/51.27	Proof:
51.08/51.27	  or(X, X)
51.08/51.27	= { by lemma 71 }
51.08/51.27	  fresh3(is_a_theorem(equiv(or(X, X), X)), true, or(X, X), X)
51.08/51.27	= { by axiom 9 (modus_ponens_2) }
51.08/51.27	  fresh3(fresh40(true, true, implies(X, or(X, X)), equiv(or(X, X), X)), true, or(X, X), X)
51.08/51.27	= { by axiom 10 (modus_ponens_2) }
51.08/51.27	  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)
51.08/51.27	= { by axiom 10 (modus_ponens_2) }
51.08/51.27	  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)
51.08/51.27	= { by axiom 1 (and_1_1) }
51.08/51.27	  fresh3(fresh40(fresh116(fresh116(fresh107(true, true, 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)
51.08/51.27	= { by axiom 40 (hilbert_and_1) }
51.08/51.27	  fresh3(fresh40(fresh116(fresh116(fresh107(and_1, true, 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)
51.08/51.27	= { by axiom 28 (and_1_1) }
51.08/51.27	  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)
51.08/51.27	= { by lemma 67 }
51.08/51.27	  fresh3(fresh40(fresh116(fresh40(is_a_theorem(implies(implies(and(X, X), X), or(not(and(X, X)), X))), true, implies(and(X, X), X), 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)
51.08/51.27	= { by lemma 72 }
51.08/51.27	  fresh3(fresh40(fresh116(fresh40(is_a_theorem(implies(implies(and(X, X), X), or(and(not(and(X, X)), not(and(X, X))), X))), true, implies(and(X, X), X), 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)
51.08/51.27	= { by lemma 75 }
51.08/51.27	  fresh3(fresh40(fresh116(fresh40(is_a_theorem(implies(implies(and(X, X), X), implies(implies(not(and(X, X)), and(X, X)), X))), true, implies(and(X, X), X), 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)
51.08/51.27	= { by lemma 73 }
51.08/51.27	  fresh3(fresh40(fresh116(fresh40(is_a_theorem(implies(implies(and(X, X), X), implies(or(and(X, X), and(X, X)), X))), true, implies(and(X, X), X), 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)
51.08/51.27	= { by lemma 70 }
51.08/51.27	  fresh3(fresh40(fresh116(fresh40(fresh116(is_a_theorem(implies(implies(and(X, X), X), implies(implies(and(X, X), X), implies(or(and(X, X), and(X, X)), X)))), true, implies(implies(and(X, X), X), implies(or(and(X, X), and(X, X)), X))), true, implies(and(X, X), X), 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)
51.08/51.27	= { by axiom 32 (or_3_1) }
51.08/51.27	  fresh3(fresh40(fresh116(fresh40(fresh116(fresh17(or_3, true, and(X, X), and(X, X), X), true, implies(implies(and(X, X), X), implies(or(and(X, X), and(X, X)), X))), true, implies(and(X, X), X), 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)
51.08/51.27	= { by axiom 45 (hilbert_or_3) }
51.08/51.27	  fresh3(fresh40(fresh116(fresh40(fresh116(fresh17(true, true, and(X, X), and(X, X), X), true, implies(implies(and(X, X), X), implies(or(and(X, X), and(X, X)), X))), true, implies(and(X, X), X), 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)
51.08/51.27	= { by axiom 21 (or_3_1) }
51.08/51.27	  fresh3(fresh40(fresh116(fresh40(fresh116(true, true, implies(implies(and(X, X), X), implies(or(and(X, X), and(X, X)), X))), true, implies(and(X, X), X), 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)
51.08/51.27	= { by axiom 10 (modus_ponens_2) }
51.08/51.27	  fresh3(fresh40(fresh116(fresh40(true, true, implies(and(X, X), X), 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)
51.08/51.27	= { by axiom 9 (modus_ponens_2) }
51.08/51.27	  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)
51.08/51.27	= { by lemma 69 }
51.08/51.27	  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)
51.08/51.27	= { by lemma 74 }
51.08/51.27	  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)
51.08/51.27	= { by lemma 72 }
51.08/51.27	  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)
51.08/51.27	= { by lemma 73 }
51.08/51.27	  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)
51.08/51.27	= { by axiom 36 (op_equiv) }
51.08/51.27	  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)
51.08/51.27	= { by axiom 47 (hilbert_op_equiv) }
51.08/51.27	  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)
51.08/51.27	= { by axiom 15 (op_equiv) }
51.08/51.27	  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)
51.08/51.27	= { by lemma 74 }
51.08/51.27	  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)
51.08/51.27	= { by lemma 67 }
51.08/51.27	  fresh3(fresh116(is_a_theorem(implies(X, or(X, X))), true, equiv(or(X, X), X)), true, or(X, X), X)
51.08/51.27	= { by axiom 34 (or_1_1) }
51.08/51.27	  fresh3(fresh116(fresh21(or_1, true, X, X), true, equiv(or(X, X), X)), true, or(X, X), X)
51.08/51.27	= { by axiom 43 (hilbert_or_1) }
51.08/51.27	  fresh3(fresh116(fresh21(true, true, X, X), true, equiv(or(X, X), X)), true, or(X, X), X)
51.08/51.27	= { by axiom 20 (or_1_1) }
51.08/51.27	  fresh3(fresh116(true, true, equiv(or(X, X), X)), true, or(X, X), X)
51.08/51.27	= { by axiom 10 (modus_ponens_2) }
51.08/51.27	  fresh3(true, true, or(X, X), X)
51.08/51.27	= { by axiom 23 (substitution_of_equivalents_2) }
51.08/51.27	  X
51.08/51.27	
51.08/51.27	Lemma 77: not(possibly(X)) = necessarily(not(X)).
51.08/51.27	Proof:
51.08/51.27	  not(possibly(X))
51.08/51.27	= { by lemma 72 }
51.08/51.27	  not(and(possibly(X), possibly(X)))
51.08/51.27	= { by lemma 64 }
51.08/51.27	  not(and(possibly(X), not(necessarily(not(X)))))
51.08/51.27	= { by lemma 65 }
51.08/51.27	  implies(possibly(X), necessarily(not(X)))
51.08/51.27	= { by lemma 64 }
51.08/51.27	  implies(not(necessarily(not(X))), necessarily(not(X)))
51.08/51.27	= { by lemma 73 }
51.08/51.27	  or(necessarily(not(X)), necessarily(not(X)))
51.08/51.27	= { by lemma 76 }
51.08/51.27	  necessarily(not(X))
51.08/51.27	
51.08/51.27	Lemma 78: necessarily(implies(X, Y)) = strict_implies(X, Y).
51.08/51.27	Proof:
51.08/51.27	  necessarily(implies(X, Y))
51.08/51.27	= { by axiom 56 (op_strict_implies) }
51.08/51.27	  fresh23(op_strict_implies, true, X, Y)
51.08/51.27	= { by axiom 63 (s1_0_op_strict_implies) }
51.08/51.27	  fresh23(true, true, X, Y)
51.08/51.27	= { by axiom 19 (op_strict_implies) }
51.08/51.27	  strict_implies(X, Y)
51.08/51.27	
51.08/51.27	Lemma 79: necessarily(or(X, Y)) = strict_implies(not(X), Y).
51.08/51.27	Proof:
51.08/51.27	  necessarily(or(X, Y))
51.08/51.27	= { by lemma 73 }
51.08/51.27	  necessarily(implies(not(X), Y))
51.08/51.27	= { by lemma 78 }
51.08/51.27	  strict_implies(not(X), Y)
51.08/51.27	
51.08/51.27	Lemma 80: is_a_theorem(implies(or(X, Y), or(Y, X))) = true.
