TSTP Solution File: LCL547+1 by Twee---2.5.0
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- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.5.0
% Problem : LCL547+1 : TPTP v8.2.0. Released v3.3.0.
% Transfm : none
% Format : tptp:raw
% Command : parallel-twee /export/starexec/sandbox2/benchmark/theBenchmark.p --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding
% Computer : n027.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Mon Jun 24 11:21:58 EDT 2024
% Result : Theorem 190.59s 24.34s
% Output : Proof 193.05s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.03/0.10 % Problem : LCL547+1 : TPTP v8.2.0. Released v3.3.0.
% 0.03/0.10 % Command : parallel-twee /export/starexec/sandbox2/benchmark/theBenchmark.p --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding
% 0.09/0.31 % Computer : n027.cluster.edu
% 0.09/0.31 % Model : x86_64 x86_64
% 0.09/0.31 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.09/0.31 % Memory : 8042.1875MB
% 0.09/0.31 % OS : Linux 3.10.0-693.el7.x86_64
% 0.09/0.31 % CPULimit : 300
% 0.09/0.31 % WCLimit : 300
% 0.09/0.31 % DateTime : Sat Jun 22 13:48:54 EDT 2024
% 0.09/0.31 % CPUTime :
% 190.59/24.34 Command-line arguments: --lhs-weight 1 --flip-ordering --normalise-queue-percent 10 --cp-renormalise-threshold 10
% 190.59/24.34
% 190.59/24.34 % SZS status Theorem
% 190.59/24.34
% 191.76/24.54 % SZS output start Proof
% 191.76/24.54 Take the following subset of the input axioms:
% 191.76/24.54 fof(and_1, axiom, and_1 <=> ![X, Y]: is_a_theorem(implies(and(X, Y), X))).
% 191.76/24.54 fof(and_2, axiom, and_2 <=> ![X2, Y2]: is_a_theorem(implies(and(X2, Y2), Y2))).
% 191.76/24.54 fof(and_3, axiom, and_3 <=> ![X2, Y2]: is_a_theorem(implies(X2, implies(Y2, and(X2, Y2))))).
% 191.76/24.54 fof(axiom_4, axiom, axiom_4 <=> ![X2]: is_a_theorem(implies(necessarily(X2), necessarily(necessarily(X2))))).
% 191.76/24.54 fof(axiom_B, axiom, axiom_B <=> ![X2]: is_a_theorem(implies(X2, necessarily(possibly(X2))))).
% 191.76/24.54 fof(axiom_K, axiom, axiom_K <=> ![X2, Y2]: is_a_theorem(implies(necessarily(implies(X2, Y2)), implies(necessarily(X2), necessarily(Y2))))).
% 191.76/24.54 fof(axiom_M, axiom, axiom_M <=> ![X2]: is_a_theorem(implies(necessarily(X2), X2))).
% 191.76/24.54 fof(axiom_s3, axiom, axiom_s3 <=> ![X2, Y2]: is_a_theorem(strict_implies(strict_implies(X2, Y2), strict_implies(not(possibly(Y2)), not(possibly(X2)))))).
% 191.76/24.54 fof(equivalence_3, axiom, equivalence_3 <=> ![X2, Y2]: is_a_theorem(implies(implies(X2, Y2), implies(implies(Y2, X2), equiv(X2, Y2))))).
% 191.76/24.54 fof(hilbert_and_1, axiom, and_1).
% 191.76/24.54 fof(hilbert_and_2, axiom, and_2).
% 191.76/24.54 fof(hilbert_and_3, axiom, and_3).
% 191.76/24.54 fof(hilbert_equivalence_3, axiom, equivalence_3).
% 191.76/24.54 fof(hilbert_implies_1, axiom, implies_1).
% 191.76/24.54 fof(hilbert_implies_2, axiom, implies_2).
% 191.76/24.54 fof(hilbert_implies_3, axiom, implies_3).
% 191.76/24.54 fof(hilbert_modus_ponens, axiom, modus_ponens).
% 191.76/24.54 fof(hilbert_modus_tollens, axiom, modus_tollens).
% 191.76/24.54 fof(hilbert_op_equiv, axiom, op_equiv).
% 191.76/24.54 fof(hilbert_op_implies_and, axiom, op_implies_and).
% 191.76/24.54 fof(hilbert_op_or, axiom, op_or).
% 191.76/24.54 fof(hilbert_or_1, axiom, or_1).
% 191.76/24.54 fof(hilbert_or_3, axiom, or_3).
% 191.76/24.54 fof(implies_1, axiom, implies_1 <=> ![X2, Y2]: is_a_theorem(implies(X2, implies(Y2, X2)))).
% 191.76/24.54 fof(implies_2, axiom, implies_2 <=> ![X2, Y2]: is_a_theorem(implies(implies(X2, implies(X2, Y2)), implies(X2, Y2)))).
% 191.76/24.54 fof(implies_3, axiom, implies_3 <=> ![Z, X2, Y2]: is_a_theorem(implies(implies(X2, Y2), implies(implies(Y2, Z), implies(X2, Z))))).
% 191.76/24.54 fof(km4b_axiom_4, axiom, axiom_4).
% 191.76/24.54 fof(km4b_axiom_B, axiom, axiom_B).
% 191.76/24.54 fof(km4b_axiom_K, axiom, axiom_K).
% 191.76/24.54 fof(km4b_axiom_M, axiom, axiom_M).
% 191.76/24.54 fof(km4b_necessitation, axiom, necessitation).
% 191.76/24.54 fof(km4b_op_possibly, axiom, op_possibly).
% 191.76/24.54 fof(modus_ponens, axiom, modus_ponens <=> ![X2, Y2]: ((is_a_theorem(X2) & is_a_theorem(implies(X2, Y2))) => is_a_theorem(Y2))).
% 191.76/24.54 fof(modus_tollens, axiom, modus_tollens <=> ![X2, Y2]: is_a_theorem(implies(implies(not(Y2), not(X2)), implies(X2, Y2)))).
% 191.76/24.54 fof(necessitation, axiom, necessitation <=> ![X2]: (is_a_theorem(X2) => is_a_theorem(necessarily(X2)))).
% 191.76/24.54 fof(op_equiv, axiom, op_equiv => ![X2, Y2]: equiv(X2, Y2)=and(implies(X2, Y2), implies(Y2, X2))).
% 191.76/24.54 fof(op_implies_and, axiom, op_implies_and => ![X2, Y2]: implies(X2, Y2)=not(and(X2, not(Y2)))).
% 191.76/24.54 fof(op_or, axiom, op_or => ![X2, Y2]: or(X2, Y2)=not(and(not(X2), not(Y2)))).
% 191.76/24.54 fof(op_possibly, axiom, op_possibly => ![X2]: possibly(X2)=not(necessarily(not(X2)))).
% 191.76/24.54 fof(op_strict_implies, axiom, op_strict_implies => ![X2, Y2]: strict_implies(X2, Y2)=necessarily(implies(X2, Y2))).
% 191.76/24.54 fof(or_1, axiom, or_1 <=> ![X2, Y2]: is_a_theorem(implies(X2, or(X2, Y2)))).
% 191.76/24.54 fof(or_3, axiom, or_3 <=> ![X2, Y2, Z2]: is_a_theorem(implies(implies(X2, Z2), implies(implies(Y2, Z2), implies(or(X2, Y2), Z2))))).
% 191.76/24.54 fof(s1_0_m6s3m9b_axiom_s3, conjecture, axiom_s3).
% 191.76/24.54 fof(s1_0_op_strict_implies, axiom, op_strict_implies).
% 191.76/24.54 fof(substitution_of_equivalents, axiom, substitution_of_equivalents <=> ![X2, Y2]: (is_a_theorem(equiv(X2, Y2)) => X2=Y2)).
% 191.76/24.54 fof(substitution_of_equivalents, axiom, substitution_of_equivalents).
% 191.76/24.54
% 191.76/24.54 Now clausify the problem and encode Horn clauses using encoding 3 of
% 191.76/24.54 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 191.76/24.54 We repeatedly replace C & s=t => u=v by the two clauses:
% 191.76/24.54 fresh(y, y, x1...xn) = u
% 191.76/24.54 C => fresh(s, t, x1...xn) = v
% 191.76/24.54 where fresh is a fresh function symbol and x1..xn are the free
% 191.76/24.54 variables of u and v.
% 191.76/24.54 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 191.76/24.54 input problem has no model of domain size 1).
% 191.76/24.54
% 191.76/24.54 The encoding turns the above axioms into the following unit equations and goals:
% 191.76/24.54
% 191.76/24.54 Axiom 1 (hilbert_modus_ponens): modus_ponens = true.
% 191.76/24.54 Axiom 2 (substitution_of_equivalents): substitution_of_equivalents = true.
% 191.76/24.54 Axiom 3 (hilbert_modus_tollens): modus_tollens = true.
% 191.76/24.54 Axiom 4 (hilbert_implies_1): implies_1 = true.
% 191.76/24.54 Axiom 5 (hilbert_implies_2): implies_2 = true.
% 191.76/24.54 Axiom 6 (hilbert_implies_3): implies_3 = true.
% 191.76/24.54 Axiom 7 (hilbert_and_1): and_1 = true.
% 191.76/24.54 Axiom 8 (hilbert_and_2): and_2 = true.
% 191.76/24.54 Axiom 9 (hilbert_and_3): and_3 = true.
% 191.76/24.54 Axiom 10 (hilbert_or_1): or_1 = true.
% 191.76/24.54 Axiom 11 (hilbert_or_3): or_3 = true.
% 191.76/24.54 Axiom 12 (hilbert_equivalence_3): equivalence_3 = true.
% 191.76/24.54 Axiom 13 (hilbert_op_or): op_or = true.
% 191.76/24.54 Axiom 14 (hilbert_op_equiv): op_equiv = true.
% 191.76/24.54 Axiom 15 (km4b_necessitation): necessitation = true.
% 191.76/24.54 Axiom 16 (km4b_axiom_K): axiom_K = true.
% 191.76/24.54 Axiom 17 (km4b_axiom_M): axiom_M = true.
% 191.76/24.54 Axiom 18 (km4b_axiom_4): axiom_4 = true.
% 191.76/24.54 Axiom 19 (km4b_axiom_B): axiom_B = true.
% 191.76/24.54 Axiom 20 (km4b_op_possibly): op_possibly = true.
% 191.76/24.54 Axiom 21 (hilbert_op_implies_and): op_implies_and = true.
% 191.76/24.54 Axiom 22 (s1_0_op_strict_implies): op_strict_implies = true.
% 191.76/24.54 Axiom 23 (axiom_s3): fresh68(X, X) = true.
% 191.76/24.54 Axiom 24 (modus_ponens_2): fresh116(X, X, Y) = true.
% 191.76/24.54 Axiom 25 (axiom_4_1): fresh101(X, X, Y) = true.
% 191.76/24.54 Axiom 26 (axiom_B_1): fresh97(X, X, Y) = true.
% 191.76/24.54 Axiom 27 (axiom_M_1): fresh93(X, X, Y) = true.
% 191.76/24.54 Axiom 28 (modus_ponens_2): fresh40(X, X, Y) = is_a_theorem(Y).
% 191.76/24.54 Axiom 29 (necessitation_1): fresh34(X, X, Y) = is_a_theorem(necessarily(Y)).
% 191.76/24.54 Axiom 30 (necessitation_1): fresh33(X, X, Y) = true.
% 191.76/24.54 Axiom 31 (op_possibly): fresh25(X, X, Y) = possibly(Y).
% 191.76/24.54 Axiom 32 (op_possibly): fresh25(op_possibly, true, X) = not(necessarily(not(X))).
% 191.76/24.54 Axiom 33 (axiom_M_1): fresh93(axiom_M, true, X) = is_a_theorem(implies(necessarily(X), X)).
% 191.76/24.54 Axiom 34 (modus_ponens_2): fresh115(X, X, Y, Z) = fresh116(modus_ponens, true, Z).
% 191.76/24.54 Axiom 35 (and_1_1): fresh107(X, X, Y, Z) = true.
% 191.76/24.54 Axiom 36 (and_2_1): fresh105(X, X, Y, Z) = true.
% 191.76/24.54 Axiom 37 (and_3_1): fresh103(X, X, Y, Z) = true.
% 191.76/24.54 Axiom 38 (axiom_K_1): fresh95(X, X, Y, Z) = true.
% 191.76/24.54 Axiom 39 (equivalence_3_1): fresh53(X, X, Y, Z) = true.
% 191.76/24.55 Axiom 40 (implies_1_1): fresh51(X, X, Y, Z) = true.
% 191.76/24.55 Axiom 41 (implies_2_1): fresh49(X, X, Y, Z) = true.
% 191.76/24.55 Axiom 42 (modus_tollens_1): fresh35(X, X, Y, Z) = true.
% 191.76/24.55 Axiom 43 (necessitation_1): fresh34(necessitation, true, X) = fresh33(is_a_theorem(X), true, X).
% 191.76/24.55 Axiom 44 (op_equiv): fresh30(X, X, Y, Z) = equiv(Y, Z).
% 191.76/24.55 Axiom 45 (op_implies_and): fresh29(X, X, Y, Z) = implies(Y, Z).
% 191.76/24.55 Axiom 46 (op_implies_and): fresh29(op_implies_and, true, X, Y) = not(and(X, not(Y))).
% 191.76/24.55 Axiom 47 (op_or): fresh26(X, X, Y, Z) = or(Y, Z).
% 191.76/24.55 Axiom 48 (op_strict_implies): fresh23(X, X, Y, Z) = strict_implies(Y, Z).
% 191.76/24.55 Axiom 49 (op_strict_implies): fresh23(op_strict_implies, true, X, Y) = necessarily(implies(X, Y)).
% 191.76/24.55 Axiom 50 (or_1_1): fresh21(X, X, Y, Z) = true.
% 191.76/24.55 Axiom 51 (substitution_of_equivalents_2): fresh4(X, X, Y, Z) = Y.
% 191.76/24.55 Axiom 52 (substitution_of_equivalents_2): fresh3(X, X, Y, Z) = Z.
% 191.76/24.55 Axiom 53 (implies_1_1): fresh51(implies_1, true, X, Y) = is_a_theorem(implies(X, implies(Y, X))).
% 191.76/24.55 Axiom 54 (axiom_B_1): fresh97(axiom_B, true, X) = is_a_theorem(implies(X, necessarily(possibly(X)))).
% 191.76/24.55 Axiom 55 (or_1_1): fresh21(or_1, true, X, Y) = is_a_theorem(implies(X, or(X, Y))).
% 191.76/24.55 Axiom 56 (and_1_1): fresh107(and_1, true, X, Y) = is_a_theorem(implies(and(X, Y), X)).
% 191.76/24.55 Axiom 57 (and_2_1): fresh105(and_2, true, X, Y) = is_a_theorem(implies(and(X, Y), Y)).
% 191.76/24.55 Axiom 58 (op_or): fresh26(op_or, true, X, Y) = not(and(not(X), not(Y))).
% 191.76/24.55 Axiom 59 (implies_3_1): fresh47(X, X, Y, Z, W) = true.
% 191.76/24.55 Axiom 60 (or_3_1): fresh17(X, X, Y, Z, W) = true.
% 191.76/24.55 Axiom 61 (axiom_4_1): fresh101(axiom_4, true, X) = is_a_theorem(implies(necessarily(X), necessarily(necessarily(X)))).
% 191.76/24.55 Axiom 62 (op_equiv): fresh30(op_equiv, true, X, Y) = and(implies(X, Y), implies(Y, X)).
% 191.76/24.55 Axiom 63 (and_3_1): fresh103(and_3, true, X, Y) = is_a_theorem(implies(X, implies(Y, and(X, Y)))).
% 191.76/24.55 Axiom 64 (modus_ponens_2): fresh115(is_a_theorem(implies(X, Y)), true, X, Y) = fresh40(is_a_theorem(X), true, Y).
% 191.76/24.55 Axiom 65 (substitution_of_equivalents_2): fresh4(substitution_of_equivalents, true, X, Y) = fresh3(is_a_theorem(equiv(X, Y)), true, X, Y).
% 191.76/24.55 Axiom 66 (implies_2_1): fresh49(implies_2, true, X, Y) = is_a_theorem(implies(implies(X, implies(X, Y)), implies(X, Y))).
% 191.76/24.55 Axiom 67 (modus_tollens_1): fresh35(modus_tollens, true, X, Y) = is_a_theorem(implies(implies(not(Y), not(X)), implies(X, Y))).
% 191.76/24.55 Axiom 68 (axiom_K_1): fresh95(axiom_K, true, X, Y) = is_a_theorem(implies(necessarily(implies(X, Y)), implies(necessarily(X), necessarily(Y)))).
% 191.76/24.55 Axiom 69 (equivalence_3_1): fresh53(equivalence_3, true, X, Y) = is_a_theorem(implies(implies(X, Y), implies(implies(Y, X), equiv(X, Y)))).
% 191.76/24.55 Axiom 70 (implies_3_1): fresh47(implies_3, true, X, Y, Z) = is_a_theorem(implies(implies(X, Y), implies(implies(Y, Z), implies(X, Z)))).
% 191.76/24.55 Axiom 71 (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)))).
% 191.76/24.55 Axiom 72 (axiom_s3): fresh68(is_a_theorem(strict_implies(strict_implies(x10, y5), strict_implies(not(possibly(y5)), not(possibly(x10))))), true) = axiom_s3.
% 191.76/24.55
% 191.76/24.55 Lemma 73: fresh115(X, X, Y, Z) = true.
% 191.76/24.55 Proof:
% 191.76/24.55 fresh115(X, X, Y, Z)
% 191.76/24.55 = { by axiom 34 (modus_ponens_2) }
% 191.76/24.55 fresh116(modus_ponens, true, Z)
% 191.76/24.55 = { by axiom 1 (hilbert_modus_ponens) }
% 191.76/24.55 fresh116(true, true, Z)
% 191.76/24.55 = { by axiom 24 (modus_ponens_2) }
% 191.76/24.55 true
% 191.76/24.55
% 191.76/24.55 Lemma 74: fresh40(is_a_theorem(implies(X, Y)), true, implies(implies(Y, X), equiv(X, Y))) = true.