51.08/51.27	Proof:
51.08/51.27	  is_a_theorem(implies(or(X, Y), or(Y, X)))
51.08/51.27	= { by axiom 30 (r3_1) }
51.08/51.27	  fresh11(r3, true, X, Y)
51.08/51.27	= { by axiom 30 (r3_1) }
51.08/51.27	  is_a_theorem(implies(or(X, Y), or(Y, X)))
51.08/51.27	= { by lemma 76 }
51.08/51.27	  is_a_theorem(implies(or(X, or(Y, Y)), or(Y, X)))
51.08/51.27	= { by axiom 9 (modus_ponens_2) }
51.08/51.27	  fresh40(true, true, necessarily(implies(or(X, or(Y, Y)), or(Y, X))), implies(or(X, or(Y, Y)), or(Y, X)))
51.08/51.27	= { by axiom 6 (axiom_M_1) }
51.08/51.27	  fresh40(fresh93(true, true, implies(or(X, or(Y, Y)), or(Y, X))), true, necessarily(implies(or(X, or(Y, Y)), or(Y, X))), implies(or(X, or(Y, Y)), or(Y, X)))
51.08/51.27	= { by axiom 60 (km5_axiom_M) }
51.08/51.27	  fresh40(fresh93(axiom_M, true, implies(or(X, or(Y, Y)), or(Y, X))), true, necessarily(implies(or(X, or(Y, Y)), or(Y, X))), implies(or(X, or(Y, Y)), or(Y, X)))
51.08/51.27	= { by axiom 53 (axiom_M_1) }
51.08/51.27	  fresh40(is_a_theorem(implies(necessarily(implies(or(X, or(Y, Y)), or(Y, X))), implies(or(X, or(Y, Y)), or(Y, X)))), true, necessarily(implies(or(X, or(Y, Y)), or(Y, X))), implies(or(X, or(Y, Y)), or(Y, X)))
51.08/51.27	= { by axiom 26 (modus_ponens_2) }
51.08/51.27	  fresh115(modus_ponens, true, necessarily(implies(or(X, or(Y, Y)), or(Y, X))), implies(or(X, or(Y, Y)), or(Y, X)))
51.08/51.27	= { by axiom 44 (hilbert_modus_ponens) }
51.08/51.27	  fresh115(true, true, necessarily(implies(or(X, or(Y, Y)), or(Y, X))), implies(or(X, or(Y, Y)), or(Y, X)))
51.08/51.27	= { by axiom 11 (modus_ponens_2) }
51.08/51.27	  fresh116(is_a_theorem(necessarily(implies(or(X, or(Y, Y)), or(Y, X)))), true, implies(or(X, or(Y, Y)), or(Y, X)))
51.08/51.27	= { by lemma 78 }
51.08/51.27	  fresh116(is_a_theorem(strict_implies(or(X, or(Y, Y)), or(Y, X))), true, implies(or(X, or(Y, Y)), or(Y, X)))
51.08/51.27	= { by lemma 74 }
51.08/51.27	  fresh116(is_a_theorem(strict_implies(or(X, not(not(Y))), or(Y, X))), true, implies(or(X, or(Y, Y)), or(Y, X)))
51.08/51.27	= { by lemma 78 }
51.08/51.27	  fresh116(is_a_theorem(necessarily(implies(or(X, not(not(Y))), or(Y, X)))), true, implies(or(X, or(Y, Y)), or(Y, X)))
51.08/51.27	= { by axiom 13 (necessitation_1) }
51.08/51.27	  fresh116(fresh34(true, true, implies(or(X, not(not(Y))), or(Y, X))), true, implies(or(X, or(Y, Y)), or(Y, X)))
51.08/51.27	= { by axiom 61 (km5_necessitation) }
51.08/51.27	  fresh116(fresh34(necessitation, true, implies(or(X, not(not(Y))), or(Y, X))), true, implies(or(X, or(Y, Y)), or(Y, X)))
51.08/51.27	= { by axiom 50 (necessitation_1) }
51.08/51.27	  fresh116(fresh33(is_a_theorem(implies(or(X, not(not(Y))), or(Y, X))), true, implies(or(X, not(not(Y))), or(Y, X))), true, implies(or(X, or(Y, Y)), or(Y, X)))
51.08/51.27	= { by lemma 73 }
51.08/51.27	  fresh116(fresh33(is_a_theorem(implies(implies(not(X), not(not(Y))), or(Y, X))), true, implies(or(X, not(not(Y))), or(Y, X))), true, implies(or(X, or(Y, Y)), or(Y, X)))
51.08/51.27	= { by lemma 73 }
51.08/51.27	  fresh116(fresh33(is_a_theorem(implies(implies(not(X), not(not(Y))), implies(not(Y), X))), true, implies(or(X, not(not(Y))), or(Y, X))), true, implies(or(X, or(Y, Y)), or(Y, X)))
51.08/51.27	= { by axiom 33 (modus_tollens_1) }
51.08/51.27	  fresh116(fresh33(fresh35(modus_tollens, true, not(Y), X), true, implies(or(X, not(not(Y))), or(Y, X))), true, implies(or(X, or(Y, Y)), or(Y, X)))
51.08/51.27	= { by axiom 38 (hilbert_modus_tollens) }
51.08/51.27	  fresh116(fresh33(fresh35(true, true, not(Y), X), true, implies(or(X, not(not(Y))), or(Y, X))), true, implies(or(X, or(Y, Y)), or(Y, X)))
51.08/51.27	= { by axiom 12 (modus_tollens_1) }
51.08/51.27	  fresh116(fresh33(true, true, implies(or(X, not(not(Y))), or(Y, X))), true, implies(or(X, or(Y, Y)), or(Y, X)))
51.08/51.27	= { by axiom 14 (necessitation_1) }
51.08/51.27	  fresh116(true, true, implies(or(X, or(Y, Y)), or(Y, X)))
51.08/51.27	= { by axiom 10 (modus_ponens_2) }
51.08/51.27	  true
51.08/51.27	
51.08/51.27	Lemma 81: strict_implies(not(X), X) = necessarily(X).
51.08/51.27	Proof:
51.08/51.27	  strict_implies(not(X), X)
51.08/51.27	= { by lemma 79 }
51.08/51.27	  necessarily(or(X, X))
51.08/51.27	= { by lemma 76 }
51.08/51.27	  necessarily(X)
51.08/51.27	
51.08/51.27	Lemma 82: possibly(and(X, not(Y))) = not(strict_implies(X, Y)).
51.08/51.27	Proof:
51.08/51.27	  possibly(and(X, not(Y)))
51.08/51.27	= { by lemma 64 }
51.08/51.27	  not(necessarily(not(and(X, not(Y)))))
51.08/51.27	= { by lemma 65 }
51.08/51.27	  not(necessarily(implies(X, Y)))
51.08/51.27	= { by lemma 78 }
51.08/51.28	  not(strict_implies(X, Y))
51.08/51.28	
51.08/51.28	Lemma 83: necessarily(possibly(X)) = possibly(X).