% 191.76/24.55 Proof:
% 191.76/24.55 fresh40(is_a_theorem(implies(X, Y)), true, implies(implies(Y, X), equiv(X, Y)))
% 191.76/24.55 = { by axiom 64 (modus_ponens_2) R->L }
% 191.76/24.55 fresh115(is_a_theorem(implies(implies(X, Y), implies(implies(Y, X), equiv(X, Y)))), true, implies(X, Y), implies(implies(Y, X), equiv(X, Y)))
% 191.76/24.55 = { by axiom 69 (equivalence_3_1) R->L }
% 191.76/24.55 fresh115(fresh53(equivalence_3, true, X, Y), true, implies(X, Y), implies(implies(Y, X), equiv(X, Y)))
% 191.76/24.55 = { by axiom 12 (hilbert_equivalence_3) }
% 191.76/24.55 fresh115(fresh53(true, true, X, Y), true, implies(X, Y), implies(implies(Y, X), equiv(X, Y)))
% 191.76/24.55 = { by axiom 39 (equivalence_3_1) }
% 191.76/24.55 fresh115(true, true, implies(X, Y), implies(implies(Y, X), equiv(X, Y)))
% 191.76/24.55 = { by lemma 73 }
% 191.76/24.55 true
% 191.76/24.55
% 191.76/24.55 Lemma 75: is_a_theorem(implies(X, implies(Y, and(X, Y)))) = true.
% 191.76/24.55 Proof:
% 191.76/24.55 is_a_theorem(implies(X, implies(Y, and(X, Y))))
% 191.76/24.55 = { by axiom 63 (and_3_1) R->L }
% 191.76/24.55 fresh103(and_3, true, X, Y)
% 191.76/24.55 = { by axiom 9 (hilbert_and_3) }
% 191.76/24.55 fresh103(true, true, X, Y)
% 191.76/24.55 = { by axiom 37 (and_3_1) }
% 191.76/24.55 true
% 191.76/24.55
% 191.76/24.55 Lemma 76: fresh40(is_a_theorem(implies(X, implies(X, Y))), true, implies(X, Y)) = true.
% 191.76/24.55 Proof:
% 191.76/24.55 fresh40(is_a_theorem(implies(X, implies(X, Y))), true, implies(X, Y))
% 191.76/24.55 = { by axiom 64 (modus_ponens_2) R->L }
% 191.76/24.55 fresh115(is_a_theorem(implies(implies(X, implies(X, Y)), implies(X, Y))), true, implies(X, implies(X, Y)), implies(X, Y))
% 191.76/24.55 = { by axiom 66 (implies_2_1) R->L }
% 191.76/24.55 fresh115(fresh49(implies_2, true, X, Y), true, implies(X, implies(X, Y)), implies(X, Y))
% 191.76/24.55 = { by axiom 5 (hilbert_implies_2) }
% 191.76/24.55 fresh115(fresh49(true, true, X, Y), true, implies(X, implies(X, Y)), implies(X, Y))
% 191.76/24.55 = { by axiom 41 (implies_2_1) }
% 191.76/24.55 fresh115(true, true, implies(X, implies(X, Y)), implies(X, Y))
% 191.76/24.55 = { by lemma 73 }
% 191.76/24.55 true
% 191.76/24.55
% 191.76/24.55 Lemma 77: fresh3(is_a_theorem(equiv(X, Y)), true, X, Y) = X.
% 191.76/24.55 Proof:
% 191.76/24.55 fresh3(is_a_theorem(equiv(X, Y)), true, X, Y)
% 191.76/24.55 = { by axiom 65 (substitution_of_equivalents_2) R->L }
% 191.76/24.55 fresh4(substitution_of_equivalents, true, X, Y)
% 191.76/24.55 = { by axiom 2 (substitution_of_equivalents) }
% 191.76/24.55 fresh4(true, true, X, Y)
% 191.76/24.55 = { by axiom 51 (substitution_of_equivalents_2) }
% 191.76/24.55 X
% 191.76/24.55
% 191.76/24.55 Lemma 78: and(X, X) = X.
% 191.76/24.55 Proof:
% 191.76/24.55 and(X, X)
% 191.76/24.55 = { by axiom 52 (substitution_of_equivalents_2) R->L }
% 191.76/24.55 fresh3(true, true, X, and(X, X))
% 191.76/24.55 = { by lemma 73 R->L }
% 191.76/24.55 fresh3(fresh115(true, true, implies(and(X, X), X), equiv(X, and(X, X))), true, X, and(X, X))
% 191.76/24.55 = { by lemma 74 R->L }
% 191.76/24.55 fresh3(fresh115(fresh40(is_a_theorem(implies(X, and(X, X))), true, 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))
% 191.76/24.55 = { by axiom 28 (modus_ponens_2) R->L }
% 191.76/24.55 fresh3(fresh115(fresh40(fresh40(true, true, implies(X, and(X, X))), true, 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))
% 191.76/24.55 = { by lemma 75 R->L }
% 191.76/24.55 fresh3(fresh115(fresh40(fresh40(is_a_theorem(implies(X, implies(X, and(X, X)))), true, implies(X, and(X, X))), true, 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))
% 191.76/24.55 = { by lemma 76 }
% 191.76/24.55 fresh3(fresh115(fresh40(true, true, 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))
% 191.76/24.55 = { by axiom 28 (modus_ponens_2) }
% 191.76/24.55 fresh3(fresh115(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))
% 191.76/24.55 = { by axiom 64 (modus_ponens_2) }
% 191.76/24.55 fresh3(fresh40(is_a_theorem(implies(and(X, X), X)), true, equiv(X, and(X, X))), true, X, and(X, X))
% 191.76/24.55 = { by axiom 56 (and_1_1) R->L }
% 191.76/24.55 fresh3(fresh40(fresh107(and_1, true, X, X), true, equiv(X, and(X, X))), true, X, and(X, X))
% 191.76/24.55 = { by axiom 7 (hilbert_and_1) }
% 191.76/24.55 fresh3(fresh40(fresh107(true, true, X, X), true, equiv(X, and(X, X))), true, X, and(X, X))
% 191.76/24.55 = { by axiom 35 (and_1_1) }
% 191.76/24.55 fresh3(fresh40(true, true, equiv(X, and(X, X))), true, X, and(X, X))
% 191.76/24.55 = { by axiom 28 (modus_ponens_2) }
% 191.76/24.55 fresh3(is_a_theorem(equiv(X, and(X, X))), true, X, and(X, X))
% 191.76/24.55 = { by lemma 77 }
% 191.76/24.55 X
% 191.76/24.55
% 191.76/24.55 Lemma 79: is_a_theorem(implies(necessarily(X), necessarily(necessarily(X)))) = true.
% 191.76/24.55 Proof:
% 191.76/24.55 is_a_theorem(implies(necessarily(X), necessarily(necessarily(X))))
% 191.76/24.55 = { by axiom 61 (axiom_4_1) R->L }
% 191.76/24.55 fresh101(axiom_4, true, X)
% 191.76/24.55 = { by axiom 18 (km4b_axiom_4) }
% 191.76/24.55 fresh101(true, true, X)
% 191.76/24.55 = { by axiom 25 (axiom_4_1) }
% 191.76/24.55 true
% 191.76/24.55
% 191.76/24.55 Lemma 80: is_a_theorem(implies(necessarily(X), X)) = true.
% 191.76/24.55 Proof:
% 191.76/24.55 is_a_theorem(implies(necessarily(X), X))
% 191.76/24.55 = { by axiom 33 (axiom_M_1) R->L }
% 191.76/24.55 fresh93(axiom_M, true, X)
% 191.76/24.55 = { by axiom 17 (km4b_axiom_M) }
% 191.76/24.55 fresh93(true, true, X)
% 191.76/24.55 = { by axiom 27 (axiom_M_1) }
% 191.76/24.55 true
% 191.76/24.55
% 191.76/24.55 Lemma 81: necessarily(necessarily(X)) = necessarily(X).
% 191.76/24.55 Proof:
% 191.76/24.55 necessarily(necessarily(X))
% 191.76/24.55 = { by axiom 52 (substitution_of_equivalents_2) R->L }
% 191.76/24.55 fresh3(true, true, necessarily(X), necessarily(necessarily(X)))
% 191.76/24.55 = { by lemma 73 R->L }
% 191.76/24.55 fresh3(fresh115(true, true, implies(necessarily(necessarily(X)), necessarily(X)), equiv(necessarily(X), necessarily(necessarily(X)))), true, necessarily(X), necessarily(necessarily(X)))
% 191.76/24.55 = { by lemma 74 R->L }
% 191.76/24.55 fresh3(fresh115(fresh40(is_a_theorem(implies(necessarily(X), necessarily(necessarily(X)))), true, implies(implies(necessarily(necessarily(X)), necessarily(X)), equiv(necessarily(X), necessarily(necessarily(X))))), true, implies(necessarily(necessarily(X)), necessarily(X)), equiv(necessarily(X), necessarily(necessarily(X)))), true, necessarily(X), necessarily(necessarily(X)))
% 191.76/24.55 = { by lemma 79 }
% 191.76/24.55 fresh3(fresh115(fresh40(true, true, implies(implies(necessarily(necessarily(X)), necessarily(X)), equiv(necessarily(X), necessarily(necessarily(X))))), true, implies(necessarily(necessarily(X)), necessarily(X)), equiv(necessarily(X), necessarily(necessarily(X)))), true, necessarily(X), necessarily(necessarily(X)))
% 191.76/24.55 = { by axiom 28 (modus_ponens_2) }
% 191.76/24.55 fresh3(fresh115(is_a_theorem(implies(implies(necessarily(necessarily(X)), necessarily(X)), equiv(necessarily(X), necessarily(necessarily(X))))), true, implies(necessarily(necessarily(X)), necessarily(X)), equiv(necessarily(X), necessarily(necessarily(X)))), true, necessarily(X), necessarily(necessarily(X)))
% 191.76/24.55 = { by axiom 64 (modus_ponens_2) }
% 191.76/24.55 fresh3(fresh40(is_a_theorem(implies(necessarily(necessarily(X)), necessarily(X))), true, equiv(necessarily(X), necessarily(necessarily(X)))), true, necessarily(X), necessarily(necessarily(X)))
% 191.76/24.55 = { by lemma 80 }
% 191.76/24.55 fresh3(fresh40(true, true, equiv(necessarily(X), necessarily(necessarily(X)))), true, necessarily(X), necessarily(necessarily(X)))
% 191.76/24.55 = { by axiom 28 (modus_ponens_2) }
% 191.76/24.56 fresh3(is_a_theorem(equiv(necessarily(X), necessarily(necessarily(X)))), true, necessarily(X), necessarily(necessarily(X)))
% 191.76/24.56 = { by lemma 77 }
% 191.76/24.56 necessarily(X)
% 191.76/24.56
% 191.76/24.56 Lemma 82: is_a_theorem(implies(X, implies(Y, X))) = true.
% 191.76/24.56 Proof:
% 191.76/24.56 is_a_theorem(implies(X, implies(Y, X)))
% 191.76/24.56 = { by axiom 53 (implies_1_1) R->L }
% 191.76/24.56 fresh51(implies_1, true, X, Y)
% 191.76/24.56 = { by axiom 4 (hilbert_implies_1) }
% 191.76/24.56 fresh51(true, true, X, Y)
% 191.76/24.56 = { by axiom 40 (implies_1_1) }
% 191.76/24.56 true
% 191.76/24.56
% 191.76/24.56 Lemma 83: or(X, X) = X.
% 191.76/24.56 Proof:
% 191.76/24.56 or(X, X)
% 191.76/24.56 = { by axiom 52 (substitution_of_equivalents_2) R->L }
% 191.76/24.56 fresh3(true, true, X, or(X, X))
% 191.76/24.56 = { by lemma 73 R->L }
% 191.76/24.56 fresh3(fresh115(true, true, implies(or(X, X), X), equiv(X, or(X, X))), true, X, or(X, X))
% 191.76/24.56 = { by lemma 74 R->L }
% 191.76/24.56 fresh3(fresh115(fresh40(is_a_theorem(implies(X, or(X, X))), true, implies(implies(or(X, X), X), equiv(X, or(X, X)))), true, implies(or(X, X), X), equiv(X, or(X, X))), true, X, or(X, X))
% 191.76/24.56 = { by axiom 55 (or_1_1) R->L }
% 191.76/24.56 fresh3(fresh115(fresh40(fresh21(or_1, true, X, X), true, implies(implies(or(X, X), X), equiv(X, or(X, X)))), true, implies(or(X, X), X), equiv(X, or(X, X))), true, X, or(X, X))
% 191.76/24.56 = { by axiom 10 (hilbert_or_1) }
% 191.76/24.56 fresh3(fresh115(fresh40(fresh21(true, true, X, X), true, implies(implies(or(X, X), X), equiv(X, or(X, X)))), true, implies(or(X, X), X), equiv(X, or(X, X))), true, X, or(X, X))
% 191.76/24.56 = { by axiom 50 (or_1_1) }
% 191.76/24.56 fresh3(fresh115(fresh40(true, true, implies(implies(or(X, X), X), equiv(X, or(X, X)))), true, implies(or(X, X), X), equiv(X, or(X, X))), true, X, or(X, X))
% 191.76/24.56 = { by axiom 28 (modus_ponens_2) }
% 191.76/24.56 fresh3(fresh115(is_a_theorem(implies(implies(or(X, X), X), equiv(X, or(X, X)))), true, implies(or(X, X), X), equiv(X, or(X, X))), true, X, or(X, X))
% 191.76/24.56 = { by axiom 64 (modus_ponens_2) }
% 191.76/24.56 fresh3(fresh40(is_a_theorem(implies(or(X, X), X)), true, equiv(X, or(X, X))), true, X, or(X, X))
% 191.76/24.56 = { by axiom 28 (modus_ponens_2) R->L }
% 191.76/24.56 fresh3(fresh40(fresh40(true, true, implies(or(X, X), X)), true, equiv(X, or(X, X))), true, X, or(X, X))
% 191.76/24.56 = { by lemma 76 R->L }
% 191.76/24.56 fresh3(fresh40(fresh40(fresh40(is_a_theorem(implies(X, implies(X, X))), true, implies(X, X)), true, implies(or(X, X), X)), true, equiv(X, or(X, X))), true, X, or(X, X))
% 191.76/24.56 = { by lemma 82 }
% 191.76/24.56 fresh3(fresh40(fresh40(fresh40(true, true, implies(X, X)), true, implies(or(X, X), X)), true, equiv(X, or(X, X))), true, X, or(X, X))
% 191.76/24.56 = { by axiom 28 (modus_ponens_2) }
% 191.76/24.56 fresh3(fresh40(fresh40(is_a_theorem(implies(X, X)), true, implies(or(X, X), X)), true, equiv(X, or(X, X))), true, X, or(X, X))
% 191.76/24.56 = { by axiom 64 (modus_ponens_2) R->L }
% 191.76/24.56 fresh3(fresh40(fresh115(is_a_theorem(implies(implies(X, X), implies(or(X, X), X))), true, implies(X, X), implies(or(X, X), X)), true, equiv(X, or(X, X))), true, X, or(X, X))
% 191.76/24.56 = { by axiom 28 (modus_ponens_2) R->L }
% 191.76/24.56 fresh3(fresh40(fresh115(fresh40(true, true, implies(implies(X, X), implies(or(X, X), X))), true, implies(X, X), implies(or(X, X), X)), true, equiv(X, or(X, X))), true, X, or(X, X))
% 191.76/24.56 = { by axiom 60 (or_3_1) R->L }
% 191.76/24.56 fresh3(fresh40(fresh115(fresh40(fresh17(true, true, X, X, X), true, implies(implies(X, X), implies(or(X, X), X))), true, implies(X, X), implies(or(X, X), X)), true, equiv(X, or(X, X))), true, X, or(X, X))
% 191.76/24.56 = { by axiom 11 (hilbert_or_3) R->L }
% 191.76/24.56 fresh3(fresh40(fresh115(fresh40(fresh17(or_3, true, X, X, X), true, implies(implies(X, X), implies(or(X, X), X))), true, implies(X, X), implies(or(X, X), X)), true, equiv(X, or(X, X))), true, X, or(X, X))
% 191.76/24.56 = { by axiom 71 (or_3_1) }
% 191.76/24.56 fresh3(fresh40(fresh115(fresh40(is_a_theorem(implies(implies(X, X), implies(implies(X, X), implies(or(X, X), X)))), true, implies(implies(X, X), implies(or(X, X), X))), true, implies(X, X), implies(or(X, X), X)), true, equiv(X, or(X, X))), true, X, or(X, X))
% 191.76/24.56 = { by lemma 76 }
% 191.76/24.56 fresh3(fresh40(fresh115(true, true, implies(X, X), implies(or(X, X), X)), true, equiv(X, or(X, X))), true, X, or(X, X))
% 191.76/24.56 = { by lemma 73 }
% 191.76/24.56 fresh3(fresh40(true, true, equiv(X, or(X, X))), true, X, or(X, X))
% 191.76/24.56 = { by axiom 28 (modus_ponens_2) }
% 191.76/24.56 fresh3(is_a_theorem(equiv(X, or(X, X))), true, X, or(X, X))
% 191.76/24.56 = { by lemma 77 }
% 191.76/24.56 X
% 191.76/24.56
% 191.76/24.56 Lemma 84: not(and(X, not(Y))) = implies(X, Y).