51.08/51.28	Proof:
51.08/51.28	  necessarily(possibly(X))
51.08/51.28	= { by axiom 23 (substitution_of_equivalents_2) }
51.08/51.28	  fresh3(true, true, possibly(X), necessarily(possibly(X)))
51.08/51.28	= { by axiom 10 (modus_ponens_2) }
51.08/51.28	  fresh3(fresh116(true, true, equiv(possibly(X), necessarily(possibly(X)))), true, possibly(X), necessarily(possibly(X)))
51.08/51.28	= { by axiom 6 (axiom_M_1) }
51.08/51.28	  fresh3(fresh116(fresh93(true, true, possibly(X)), true, equiv(possibly(X), necessarily(possibly(X)))), true, possibly(X), necessarily(possibly(X)))
51.08/51.28	= { by axiom 60 (km5_axiom_M) }
51.08/51.28	  fresh3(fresh116(fresh93(axiom_M, true, possibly(X)), true, equiv(possibly(X), necessarily(possibly(X)))), true, possibly(X), necessarily(possibly(X)))
51.08/51.28	= { by axiom 53 (axiom_M_1) }
51.08/51.28	  fresh3(fresh116(is_a_theorem(implies(necessarily(possibly(X)), possibly(X))), true, equiv(possibly(X), necessarily(possibly(X)))), true, possibly(X), necessarily(possibly(X)))
51.08/51.28	= { by lemma 67 }
51.08/51.28	  fresh3(fresh40(is_a_theorem(implies(implies(necessarily(possibly(X)), possibly(X)), equiv(possibly(X), necessarily(possibly(X))))), true, implies(necessarily(possibly(X)), possibly(X)), equiv(possibly(X), necessarily(possibly(X)))), true, possibly(X), necessarily(possibly(X)))
51.08/51.28	= { by lemma 66 }
51.08/51.28	  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))))), true, implies(necessarily(possibly(X)), possibly(X)), equiv(possibly(X), necessarily(possibly(X)))), true, possibly(X), necessarily(possibly(X)))
51.08/51.28	= { by lemma 69 }
51.08/51.28	  fresh3(fresh40(fresh116(is_a_theorem(implies(possibly(X), necessarily(possibly(X)))), true, implies(implies(necessarily(possibly(X)), possibly(X)), and(implies(possibly(X), necessarily(possibly(X))), implies(necessarily(possibly(X)), possibly(X))))), true, implies(necessarily(possibly(X)), possibly(X)), equiv(possibly(X), necessarily(possibly(X)))), true, possibly(X), necessarily(possibly(X)))
51.08/51.28	= { by axiom 52 (axiom_5_1) }
51.08/51.28	  fresh3(fresh40(fresh116(fresh99(axiom_5, true, X), true, implies(implies(necessarily(possibly(X)), possibly(X)), and(implies(possibly(X), necessarily(possibly(X))), implies(necessarily(possibly(X)), possibly(X))))), true, implies(necessarily(possibly(X)), possibly(X)), equiv(possibly(X), necessarily(possibly(X)))), true, possibly(X), necessarily(possibly(X)))
51.08/51.28	= { by axiom 59 (km5_axiom_5) }
51.08/51.28	  fresh3(fresh40(fresh116(fresh99(true, true, X), true, implies(implies(necessarily(possibly(X)), possibly(X)), and(implies(possibly(X), necessarily(possibly(X))), implies(necessarily(possibly(X)), possibly(X))))), true, implies(necessarily(possibly(X)), possibly(X)), equiv(possibly(X), necessarily(possibly(X)))), true, possibly(X), necessarily(possibly(X)))
51.08/51.28	= { by axiom 4 (axiom_5_1) }
51.08/51.28	  fresh3(fresh40(fresh116(true, true, implies(implies(necessarily(possibly(X)), possibly(X)), and(implies(possibly(X), necessarily(possibly(X))), implies(necessarily(possibly(X)), possibly(X))))), true, implies(necessarily(possibly(X)), possibly(X)), equiv(possibly(X), necessarily(possibly(X)))), true, possibly(X), necessarily(possibly(X)))
51.08/51.28	= { by axiom 10 (modus_ponens_2) }
51.08/51.28	  fresh3(fresh40(true, true, implies(necessarily(possibly(X)), possibly(X)), equiv(possibly(X), necessarily(possibly(X)))), true, possibly(X), necessarily(possibly(X)))
51.08/51.28	= { by axiom 9 (modus_ponens_2) }
51.08/51.28	  fresh3(is_a_theorem(equiv(possibly(X), necessarily(possibly(X)))), true, possibly(X), necessarily(possibly(X)))
51.08/51.28	= { by lemma 71 }
51.08/51.28	  possibly(X)
51.08/51.28	
51.08/51.28	Lemma 84: possibly(necessarily(not(X))) = not(possibly(X)).
51.08/51.28	Proof:
51.08/51.28	  possibly(necessarily(not(X)))
51.08/51.28	= { by lemma 64 }
51.08/51.28	  not(necessarily(not(necessarily(not(X)))))
51.08/51.28	= { by lemma 64 }
51.08/51.28	  not(necessarily(possibly(X)))
51.08/51.28	= { by lemma 83 }
51.08/51.28	  not(possibly(X))
51.08/51.28	
51.08/51.28	Lemma 85: is_a_theorem(implies(strict_implies(X, Y), implies(necessarily(X), necessarily(Y)))) = true.
51.08/51.28	Proof:
51.08/51.28	  is_a_theorem(implies(strict_implies(X, Y), implies(necessarily(X), necessarily(Y))))
51.08/51.28	= { by lemma 78 }
51.08/51.28	  is_a_theorem(implies(necessarily(implies(X, Y)), implies(necessarily(X), necessarily(Y))))
51.08/51.28	= { by axiom 51 (axiom_K_1) }
51.08/51.28	  fresh95(axiom_K, true, X, Y)
51.08/51.28	= { by axiom 58 (km5_axiom_K) }
51.08/51.28	  fresh95(true, true, X, Y)
51.08/51.28	= { by axiom 5 (axiom_K_1) }
51.44/51.63	  true
51.44/51.63	
51.44/51.63	Goal 1 (s1_0_m6s3m9b_axiom_s3): axiom_s3 = true.
51.44/51.63	Proof:
51.44/51.63	  axiom_s3
51.44/51.63	= { by axiom 55 (axiom_s3) }
51.44/51.63	  fresh68(is_a_theorem(strict_implies(strict_implies(sK5_axiom_s3_X, sK4_axiom_s3_Y), strict_implies(not(possibly(sK4_axiom_s3_Y)), not(possibly(sK5_axiom_s3_X))))), true)
51.44/51.63	= { by lemma 78 }
51.44/51.63	  fresh68(is_a_theorem(strict_implies(necessarily(implies(sK5_axiom_s3_X, sK4_axiom_s3_Y)), strict_implies(not(possibly(sK4_axiom_s3_Y)), not(possibly(sK5_axiom_s3_X))))), true)
51.44/51.63	= { by lemma 76 }
51.44/51.63	  fresh68(is_a_theorem(strict_implies(necessarily(implies(or(sK5_axiom_s3_X, sK5_axiom_s3_X), sK4_axiom_s3_Y)), strict_implies(not(possibly(sK4_axiom_s3_Y)), not(possibly(sK5_axiom_s3_X))))), true)
51.44/51.63	= { by lemma 73 }
51.44/51.63	  fresh68(is_a_theorem(strict_implies(necessarily(implies(implies(not(sK5_axiom_s3_X), sK5_axiom_s3_X), sK4_axiom_s3_Y)), strict_implies(not(possibly(sK4_axiom_s3_Y)), not(possibly(sK5_axiom_s3_X))))), true)
51.44/51.63	= { by lemma 75 }
51.44/51.63	  fresh68(is_a_theorem(strict_implies(necessarily(or(and(not(sK5_axiom_s3_X), not(sK5_axiom_s3_X)), sK4_axiom_s3_Y)), strict_implies(not(possibly(sK4_axiom_s3_Y)), not(possibly(sK5_axiom_s3_X))))), true)
51.44/51.63	= { by lemma 71 }
51.44/51.63	  fresh68(is_a_theorem(strict_implies(necessarily(fresh3(is_a_theorem(equiv(or(and(not(sK5_axiom_s3_X), not(sK5_axiom_s3_X)), sK4_axiom_s3_Y), or(sK4_axiom_s3_Y, and(not(sK5_axiom_s3_X), not(sK5_axiom_s3_X))))), true, or(and(not(sK5_axiom_s3_X), not(sK5_axiom_s3_X)), sK4_axiom_s3_Y), or(sK4_axiom_s3_Y, and(not(sK5_axiom_s3_X), not(sK5_axiom_s3_X))))), strict_implies(not(possibly(sK4_axiom_s3_Y)), not(possibly(sK5_axiom_s3_X))))), true)
51.44/51.63	= { by axiom 9 (modus_ponens_2) }
51.44/51.63	  fresh68(is_a_theorem(strict_implies(necessarily(fresh3(fresh40(true, true, implies(or(sK4_axiom_s3_Y, and(not(sK5_axiom_s3_X), not(sK5_axiom_s3_X))), or(and(not(sK5_axiom_s3_X), not(sK5_axiom_s3_X)), sK4_axiom_s3_Y)), equiv(or(and(not(sK5_axiom_s3_X), not(sK5_axiom_s3_X)), sK4_axiom_s3_Y), or(sK4_axiom_s3_Y, and(not(sK5_axiom_s3_X), not(sK5_axiom_s3_X))))), true, or(and(not(sK5_axiom_s3_X), not(sK5_axiom_s3_X)), sK4_axiom_s3_Y), or(sK4_axiom_s3_Y, and(not(sK5_axiom_s3_X), not(sK5_axiom_s3_X))))), strict_implies(not(possibly(sK4_axiom_s3_Y)), not(possibly(sK5_axiom_s3_X))))), true)
51.44/51.63	= { by axiom 10 (modus_ponens_2) }
51.44/51.63	  fresh68(is_a_theorem(strict_implies(necessarily(fresh3(fresh40(fresh116(true, true, implies(implies(or(sK4_axiom_s3_Y, and(not(sK5_axiom_s3_X), not(sK5_axiom_s3_X))), or(and(not(sK5_axiom_s3_X), not(sK5_axiom_s3_X)), sK4_axiom_s3_Y)), and(implies(or(and(not(sK5_axiom_s3_X), not(sK5_axiom_s3_X)), sK4_axiom_s3_Y), or(sK4_axiom_s3_Y, and(not(sK5_axiom_s3_X), not(sK5_axiom_s3_X)))), implies(or(sK4_axiom_s3_Y, and(not(sK5_axiom_s3_X), not(sK5_axiom_s3_X))), or(and(not(sK5_axiom_s3_X), not(sK5_axiom_s3_X)), sK4_axiom_s3_Y))))), true, implies(or(sK4_axiom_s3_Y, and(not(sK5_axiom_s3_X), not(sK5_axiom_s3_X))), or(and(not(sK5_axiom_s3_X), not(sK5_axiom_s3_X)), sK4_axiom_s3_Y)), equiv(or(and(not(sK5_axiom_s3_X), not(sK5_axiom_s3_X)), sK4_axiom_s3_Y), or(sK4_axiom_s3_Y, and(not(sK5_axiom_s3_X), not(sK5_axiom_s3_X))))), true, or(and(not(sK5_axiom_s3_X), not(sK5_axiom_s3_X)), sK4_axiom_s3_Y), or(sK4_axiom_s3_Y, and(not(sK5_axiom_s3_X), not(sK5_axiom_s3_X))))), strict_implies(not(possibly(sK4_axiom_s3_Y)), not(possibly(sK5_axiom_s3_X))))), true)
51.44/51.63	= { by lemma 80 }
51.44/51.63	  fresh68(is_a_theorem(strict_implies(necessarily(fresh3(fresh40(fresh116(is_a_theorem(implies(or(and(not(sK5_axiom_s3_X), not(sK5_axiom_s3_X)), sK4_axiom_s3_Y), or(sK4_axiom_s3_Y, and(not(sK5_axiom_s3_X), not(sK5_axiom_s3_X))))), true, implies(implies(or(sK4_axiom_s3_Y, and(not(sK5_axiom_s3_X), not(sK5_axiom_s3_X))), or(and(not(sK5_axiom_s3_X), not(sK5_axiom_s3_X)), sK4_axiom_s3_Y)), and(implies(or(and(not(sK5_axiom_s3_X), not(sK5_axiom_s3_X)), sK4_axiom_s3_Y), or(sK4_axiom_s3_Y, and(not(sK5_axiom_s3_X), not(sK5_axiom_s3_X)))), implies(or(sK4_axiom_s3_Y, and(not(sK5_axiom_s3_X), not(sK5_axiom_s3_X))), or(and(not(sK5_axiom_s3_X), not(sK5_axiom_s3_X)), sK4_axiom_s3_Y))))), true, implies(or(sK4_axiom_s3_Y, and(not(sK5_axiom_s3_X), not(sK5_axiom_s3_X))), or(and(not(sK5_axiom_s3_X), not(sK5_axiom_s3_X)), sK4_axiom_s3_Y)), equiv(or(and(not(sK5_axiom_s3_X), not(sK5_axiom_s3_X)), sK4_axiom_s3_Y), or(sK4_axiom_s3_Y, and(not(sK5_axiom_s3_X), not(sK5_axiom_s3_X))))), true, or(and(not(sK5_axiom_s3_X), not(sK5_axiom_s3_X)), sK4_axiom_s3_Y), or(sK4_axiom_s3_Y, and(not(sK5_axiom_s3_X), not(sK5_axiom_s3_X))))), strict_implies(not(possibly(sK4_axiom_s3_Y)), not(possibly(sK5_axiom_s3_X))))), true)
51.44/51.63	= { by lemma 69 }
51.44/51.63	  fresh68(is_a_theorem(strict_implies(necessarily(fresh3(fresh40(is_a_theorem(implies(implies(or(sK4_axiom_s3_Y, and(not(sK5_axiom_s3_X), not(sK5_axiom_s3_X))), or(and(not(sK5_axiom_s3_X), not(sK5_axiom_s3_X)), sK4_axiom_s3_Y)), and(implies(or(and(not(sK5_axiom_s3_X), not(sK5_axiom_s3_X)), sK4_axiom_s3_Y), or(sK4_axiom_s3_Y, and(not(sK5_axiom_s3_X), not(sK5_axiom_s3_X)))), implies(or(sK4_axiom_s3_Y, and(not(sK5_axiom_s3_X), not(sK5_axiom_s3_X))), or(and(not(sK5_axiom_s3_X), not(sK5_axiom_s3_X)), sK4_axiom_s3_Y))))), true, implies(or(sK4_axiom_s3_Y, and(not(sK5_axiom_s3_X), not(sK5_axiom_s3_X))), or(and(not(sK5_axiom_s3_X), not(sK5_axiom_s3_X)), sK4_axiom_s3_Y)), equiv(or(and(not(sK5_axiom_s3_X), not(sK5_axiom_s3_X)), sK4_axiom_s3_Y), or(sK4_axiom_s3_Y, and(not(sK5_axiom_s3_X), not(sK5_axiom_s3_X))))), true, or(and(not(sK5_axiom_s3_X), not(sK5_axiom_s3_X)), sK4_axiom_s3_Y), or(sK4_axiom_s3_Y, and(not(sK5_axiom_s3_X), not(sK5_axiom_s3_X))))), strict_implies(not(possibly(sK4_axiom_s3_Y)), not(possibly(sK5_axiom_s3_X))))), true)
51.44/51.63	= { by lemma 66 }