% 191.76/24.56 Proof:
% 191.76/24.56 not(and(X, not(Y)))
% 191.76/24.56 = { by axiom 46 (op_implies_and) R->L }
% 191.76/24.56 fresh29(op_implies_and, true, X, Y)
% 191.76/24.56 = { by axiom 21 (hilbert_op_implies_and) }
% 191.76/24.56 fresh29(true, true, X, Y)
% 191.76/24.56 = { by axiom 45 (op_implies_and) }
% 191.76/24.56 implies(X, Y)
% 191.76/24.56
% 191.76/24.56 Lemma 85: implies(not(X), Y) = or(X, Y).
% 191.76/24.56 Proof:
% 191.76/24.56 implies(not(X), Y)
% 191.76/24.56 = { by lemma 84 R->L }
% 191.76/24.56 not(and(not(X), not(Y)))
% 191.76/24.56 = { by axiom 58 (op_or) R->L }
% 191.76/24.56 fresh26(op_or, true, X, Y)
% 191.76/24.56 = { by axiom 13 (hilbert_op_or) }
% 191.76/24.56 fresh26(true, true, X, Y)
% 191.76/24.56 = { by axiom 47 (op_or) }
% 191.76/24.56 or(X, Y)
% 191.76/24.56
% 191.76/24.56 Lemma 86: is_a_theorem(implies(or(X, not(Y)), implies(Y, X))) = true.
% 191.76/24.56 Proof:
% 191.76/24.56 is_a_theorem(implies(or(X, not(Y)), implies(Y, X)))
% 191.76/24.56 = { by lemma 85 R->L }
% 191.76/24.56 is_a_theorem(implies(implies(not(X), not(Y)), implies(Y, X)))
% 191.76/24.56 = { by axiom 67 (modus_tollens_1) R->L }
% 191.76/24.56 fresh35(modus_tollens, true, Y, X)
% 191.76/24.56 = { by axiom 3 (hilbert_modus_tollens) }
% 191.76/24.56 fresh35(true, true, Y, X)
% 191.76/24.56 = { by axiom 42 (modus_tollens_1) }
% 191.76/24.56 true
% 191.76/24.56
% 191.76/24.56 Lemma 87: is_a_theorem(implies(or(X, Y), or(Y, X))) = true.
% 191.76/24.56 Proof:
% 191.76/24.56 is_a_theorem(implies(or(X, Y), or(Y, X)))
% 191.76/24.56 = { by lemma 85 R->L }
% 191.76/24.56 is_a_theorem(implies(or(X, Y), implies(not(Y), X)))
% 191.76/24.56 = { by lemma 83 R->L }
% 191.76/24.56 is_a_theorem(implies(or(X, or(Y, Y)), implies(not(Y), X)))
% 191.76/24.56 = { by lemma 78 R->L }
% 191.76/24.56 is_a_theorem(implies(or(X, or(Y, Y)), implies(and(not(Y), not(Y)), X)))
% 191.76/24.56 = { by lemma 85 R->L }
% 191.76/24.56 is_a_theorem(implies(or(X, implies(not(Y), Y)), implies(and(not(Y), not(Y)), X)))
% 191.76/24.56 = { by lemma 84 R->L }
% 191.76/24.56 is_a_theorem(implies(or(X, not(and(not(Y), not(Y)))), implies(and(not(Y), not(Y)), X)))
% 191.76/24.56 = { by lemma 86 }
% 191.76/24.56 true
% 191.76/24.56
% 191.76/24.56 Lemma 88: or(Y, X) = or(X, Y).
% 191.76/24.56 Proof:
% 191.76/24.56 or(Y, X)
% 191.76/24.56 = { by axiom 52 (substitution_of_equivalents_2) R->L }
% 191.76/24.56 fresh3(true, true, or(X, Y), or(Y, X))
% 191.76/24.56 = { by lemma 73 R->L }
% 191.76/24.56 fresh3(fresh115(true, true, implies(or(Y, X), or(X, Y)), equiv(or(X, Y), or(Y, X))), true, or(X, Y), or(Y, X))
% 191.76/24.56 = { by lemma 74 R->L }
% 191.76/24.56 fresh3(fresh115(fresh40(is_a_theorem(implies(or(X, Y), or(Y, X))), true, implies(implies(or(Y, X), or(X, Y)), equiv(or(X, Y), or(Y, X)))), true, implies(or(Y, X), or(X, Y)), equiv(or(X, Y), or(Y, X))), true, or(X, Y), or(Y, X))
% 191.76/24.56 = { by lemma 87 }
% 191.76/24.56 fresh3(fresh115(fresh40(true, true, implies(implies(or(Y, X), or(X, Y)), equiv(or(X, Y), or(Y, X)))), true, implies(or(Y, X), or(X, Y)), equiv(or(X, Y), or(Y, X))), true, or(X, Y), or(Y, X))
% 191.76/24.56 = { by axiom 28 (modus_ponens_2) }
% 191.76/24.56 fresh3(fresh115(is_a_theorem(implies(implies(or(Y, X), or(X, Y)), equiv(or(X, Y), or(Y, X)))), true, implies(or(Y, X), or(X, Y)), equiv(or(X, Y), or(Y, X))), true, or(X, Y), or(Y, X))
% 191.76/24.56 = { by axiom 64 (modus_ponens_2) }
% 191.76/24.56 fresh3(fresh40(is_a_theorem(implies(or(Y, X), or(X, Y))), true, equiv(or(X, Y), or(Y, X))), true, or(X, Y), or(Y, X))
% 191.76/24.56 = { by lemma 87 }
% 191.76/24.56 fresh3(fresh40(true, true, equiv(or(X, Y), or(Y, X))), true, or(X, Y), or(Y, X))
% 191.76/24.56 = { by axiom 28 (modus_ponens_2) }
% 191.76/24.56 fresh3(is_a_theorem(equiv(or(X, Y), or(Y, X))), true, or(X, Y), or(Y, X))
% 191.76/24.56 = { by lemma 77 }
% 191.76/24.57 or(X, Y)
% 191.76/24.57
% 191.76/24.57 Lemma 89: or(X, and(Y, not(Z))) = implies(implies(Y, Z), X).
% 191.76/24.57 Proof:
% 191.76/24.57 or(X, and(Y, not(Z)))
% 191.76/24.57 = { by lemma 88 }
% 191.76/24.57 or(and(Y, not(Z)), X)
% 191.76/24.57 = { by lemma 85 R->L }
% 191.76/24.57 implies(not(and(Y, not(Z))), X)
% 191.76/24.57 = { by lemma 84 }
% 191.76/24.57 implies(implies(Y, Z), X)
% 191.76/24.57
% 191.76/24.57 Lemma 90: fresh40(is_a_theorem(X), true, implies(implies(X, Y), Y)) = true.
% 191.76/24.57 Proof:
% 191.76/24.57 fresh40(is_a_theorem(X), true, implies(implies(X, Y), Y))
% 191.76/24.57 = { by lemma 89 R->L }
% 191.76/24.57 fresh40(is_a_theorem(X), true, or(Y, and(X, not(Y))))
% 191.76/24.57 = { by axiom 64 (modus_ponens_2) R->L }
% 191.76/24.57 fresh115(is_a_theorem(implies(X, or(Y, and(X, not(Y))))), true, X, or(Y, and(X, not(Y))))
% 191.76/24.57 = { by lemma 85 R->L }
% 191.76/24.57 fresh115(is_a_theorem(implies(X, implies(not(Y), and(X, not(Y))))), true, X, or(Y, and(X, not(Y))))
% 191.76/24.57 = { by lemma 75 }
% 191.76/24.57 fresh115(true, true, X, or(Y, and(X, not(Y))))
% 191.76/24.57 = { by lemma 73 }
% 191.76/24.57 true
% 191.76/24.57
% 191.76/24.57 Lemma 91: necessarily(implies(X, Y)) = strict_implies(X, Y).
% 191.76/24.57 Proof:
% 191.76/24.57 necessarily(implies(X, Y))
% 191.76/24.57 = { by axiom 49 (op_strict_implies) R->L }
% 191.76/24.57 fresh23(op_strict_implies, true, X, Y)
% 191.76/24.57 = { by axiom 22 (s1_0_op_strict_implies) }
% 191.76/24.57 fresh23(true, true, X, Y)
% 191.76/24.57 = { by axiom 48 (op_strict_implies) }
% 191.76/24.57 strict_implies(X, Y)
% 191.76/24.57
% 191.76/24.57 Lemma 92: fresh33(is_a_theorem(X), true, X) = is_a_theorem(necessarily(X)).
% 191.76/24.57 Proof:
% 191.76/24.57 fresh33(is_a_theorem(X), true, X)
% 191.76/24.57 = { by axiom 43 (necessitation_1) R->L }
% 191.76/24.57 fresh34(necessitation, true, X)
% 191.76/24.57 = { by axiom 15 (km4b_necessitation) }
% 191.76/24.57 fresh34(true, true, X)
% 191.76/24.57 = { by axiom 29 (necessitation_1) }
% 191.76/24.57 is_a_theorem(necessarily(X))
% 191.76/24.57
% 191.76/24.57 Lemma 93: is_a_theorem(implies(X, necessarily(possibly(X)))) = true.
% 191.76/24.57 Proof:
% 191.76/24.57 is_a_theorem(implies(X, necessarily(possibly(X))))
% 191.76/24.57 = { by axiom 54 (axiom_B_1) R->L }
% 191.76/24.57 fresh97(axiom_B, true, X)
% 191.76/24.57 = { by axiom 19 (km4b_axiom_B) }
% 191.76/24.57 fresh97(true, true, X)
% 191.76/24.57 = { by axiom 26 (axiom_B_1) }
% 191.76/24.57 true
% 191.76/24.57
% 191.76/24.57 Lemma 94: fresh40(is_a_theorem(implies(X, Y)), true, implies(implies(Y, Z), implies(X, Z))) = true.
% 191.76/24.57 Proof:
% 191.76/24.57 fresh40(is_a_theorem(implies(X, Y)), true, implies(implies(Y, Z), implies(X, Z)))
% 191.76/24.57 = { by axiom 64 (modus_ponens_2) R->L }
% 191.76/24.57 fresh115(is_a_theorem(implies(implies(X, Y), implies(implies(Y, Z), implies(X, Z)))), true, implies(X, Y), implies(implies(Y, Z), implies(X, Z)))
% 191.76/24.57 = { by axiom 70 (implies_3_1) R->L }
% 191.76/24.57 fresh115(fresh47(implies_3, true, X, Y, Z), true, implies(X, Y), implies(implies(Y, Z), implies(X, Z)))
% 191.76/24.57 = { by axiom 6 (hilbert_implies_3) }
% 191.76/24.57 fresh115(fresh47(true, true, X, Y, Z), true, implies(X, Y), implies(implies(Y, Z), implies(X, Z)))
% 191.76/24.57 = { by axiom 59 (implies_3_1) }
% 191.76/24.57 fresh115(true, true, implies(X, Y), implies(implies(Y, Z), implies(X, Z)))
% 191.76/24.57 = { by lemma 73 }
% 191.76/24.57 true
% 191.76/24.57
% 191.76/24.57 Lemma 95: fresh40(is_a_theorem(implies(implies(X, Y), Y)), true, equiv(Y, implies(X, Y))) = true.
% 191.76/24.57 Proof:
% 191.76/24.57 fresh40(is_a_theorem(implies(implies(X, Y), Y)), true, equiv(Y, implies(X, Y)))
% 191.76/24.57 = { by axiom 64 (modus_ponens_2) R->L }
% 191.76/24.57 fresh115(is_a_theorem(implies(implies(implies(X, Y), Y), equiv(Y, implies(X, Y)))), true, implies(implies(X, Y), Y), equiv(Y, implies(X, Y)))
% 191.76/24.57 = { by axiom 28 (modus_ponens_2) R->L }
% 191.76/24.57 fresh115(fresh40(true, true, implies(implies(implies(X, Y), Y), equiv(Y, implies(X, Y)))), true, implies(implies(X, Y), Y), equiv(Y, implies(X, Y)))
% 191.76/24.57 = { by lemma 82 R->L }
% 191.76/24.57 fresh115(fresh40(is_a_theorem(implies(Y, implies(X, Y))), true, implies(implies(implies(X, Y), Y), equiv(Y, implies(X, Y)))), true, implies(implies(X, Y), Y), equiv(Y, implies(X, Y)))
% 191.76/24.57 = { by lemma 74 }
% 191.76/24.57 fresh115(true, true, implies(implies(X, Y), Y), equiv(Y, implies(X, Y)))
% 191.76/24.57 = { by lemma 73 }
% 191.76/24.57 true
% 191.76/24.57
% 191.76/24.57 Lemma 96: fresh40(is_a_theorem(X), true, implies(Y, and(X, Y))) = true.
% 191.76/24.57 Proof:
% 191.76/24.57 fresh40(is_a_theorem(X), true, implies(Y, and(X, Y)))
% 191.76/24.57 = { by axiom 64 (modus_ponens_2) R->L }
% 191.76/24.57 fresh115(is_a_theorem(implies(X, implies(Y, and(X, Y)))), true, X, implies(Y, and(X, Y)))
% 191.76/24.57 = { by lemma 75 }
% 191.76/24.57 fresh115(true, true, X, implies(Y, and(X, Y)))
% 191.76/24.57 = { by lemma 73 }
% 191.76/24.57 true
% 191.76/24.57
% 191.76/24.57 Lemma 97: is_a_theorem(or(necessarily(possibly(not(X))), X)) = true.
% 191.76/24.57 Proof:
% 191.76/24.57 is_a_theorem(or(necessarily(possibly(not(X))), X))
% 191.76/24.57 = { by axiom 28 (modus_ponens_2) R->L }
% 191.76/24.57 fresh40(true, true, or(necessarily(possibly(not(X))), X))
% 191.76/24.57 = { by lemma 93 R->L }
% 191.76/24.57 fresh40(is_a_theorem(implies(not(X), necessarily(possibly(not(X))))), true, or(necessarily(possibly(not(X))), X))
% 191.76/24.57 = { by lemma 85 }
% 191.76/24.57 fresh40(is_a_theorem(or(X, necessarily(possibly(not(X))))), true, or(necessarily(possibly(not(X))), X))
% 191.76/24.57 = { by axiom 64 (modus_ponens_2) R->L }
% 191.76/24.57 fresh115(is_a_theorem(implies(or(X, necessarily(possibly(not(X)))), or(necessarily(possibly(not(X))), X))), true, or(X, necessarily(possibly(not(X)))), or(necessarily(possibly(not(X))), X))
% 191.76/24.57 = { by lemma 87 }
% 191.76/24.57 fresh115(true, true, or(X, necessarily(possibly(not(X)))), or(necessarily(possibly(not(X))), X))
% 191.76/24.57 = { by lemma 73 }
% 191.76/24.57 true
% 191.76/24.57
% 191.76/24.57 Lemma 98: not(necessarily(not(X))) = possibly(X).
% 191.76/24.57 Proof:
% 191.76/24.57 not(necessarily(not(X)))
% 191.76/24.57 = { by axiom 32 (op_possibly) R->L }
% 191.76/24.57 fresh25(op_possibly, true, X)
% 191.76/24.57 = { by axiom 20 (km4b_op_possibly) }
% 191.76/24.57 fresh25(true, true, X)
% 191.76/24.57 = { by axiom 31 (op_possibly) }
% 191.76/24.57 possibly(X)
% 191.76/24.57
% 191.76/24.57 Lemma 99: not(necessarily(possibly(X))) = possibly(necessarily(not(X))).
% 191.76/24.57 Proof:
% 191.76/24.57 not(necessarily(possibly(X)))
% 191.76/24.57 = { by lemma 98 R->L }
% 191.76/24.57 not(necessarily(not(necessarily(not(X)))))
% 191.76/24.57 = { by lemma 98 }
% 191.76/24.57 possibly(necessarily(not(X)))
% 191.76/24.57
% 191.76/24.57 Lemma 100: not(not(X)) = X.
% 191.76/24.57 Proof:
% 191.76/24.57 not(not(X))
% 191.76/24.57 = { by lemma 78 R->L }
% 191.76/24.57 not(and(not(X), not(X)))
% 191.76/24.57 = { by lemma 84 }
% 191.76/24.57 implies(not(X), X)
% 191.76/24.57 = { by lemma 85 }
% 191.76/24.57 or(X, X)
% 191.76/24.57 = { by lemma 83 }
% 191.76/24.57 X
% 191.76/24.57
% 191.76/24.57 Lemma 101: possibly(necessarily(X)) = necessarily(X).