51.44/51.63	  fresh68(is_a_theorem(strict_implies(necessarily(fresh3(fresh40(is_a_theorem(implies(implies(or(sK4_axiom_s3_Y, and(not(sK5_axiom_s3_X), not(sK5_axiom_s3_X))), or(and(not(sK5_axiom_s3_X), not(sK5_axiom_s3_X)), sK4_axiom_s3_Y)), equiv(or(and(not(sK5_axiom_s3_X), not(sK5_axiom_s3_X)), sK4_axiom_s3_Y), or(sK4_axiom_s3_Y, and(not(sK5_axiom_s3_X), not(sK5_axiom_s3_X)))))), true, implies(or(sK4_axiom_s3_Y, and(not(sK5_axiom_s3_X), not(sK5_axiom_s3_X))), or(and(not(sK5_axiom_s3_X), not(sK5_axiom_s3_X)), sK4_axiom_s3_Y)), equiv(or(and(not(sK5_axiom_s3_X), not(sK5_axiom_s3_X)), sK4_axiom_s3_Y), or(sK4_axiom_s3_Y, and(not(sK5_axiom_s3_X), not(sK5_axiom_s3_X))))), true, or(and(not(sK5_axiom_s3_X), not(sK5_axiom_s3_X)), sK4_axiom_s3_Y), or(sK4_axiom_s3_Y, and(not(sK5_axiom_s3_X), not(sK5_axiom_s3_X))))), strict_implies(not(possibly(sK4_axiom_s3_Y)), not(possibly(sK5_axiom_s3_X))))), true)
51.44/51.63	= { by lemma 67 }
51.44/51.63	  fresh68(is_a_theorem(strict_implies(necessarily(fresh3(fresh116(is_a_theorem(implies(or(sK4_axiom_s3_Y, and(not(sK5_axiom_s3_X), not(sK5_axiom_s3_X))), or(and(not(sK5_axiom_s3_X), not(sK5_axiom_s3_X)), sK4_axiom_s3_Y))), true, equiv(or(and(not(sK5_axiom_s3_X), not(sK5_axiom_s3_X)), sK4_axiom_s3_Y), or(sK4_axiom_s3_Y, and(not(sK5_axiom_s3_X), not(sK5_axiom_s3_X))))), true, or(and(not(sK5_axiom_s3_X), not(sK5_axiom_s3_X)), sK4_axiom_s3_Y), or(sK4_axiom_s3_Y, and(not(sK5_axiom_s3_X), not(sK5_axiom_s3_X))))), strict_implies(not(possibly(sK4_axiom_s3_Y)), not(possibly(sK5_axiom_s3_X))))), true)
51.44/51.63	= { by lemma 80 }
51.44/51.63	  fresh68(is_a_theorem(strict_implies(necessarily(fresh3(fresh116(true, true, equiv(or(and(not(sK5_axiom_s3_X), not(sK5_axiom_s3_X)), sK4_axiom_s3_Y), or(sK4_axiom_s3_Y, and(not(sK5_axiom_s3_X), not(sK5_axiom_s3_X))))), true, or(and(not(sK5_axiom_s3_X), not(sK5_axiom_s3_X)), sK4_axiom_s3_Y), or(sK4_axiom_s3_Y, and(not(sK5_axiom_s3_X), not(sK5_axiom_s3_X))))), strict_implies(not(possibly(sK4_axiom_s3_Y)), not(possibly(sK5_axiom_s3_X))))), true)
51.44/51.63	= { by axiom 10 (modus_ponens_2) }
51.44/51.63	  fresh68(is_a_theorem(strict_implies(necessarily(fresh3(true, true, or(and(not(sK5_axiom_s3_X), not(sK5_axiom_s3_X)), sK4_axiom_s3_Y), or(sK4_axiom_s3_Y, and(not(sK5_axiom_s3_X), not(sK5_axiom_s3_X))))), strict_implies(not(possibly(sK4_axiom_s3_Y)), not(possibly(sK5_axiom_s3_X))))), true)
51.44/51.63	= { by axiom 23 (substitution_of_equivalents_2) }
51.44/51.63	  fresh68(is_a_theorem(strict_implies(necessarily(or(sK4_axiom_s3_Y, and(not(sK5_axiom_s3_X), not(sK5_axiom_s3_X)))), strict_implies(not(possibly(sK4_axiom_s3_Y)), not(possibly(sK5_axiom_s3_X))))), true)
51.44/51.63	= { by lemma 72 }
51.44/51.63	  fresh68(is_a_theorem(strict_implies(necessarily(or(sK4_axiom_s3_Y, not(sK5_axiom_s3_X))), strict_implies(not(possibly(sK4_axiom_s3_Y)), not(possibly(sK5_axiom_s3_X))))), true)
51.44/51.63	= { by lemma 79 }
51.44/51.63	  fresh68(is_a_theorem(strict_implies(strict_implies(not(sK4_axiom_s3_Y), not(sK5_axiom_s3_X)), strict_implies(not(possibly(sK4_axiom_s3_Y)), not(possibly(sK5_axiom_s3_X))))), true)
51.44/51.63	= { by lemma 78 }
51.44/51.63	  fresh68(is_a_theorem(strict_implies(necessarily(implies(not(sK4_axiom_s3_Y), not(sK5_axiom_s3_X))), strict_implies(not(possibly(sK4_axiom_s3_Y)), not(possibly(sK5_axiom_s3_X))))), true)
51.44/51.63	= { by lemma 76 }
51.44/51.63	  fresh68(is_a_theorem(strict_implies(necessarily(or(implies(not(sK4_axiom_s3_Y), not(sK5_axiom_s3_X)), implies(not(sK4_axiom_s3_Y), not(sK5_axiom_s3_X)))), strict_implies(not(possibly(sK4_axiom_s3_Y)), not(possibly(sK5_axiom_s3_X))))), true)
51.44/51.63	= { by lemma 73 }
51.44/51.63	  fresh68(is_a_theorem(strict_implies(necessarily(implies(not(implies(not(sK4_axiom_s3_Y), not(sK5_axiom_s3_X))), implies(not(sK4_axiom_s3_Y), not(sK5_axiom_s3_X)))), strict_implies(not(possibly(sK4_axiom_s3_Y)), not(possibly(sK5_axiom_s3_X))))), true)
51.44/51.63	= { by lemma 65 }
51.44/51.63	  fresh68(is_a_theorem(strict_implies(necessarily(not(and(not(implies(not(sK4_axiom_s3_Y), not(sK5_axiom_s3_X))), not(implies(not(sK4_axiom_s3_Y), not(sK5_axiom_s3_X)))))), strict_implies(not(possibly(sK4_axiom_s3_Y)), not(possibly(sK5_axiom_s3_X))))), true)
51.44/51.63	= { by lemma 72 }
51.44/51.63	  fresh68(is_a_theorem(strict_implies(necessarily(not(not(implies(not(sK4_axiom_s3_Y), not(sK5_axiom_s3_X))))), strict_implies(not(possibly(sK4_axiom_s3_Y)), not(possibly(sK5_axiom_s3_X))))), true)
51.44/51.63	= { by lemma 77 }
51.44/51.63	  fresh68(is_a_theorem(strict_implies(not(possibly(not(implies(not(sK4_axiom_s3_Y), not(sK5_axiom_s3_X))))), strict_implies(not(possibly(sK4_axiom_s3_Y)), not(possibly(sK5_axiom_s3_X))))), true)
51.44/51.63	= { by lemma 84 }
51.44/51.63	  fresh68(is_a_theorem(strict_implies(possibly(necessarily(not(not(implies(not(sK4_axiom_s3_Y), not(sK5_axiom_s3_X)))))), strict_implies(not(possibly(sK4_axiom_s3_Y)), not(possibly(sK5_axiom_s3_X))))), true)
51.44/51.63	= { by lemma 74 }
51.44/51.63	  fresh68(is_a_theorem(strict_implies(possibly(necessarily(or(implies(not(sK4_axiom_s3_Y), not(sK5_axiom_s3_X)), implies(not(sK4_axiom_s3_Y), not(sK5_axiom_s3_X))))), strict_implies(not(possibly(sK4_axiom_s3_Y)), not(possibly(sK5_axiom_s3_X))))), true)
51.44/51.63	= { by lemma 79 }