% 191.76/24.57 Proof:
% 191.76/24.57 possibly(necessarily(X))
% 191.76/24.57 = { by lemma 77 R->L }
% 191.76/24.57 fresh3(is_a_theorem(equiv(possibly(necessarily(X)), necessarily(X))), true, possibly(necessarily(X)), necessarily(X))
% 191.76/24.57 = { by lemma 81 R->L }
% 191.76/24.57 fresh3(is_a_theorem(equiv(possibly(necessarily(necessarily(X))), necessarily(X))), true, possibly(necessarily(X)), necessarily(X))
% 191.76/24.57 = { by lemma 77 R->L }
% 191.76/24.57 fresh3(is_a_theorem(fresh3(is_a_theorem(equiv(equiv(possibly(necessarily(necessarily(X))), necessarily(X)), implies(strict_implies(necessarily(X), possibly(necessarily(X))), equiv(possibly(necessarily(necessarily(X))), necessarily(X))))), true, equiv(possibly(necessarily(necessarily(X))), necessarily(X)), implies(strict_implies(necessarily(X), possibly(necessarily(X))), equiv(possibly(necessarily(necessarily(X))), necessarily(X))))), true, possibly(necessarily(X)), necessarily(X))
% 191.76/24.57 = { by axiom 28 (modus_ponens_2) R->L }
% 191.76/24.57 fresh3(is_a_theorem(fresh3(fresh40(true, true, equiv(equiv(possibly(necessarily(necessarily(X))), necessarily(X)), implies(strict_implies(necessarily(X), possibly(necessarily(X))), equiv(possibly(necessarily(necessarily(X))), necessarily(X))))), true, equiv(possibly(necessarily(necessarily(X))), necessarily(X)), implies(strict_implies(necessarily(X), possibly(necessarily(X))), equiv(possibly(necessarily(necessarily(X))), necessarily(X))))), true, possibly(necessarily(X)), necessarily(X))
% 191.76/24.58 = { by lemma 90 R->L }
% 191.76/24.58 fresh3(is_a_theorem(fresh3(fresh40(fresh40(is_a_theorem(strict_implies(necessarily(X), possibly(necessarily(X)))), true, implies(implies(strict_implies(necessarily(X), possibly(necessarily(X))), equiv(possibly(necessarily(necessarily(X))), necessarily(X))), equiv(possibly(necessarily(necessarily(X))), necessarily(X)))), true, equiv(equiv(possibly(necessarily(necessarily(X))), necessarily(X)), implies(strict_implies(necessarily(X), possibly(necessarily(X))), equiv(possibly(necessarily(necessarily(X))), necessarily(X))))), true, equiv(possibly(necessarily(necessarily(X))), necessarily(X)), implies(strict_implies(necessarily(X), possibly(necessarily(X))), equiv(possibly(necessarily(necessarily(X))), necessarily(X))))), true, possibly(necessarily(X)), necessarily(X))
% 191.76/24.58 = { by lemma 91 R->L }
% 191.76/24.58 fresh3(is_a_theorem(fresh3(fresh40(fresh40(is_a_theorem(necessarily(implies(necessarily(X), possibly(necessarily(X))))), true, implies(implies(strict_implies(necessarily(X), possibly(necessarily(X))), equiv(possibly(necessarily(necessarily(X))), necessarily(X))), equiv(possibly(necessarily(necessarily(X))), necessarily(X)))), true, equiv(equiv(possibly(necessarily(necessarily(X))), necessarily(X)), implies(strict_implies(necessarily(X), possibly(necessarily(X))), equiv(possibly(necessarily(necessarily(X))), necessarily(X))))), true, equiv(possibly(necessarily(necessarily(X))), necessarily(X)), implies(strict_implies(necessarily(X), possibly(necessarily(X))), equiv(possibly(necessarily(necessarily(X))), necessarily(X))))), true, possibly(necessarily(X)), necessarily(X))
% 191.76/24.58 = { by lemma 92 R->L }
% 191.76/24.58 fresh3(is_a_theorem(fresh3(fresh40(fresh40(fresh33(is_a_theorem(implies(necessarily(X), possibly(necessarily(X)))), true, implies(necessarily(X), possibly(necessarily(X)))), true, implies(implies(strict_implies(necessarily(X), possibly(necessarily(X))), equiv(possibly(necessarily(necessarily(X))), necessarily(X))), equiv(possibly(necessarily(necessarily(X))), necessarily(X)))), true, equiv(equiv(possibly(necessarily(necessarily(X))), necessarily(X)), implies(strict_implies(necessarily(X), possibly(necessarily(X))), equiv(possibly(necessarily(necessarily(X))), necessarily(X))))), true, equiv(possibly(necessarily(necessarily(X))), necessarily(X)), implies(strict_implies(necessarily(X), possibly(necessarily(X))), equiv(possibly(necessarily(necessarily(X))), necessarily(X))))), true, possibly(necessarily(X)), necessarily(X))
% 191.76/24.58 = { by axiom 28 (modus_ponens_2) R->L }
% 191.76/24.58 fresh3(is_a_theorem(fresh3(fresh40(fresh40(fresh33(fresh40(true, true, implies(necessarily(X), possibly(necessarily(X)))), true, implies(necessarily(X), possibly(necessarily(X)))), true, implies(implies(strict_implies(necessarily(X), possibly(necessarily(X))), equiv(possibly(necessarily(necessarily(X))), necessarily(X))), equiv(possibly(necessarily(necessarily(X))), necessarily(X)))), true, equiv(equiv(possibly(necessarily(necessarily(X))), necessarily(X)), implies(strict_implies(necessarily(X), possibly(necessarily(X))), equiv(possibly(necessarily(necessarily(X))), necessarily(X))))), true, equiv(possibly(necessarily(necessarily(X))), necessarily(X)), implies(strict_implies(necessarily(X), possibly(necessarily(X))), equiv(possibly(necessarily(necessarily(X))), necessarily(X))))), true, possibly(necessarily(X)), necessarily(X))
% 191.76/24.58 = { by lemma 80 R->L }
% 191.76/24.58 fresh3(is_a_theorem(fresh3(fresh40(fresh40(fresh33(fresh40(is_a_theorem(implies(necessarily(possibly(necessarily(X))), possibly(necessarily(X)))), true, implies(necessarily(X), possibly(necessarily(X)))), true, implies(necessarily(X), possibly(necessarily(X)))), true, implies(implies(strict_implies(necessarily(X), possibly(necessarily(X))), equiv(possibly(necessarily(necessarily(X))), necessarily(X))), equiv(possibly(necessarily(necessarily(X))), necessarily(X)))), true, equiv(equiv(possibly(necessarily(necessarily(X))), necessarily(X)), implies(strict_implies(necessarily(X), possibly(necessarily(X))), equiv(possibly(necessarily(necessarily(X))), necessarily(X))))), true, equiv(possibly(necessarily(necessarily(X))), necessarily(X)), implies(strict_implies(necessarily(X), possibly(necessarily(X))), equiv(possibly(necessarily(necessarily(X))), necessarily(X))))), true, possibly(necessarily(X)), necessarily(X))
% 191.76/24.58 = { by axiom 64 (modus_ponens_2) R->L }
% 191.76/24.58 fresh3(is_a_theorem(fresh3(fresh40(fresh40(fresh33(fresh115(is_a_theorem(implies(implies(necessarily(possibly(necessarily(X))), possibly(necessarily(X))), implies(necessarily(X), possibly(necessarily(X))))), true, implies(necessarily(possibly(necessarily(X))), possibly(necessarily(X))), implies(necessarily(X), possibly(necessarily(X)))), true, implies(necessarily(X), possibly(necessarily(X)))), true, implies(implies(strict_implies(necessarily(X), possibly(necessarily(X))), equiv(possibly(necessarily(necessarily(X))), necessarily(X))), equiv(possibly(necessarily(necessarily(X))), necessarily(X)))), true, equiv(equiv(possibly(necessarily(necessarily(X))), necessarily(X)), implies(strict_implies(necessarily(X), possibly(necessarily(X))), equiv(possibly(necessarily(necessarily(X))), necessarily(X))))), true, equiv(possibly(necessarily(necessarily(X))), necessarily(X)), implies(strict_implies(necessarily(X), possibly(necessarily(X))), equiv(possibly(necessarily(necessarily(X))), necessarily(X))))), true, possibly(necessarily(X)), necessarily(X))
% 191.76/24.58 = { by axiom 28 (modus_ponens_2) R->L }
% 191.76/24.58 fresh3(is_a_theorem(fresh3(fresh40(fresh40(fresh33(fresh115(fresh40(true, true, implies(implies(necessarily(possibly(necessarily(X))), possibly(necessarily(X))), implies(necessarily(X), possibly(necessarily(X))))), true, implies(necessarily(possibly(necessarily(X))), possibly(necessarily(X))), implies(necessarily(X), possibly(necessarily(X)))), true, implies(necessarily(X), possibly(necessarily(X)))), true, implies(implies(strict_implies(necessarily(X), possibly(necessarily(X))), equiv(possibly(necessarily(necessarily(X))), necessarily(X))), equiv(possibly(necessarily(necessarily(X))), necessarily(X)))), true, equiv(equiv(possibly(necessarily(necessarily(X))), necessarily(X)), implies(strict_implies(necessarily(X), possibly(necessarily(X))), equiv(possibly(necessarily(necessarily(X))), necessarily(X))))), true, equiv(possibly(necessarily(necessarily(X))), necessarily(X)), implies(strict_implies(necessarily(X), possibly(necessarily(X))), equiv(possibly(necessarily(necessarily(X))), necessarily(X))))), true, possibly(necessarily(X)), necessarily(X))
% 191.76/24.58 = { by lemma 93 R->L }
% 191.76/24.58 fresh3(is_a_theorem(fresh3(fresh40(fresh40(fresh33(fresh115(fresh40(is_a_theorem(implies(necessarily(X), necessarily(possibly(necessarily(X))))), true, implies(implies(necessarily(possibly(necessarily(X))), possibly(necessarily(X))), implies(necessarily(X), possibly(necessarily(X))))), true, implies(necessarily(possibly(necessarily(X))), possibly(necessarily(X))), implies(necessarily(X), possibly(necessarily(X)))), true, implies(necessarily(X), possibly(necessarily(X)))), true, implies(implies(strict_implies(necessarily(X), possibly(necessarily(X))), equiv(possibly(necessarily(necessarily(X))), necessarily(X))), equiv(possibly(necessarily(necessarily(X))), necessarily(X)))), true, equiv(equiv(possibly(necessarily(necessarily(X))), necessarily(X)), implies(strict_implies(necessarily(X), possibly(necessarily(X))), equiv(possibly(necessarily(necessarily(X))), necessarily(X))))), true, equiv(possibly(necessarily(necessarily(X))), necessarily(X)), implies(strict_implies(necessarily(X), possibly(necessarily(X))), equiv(possibly(necessarily(necessarily(X))), necessarily(X))))), true, possibly(necessarily(X)), necessarily(X))
% 191.76/24.58 = { by lemma 94 }
% 191.76/24.58 fresh3(is_a_theorem(fresh3(fresh40(fresh40(fresh33(fresh115(true, true, implies(necessarily(possibly(necessarily(X))), possibly(necessarily(X))), implies(necessarily(X), possibly(necessarily(X)))), true, implies(necessarily(X), possibly(necessarily(X)))), true, implies(implies(strict_implies(necessarily(X), possibly(necessarily(X))), equiv(possibly(necessarily(necessarily(X))), necessarily(X))), equiv(possibly(necessarily(necessarily(X))), necessarily(X)))), true, equiv(equiv(possibly(necessarily(necessarily(X))), necessarily(X)), implies(strict_implies(necessarily(X), possibly(necessarily(X))), equiv(possibly(necessarily(necessarily(X))), necessarily(X))))), true, equiv(possibly(necessarily(necessarily(X))), necessarily(X)), implies(strict_implies(necessarily(X), possibly(necessarily(X))), equiv(possibly(necessarily(necessarily(X))), necessarily(X))))), true, possibly(necessarily(X)), necessarily(X))
% 191.76/24.58 = { by lemma 73 }
% 191.76/24.58 fresh3(is_a_theorem(fresh3(fresh40(fresh40(fresh33(true, true, implies(necessarily(X), possibly(necessarily(X)))), true, implies(implies(strict_implies(necessarily(X), possibly(necessarily(X))), equiv(possibly(necessarily(necessarily(X))), necessarily(X))), equiv(possibly(necessarily(necessarily(X))), necessarily(X)))), true, equiv(equiv(possibly(necessarily(necessarily(X))), necessarily(X)), implies(strict_implies(necessarily(X), possibly(necessarily(X))), equiv(possibly(necessarily(necessarily(X))), necessarily(X))))), true, equiv(possibly(necessarily(necessarily(X))), necessarily(X)), implies(strict_implies(necessarily(X), possibly(necessarily(X))), equiv(possibly(necessarily(necessarily(X))), necessarily(X))))), true, possibly(necessarily(X)), necessarily(X))
% 191.76/24.58 = { by axiom 30 (necessitation_1) }
% 191.76/24.58 fresh3(is_a_theorem(fresh3(fresh40(fresh40(true, true, implies(implies(strict_implies(necessarily(X), possibly(necessarily(X))), equiv(possibly(necessarily(necessarily(X))), necessarily(X))), equiv(possibly(necessarily(necessarily(X))), necessarily(X)))), true, equiv(equiv(possibly(necessarily(necessarily(X))), necessarily(X)), implies(strict_implies(necessarily(X), possibly(necessarily(X))), equiv(possibly(necessarily(necessarily(X))), necessarily(X))))), true, equiv(possibly(necessarily(necessarily(X))), necessarily(X)), implies(strict_implies(necessarily(X), possibly(necessarily(X))), equiv(possibly(necessarily(necessarily(X))), necessarily(X))))), true, possibly(necessarily(X)), necessarily(X))
% 191.76/24.58 = { by axiom 28 (modus_ponens_2) }
% 191.76/24.58 fresh3(is_a_theorem(fresh3(fresh40(is_a_theorem(implies(implies(strict_implies(necessarily(X), possibly(necessarily(X))), equiv(possibly(necessarily(necessarily(X))), necessarily(X))), equiv(possibly(necessarily(necessarily(X))), necessarily(X)))), true, equiv(equiv(possibly(necessarily(necessarily(X))), necessarily(X)), implies(strict_implies(necessarily(X), possibly(necessarily(X))), equiv(possibly(necessarily(necessarily(X))), necessarily(X))))), true, equiv(possibly(necessarily(necessarily(X))), necessarily(X)), implies(strict_implies(necessarily(X), possibly(necessarily(X))), equiv(possibly(necessarily(necessarily(X))), necessarily(X))))), true, possibly(necessarily(X)), necessarily(X))
% 191.76/24.58 = { by lemma 95 }
% 191.76/24.58 fresh3(is_a_theorem(fresh3(true, true, equiv(possibly(necessarily(necessarily(X))), necessarily(X)), implies(strict_implies(necessarily(X), possibly(necessarily(X))), equiv(possibly(necessarily(necessarily(X))), necessarily(X))))), true, possibly(necessarily(X)), necessarily(X))
% 191.76/24.58 = { by axiom 52 (substitution_of_equivalents_2) }
% 191.76/24.58 fresh3(is_a_theorem(implies(strict_implies(necessarily(X), possibly(necessarily(X))), equiv(possibly(necessarily(necessarily(X))), necessarily(X)))), true, possibly(necessarily(X)), necessarily(X))
% 191.76/24.58 = { by lemma 81 R->L }
% 191.76/24.58 fresh3(is_a_theorem(implies(strict_implies(necessarily(X), possibly(necessarily(necessarily(X)))), equiv(possibly(necessarily(necessarily(X))), necessarily(X)))), true, possibly(necessarily(X)), necessarily(X))
% 191.76/24.58 = { by lemma 91 R->L }
% 191.76/24.58 fresh3(is_a_theorem(implies(necessarily(implies(necessarily(X), possibly(necessarily(necessarily(X))))), equiv(possibly(necessarily(necessarily(X))), necessarily(X)))), true, possibly(necessarily(X)), necessarily(X))
% 191.76/24.58 = { by axiom 28 (modus_ponens_2) R->L }
% 191.76/24.58 fresh3(fresh40(true, true, implies(necessarily(implies(necessarily(X), possibly(necessarily(necessarily(X))))), equiv(possibly(necessarily(necessarily(X))), necessarily(X)))), true, possibly(necessarily(X)), necessarily(X))
% 191.76/24.58 = { by lemma 96 R->L }
% 191.76/24.58 fresh3(fresh40(fresh40(is_a_theorem(or(necessarily(possibly(not(necessarily(X)))), necessarily(X))), true, implies(implies(necessarily(X), not(necessarily(possibly(not(necessarily(X)))))), and(or(necessarily(possibly(not(necessarily(X)))), necessarily(X)), implies(necessarily(X), not(necessarily(possibly(not(necessarily(X))))))))), true, implies(necessarily(implies(necessarily(X), possibly(necessarily(necessarily(X))))), equiv(possibly(necessarily(necessarily(X))), necessarily(X)))), true, possibly(necessarily(X)), necessarily(X))
% 191.76/24.58 = { by lemma 85 R->L }
% 191.76/24.58 fresh3(fresh40(fresh40(is_a_theorem(or(necessarily(possibly(not(necessarily(X)))), necessarily(X))), true, implies(implies(necessarily(X), not(necessarily(possibly(not(necessarily(X)))))), and(implies(not(necessarily(possibly(not(necessarily(X))))), necessarily(X)), implies(necessarily(X), not(necessarily(possibly(not(necessarily(X))))))))), true, implies(necessarily(implies(necessarily(X), possibly(necessarily(necessarily(X))))), equiv(possibly(necessarily(necessarily(X))), necessarily(X)))), true, possibly(necessarily(X)), necessarily(X))