51.44/51.63	  fresh68(is_a_theorem(strict_implies(possibly(strict_implies(not(implies(not(sK4_axiom_s3_Y), not(sK5_axiom_s3_X))), implies(not(sK4_axiom_s3_Y), not(sK5_axiom_s3_X)))), strict_implies(not(possibly(sK4_axiom_s3_Y)), not(possibly(sK5_axiom_s3_X))))), true)
51.44/51.63	= { by lemma 83 }
51.44/51.63	  fresh68(is_a_theorem(strict_implies(necessarily(possibly(strict_implies(not(implies(not(sK4_axiom_s3_Y), not(sK5_axiom_s3_X))), implies(not(sK4_axiom_s3_Y), not(sK5_axiom_s3_X))))), strict_implies(not(possibly(sK4_axiom_s3_Y)), not(possibly(sK5_axiom_s3_X))))), true)
51.44/51.63	= { by lemma 78 }
51.44/51.63	  fresh68(is_a_theorem(strict_implies(necessarily(possibly(necessarily(implies(not(implies(not(sK4_axiom_s3_Y), not(sK5_axiom_s3_X))), implies(not(sK4_axiom_s3_Y), not(sK5_axiom_s3_X)))))), strict_implies(not(possibly(sK4_axiom_s3_Y)), not(possibly(sK5_axiom_s3_X))))), true)
51.44/51.63	= { by lemma 65 }
51.44/51.63	  fresh68(is_a_theorem(strict_implies(necessarily(possibly(necessarily(not(and(not(implies(not(sK4_axiom_s3_Y), not(sK5_axiom_s3_X))), not(implies(not(sK4_axiom_s3_Y), not(sK5_axiom_s3_X)))))))), strict_implies(not(possibly(sK4_axiom_s3_Y)), not(possibly(sK5_axiom_s3_X))))), true)
51.44/51.63	= { by lemma 84 }
51.44/51.63	  fresh68(is_a_theorem(strict_implies(necessarily(not(possibly(and(not(implies(not(sK4_axiom_s3_Y), not(sK5_axiom_s3_X))), not(implies(not(sK4_axiom_s3_Y), not(sK5_axiom_s3_X))))))), strict_implies(not(possibly(sK4_axiom_s3_Y)), not(possibly(sK5_axiom_s3_X))))), true)
51.44/51.63	= { by lemma 82 }
51.44/51.63	  fresh68(is_a_theorem(strict_implies(necessarily(not(not(strict_implies(not(implies(not(sK4_axiom_s3_Y), not(sK5_axiom_s3_X))), implies(not(sK4_axiom_s3_Y), not(sK5_axiom_s3_X)))))), strict_implies(not(possibly(sK4_axiom_s3_Y)), not(possibly(sK5_axiom_s3_X))))), true)
51.44/51.63	= { by lemma 72 }
51.44/51.63	  fresh68(is_a_theorem(strict_implies(necessarily(not(and(not(strict_implies(not(implies(not(sK4_axiom_s3_Y), not(sK5_axiom_s3_X))), implies(not(sK4_axiom_s3_Y), not(sK5_axiom_s3_X)))), not(strict_implies(not(implies(not(sK4_axiom_s3_Y), not(sK5_axiom_s3_X))), implies(not(sK4_axiom_s3_Y), not(sK5_axiom_s3_X))))))), strict_implies(not(possibly(sK4_axiom_s3_Y)), not(possibly(sK5_axiom_s3_X))))), true)
51.44/51.63	= { by lemma 65 }
51.44/51.63	  fresh68(is_a_theorem(strict_implies(necessarily(implies(not(strict_implies(not(implies(not(sK4_axiom_s3_Y), not(sK5_axiom_s3_X))), implies(not(sK4_axiom_s3_Y), not(sK5_axiom_s3_X)))), strict_implies(not(implies(not(sK4_axiom_s3_Y), not(sK5_axiom_s3_X))), implies(not(sK4_axiom_s3_Y), not(sK5_axiom_s3_X))))), strict_implies(not(possibly(sK4_axiom_s3_Y)), not(possibly(sK5_axiom_s3_X))))), true)
51.44/51.63	= { by lemma 73 }
51.44/51.63	  fresh68(is_a_theorem(strict_implies(necessarily(or(strict_implies(not(implies(not(sK4_axiom_s3_Y), not(sK5_axiom_s3_X))), implies(not(sK4_axiom_s3_Y), not(sK5_axiom_s3_X))), strict_implies(not(implies(not(sK4_axiom_s3_Y), not(sK5_axiom_s3_X))), implies(not(sK4_axiom_s3_Y), not(sK5_axiom_s3_X))))), strict_implies(not(possibly(sK4_axiom_s3_Y)), not(possibly(sK5_axiom_s3_X))))), true)
51.44/51.63	= { by lemma 79 }
51.44/51.63	  fresh68(is_a_theorem(strict_implies(strict_implies(not(strict_implies(not(implies(not(sK4_axiom_s3_Y), not(sK5_axiom_s3_X))), implies(not(sK4_axiom_s3_Y), not(sK5_axiom_s3_X)))), strict_implies(not(implies(not(sK4_axiom_s3_Y), not(sK5_axiom_s3_X))), implies(not(sK4_axiom_s3_Y), not(sK5_axiom_s3_X)))), strict_implies(not(possibly(sK4_axiom_s3_Y)), not(possibly(sK5_axiom_s3_X))))), true)
51.44/51.63	= { by lemma 82 }
51.44/51.63	  fresh68(is_a_theorem(strict_implies(strict_implies(possibly(and(not(implies(not(sK4_axiom_s3_Y), not(sK5_axiom_s3_X))), not(implies(not(sK4_axiom_s3_Y), not(sK5_axiom_s3_X))))), strict_implies(not(implies(not(sK4_axiom_s3_Y), not(sK5_axiom_s3_X))), implies(not(sK4_axiom_s3_Y), not(sK5_axiom_s3_X)))), strict_implies(not(possibly(sK4_axiom_s3_Y)), not(possibly(sK5_axiom_s3_X))))), true)
51.44/51.63	= { by lemma 72 }
51.44/51.63	  fresh68(is_a_theorem(strict_implies(strict_implies(possibly(not(implies(not(sK4_axiom_s3_Y), not(sK5_axiom_s3_X)))), strict_implies(not(implies(not(sK4_axiom_s3_Y), not(sK5_axiom_s3_X))), implies(not(sK4_axiom_s3_Y), not(sK5_axiom_s3_X)))), strict_implies(not(possibly(sK4_axiom_s3_Y)), not(possibly(sK5_axiom_s3_X))))), true)
51.44/51.63	= { by lemma 72 }
51.44/51.63	  fresh68(is_a_theorem(strict_implies(strict_implies(possibly(not(and(implies(not(sK4_axiom_s3_Y), not(sK5_axiom_s3_X)), implies(not(sK4_axiom_s3_Y), not(sK5_axiom_s3_X))))), strict_implies(not(implies(not(sK4_axiom_s3_Y), not(sK5_axiom_s3_X))), implies(not(sK4_axiom_s3_Y), not(sK5_axiom_s3_X)))), strict_implies(not(possibly(sK4_axiom_s3_Y)), not(possibly(sK5_axiom_s3_X))))), true)
51.44/51.63	= { by lemma 65 }
51.44/51.63	  fresh68(is_a_theorem(strict_implies(strict_implies(possibly(not(and(implies(not(sK4_axiom_s3_Y), not(sK5_axiom_s3_X)), not(and(not(sK4_axiom_s3_Y), not(not(sK5_axiom_s3_X))))))), strict_implies(not(implies(not(sK4_axiom_s3_Y), not(sK5_axiom_s3_X))), implies(not(sK4_axiom_s3_Y), not(sK5_axiom_s3_X)))), strict_implies(not(possibly(sK4_axiom_s3_Y)), not(possibly(sK5_axiom_s3_X))))), true)
51.44/51.63	= { by lemma 65 }