% 191.76/24.58 = { by axiom 62 (op_equiv) R->L }
% 191.76/24.58 fresh3(fresh40(fresh40(is_a_theorem(or(necessarily(possibly(not(necessarily(X)))), necessarily(X))), true, implies(implies(necessarily(X), not(necessarily(possibly(not(necessarily(X)))))), fresh30(op_equiv, true, not(necessarily(possibly(not(necessarily(X))))), necessarily(X)))), true, implies(necessarily(implies(necessarily(X), possibly(necessarily(necessarily(X))))), equiv(possibly(necessarily(necessarily(X))), necessarily(X)))), true, possibly(necessarily(X)), necessarily(X))
% 191.76/24.58 = { by axiom 14 (hilbert_op_equiv) }
% 191.76/24.58 fresh3(fresh40(fresh40(is_a_theorem(or(necessarily(possibly(not(necessarily(X)))), necessarily(X))), true, implies(implies(necessarily(X), not(necessarily(possibly(not(necessarily(X)))))), fresh30(true, true, not(necessarily(possibly(not(necessarily(X))))), necessarily(X)))), true, implies(necessarily(implies(necessarily(X), possibly(necessarily(necessarily(X))))), equiv(possibly(necessarily(necessarily(X))), necessarily(X)))), true, possibly(necessarily(X)), necessarily(X))
% 191.76/24.58 = { by axiom 44 (op_equiv) }
% 191.76/24.58 fresh3(fresh40(fresh40(is_a_theorem(or(necessarily(possibly(not(necessarily(X)))), necessarily(X))), true, implies(implies(necessarily(X), not(necessarily(possibly(not(necessarily(X)))))), equiv(not(necessarily(possibly(not(necessarily(X))))), necessarily(X)))), true, implies(necessarily(implies(necessarily(X), possibly(necessarily(necessarily(X))))), equiv(possibly(necessarily(necessarily(X))), necessarily(X)))), true, possibly(necessarily(X)), necessarily(X))
% 191.76/24.58 = { by lemma 97 }
% 191.76/24.59 fresh3(fresh40(fresh40(true, true, implies(implies(necessarily(X), not(necessarily(possibly(not(necessarily(X)))))), equiv(not(necessarily(possibly(not(necessarily(X))))), necessarily(X)))), true, implies(necessarily(implies(necessarily(X), possibly(necessarily(necessarily(X))))), equiv(possibly(necessarily(necessarily(X))), necessarily(X)))), true, possibly(necessarily(X)), necessarily(X))
% 191.76/24.59 = { by axiom 28 (modus_ponens_2) }
% 191.76/24.59 fresh3(fresh40(is_a_theorem(implies(implies(necessarily(X), not(necessarily(possibly(not(necessarily(X)))))), equiv(not(necessarily(possibly(not(necessarily(X))))), necessarily(X)))), true, implies(necessarily(implies(necessarily(X), possibly(necessarily(necessarily(X))))), equiv(possibly(necessarily(necessarily(X))), necessarily(X)))), true, possibly(necessarily(X)), necessarily(X))
% 191.76/24.59 = { by lemma 99 }
% 191.76/24.59 fresh3(fresh40(is_a_theorem(implies(implies(necessarily(X), not(necessarily(possibly(not(necessarily(X)))))), equiv(possibly(necessarily(not(not(necessarily(X))))), necessarily(X)))), true, implies(necessarily(implies(necessarily(X), possibly(necessarily(necessarily(X))))), equiv(possibly(necessarily(necessarily(X))), necessarily(X)))), true, possibly(necessarily(X)), necessarily(X))
% 191.76/24.59 = { by lemma 99 }
% 191.76/24.59 fresh3(fresh40(is_a_theorem(implies(implies(necessarily(X), possibly(necessarily(not(not(necessarily(X)))))), equiv(possibly(necessarily(not(not(necessarily(X))))), necessarily(X)))), true, implies(necessarily(implies(necessarily(X), possibly(necessarily(necessarily(X))))), equiv(possibly(necessarily(necessarily(X))), necessarily(X)))), true, possibly(necessarily(X)), necessarily(X))
% 191.76/24.59 = { by lemma 100 }
% 191.76/24.59 fresh3(fresh40(is_a_theorem(implies(implies(necessarily(X), possibly(necessarily(not(not(necessarily(X)))))), equiv(possibly(necessarily(necessarily(X))), necessarily(X)))), true, implies(necessarily(implies(necessarily(X), possibly(necessarily(necessarily(X))))), equiv(possibly(necessarily(necessarily(X))), necessarily(X)))), true, possibly(necessarily(X)), necessarily(X))
% 191.76/24.59 = { by lemma 100 }
% 191.76/24.59 fresh3(fresh40(is_a_theorem(implies(implies(necessarily(X), possibly(necessarily(necessarily(X)))), equiv(possibly(necessarily(necessarily(X))), necessarily(X)))), true, implies(necessarily(implies(necessarily(X), possibly(necessarily(necessarily(X))))), equiv(possibly(necessarily(necessarily(X))), necessarily(X)))), true, possibly(necessarily(X)), necessarily(X))
% 191.76/24.59 = { by axiom 64 (modus_ponens_2) R->L }
% 191.76/24.59 fresh3(fresh115(is_a_theorem(implies(implies(implies(necessarily(X), possibly(necessarily(necessarily(X)))), equiv(possibly(necessarily(necessarily(X))), necessarily(X))), implies(necessarily(implies(necessarily(X), possibly(necessarily(necessarily(X))))), equiv(possibly(necessarily(necessarily(X))), necessarily(X))))), true, implies(implies(necessarily(X), possibly(necessarily(necessarily(X)))), equiv(possibly(necessarily(necessarily(X))), necessarily(X))), implies(necessarily(implies(necessarily(X), possibly(necessarily(necessarily(X))))), equiv(possibly(necessarily(necessarily(X))), necessarily(X)))), true, possibly(necessarily(X)), necessarily(X))
% 191.76/24.59 = { by axiom 28 (modus_ponens_2) R->L }
% 191.76/24.59 fresh3(fresh115(fresh40(true, true, implies(implies(implies(necessarily(X), possibly(necessarily(necessarily(X)))), equiv(possibly(necessarily(necessarily(X))), necessarily(X))), implies(necessarily(implies(necessarily(X), possibly(necessarily(necessarily(X))))), equiv(possibly(necessarily(necessarily(X))), necessarily(X))))), true, implies(implies(necessarily(X), possibly(necessarily(necessarily(X)))), equiv(possibly(necessarily(necessarily(X))), necessarily(X))), implies(necessarily(implies(necessarily(X), possibly(necessarily(necessarily(X))))), equiv(possibly(necessarily(necessarily(X))), necessarily(X)))), true, possibly(necessarily(X)), necessarily(X))
% 191.76/24.59 = { by lemma 80 R->L }
% 192.59/24.59 fresh3(fresh115(fresh40(is_a_theorem(implies(necessarily(implies(necessarily(X), possibly(necessarily(necessarily(X))))), implies(necessarily(X), possibly(necessarily(necessarily(X)))))), true, implies(implies(implies(necessarily(X), possibly(necessarily(necessarily(X)))), equiv(possibly(necessarily(necessarily(X))), necessarily(X))), implies(necessarily(implies(necessarily(X), possibly(necessarily(necessarily(X))))), equiv(possibly(necessarily(necessarily(X))), necessarily(X))))), true, implies(implies(necessarily(X), possibly(necessarily(necessarily(X)))), equiv(possibly(necessarily(necessarily(X))), necessarily(X))), implies(necessarily(implies(necessarily(X), possibly(necessarily(necessarily(X))))), equiv(possibly(necessarily(necessarily(X))), necessarily(X)))), true, possibly(necessarily(X)), necessarily(X))
% 192.59/24.59 = { by lemma 94 }
% 192.59/24.59 fresh3(fresh115(true, true, implies(implies(necessarily(X), possibly(necessarily(necessarily(X)))), equiv(possibly(necessarily(necessarily(X))), necessarily(X))), implies(necessarily(implies(necessarily(X), possibly(necessarily(necessarily(X))))), equiv(possibly(necessarily(necessarily(X))), necessarily(X)))), true, possibly(necessarily(X)), necessarily(X))
% 192.59/24.59 = { by lemma 73 }
% 192.59/24.59 fresh3(true, true, possibly(necessarily(X)), necessarily(X))
% 192.59/24.59 = { by axiom 52 (substitution_of_equivalents_2) }
% 192.59/24.59 necessarily(X)
% 192.59/24.59
% 192.59/24.59 Lemma 102: necessarily(or(X, Y)) = strict_implies(not(X), Y).
% 192.59/24.59 Proof:
% 192.59/24.59 necessarily(or(X, Y))
% 192.59/24.59 = { by lemma 85 R->L }
% 192.59/24.59 necessarily(implies(not(X), Y))
% 192.59/24.59 = { by lemma 91 }
% 192.59/24.59 strict_implies(not(X), Y)
% 192.59/24.59
% 192.59/24.59 Lemma 103: or(X, necessarily(not(Y))) = implies(possibly(Y), X).
% 192.59/24.59 Proof:
% 192.59/24.59 or(X, necessarily(not(Y)))
% 192.59/24.59 = { by lemma 88 }
% 192.59/24.59 or(necessarily(not(Y)), X)
% 192.59/24.59 = { by lemma 85 R->L }
% 192.59/24.59 implies(not(necessarily(not(Y))), X)
% 192.59/24.59 = { by lemma 98 }
% 192.59/24.59 implies(possibly(Y), X)
% 192.59/24.59
% 192.59/24.59 Lemma 104: strict_implies(not(X), necessarily(not(Y))) = strict_implies(possibly(Y), X).
% 192.59/24.59 Proof:
% 192.59/24.59 strict_implies(not(X), necessarily(not(Y)))
% 192.59/24.59 = { by lemma 102 R->L }
% 192.59/24.59 necessarily(or(X, necessarily(not(Y))))
% 192.59/24.59 = { by lemma 103 }
% 192.59/24.59 necessarily(implies(possibly(Y), X))
% 192.59/24.59 = { by lemma 91 }
% 192.59/24.59 strict_implies(possibly(Y), X)
% 192.59/24.59
% 192.59/24.59 Lemma 105: is_a_theorem(implies(strict_implies(X, Y), implies(necessarily(X), necessarily(Y)))) = true.
% 192.59/24.59 Proof:
% 192.59/24.59 is_a_theorem(implies(strict_implies(X, Y), implies(necessarily(X), necessarily(Y))))
% 192.59/24.59 = { by lemma 91 R->L }
% 192.59/24.59 is_a_theorem(implies(necessarily(implies(X, Y)), implies(necessarily(X), necessarily(Y))))
% 192.59/24.59 = { by axiom 68 (axiom_K_1) R->L }
% 192.59/24.59 fresh95(axiom_K, true, X, Y)
% 192.59/24.59 = { by axiom 16 (km4b_axiom_K) }
% 192.59/24.59 fresh95(true, true, X, Y)
% 192.59/24.59 = { by axiom 38 (axiom_K_1) }
% 192.59/24.59 true
% 192.59/24.59
% 192.59/24.59 Goal 1 (s1_0_m6s3m9b_axiom_s3): axiom_s3 = true.
% 192.59/24.59 Proof:
% 192.59/24.59 axiom_s3
% 192.59/24.59 = { by axiom 72 (axiom_s3) R->L }
% 192.59/24.59 fresh68(is_a_theorem(strict_implies(strict_implies(x10, y5), strict_implies(not(possibly(y5)), not(possibly(x10))))), true)
% 192.59/24.60 = { by lemma 91 R->L }
% 192.59/24.60 fresh68(is_a_theorem(strict_implies(strict_implies(x10, y5), necessarily(implies(not(possibly(y5)), not(possibly(x10)))))), true)
% 192.59/24.60 = { by lemma 78 R->L }
% 192.59/24.60 fresh68(is_a_theorem(strict_implies(strict_implies(x10, y5), necessarily(implies(not(possibly(y5)), and(not(possibly(x10)), not(possibly(x10))))))), true)
% 192.59/24.60 = { by lemma 84 R->L }
% 192.59/24.60 fresh68(is_a_theorem(strict_implies(strict_implies(x10, y5), necessarily(not(and(not(possibly(y5)), not(and(not(possibly(x10)), not(possibly(x10))))))))), true)
% 192.59/24.60 = { by lemma 84 }
% 192.59/24.60 fresh68(is_a_theorem(strict_implies(strict_implies(x10, y5), necessarily(not(and(not(possibly(y5)), implies(not(possibly(x10)), possibly(x10))))))), true)
% 192.59/24.60 = { by lemma 85 }
% 192.59/24.60 fresh68(is_a_theorem(strict_implies(strict_implies(x10, y5), necessarily(not(and(not(possibly(y5)), or(possibly(x10), possibly(x10))))))), true)
% 192.59/24.60 = { by lemma 83 }
% 192.59/24.60 fresh68(is_a_theorem(strict_implies(strict_implies(x10, y5), necessarily(not(and(not(possibly(y5)), possibly(x10)))))), true)
% 192.59/24.60 = { by lemma 98 R->L }
% 192.59/24.60 fresh68(is_a_theorem(strict_implies(strict_implies(x10, y5), necessarily(not(and(not(possibly(y5)), not(necessarily(not(x10)))))))), true)
% 192.59/24.60 = { by lemma 84 }
% 192.59/24.60 fresh68(is_a_theorem(strict_implies(strict_implies(x10, y5), necessarily(implies(not(possibly(y5)), necessarily(not(x10)))))), true)
% 192.59/24.60 = { by lemma 91 }
% 192.59/24.60 fresh68(is_a_theorem(strict_implies(strict_implies(x10, y5), strict_implies(not(possibly(y5)), necessarily(not(x10))))), true)
% 192.59/24.60 = { by lemma 104 }
% 192.59/24.60 fresh68(is_a_theorem(strict_implies(strict_implies(x10, y5), strict_implies(possibly(x10), possibly(y5)))), true)
% 192.59/24.60 = { by lemma 100 R->L }
% 192.59/24.60 fresh68(is_a_theorem(strict_implies(strict_implies(not(not(x10)), y5), strict_implies(possibly(x10), possibly(y5)))), true)
% 192.59/24.60 = { by lemma 78 R->L }
% 192.59/24.60 fresh68(is_a_theorem(strict_implies(strict_implies(not(and(not(x10), not(x10))), y5), strict_implies(possibly(x10), possibly(y5)))), true)
% 192.59/24.60 = { by lemma 84 }
% 192.59/24.60 fresh68(is_a_theorem(strict_implies(strict_implies(implies(not(x10), x10), y5), strict_implies(possibly(x10), possibly(y5)))), true)
% 192.59/24.60 = { by lemma 91 R->L }
% 192.59/24.60 fresh68(is_a_theorem(strict_implies(necessarily(implies(implies(not(x10), x10), y5)), strict_implies(possibly(x10), possibly(y5)))), true)
% 192.59/24.60 = { by lemma 89 R->L }
% 192.59/24.60 fresh68(is_a_theorem(strict_implies(necessarily(or(y5, and(not(x10), not(x10)))), strict_implies(possibly(x10), possibly(y5)))), true)
% 192.59/24.60 = { by lemma 102 }
% 192.59/24.60 fresh68(is_a_theorem(strict_implies(strict_implies(not(y5), and(not(x10), not(x10))), strict_implies(possibly(x10), possibly(y5)))), true)
% 192.59/24.60 = { by lemma 78 }
% 192.59/24.60 fresh68(is_a_theorem(strict_implies(strict_implies(not(y5), not(x10)), strict_implies(possibly(x10), possibly(y5)))), true)
% 192.59/24.60 = { by lemma 77 R->L }
% 192.59/24.60 fresh68(is_a_theorem(strict_implies(strict_implies(not(y5), not(x10)), strict_implies(possibly(x10), fresh3(is_a_theorem(equiv(possibly(y5), necessarily(possibly(y5)))), true, possibly(y5), necessarily(possibly(y5)))))), true)
% 192.59/24.60 = { by lemma 77 R->L }
% 192.59/24.60 fresh68(is_a_theorem(strict_implies(strict_implies(not(y5), not(x10)), strict_implies(possibly(x10), fresh3(is_a_theorem(fresh3(is_a_theorem(equiv(equiv(possibly(y5), necessarily(possibly(y5))), implies(implies(necessarily(possibly(y5)), possibly(y5)), equiv(possibly(y5), necessarily(possibly(y5)))))), true, equiv(possibly(y5), necessarily(possibly(y5))), implies(implies(necessarily(possibly(y5)), possibly(y5)), equiv(possibly(y5), necessarily(possibly(y5)))))), true, possibly(y5), necessarily(possibly(y5)))))), true)
% 192.59/24.60 = { by axiom 28 (modus_ponens_2) R->L }
% 192.59/24.60 fresh68(is_a_theorem(strict_implies(strict_implies(not(y5), not(x10)), strict_implies(possibly(x10), fresh3(is_a_theorem(fresh3(fresh40(true, true, equiv(equiv(possibly(y5), necessarily(possibly(y5))), implies(implies(necessarily(possibly(y5)), possibly(y5)), equiv(possibly(y5), necessarily(possibly(y5)))))), true, equiv(possibly(y5), necessarily(possibly(y5))), implies(implies(necessarily(possibly(y5)), possibly(y5)), equiv(possibly(y5), necessarily(possibly(y5)))))), true, possibly(y5), necessarily(possibly(y5)))))), true)
% 192.59/24.60 = { by lemma 90 R->L }
% 192.59/24.60 fresh68(is_a_theorem(strict_implies(strict_implies(not(y5), not(x10)), strict_implies(possibly(x10), fresh3(is_a_theorem(fresh3(fresh40(fresh40(is_a_theorem(implies(necessarily(possibly(y5)), possibly(y5))), true, implies(implies(implies(necessarily(possibly(y5)), possibly(y5)), equiv(possibly(y5), necessarily(possibly(y5)))), equiv(possibly(y5), necessarily(possibly(y5))))), true, equiv(equiv(possibly(y5), necessarily(possibly(y5))), implies(implies(necessarily(possibly(y5)), possibly(y5)), equiv(possibly(y5), necessarily(possibly(y5)))))), true, equiv(possibly(y5), necessarily(possibly(y5))), implies(implies(necessarily(possibly(y5)), possibly(y5)), equiv(possibly(y5), necessarily(possibly(y5)))))), true, possibly(y5), necessarily(possibly(y5)))))), true)
% 192.59/24.60 = { by lemma 80 }