51.44/51.63	  fresh68(is_a_theorem(strict_implies(strict_implies(possibly(implies(implies(not(sK4_axiom_s3_Y), not(sK5_axiom_s3_X)), and(not(sK4_axiom_s3_Y), not(not(sK5_axiom_s3_X))))), strict_implies(not(implies(not(sK4_axiom_s3_Y), not(sK5_axiom_s3_X))), implies(not(sK4_axiom_s3_Y), not(sK5_axiom_s3_X)))), strict_implies(not(possibly(sK4_axiom_s3_Y)), not(possibly(sK5_axiom_s3_X))))), true)
51.44/51.63	= { by lemma 75 }
51.44/51.63	  fresh68(is_a_theorem(strict_implies(strict_implies(possibly(or(and(not(sK4_axiom_s3_Y), not(not(sK5_axiom_s3_X))), and(not(sK4_axiom_s3_Y), not(not(sK5_axiom_s3_X))))), strict_implies(not(implies(not(sK4_axiom_s3_Y), not(sK5_axiom_s3_X))), implies(not(sK4_axiom_s3_Y), not(sK5_axiom_s3_X)))), strict_implies(not(possibly(sK4_axiom_s3_Y)), not(possibly(sK5_axiom_s3_X))))), true)
51.44/51.63	= { by lemma 76 }
51.44/51.63	  fresh68(is_a_theorem(strict_implies(strict_implies(possibly(and(not(sK4_axiom_s3_Y), not(not(sK5_axiom_s3_X)))), strict_implies(not(implies(not(sK4_axiom_s3_Y), not(sK5_axiom_s3_X))), implies(not(sK4_axiom_s3_Y), not(sK5_axiom_s3_X)))), strict_implies(not(possibly(sK4_axiom_s3_Y)), not(possibly(sK5_axiom_s3_X))))), true)
51.44/51.63	= { by lemma 82 }
51.44/51.63	  fresh68(is_a_theorem(strict_implies(strict_implies(not(strict_implies(not(sK4_axiom_s3_Y), not(sK5_axiom_s3_X))), strict_implies(not(implies(not(sK4_axiom_s3_Y), not(sK5_axiom_s3_X))), implies(not(sK4_axiom_s3_Y), not(sK5_axiom_s3_X)))), strict_implies(not(possibly(sK4_axiom_s3_Y)), not(possibly(sK5_axiom_s3_X))))), true)
51.44/51.63	= { by lemma 81 }
51.44/51.63	  fresh68(is_a_theorem(strict_implies(strict_implies(not(strict_implies(not(sK4_axiom_s3_Y), not(sK5_axiom_s3_X))), necessarily(implies(not(sK4_axiom_s3_Y), not(sK5_axiom_s3_X)))), strict_implies(not(possibly(sK4_axiom_s3_Y)), not(possibly(sK5_axiom_s3_X))))), true)
51.44/51.63	= { by lemma 78 }
51.44/51.63	  fresh68(is_a_theorem(strict_implies(strict_implies(not(strict_implies(not(sK4_axiom_s3_Y), not(sK5_axiom_s3_X))), strict_implies(not(sK4_axiom_s3_Y), not(sK5_axiom_s3_X))), strict_implies(not(possibly(sK4_axiom_s3_Y)), not(possibly(sK5_axiom_s3_X))))), true)
51.44/51.63	= { by lemma 81 }
51.44/51.63	  fresh68(is_a_theorem(strict_implies(necessarily(strict_implies(not(sK4_axiom_s3_Y), not(sK5_axiom_s3_X))), strict_implies(not(possibly(sK4_axiom_s3_Y)), not(possibly(sK5_axiom_s3_X))))), true)
51.44/51.63	= { by lemma 77 }
51.44/51.63	  fresh68(is_a_theorem(strict_implies(necessarily(strict_implies(not(sK4_axiom_s3_Y), not(sK5_axiom_s3_X))), strict_implies(necessarily(not(sK4_axiom_s3_Y)), not(possibly(sK5_axiom_s3_X))))), true)
51.44/51.63	= { by lemma 77 }
51.44/51.63	  fresh68(is_a_theorem(strict_implies(necessarily(strict_implies(not(sK4_axiom_s3_Y), not(sK5_axiom_s3_X))), strict_implies(necessarily(not(sK4_axiom_s3_Y)), necessarily(not(sK5_axiom_s3_X))))), true)
51.44/51.63	= { by lemma 78 }
51.44/51.63	  fresh68(is_a_theorem(necessarily(implies(necessarily(strict_implies(not(sK4_axiom_s3_Y), not(sK5_axiom_s3_X))), strict_implies(necessarily(not(sK4_axiom_s3_Y)), necessarily(not(sK5_axiom_s3_X)))))), true)
51.44/51.63	= { by axiom 13 (necessitation_1) }
51.44/51.63	  fresh68(fresh34(true, true, implies(necessarily(strict_implies(not(sK4_axiom_s3_Y), not(sK5_axiom_s3_X))), strict_implies(necessarily(not(sK4_axiom_s3_Y)), necessarily(not(sK5_axiom_s3_X))))), true)
51.44/51.63	= { by axiom 61 (km5_necessitation) }
51.44/51.63	  fresh68(fresh34(necessitation, true, implies(necessarily(strict_implies(not(sK4_axiom_s3_Y), not(sK5_axiom_s3_X))), strict_implies(necessarily(not(sK4_axiom_s3_Y)), necessarily(not(sK5_axiom_s3_X))))), true)
51.44/51.63	= { by axiom 50 (necessitation_1) }
51.44/51.63	  fresh68(fresh33(is_a_theorem(implies(necessarily(strict_implies(not(sK4_axiom_s3_Y), not(sK5_axiom_s3_X))), strict_implies(necessarily(not(sK4_axiom_s3_Y)), necessarily(not(sK5_axiom_s3_X))))), true, implies(necessarily(strict_implies(not(sK4_axiom_s3_Y), not(sK5_axiom_s3_X))), strict_implies(necessarily(not(sK4_axiom_s3_Y)), necessarily(not(sK5_axiom_s3_X))))), true)
51.44/51.63	= { by lemma 78 }
51.44/51.63	  fresh68(fresh33(is_a_theorem(implies(necessarily(strict_implies(not(sK4_axiom_s3_Y), not(sK5_axiom_s3_X))), necessarily(implies(necessarily(not(sK4_axiom_s3_Y)), necessarily(not(sK5_axiom_s3_X)))))), true, implies(necessarily(strict_implies(not(sK4_axiom_s3_Y), not(sK5_axiom_s3_X))), strict_implies(necessarily(not(sK4_axiom_s3_Y)), necessarily(not(sK5_axiom_s3_X))))), true)
51.44/51.63	= { by axiom 9 (modus_ponens_2) }
51.44/51.63	  fresh68(fresh33(fresh40(true, true, strict_implies(strict_implies(not(sK4_axiom_s3_Y), not(sK5_axiom_s3_X)), implies(necessarily(not(sK4_axiom_s3_Y)), necessarily(not(sK5_axiom_s3_X)))), implies(necessarily(strict_implies(not(sK4_axiom_s3_Y), not(sK5_axiom_s3_X))), necessarily(implies(necessarily(not(sK4_axiom_s3_Y)), necessarily(not(sK5_axiom_s3_X)))))), true, implies(necessarily(strict_implies(not(sK4_axiom_s3_Y), not(sK5_axiom_s3_X))), strict_implies(necessarily(not(sK4_axiom_s3_Y)), necessarily(not(sK5_axiom_s3_X))))), true)
51.44/51.63	= { by lemma 85 }