% 192.59/24.60 fresh68(is_a_theorem(strict_implies(strict_implies(not(y5), not(x10)), strict_implies(possibly(x10), fresh3(is_a_theorem(fresh3(fresh40(fresh40(true, true, implies(implies(implies(necessarily(possibly(y5)), possibly(y5)), equiv(possibly(y5), necessarily(possibly(y5)))), equiv(possibly(y5), necessarily(possibly(y5))))), true, equiv(equiv(possibly(y5), necessarily(possibly(y5))), implies(implies(necessarily(possibly(y5)), possibly(y5)), equiv(possibly(y5), necessarily(possibly(y5)))))), true, equiv(possibly(y5), necessarily(possibly(y5))), implies(implies(necessarily(possibly(y5)), possibly(y5)), equiv(possibly(y5), necessarily(possibly(y5)))))), true, possibly(y5), necessarily(possibly(y5)))))), true)
% 192.59/24.60 = { by axiom 28 (modus_ponens_2) }
% 192.59/24.60 fresh68(is_a_theorem(strict_implies(strict_implies(not(y5), not(x10)), strict_implies(possibly(x10), fresh3(is_a_theorem(fresh3(fresh40(is_a_theorem(implies(implies(implies(necessarily(possibly(y5)), possibly(y5)), equiv(possibly(y5), necessarily(possibly(y5)))), equiv(possibly(y5), necessarily(possibly(y5))))), true, equiv(equiv(possibly(y5), necessarily(possibly(y5))), implies(implies(necessarily(possibly(y5)), possibly(y5)), equiv(possibly(y5), necessarily(possibly(y5)))))), true, equiv(possibly(y5), necessarily(possibly(y5))), implies(implies(necessarily(possibly(y5)), possibly(y5)), equiv(possibly(y5), necessarily(possibly(y5)))))), true, possibly(y5), necessarily(possibly(y5)))))), true)
% 192.59/24.60 = { by lemma 95 }
% 192.59/24.60 fresh68(is_a_theorem(strict_implies(strict_implies(not(y5), not(x10)), strict_implies(possibly(x10), fresh3(is_a_theorem(fresh3(true, true, equiv(possibly(y5), necessarily(possibly(y5))), implies(implies(necessarily(possibly(y5)), possibly(y5)), equiv(possibly(y5), necessarily(possibly(y5)))))), true, possibly(y5), necessarily(possibly(y5)))))), true)
% 192.59/24.60 = { by axiom 52 (substitution_of_equivalents_2) }
% 192.59/24.60 fresh68(is_a_theorem(strict_implies(strict_implies(not(y5), not(x10)), strict_implies(possibly(x10), fresh3(is_a_theorem(implies(implies(necessarily(possibly(y5)), possibly(y5)), equiv(possibly(y5), necessarily(possibly(y5))))), true, possibly(y5), necessarily(possibly(y5)))))), true)
% 192.59/24.60 = { by axiom 28 (modus_ponens_2) R->L }
% 192.59/24.60 fresh68(is_a_theorem(strict_implies(strict_implies(not(y5), not(x10)), strict_implies(possibly(x10), fresh3(fresh40(true, true, implies(implies(necessarily(possibly(y5)), possibly(y5)), equiv(possibly(y5), necessarily(possibly(y5))))), true, possibly(y5), necessarily(possibly(y5)))))), true)
% 192.59/24.60 = { by lemma 73 R->L }
% 192.59/24.60 fresh68(is_a_theorem(strict_implies(strict_implies(not(y5), not(x10)), strict_implies(possibly(x10), fresh3(fresh40(fresh115(true, true, or(necessarily(possibly(y5)), not(and(strict_implies(possibly(X), necessarily(possibly(X))), not(possibly(necessarily(not(y5))))))), implies(and(strict_implies(possibly(X), necessarily(possibly(X))), not(possibly(necessarily(not(y5))))), necessarily(possibly(y5)))), true, implies(implies(necessarily(possibly(y5)), possibly(y5)), equiv(possibly(y5), necessarily(possibly(y5))))), true, possibly(y5), necessarily(possibly(y5)))))), true)
% 192.59/24.60 = { by lemma 86 R->L }
% 192.59/24.60 fresh68(is_a_theorem(strict_implies(strict_implies(not(y5), not(x10)), strict_implies(possibly(x10), fresh3(fresh40(fresh115(is_a_theorem(implies(or(necessarily(possibly(y5)), not(and(strict_implies(possibly(X), necessarily(possibly(X))), not(possibly(necessarily(not(y5))))))), implies(and(strict_implies(possibly(X), necessarily(possibly(X))), not(possibly(necessarily(not(y5))))), necessarily(possibly(y5))))), true, or(necessarily(possibly(y5)), not(and(strict_implies(possibly(X), necessarily(possibly(X))), not(possibly(necessarily(not(y5))))))), implies(and(strict_implies(possibly(X), necessarily(possibly(X))), not(possibly(necessarily(not(y5))))), necessarily(possibly(y5)))), true, implies(implies(necessarily(possibly(y5)), possibly(y5)), equiv(possibly(y5), necessarily(possibly(y5))))), true, possibly(y5), necessarily(possibly(y5)))))), true)
% 192.59/24.61 = { by axiom 64 (modus_ponens_2) }
% 192.59/24.61 fresh68(is_a_theorem(strict_implies(strict_implies(not(y5), not(x10)), strict_implies(possibly(x10), fresh3(fresh40(fresh40(is_a_theorem(or(necessarily(possibly(y5)), not(and(strict_implies(possibly(X), necessarily(possibly(X))), not(possibly(necessarily(not(y5)))))))), true, implies(and(strict_implies(possibly(X), necessarily(possibly(X))), not(possibly(necessarily(not(y5))))), necessarily(possibly(y5)))), true, implies(implies(necessarily(possibly(y5)), possibly(y5)), equiv(possibly(y5), necessarily(possibly(y5))))), true, possibly(y5), necessarily(possibly(y5)))))), true)
% 192.59/24.61 = { by lemma 84 }
% 192.59/24.61 fresh68(is_a_theorem(strict_implies(strict_implies(not(y5), not(x10)), strict_implies(possibly(x10), fresh3(fresh40(fresh40(is_a_theorem(or(necessarily(possibly(y5)), implies(strict_implies(possibly(X), necessarily(possibly(X))), possibly(necessarily(not(y5)))))), true, implies(and(strict_implies(possibly(X), necessarily(possibly(X))), not(possibly(necessarily(not(y5))))), necessarily(possibly(y5)))), true, implies(implies(necessarily(possibly(y5)), possibly(y5)), equiv(possibly(y5), necessarily(possibly(y5))))), true, possibly(y5), necessarily(possibly(y5)))))), true)
% 192.59/24.61 = { by lemma 99 R->L }
% 192.59/24.61 fresh68(is_a_theorem(strict_implies(strict_implies(not(y5), not(x10)), strict_implies(possibly(x10), fresh3(fresh40(fresh40(is_a_theorem(or(necessarily(possibly(y5)), implies(strict_implies(possibly(X), necessarily(possibly(X))), not(necessarily(possibly(y5)))))), true, implies(and(strict_implies(possibly(X), necessarily(possibly(X))), not(possibly(necessarily(not(y5))))), necessarily(possibly(y5)))), true, implies(implies(necessarily(possibly(y5)), possibly(y5)), equiv(possibly(y5), necessarily(possibly(y5))))), true, possibly(y5), necessarily(possibly(y5)))))), true)
% 192.59/24.61 = { by lemma 85 R->L }
% 192.59/24.61 fresh68(is_a_theorem(strict_implies(strict_implies(not(y5), not(x10)), strict_implies(possibly(x10), fresh3(fresh40(fresh40(is_a_theorem(implies(not(necessarily(possibly(y5))), implies(strict_implies(possibly(X), necessarily(possibly(X))), not(necessarily(possibly(y5)))))), true, implies(and(strict_implies(possibly(X), necessarily(possibly(X))), not(possibly(necessarily(not(y5))))), necessarily(possibly(y5)))), true, implies(implies(necessarily(possibly(y5)), possibly(y5)), equiv(possibly(y5), necessarily(possibly(y5))))), true, possibly(y5), necessarily(possibly(y5)))))), true)
% 192.59/24.61 = { by lemma 82 }
% 192.59/24.61 fresh68(is_a_theorem(strict_implies(strict_implies(not(y5), not(x10)), strict_implies(possibly(x10), fresh3(fresh40(fresh40(true, true, implies(and(strict_implies(possibly(X), necessarily(possibly(X))), not(possibly(necessarily(not(y5))))), necessarily(possibly(y5)))), true, implies(implies(necessarily(possibly(y5)), possibly(y5)), equiv(possibly(y5), necessarily(possibly(y5))))), true, possibly(y5), necessarily(possibly(y5)))))), true)
% 192.59/24.61 = { by axiom 28 (modus_ponens_2) }
% 192.59/24.61 fresh68(is_a_theorem(strict_implies(strict_implies(not(y5), not(x10)), strict_implies(possibly(x10), fresh3(fresh40(is_a_theorem(implies(and(strict_implies(possibly(X), necessarily(possibly(X))), not(possibly(necessarily(not(y5))))), necessarily(possibly(y5)))), true, implies(implies(necessarily(possibly(y5)), possibly(y5)), equiv(possibly(y5), necessarily(possibly(y5))))), true, possibly(y5), necessarily(possibly(y5)))))), true)
% 192.59/24.61 = { by lemma 77 R->L }
% 192.59/24.61 fresh68(is_a_theorem(strict_implies(strict_implies(not(y5), not(x10)), strict_implies(possibly(x10), fresh3(fresh40(is_a_theorem(implies(fresh3(is_a_theorem(equiv(and(strict_implies(possibly(X), necessarily(possibly(X))), not(possibly(necessarily(not(y5))))), not(possibly(necessarily(not(y5)))))), true, and(strict_implies(possibly(X), necessarily(possibly(X))), not(possibly(necessarily(not(y5))))), not(possibly(necessarily(not(y5))))), necessarily(possibly(y5)))), true, implies(implies(necessarily(possibly(y5)), possibly(y5)), equiv(possibly(y5), necessarily(possibly(y5))))), true, possibly(y5), necessarily(possibly(y5)))))), true)
% 192.59/24.61 = { by axiom 28 (modus_ponens_2) R->L }
% 192.59/24.61 fresh68(is_a_theorem(strict_implies(strict_implies(not(y5), not(x10)), strict_implies(possibly(x10), fresh3(fresh40(is_a_theorem(implies(fresh3(fresh40(true, true, equiv(and(strict_implies(possibly(X), necessarily(possibly(X))), not(possibly(necessarily(not(y5))))), not(possibly(necessarily(not(y5)))))), true, and(strict_implies(possibly(X), necessarily(possibly(X))), not(possibly(necessarily(not(y5))))), not(possibly(necessarily(not(y5))))), necessarily(possibly(y5)))), true, implies(implies(necessarily(possibly(y5)), possibly(y5)), equiv(possibly(y5), necessarily(possibly(y5))))), true, possibly(y5), necessarily(possibly(y5)))))), true)
% 192.82/24.61 = { by lemma 96 R->L }
% 192.82/24.61 fresh68(is_a_theorem(strict_implies(strict_implies(not(y5), not(x10)), strict_implies(possibly(x10), fresh3(fresh40(is_a_theorem(implies(fresh3(fresh40(fresh40(is_a_theorem(strict_implies(possibly(X), necessarily(possibly(X)))), true, implies(not(possibly(necessarily(not(y5)))), and(strict_implies(possibly(X), necessarily(possibly(X))), not(possibly(necessarily(not(y5))))))), true, equiv(and(strict_implies(possibly(X), necessarily(possibly(X))), not(possibly(necessarily(not(y5))))), not(possibly(necessarily(not(y5)))))), true, and(strict_implies(possibly(X), necessarily(possibly(X))), not(possibly(necessarily(not(y5))))), not(possibly(necessarily(not(y5))))), necessarily(possibly(y5)))), true, implies(implies(necessarily(possibly(y5)), possibly(y5)), equiv(possibly(y5), necessarily(possibly(y5))))), true, possibly(y5), necessarily(possibly(y5)))))), true)
% 192.82/24.61 = { by lemma 98 R->L }
% 192.82/24.61 fresh68(is_a_theorem(strict_implies(strict_implies(not(y5), not(x10)), strict_implies(possibly(x10), fresh3(fresh40(is_a_theorem(implies(fresh3(fresh40(fresh40(is_a_theorem(strict_implies(not(necessarily(not(X))), necessarily(possibly(X)))), true, implies(not(possibly(necessarily(not(y5)))), and(strict_implies(possibly(X), necessarily(possibly(X))), not(possibly(necessarily(not(y5))))))), true, equiv(and(strict_implies(possibly(X), necessarily(possibly(X))), not(possibly(necessarily(not(y5))))), not(possibly(necessarily(not(y5)))))), true, and(strict_implies(possibly(X), necessarily(possibly(X))), not(possibly(necessarily(not(y5))))), not(possibly(necessarily(not(y5))))), necessarily(possibly(y5)))), true, implies(implies(necessarily(possibly(y5)), possibly(y5)), equiv(possibly(y5), necessarily(possibly(y5))))), true, possibly(y5), necessarily(possibly(y5)))))), true)
% 192.82/24.61 = { by lemma 102 R->L }
% 192.82/24.61 fresh68(is_a_theorem(strict_implies(strict_implies(not(y5), not(x10)), strict_implies(possibly(x10), fresh3(fresh40(is_a_theorem(implies(fresh3(fresh40(fresh40(is_a_theorem(necessarily(or(necessarily(not(X)), necessarily(possibly(X))))), true, implies(not(possibly(necessarily(not(y5)))), and(strict_implies(possibly(X), necessarily(possibly(X))), not(possibly(necessarily(not(y5))))))), true, equiv(and(strict_implies(possibly(X), necessarily(possibly(X))), not(possibly(necessarily(not(y5))))), not(possibly(necessarily(not(y5)))))), true, and(strict_implies(possibly(X), necessarily(possibly(X))), not(possibly(necessarily(not(y5))))), not(possibly(necessarily(not(y5))))), necessarily(possibly(y5)))), true, implies(implies(necessarily(possibly(y5)), possibly(y5)), equiv(possibly(y5), necessarily(possibly(y5))))), true, possibly(y5), necessarily(possibly(y5)))))), true)
% 192.82/24.61 = { by lemma 88 }
% 192.82/24.62 fresh68(is_a_theorem(strict_implies(strict_implies(not(y5), not(x10)), strict_implies(possibly(x10), fresh3(fresh40(is_a_theorem(implies(fresh3(fresh40(fresh40(is_a_theorem(necessarily(or(necessarily(possibly(X)), necessarily(not(X))))), true, implies(not(possibly(necessarily(not(y5)))), and(strict_implies(possibly(X), necessarily(possibly(X))), not(possibly(necessarily(not(y5))))))), true, equiv(and(strict_implies(possibly(X), necessarily(possibly(X))), not(possibly(necessarily(not(y5))))), not(possibly(necessarily(not(y5)))))), true, and(strict_implies(possibly(X), necessarily(possibly(X))), not(possibly(necessarily(not(y5))))), not(possibly(necessarily(not(y5))))), necessarily(possibly(y5)))), true, implies(implies(necessarily(possibly(y5)), possibly(y5)), equiv(possibly(y5), necessarily(possibly(y5))))), true, possibly(y5), necessarily(possibly(y5)))))), true)
% 192.82/24.62 = { by lemma 85 R->L }
% 192.82/24.62 fresh68(is_a_theorem(strict_implies(strict_implies(not(y5), not(x10)), strict_implies(possibly(x10), fresh3(fresh40(is_a_theorem(implies(fresh3(fresh40(fresh40(is_a_theorem(necessarily(implies(not(necessarily(possibly(X))), necessarily(not(X))))), true, implies(not(possibly(necessarily(not(y5)))), and(strict_implies(possibly(X), necessarily(possibly(X))), not(possibly(necessarily(not(y5))))))), true, equiv(and(strict_implies(possibly(X), necessarily(possibly(X))), not(possibly(necessarily(not(y5))))), not(possibly(necessarily(not(y5)))))), true, and(strict_implies(possibly(X), necessarily(possibly(X))), not(possibly(necessarily(not(y5))))), not(possibly(necessarily(not(y5))))), necessarily(possibly(y5)))), true, implies(implies(necessarily(possibly(y5)), possibly(y5)), equiv(possibly(y5), necessarily(possibly(y5))))), true, possibly(y5), necessarily(possibly(y5)))))), true)
% 192.82/24.62 = { by lemma 99 }
% 192.82/24.62 fresh68(is_a_theorem(strict_implies(strict_implies(not(y5), not(x10)), strict_implies(possibly(x10), fresh3(fresh40(is_a_theorem(implies(fresh3(fresh40(fresh40(is_a_theorem(necessarily(implies(possibly(necessarily(not(X))), necessarily(not(X))))), true, implies(not(possibly(necessarily(not(y5)))), and(strict_implies(possibly(X), necessarily(possibly(X))), not(possibly(necessarily(not(y5))))))), true, equiv(and(strict_implies(possibly(X), necessarily(possibly(X))), not(possibly(necessarily(not(y5))))), not(possibly(necessarily(not(y5)))))), true, and(strict_implies(possibly(X), necessarily(possibly(X))), not(possibly(necessarily(not(y5))))), not(possibly(necessarily(not(y5))))), necessarily(possibly(y5)))), true, implies(implies(necessarily(possibly(y5)), possibly(y5)), equiv(possibly(y5), necessarily(possibly(y5))))), true, possibly(y5), necessarily(possibly(y5)))))), true)
% 192.82/24.62 = { by lemma 91 }
% 192.82/24.62 fresh68(is_a_theorem(strict_implies(strict_implies(not(y5), not(x10)), strict_implies(possibly(x10), fresh3(fresh40(is_a_theorem(implies(fresh3(fresh40(fresh40(is_a_theorem(strict_implies(possibly(necessarily(not(X))), necessarily(not(X)))), true, implies(not(possibly(necessarily(not(y5)))), and(strict_implies(possibly(X), necessarily(possibly(X))), not(possibly(necessarily(not(y5))))))), true, equiv(and(strict_implies(possibly(X), necessarily(possibly(X))), not(possibly(necessarily(not(y5))))), not(possibly(necessarily(not(y5)))))), true, and(strict_implies(possibly(X), necessarily(possibly(X))), not(possibly(necessarily(not(y5))))), not(possibly(necessarily(not(y5))))), necessarily(possibly(y5)))), true, implies(implies(necessarily(possibly(y5)), possibly(y5)), equiv(possibly(y5), necessarily(possibly(y5))))), true, possibly(y5), necessarily(possibly(y5)))))), true)
% 192.82/24.62 = { by lemma 81 R->L }
% 192.82/24.62 fresh68(is_a_theorem(strict_implies(strict_implies(not(y5), not(x10)), strict_implies(possibly(x10), fresh3(fresh40(is_a_theorem(implies(fresh3(fresh40(fresh40(is_a_theorem(strict_implies(possibly(necessarily(necessarily(not(X)))), necessarily(not(X)))), true, implies(not(possibly(necessarily(not(y5)))), and(strict_implies(possibly(X), necessarily(possibly(X))), not(possibly(necessarily(not(y5))))))), true, equiv(and(strict_implies(possibly(X), necessarily(possibly(X))), not(possibly(necessarily(not(y5))))), not(possibly(necessarily(not(y5)))))), true, and(strict_implies(possibly(X), necessarily(possibly(X))), not(possibly(necessarily(not(y5))))), not(possibly(necessarily(not(y5))))), necessarily(possibly(y5)))), true, implies(implies(necessarily(possibly(y5)), possibly(y5)), equiv(possibly(y5), necessarily(possibly(y5))))), true, possibly(y5), necessarily(possibly(y5)))))), true)
% 192.82/24.62 = { by lemma 100 R->L }
% 192.82/24.62 fresh68(is_a_theorem(strict_implies(strict_implies(not(y5), not(x10)), strict_implies(possibly(x10), fresh3(fresh40(is_a_theorem(implies(fresh3(fresh40(fresh40(is_a_theorem(strict_implies(possibly(necessarily(not(not(necessarily(not(X)))))), necessarily(not(X)))), true, implies(not(possibly(necessarily(not(y5)))), and(strict_implies(possibly(X), necessarily(possibly(X))), not(possibly(necessarily(not(y5))))))), true, equiv(and(strict_implies(possibly(X), necessarily(possibly(X))), not(possibly(necessarily(not(y5))))), not(possibly(necessarily(not(y5)))))), true, and(strict_implies(possibly(X), necessarily(possibly(X))), not(possibly(necessarily(not(y5))))), not(possibly(necessarily(not(y5))))), necessarily(possibly(y5)))), true, implies(implies(necessarily(possibly(y5)), possibly(y5)), equiv(possibly(y5), necessarily(possibly(y5))))), true, possibly(y5), necessarily(possibly(y5)))))), true)
% 192.82/24.62 = { by lemma 99 R->L }
% 192.82/24.62 fresh68(is_a_theorem(strict_implies(strict_implies(not(y5), not(x10)), strict_implies(possibly(x10), fresh3(fresh40(is_a_theorem(implies(fresh3(fresh40(fresh40(is_a_theorem(strict_implies(not(necessarily(possibly(not(necessarily(not(X)))))), necessarily(not(X)))), true, implies(not(possibly(necessarily(not(y5)))), and(strict_implies(possibly(X), necessarily(possibly(X))), not(possibly(necessarily(not(y5))))))), true, equiv(and(strict_implies(possibly(X), necessarily(possibly(X))), not(possibly(necessarily(not(y5))))), not(possibly(necessarily(not(y5)))))), true, and(strict_implies(possibly(X), necessarily(possibly(X))), not(possibly(necessarily(not(y5))))), not(possibly(necessarily(not(y5))))), necessarily(possibly(y5)))), true, implies(implies(necessarily(possibly(y5)), possibly(y5)), equiv(possibly(y5), necessarily(possibly(y5))))), true, possibly(y5), necessarily(possibly(y5)))))), true)
% 192.82/24.62 = { by lemma 102 R->L }
% 192.82/24.62 fresh68(is_a_theorem(strict_implies(strict_implies(not(y5), not(x10)), strict_implies(possibly(x10), fresh3(fresh40(is_a_theorem(implies(fresh3(fresh40(fresh40(is_a_theorem(necessarily(or(necessarily(possibly(not(necessarily(not(X))))), necessarily(not(X))))), true, implies(not(possibly(necessarily(not(y5)))), and(strict_implies(possibly(X), necessarily(possibly(X))), not(possibly(necessarily(not(y5))))))), true, equiv(and(strict_implies(possibly(X), necessarily(possibly(X))), not(possibly(necessarily(not(y5))))), not(possibly(necessarily(not(y5)))))), true, and(strict_implies(possibly(X), necessarily(possibly(X))), not(possibly(necessarily(not(y5))))), not(possibly(necessarily(not(y5))))), necessarily(possibly(y5)))), true, implies(implies(necessarily(possibly(y5)), possibly(y5)), equiv(possibly(y5), necessarily(possibly(y5))))), true, possibly(y5), necessarily(possibly(y5)))))), true)
% 192.82/24.62 = { by lemma 92 R->L }
% 192.82/24.62 fresh68(is_a_theorem(strict_implies(strict_implies(not(y5), not(x10)), strict_implies(possibly(x10), fresh3(fresh40(is_a_theorem(implies(fresh3(fresh40(fresh40(fresh33(is_a_theorem(or(necessarily(possibly(not(necessarily(not(X))))), necessarily(not(X)))), true, or(necessarily(possibly(not(necessarily(not(X))))), necessarily(not(X)))), true, implies(not(possibly(necessarily(not(y5)))), and(strict_implies(possibly(X), necessarily(possibly(X))), not(possibly(necessarily(not(y5))))))), true, equiv(and(strict_implies(possibly(X), necessarily(possibly(X))), not(possibly(necessarily(not(y5))))), not(possibly(necessarily(not(y5)))))), true, and(strict_implies(possibly(X), necessarily(possibly(X))), not(possibly(necessarily(not(y5))))), not(possibly(necessarily(not(y5))))), necessarily(possibly(y5)))), true, implies(implies(necessarily(possibly(y5)), possibly(y5)), equiv(possibly(y5), necessarily(possibly(y5))))), true, possibly(y5), necessarily(possibly(y5)))))), true)
% 192.82/24.62 = { by lemma 97 }
% 192.82/24.62 fresh68(is_a_theorem(strict_implies(strict_implies(not(y5), not(x10)), strict_implies(possibly(x10), fresh3(fresh40(is_a_theorem(implies(fresh3(fresh40(fresh40(fresh33(true, true, or(necessarily(possibly(not(necessarily(not(X))))), necessarily(not(X)))), true, implies(not(possibly(necessarily(not(y5)))), and(strict_implies(possibly(X), necessarily(possibly(X))), not(possibly(necessarily(not(y5))))))), true, equiv(and(strict_implies(possibly(X), necessarily(possibly(X))), not(possibly(necessarily(not(y5))))), not(possibly(necessarily(not(y5)))))), true, and(strict_implies(possibly(X), necessarily(possibly(X))), not(possibly(necessarily(not(y5))))), not(possibly(necessarily(not(y5))))), necessarily(possibly(y5)))), true, implies(implies(necessarily(possibly(y5)), possibly(y5)), equiv(possibly(y5), necessarily(possibly(y5))))), true, possibly(y5), necessarily(possibly(y5)))))), true)
% 192.82/24.63 = { by axiom 30 (necessitation_1) }
% 192.82/24.63 fresh68(is_a_theorem(strict_implies(strict_implies(not(y5), not(x10)), strict_implies(possibly(x10), fresh3(fresh40(is_a_theorem(implies(fresh3(fresh40(fresh40(true, true, implies(not(possibly(necessarily(not(y5)))), and(strict_implies(possibly(X), necessarily(possibly(X))), not(possibly(necessarily(not(y5))))))), true, equiv(and(strict_implies(possibly(X), necessarily(possibly(X))), not(possibly(necessarily(not(y5))))), not(possibly(necessarily(not(y5)))))), true, and(strict_implies(possibly(X), necessarily(possibly(X))), not(possibly(necessarily(not(y5))))), not(possibly(necessarily(not(y5))))), necessarily(possibly(y5)))), true, implies(implies(necessarily(possibly(y5)), possibly(y5)), equiv(possibly(y5), necessarily(possibly(y5))))), true, possibly(y5), necessarily(possibly(y5)))))), true)
% 192.82/24.63 = { by axiom 28 (modus_ponens_2) }
% 192.82/24.63 fresh68(is_a_theorem(strict_implies(strict_implies(not(y5), not(x10)), strict_implies(possibly(x10), fresh3(fresh40(is_a_theorem(implies(fresh3(fresh40(is_a_theorem(implies(not(possibly(necessarily(not(y5)))), and(strict_implies(possibly(X), necessarily(possibly(X))), not(possibly(necessarily(not(y5))))))), true, equiv(and(strict_implies(possibly(X), necessarily(possibly(X))), not(possibly(necessarily(not(y5))))), not(possibly(necessarily(not(y5)))))), true, and(strict_implies(possibly(X), necessarily(possibly(X))), not(possibly(necessarily(not(y5))))), not(possibly(necessarily(not(y5))))), necessarily(possibly(y5)))), true, implies(implies(necessarily(possibly(y5)), possibly(y5)), equiv(possibly(y5), necessarily(possibly(y5))))), true, possibly(y5), necessarily(possibly(y5)))))), true)
% 192.82/24.63 = { by axiom 64 (modus_ponens_2) R->L }
% 192.82/24.63 fresh68(is_a_theorem(strict_implies(strict_implies(not(y5), not(x10)), strict_implies(possibly(x10), fresh3(fresh40(is_a_theorem(implies(fresh3(fresh115(is_a_theorem(implies(implies(not(possibly(necessarily(not(y5)))), and(strict_implies(possibly(X), necessarily(possibly(X))), not(possibly(necessarily(not(y5)))))), equiv(and(strict_implies(possibly(X), necessarily(possibly(X))), not(possibly(necessarily(not(y5))))), not(possibly(necessarily(not(y5))))))), true, implies(not(possibly(necessarily(not(y5)))), and(strict_implies(possibly(X), necessarily(possibly(X))), not(possibly(necessarily(not(y5)))))), equiv(and(strict_implies(possibly(X), necessarily(possibly(X))), not(possibly(necessarily(not(y5))))), not(possibly(necessarily(not(y5)))))), true, and(strict_implies(possibly(X), necessarily(possibly(X))), not(possibly(necessarily(not(y5))))), not(possibly(necessarily(not(y5))))), necessarily(possibly(y5)))), true, implies(implies(necessarily(possibly(y5)), possibly(y5)), equiv(possibly(y5), necessarily(possibly(y5))))), true, possibly(y5), necessarily(possibly(y5)))))), true)
% 192.82/24.63 = { by axiom 28 (modus_ponens_2) R->L }
% 192.82/24.63 fresh68(is_a_theorem(strict_implies(strict_implies(not(y5), not(x10)), strict_implies(possibly(x10), fresh3(fresh40(is_a_theorem(implies(fresh3(fresh115(fresh40(true, true, implies(implies(not(possibly(necessarily(not(y5)))), and(strict_implies(possibly(X), necessarily(possibly(X))), not(possibly(necessarily(not(y5)))))), equiv(and(strict_implies(possibly(X), necessarily(possibly(X))), not(possibly(necessarily(not(y5))))), not(possibly(necessarily(not(y5))))))), true, implies(not(possibly(necessarily(not(y5)))), and(strict_implies(possibly(X), necessarily(possibly(X))), not(possibly(necessarily(not(y5)))))), equiv(and(strict_implies(possibly(X), necessarily(possibly(X))), not(possibly(necessarily(not(y5))))), not(possibly(necessarily(not(y5)))))), true, and(strict_implies(possibly(X), necessarily(possibly(X))), not(possibly(necessarily(not(y5))))), not(possibly(necessarily(not(y5))))), necessarily(possibly(y5)))), true, implies(implies(necessarily(possibly(y5)), possibly(y5)), equiv(possibly(y5), necessarily(possibly(y5))))), true, possibly(y5), necessarily(possibly(y5)))))), true)
% 192.82/24.63 = { by axiom 36 (and_2_1) R->L }
% 192.82/24.63 fresh68(is_a_theorem(strict_implies(strict_implies(not(y5), not(x10)), strict_implies(possibly(x10), fresh3(fresh40(is_a_theorem(implies(fresh3(fresh115(fresh40(fresh105(true, true, strict_implies(possibly(X), necessarily(possibly(X))), not(possibly(necessarily(not(y5))))), true, implies(implies(not(possibly(necessarily(not(y5)))), and(strict_implies(possibly(X), necessarily(possibly(X))), not(possibly(necessarily(not(y5)))))), equiv(and(strict_implies(possibly(X), necessarily(possibly(X))), not(possibly(necessarily(not(y5))))), not(possibly(necessarily(not(y5))))))), true, implies(not(possibly(necessarily(not(y5)))), and(strict_implies(possibly(X), necessarily(possibly(X))), not(possibly(necessarily(not(y5)))))), equiv(and(strict_implies(possibly(X), necessarily(possibly(X))), not(possibly(necessarily(not(y5))))), not(possibly(necessarily(not(y5)))))), true, and(strict_implies(possibly(X), necessarily(possibly(X))), not(possibly(necessarily(not(y5))))), not(possibly(necessarily(not(y5))))), necessarily(possibly(y5)))), true, implies(implies(necessarily(possibly(y5)), possibly(y5)), equiv(possibly(y5), necessarily(possibly(y5))))), true, possibly(y5), necessarily(possibly(y5)))))), true)
% 192.82/24.63 = { by axiom 8 (hilbert_and_2) R->L }
% 192.82/24.63 fresh68(is_a_theorem(strict_implies(strict_implies(not(y5), not(x10)), strict_implies(possibly(x10), fresh3(fresh40(is_a_theorem(implies(fresh3(fresh115(fresh40(fresh105(and_2, true, strict_implies(possibly(X), necessarily(possibly(X))), not(possibly(necessarily(not(y5))))), true, implies(implies(not(possibly(necessarily(not(y5)))), and(strict_implies(possibly(X), necessarily(possibly(X))), not(possibly(necessarily(not(y5)))))), equiv(and(strict_implies(possibly(X), necessarily(possibly(X))), not(possibly(necessarily(not(y5))))), not(possibly(necessarily(not(y5))))))), true, implies(not(possibly(necessarily(not(y5)))), and(strict_implies(possibly(X), necessarily(possibly(X))), not(possibly(necessarily(not(y5)))))), equiv(and(strict_implies(possibly(X), necessarily(possibly(X))), not(possibly(necessarily(not(y5))))), not(possibly(necessarily(not(y5)))))), true, and(strict_implies(possibly(X), necessarily(possibly(X))), not(possibly(necessarily(not(y5))))), not(possibly(necessarily(not(y5))))), necessarily(possibly(y5)))), true, implies(implies(necessarily(possibly(y5)), possibly(y5)), equiv(possibly(y5), necessarily(possibly(y5))))), true, possibly(y5), necessarily(possibly(y5)))))), true)
% 192.82/24.63 = { by axiom 57 (and_2_1) }
% 192.82/24.63 fresh68(is_a_theorem(strict_implies(strict_implies(not(y5), not(x10)), strict_implies(possibly(x10), fresh3(fresh40(is_a_theorem(implies(fresh3(fresh115(fresh40(is_a_theorem(implies(and(strict_implies(possibly(X), necessarily(possibly(X))), not(possibly(necessarily(not(y5))))), not(possibly(necessarily(not(y5)))))), true, implies(implies(not(possibly(necessarily(not(y5)))), and(strict_implies(possibly(X), necessarily(possibly(X))), not(possibly(necessarily(not(y5)))))), equiv(and(strict_implies(possibly(X), necessarily(possibly(X))), not(possibly(necessarily(not(y5))))), not(possibly(necessarily(not(y5))))))), true, implies(not(possibly(necessarily(not(y5)))), and(strict_implies(possibly(X), necessarily(possibly(X))), not(possibly(necessarily(not(y5)))))), equiv(and(strict_implies(possibly(X), necessarily(possibly(X))), not(possibly(necessarily(not(y5))))), not(possibly(necessarily(not(y5)))))), true, and(strict_implies(possibly(X), necessarily(possibly(X))), not(possibly(necessarily(not(y5))))), not(possibly(necessarily(not(y5))))), necessarily(possibly(y5)))), true, implies(implies(necessarily(possibly(y5)), possibly(y5)), equiv(possibly(y5), necessarily(possibly(y5))))), true, possibly(y5), necessarily(possibly(y5)))))), true)
% 192.82/24.63 = { by lemma 74 }
% 192.82/24.64 fresh68(is_a_theorem(strict_implies(strict_implies(not(y5), not(x10)), strict_implies(possibly(x10), fresh3(fresh40(is_a_theorem(implies(fresh3(fresh115(true, true, implies(not(possibly(necessarily(not(y5)))), and(strict_implies(possibly(X), necessarily(possibly(X))), not(possibly(necessarily(not(y5)))))), equiv(and(strict_implies(possibly(X), necessarily(possibly(X))), not(possibly(necessarily(not(y5))))), not(possibly(necessarily(not(y5)))))), true, and(strict_implies(possibly(X), necessarily(possibly(X))), not(possibly(necessarily(not(y5))))), not(possibly(necessarily(not(y5))))), necessarily(possibly(y5)))), true, implies(implies(necessarily(possibly(y5)), possibly(y5)), equiv(possibly(y5), necessarily(possibly(y5))))), true, possibly(y5), necessarily(possibly(y5)))))), true)
% 192.82/24.64 = { by lemma 73 }