51.44/51.63	  fresh68(fresh33(fresh40(is_a_theorem(implies(strict_implies(strict_implies(not(sK4_axiom_s3_Y), not(sK5_axiom_s3_X)), implies(necessarily(not(sK4_axiom_s3_Y)), necessarily(not(sK5_axiom_s3_X)))), implies(necessarily(strict_implies(not(sK4_axiom_s3_Y), not(sK5_axiom_s3_X))), necessarily(implies(necessarily(not(sK4_axiom_s3_Y)), necessarily(not(sK5_axiom_s3_X))))))), true, strict_implies(strict_implies(not(sK4_axiom_s3_Y), not(sK5_axiom_s3_X)), implies(necessarily(not(sK4_axiom_s3_Y)), necessarily(not(sK5_axiom_s3_X)))), implies(necessarily(strict_implies(not(sK4_axiom_s3_Y), not(sK5_axiom_s3_X))), necessarily(implies(necessarily(not(sK4_axiom_s3_Y)), necessarily(not(sK5_axiom_s3_X)))))), true, implies(necessarily(strict_implies(not(sK4_axiom_s3_Y), not(sK5_axiom_s3_X))), strict_implies(necessarily(not(sK4_axiom_s3_Y)), necessarily(not(sK5_axiom_s3_X))))), true)
51.44/51.63	= { by lemma 67 }
51.44/51.63	  fresh68(fresh33(fresh116(is_a_theorem(strict_implies(strict_implies(not(sK4_axiom_s3_Y), not(sK5_axiom_s3_X)), implies(necessarily(not(sK4_axiom_s3_Y)), necessarily(not(sK5_axiom_s3_X))))), true, implies(necessarily(strict_implies(not(sK4_axiom_s3_Y), not(sK5_axiom_s3_X))), necessarily(implies(necessarily(not(sK4_axiom_s3_Y)), necessarily(not(sK5_axiom_s3_X)))))), true, implies(necessarily(strict_implies(not(sK4_axiom_s3_Y), not(sK5_axiom_s3_X))), strict_implies(necessarily(not(sK4_axiom_s3_Y)), necessarily(not(sK5_axiom_s3_X))))), true)
51.44/51.63	= { by lemma 78 }
51.44/51.63	  fresh68(fresh33(fresh116(is_a_theorem(necessarily(implies(strict_implies(not(sK4_axiom_s3_Y), not(sK5_axiom_s3_X)), implies(necessarily(not(sK4_axiom_s3_Y)), necessarily(not(sK5_axiom_s3_X)))))), true, implies(necessarily(strict_implies(not(sK4_axiom_s3_Y), not(sK5_axiom_s3_X))), necessarily(implies(necessarily(not(sK4_axiom_s3_Y)), necessarily(not(sK5_axiom_s3_X)))))), true, implies(necessarily(strict_implies(not(sK4_axiom_s3_Y), not(sK5_axiom_s3_X))), strict_implies(necessarily(not(sK4_axiom_s3_Y)), necessarily(not(sK5_axiom_s3_X))))), true)
51.44/51.63	= { by axiom 13 (necessitation_1) }
51.44/51.63	  fresh68(fresh33(fresh116(fresh34(true, true, implies(strict_implies(not(sK4_axiom_s3_Y), not(sK5_axiom_s3_X)), implies(necessarily(not(sK4_axiom_s3_Y)), necessarily(not(sK5_axiom_s3_X))))), true, implies(necessarily(strict_implies(not(sK4_axiom_s3_Y), not(sK5_axiom_s3_X))), necessarily(implies(necessarily(not(sK4_axiom_s3_Y)), necessarily(not(sK5_axiom_s3_X)))))), true, implies(necessarily(strict_implies(not(sK4_axiom_s3_Y), not(sK5_axiom_s3_X))), strict_implies(necessarily(not(sK4_axiom_s3_Y)), necessarily(not(sK5_axiom_s3_X))))), true)
51.44/51.63	= { by axiom 61 (km5_necessitation) }
51.44/51.63	  fresh68(fresh33(fresh116(fresh34(necessitation, true, implies(strict_implies(not(sK4_axiom_s3_Y), not(sK5_axiom_s3_X)), implies(necessarily(not(sK4_axiom_s3_Y)), necessarily(not(sK5_axiom_s3_X))))), true, implies(necessarily(strict_implies(not(sK4_axiom_s3_Y), not(sK5_axiom_s3_X))), necessarily(implies(necessarily(not(sK4_axiom_s3_Y)), necessarily(not(sK5_axiom_s3_X)))))), true, implies(necessarily(strict_implies(not(sK4_axiom_s3_Y), not(sK5_axiom_s3_X))), strict_implies(necessarily(not(sK4_axiom_s3_Y)), necessarily(not(sK5_axiom_s3_X))))), true)
51.44/51.63	= { by axiom 50 (necessitation_1) }
51.44/51.63	  fresh68(fresh33(fresh116(fresh33(is_a_theorem(implies(strict_implies(not(sK4_axiom_s3_Y), not(sK5_axiom_s3_X)), implies(necessarily(not(sK4_axiom_s3_Y)), necessarily(not(sK5_axiom_s3_X))))), true, implies(strict_implies(not(sK4_axiom_s3_Y), not(sK5_axiom_s3_X)), implies(necessarily(not(sK4_axiom_s3_Y)), necessarily(not(sK5_axiom_s3_X))))), true, implies(necessarily(strict_implies(not(sK4_axiom_s3_Y), not(sK5_axiom_s3_X))), necessarily(implies(necessarily(not(sK4_axiom_s3_Y)), necessarily(not(sK5_axiom_s3_X)))))), true, implies(necessarily(strict_implies(not(sK4_axiom_s3_Y), not(sK5_axiom_s3_X))), strict_implies(necessarily(not(sK4_axiom_s3_Y)), necessarily(not(sK5_axiom_s3_X))))), true)
51.44/51.63	= { by lemma 85 }
51.44/51.63	  fresh68(fresh33(fresh116(fresh33(true, true, implies(strict_implies(not(sK4_axiom_s3_Y), not(sK5_axiom_s3_X)), implies(necessarily(not(sK4_axiom_s3_Y)), necessarily(not(sK5_axiom_s3_X))))), true, implies(necessarily(strict_implies(not(sK4_axiom_s3_Y), not(sK5_axiom_s3_X))), necessarily(implies(necessarily(not(sK4_axiom_s3_Y)), necessarily(not(sK5_axiom_s3_X)))))), true, implies(necessarily(strict_implies(not(sK4_axiom_s3_Y), not(sK5_axiom_s3_X))), strict_implies(necessarily(not(sK4_axiom_s3_Y)), necessarily(not(sK5_axiom_s3_X))))), true)
51.44/51.63	= { by axiom 14 (necessitation_1) }
51.44/51.63	  fresh68(fresh33(fresh116(true, true, implies(necessarily(strict_implies(not(sK4_axiom_s3_Y), not(sK5_axiom_s3_X))), necessarily(implies(necessarily(not(sK4_axiom_s3_Y)), necessarily(not(sK5_axiom_s3_X)))))), true, implies(necessarily(strict_implies(not(sK4_axiom_s3_Y), not(sK5_axiom_s3_X))), strict_implies(necessarily(not(sK4_axiom_s3_Y)), necessarily(not(sK5_axiom_s3_X))))), true)
51.44/51.63	= { by axiom 10 (modus_ponens_2) }
51.44/51.63	  fresh68(fresh33(true, true, implies(necessarily(strict_implies(not(sK4_axiom_s3_Y), not(sK5_axiom_s3_X))), strict_implies(necessarily(not(sK4_axiom_s3_Y)), necessarily(not(sK5_axiom_s3_X))))), true)
51.44/51.63	= { by axiom 14 (necessitation_1) }
51.44/51.63	  fresh68(true, true)
51.44/51.63	= { by axiom 7 (axiom_s3) }
51.44/51.63	  true
51.44/51.63	% SZS output end Proof
51.44/51.63	
51.44/51.63	RESULT: Theorem (the conjecture is true).
51.44/51.65	EOF