% 192.82/24.64 fresh68(is_a_theorem(strict_implies(strict_implies(not(y5), not(x10)), strict_implies(possibly(x10), fresh3(fresh40(is_a_theorem(implies(fresh3(true, true, and(strict_implies(possibly(X), necessarily(possibly(X))), not(possibly(necessarily(not(y5))))), not(possibly(necessarily(not(y5))))), necessarily(possibly(y5)))), true, implies(implies(necessarily(possibly(y5)), possibly(y5)), equiv(possibly(y5), necessarily(possibly(y5))))), true, possibly(y5), necessarily(possibly(y5)))))), true)
% 192.82/24.64 = { by axiom 52 (substitution_of_equivalents_2) }
% 192.82/24.64 fresh68(is_a_theorem(strict_implies(strict_implies(not(y5), not(x10)), strict_implies(possibly(x10), fresh3(fresh40(is_a_theorem(implies(not(possibly(necessarily(not(y5)))), necessarily(possibly(y5)))), true, implies(implies(necessarily(possibly(y5)), possibly(y5)), equiv(possibly(y5), necessarily(possibly(y5))))), true, possibly(y5), necessarily(possibly(y5)))))), true)
% 192.82/24.64 = { by lemma 85 }
% 192.82/24.64 fresh68(is_a_theorem(strict_implies(strict_implies(not(y5), not(x10)), strict_implies(possibly(x10), fresh3(fresh40(is_a_theorem(or(possibly(necessarily(not(y5))), necessarily(possibly(y5)))), true, implies(implies(necessarily(possibly(y5)), possibly(y5)), equiv(possibly(y5), necessarily(possibly(y5))))), true, possibly(y5), necessarily(possibly(y5)))))), true)
% 192.82/24.64 = { by lemma 101 }
% 192.82/24.64 fresh68(is_a_theorem(strict_implies(strict_implies(not(y5), not(x10)), strict_implies(possibly(x10), fresh3(fresh40(is_a_theorem(or(necessarily(not(y5)), necessarily(possibly(y5)))), true, implies(implies(necessarily(possibly(y5)), possibly(y5)), equiv(possibly(y5), necessarily(possibly(y5))))), true, possibly(y5), necessarily(possibly(y5)))))), true)
% 192.82/24.64 = { by lemma 88 }
% 192.82/24.64 fresh68(is_a_theorem(strict_implies(strict_implies(not(y5), not(x10)), strict_implies(possibly(x10), fresh3(fresh40(is_a_theorem(or(necessarily(possibly(y5)), necessarily(not(y5)))), true, implies(implies(necessarily(possibly(y5)), possibly(y5)), equiv(possibly(y5), necessarily(possibly(y5))))), true, possibly(y5), necessarily(possibly(y5)))))), true)
% 192.82/24.64 = { by lemma 103 }
% 192.82/24.64 fresh68(is_a_theorem(strict_implies(strict_implies(not(y5), not(x10)), strict_implies(possibly(x10), fresh3(fresh40(is_a_theorem(implies(possibly(y5), necessarily(possibly(y5)))), true, implies(implies(necessarily(possibly(y5)), possibly(y5)), equiv(possibly(y5), necessarily(possibly(y5))))), true, possibly(y5), necessarily(possibly(y5)))))), true)
% 192.82/24.64 = { by lemma 74 }
% 192.82/24.64 fresh68(is_a_theorem(strict_implies(strict_implies(not(y5), not(x10)), strict_implies(possibly(x10), fresh3(true, true, possibly(y5), necessarily(possibly(y5)))))), true)
% 192.82/24.64 = { by axiom 52 (substitution_of_equivalents_2) }
% 192.82/24.64 fresh68(is_a_theorem(strict_implies(strict_implies(not(y5), not(x10)), strict_implies(possibly(x10), necessarily(possibly(y5))))), true)
% 192.82/24.64 = { by lemma 104 R->L }
% 192.82/24.64 fresh68(is_a_theorem(strict_implies(strict_implies(not(y5), not(x10)), strict_implies(not(necessarily(possibly(y5))), necessarily(not(x10))))), true)
% 192.82/24.64 = { by lemma 99 }
% 192.82/24.64 fresh68(is_a_theorem(strict_implies(strict_implies(not(y5), not(x10)), strict_implies(possibly(necessarily(not(y5))), necessarily(not(x10))))), true)
% 192.82/24.64 = { by lemma 101 }
% 192.82/24.64 fresh68(is_a_theorem(strict_implies(strict_implies(not(y5), not(x10)), strict_implies(necessarily(not(y5)), necessarily(not(x10))))), true)
% 192.82/24.64 = { by lemma 91 R->L }
% 192.82/24.64 fresh68(is_a_theorem(necessarily(implies(strict_implies(not(y5), not(x10)), strict_implies(necessarily(not(y5)), necessarily(not(x10)))))), true)
% 192.82/24.64 = { by lemma 92 R->L }
% 192.82/24.64 fresh68(fresh33(is_a_theorem(implies(strict_implies(not(y5), not(x10)), strict_implies(necessarily(not(y5)), necessarily(not(x10))))), true, implies(strict_implies(not(y5), not(x10)), strict_implies(necessarily(not(y5)), necessarily(not(x10))))), true)
% 192.82/24.64 = { by lemma 77 R->L }
% 192.82/24.64 fresh68(fresh33(is_a_theorem(implies(fresh3(is_a_theorem(equiv(strict_implies(not(y5), not(x10)), necessarily(strict_implies(not(y5), not(x10))))), true, strict_implies(not(y5), not(x10)), necessarily(strict_implies(not(y5), not(x10)))), strict_implies(necessarily(not(y5)), necessarily(not(x10))))), true, implies(strict_implies(not(y5), not(x10)), strict_implies(necessarily(not(y5)), necessarily(not(x10))))), true)
% 193.05/24.64 = { by axiom 28 (modus_ponens_2) R->L }
% 193.05/24.64 fresh68(fresh33(is_a_theorem(implies(fresh3(fresh40(true, true, equiv(strict_implies(not(y5), not(x10)), necessarily(strict_implies(not(y5), not(x10))))), true, strict_implies(not(y5), not(x10)), necessarily(strict_implies(not(y5), not(x10)))), strict_implies(necessarily(not(y5)), necessarily(not(x10))))), true, implies(strict_implies(not(y5), not(x10)), strict_implies(necessarily(not(y5)), necessarily(not(x10))))), true)
% 193.05/24.64 = { by lemma 80 R->L }
% 193.05/24.64 fresh68(fresh33(is_a_theorem(implies(fresh3(fresh40(is_a_theorem(implies(necessarily(strict_implies(not(y5), not(x10))), strict_implies(not(y5), not(x10)))), true, equiv(strict_implies(not(y5), not(x10)), necessarily(strict_implies(not(y5), not(x10))))), true, strict_implies(not(y5), not(x10)), necessarily(strict_implies(not(y5), not(x10)))), strict_implies(necessarily(not(y5)), necessarily(not(x10))))), true, implies(strict_implies(not(y5), not(x10)), strict_implies(necessarily(not(y5)), necessarily(not(x10))))), true)
% 193.05/24.64 = { by axiom 64 (modus_ponens_2) R->L }
% 193.05/24.64 fresh68(fresh33(is_a_theorem(implies(fresh3(fresh115(is_a_theorem(implies(implies(necessarily(strict_implies(not(y5), not(x10))), strict_implies(not(y5), not(x10))), equiv(strict_implies(not(y5), not(x10)), necessarily(strict_implies(not(y5), not(x10)))))), true, implies(necessarily(strict_implies(not(y5), not(x10))), strict_implies(not(y5), not(x10))), equiv(strict_implies(not(y5), not(x10)), necessarily(strict_implies(not(y5), not(x10))))), true, strict_implies(not(y5), not(x10)), necessarily(strict_implies(not(y5), not(x10)))), strict_implies(necessarily(not(y5)), necessarily(not(x10))))), true, implies(strict_implies(not(y5), not(x10)), strict_implies(necessarily(not(y5)), necessarily(not(x10))))), true)
% 193.05/24.64 = { by axiom 28 (modus_ponens_2) R->L }
% 193.05/24.64 fresh68(fresh33(is_a_theorem(implies(fresh3(fresh115(fresh40(true, true, implies(implies(necessarily(strict_implies(not(y5), not(x10))), strict_implies(not(y5), not(x10))), equiv(strict_implies(not(y5), not(x10)), necessarily(strict_implies(not(y5), not(x10)))))), true, implies(necessarily(strict_implies(not(y5), not(x10))), strict_implies(not(y5), not(x10))), equiv(strict_implies(not(y5), not(x10)), necessarily(strict_implies(not(y5), not(x10))))), true, strict_implies(not(y5), not(x10)), necessarily(strict_implies(not(y5), not(x10)))), strict_implies(necessarily(not(y5)), necessarily(not(x10))))), true, implies(strict_implies(not(y5), not(x10)), strict_implies(necessarily(not(y5)), necessarily(not(x10))))), true)
% 193.05/24.64 = { by lemma 79 R->L }
% 193.05/24.64 fresh68(fresh33(is_a_theorem(implies(fresh3(fresh115(fresh40(is_a_theorem(implies(necessarily(implies(not(y5), not(x10))), necessarily(necessarily(implies(not(y5), not(x10)))))), true, implies(implies(necessarily(strict_implies(not(y5), not(x10))), strict_implies(not(y5), not(x10))), equiv(strict_implies(not(y5), not(x10)), necessarily(strict_implies(not(y5), not(x10)))))), true, implies(necessarily(strict_implies(not(y5), not(x10))), strict_implies(not(y5), not(x10))), equiv(strict_implies(not(y5), not(x10)), necessarily(strict_implies(not(y5), not(x10))))), true, strict_implies(not(y5), not(x10)), necessarily(strict_implies(not(y5), not(x10)))), strict_implies(necessarily(not(y5)), necessarily(not(x10))))), true, implies(strict_implies(not(y5), not(x10)), strict_implies(necessarily(not(y5)), necessarily(not(x10))))), true)
% 193.05/24.64 = { by lemma 91 }
% 193.05/24.64 fresh68(fresh33(is_a_theorem(implies(fresh3(fresh115(fresh40(is_a_theorem(implies(strict_implies(not(y5), not(x10)), necessarily(necessarily(implies(not(y5), not(x10)))))), true, implies(implies(necessarily(strict_implies(not(y5), not(x10))), strict_implies(not(y5), not(x10))), equiv(strict_implies(not(y5), not(x10)), necessarily(strict_implies(not(y5), not(x10)))))), true, implies(necessarily(strict_implies(not(y5), not(x10))), strict_implies(not(y5), not(x10))), equiv(strict_implies(not(y5), not(x10)), necessarily(strict_implies(not(y5), not(x10))))), true, strict_implies(not(y5), not(x10)), necessarily(strict_implies(not(y5), not(x10)))), strict_implies(necessarily(not(y5)), necessarily(not(x10))))), true, implies(strict_implies(not(y5), not(x10)), strict_implies(necessarily(not(y5)), necessarily(not(x10))))), true)
% 193.05/24.64 = { by lemma 91 }
% 193.05/24.64 fresh68(fresh33(is_a_theorem(implies(fresh3(fresh115(fresh40(is_a_theorem(implies(strict_implies(not(y5), not(x10)), necessarily(strict_implies(not(y5), not(x10))))), true, implies(implies(necessarily(strict_implies(not(y5), not(x10))), strict_implies(not(y5), not(x10))), equiv(strict_implies(not(y5), not(x10)), necessarily(strict_implies(not(y5), not(x10)))))), true, implies(necessarily(strict_implies(not(y5), not(x10))), strict_implies(not(y5), not(x10))), equiv(strict_implies(not(y5), not(x10)), necessarily(strict_implies(not(y5), not(x10))))), true, strict_implies(not(y5), not(x10)), necessarily(strict_implies(not(y5), not(x10)))), strict_implies(necessarily(not(y5)), necessarily(not(x10))))), true, implies(strict_implies(not(y5), not(x10)), strict_implies(necessarily(not(y5)), necessarily(not(x10))))), true)
% 193.05/24.65 = { by lemma 74 }
% 193.05/24.65 fresh68(fresh33(is_a_theorem(implies(fresh3(fresh115(true, true, implies(necessarily(strict_implies(not(y5), not(x10))), strict_implies(not(y5), not(x10))), equiv(strict_implies(not(y5), not(x10)), necessarily(strict_implies(not(y5), not(x10))))), true, strict_implies(not(y5), not(x10)), necessarily(strict_implies(not(y5), not(x10)))), strict_implies(necessarily(not(y5)), necessarily(not(x10))))), true, implies(strict_implies(not(y5), not(x10)), strict_implies(necessarily(not(y5)), necessarily(not(x10))))), true)
% 193.05/24.65 = { by lemma 73 }
% 193.05/24.65 fresh68(fresh33(is_a_theorem(implies(fresh3(true, true, strict_implies(not(y5), not(x10)), necessarily(strict_implies(not(y5), not(x10)))), strict_implies(necessarily(not(y5)), necessarily(not(x10))))), true, implies(strict_implies(not(y5), not(x10)), strict_implies(necessarily(not(y5)), necessarily(not(x10))))), true)
% 193.05/24.65 = { by axiom 52 (substitution_of_equivalents_2) }
% 193.05/24.65 fresh68(fresh33(is_a_theorem(implies(necessarily(strict_implies(not(y5), not(x10))), strict_implies(necessarily(not(y5)), necessarily(not(x10))))), true, implies(strict_implies(not(y5), not(x10)), strict_implies(necessarily(not(y5)), necessarily(not(x10))))), true)
% 193.05/24.65 = { by lemma 91 R->L }
% 193.05/24.65 fresh68(fresh33(is_a_theorem(implies(necessarily(strict_implies(not(y5), not(x10))), necessarily(implies(necessarily(not(y5)), necessarily(not(x10)))))), true, implies(strict_implies(not(y5), not(x10)), strict_implies(necessarily(not(y5)), necessarily(not(x10))))), true)
% 193.05/24.65 = { by axiom 28 (modus_ponens_2) R->L }
% 193.05/24.65 fresh68(fresh33(fresh40(true, true, implies(necessarily(strict_implies(not(y5), not(x10))), necessarily(implies(necessarily(not(y5)), necessarily(not(x10)))))), true, implies(strict_implies(not(y5), not(x10)), strict_implies(necessarily(not(y5)), necessarily(not(x10))))), true)
% 193.05/24.65 = { by axiom 30 (necessitation_1) R->L }
% 193.05/24.65 fresh68(fresh33(fresh40(fresh33(true, true, implies(strict_implies(not(y5), not(x10)), implies(necessarily(not(y5)), necessarily(not(x10))))), true, implies(necessarily(strict_implies(not(y5), not(x10))), necessarily(implies(necessarily(not(y5)), necessarily(not(x10)))))), true, implies(strict_implies(not(y5), not(x10)), strict_implies(necessarily(not(y5)), necessarily(not(x10))))), true)
% 193.05/24.65 = { by lemma 105 R->L }
% 193.05/24.65 fresh68(fresh33(fresh40(fresh33(is_a_theorem(implies(strict_implies(not(y5), not(x10)), implies(necessarily(not(y5)), necessarily(not(x10))))), true, implies(strict_implies(not(y5), not(x10)), implies(necessarily(not(y5)), necessarily(not(x10))))), true, implies(necessarily(strict_implies(not(y5), not(x10))), necessarily(implies(necessarily(not(y5)), necessarily(not(x10)))))), true, implies(strict_implies(not(y5), not(x10)), strict_implies(necessarily(not(y5)), necessarily(not(x10))))), true)
% 193.05/24.65 = { by lemma 92 }
% 193.05/24.65 fresh68(fresh33(fresh40(is_a_theorem(necessarily(implies(strict_implies(not(y5), not(x10)), implies(necessarily(not(y5)), necessarily(not(x10)))))), true, implies(necessarily(strict_implies(not(y5), not(x10))), necessarily(implies(necessarily(not(y5)), necessarily(not(x10)))))), true, implies(strict_implies(not(y5), not(x10)), strict_implies(necessarily(not(y5)), necessarily(not(x10))))), true)
% 193.05/24.65 = { by lemma 91 }
% 193.05/24.65 fresh68(fresh33(fresh40(is_a_theorem(strict_implies(strict_implies(not(y5), not(x10)), implies(necessarily(not(y5)), necessarily(not(x10))))), true, implies(necessarily(strict_implies(not(y5), not(x10))), necessarily(implies(necessarily(not(y5)), necessarily(not(x10)))))), true, implies(strict_implies(not(y5), not(x10)), strict_implies(necessarily(not(y5)), necessarily(not(x10))))), true)
% 193.05/24.65 = { by axiom 64 (modus_ponens_2) R->L }
% 193.05/24.65 fresh68(fresh33(fresh115(is_a_theorem(implies(strict_implies(strict_implies(not(y5), not(x10)), implies(necessarily(not(y5)), necessarily(not(x10)))), implies(necessarily(strict_implies(not(y5), not(x10))), necessarily(implies(necessarily(not(y5)), necessarily(not(x10))))))), true, strict_implies(strict_implies(not(y5), not(x10)), implies(necessarily(not(y5)), necessarily(not(x10)))), implies(necessarily(strict_implies(not(y5), not(x10))), necessarily(implies(necessarily(not(y5)), necessarily(not(x10)))))), true, implies(strict_implies(not(y5), not(x10)), strict_implies(necessarily(not(y5)), necessarily(not(x10))))), true)
% 193.05/24.65 = { by lemma 105 }
% 193.05/24.65 fresh68(fresh33(fresh115(true, true, strict_implies(strict_implies(not(y5), not(x10)), implies(necessarily(not(y5)), necessarily(not(x10)))), implies(necessarily(strict_implies(not(y5), not(x10))), necessarily(implies(necessarily(not(y5)), necessarily(not(x10)))))), true, implies(strict_implies(not(y5), not(x10)), strict_implies(necessarily(not(y5)), necessarily(not(x10))))), true)
% 193.05/24.65 = { by lemma 73 }
% 193.05/24.65 fresh68(fresh33(true, true, implies(strict_implies(not(y5), not(x10)), strict_implies(necessarily(not(y5)), necessarily(not(x10))))), true)
% 193.05/24.65 = { by axiom 30 (necessitation_1) }
% 193.05/24.65 fresh68(true, true)
% 193.05/24.65 = { by axiom 23 (axiom_s3) }
% 193.05/24.65 true
% 193.05/24.65 % SZS output end Proof
% 193.05/24.65
% 193.05/24.65 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------