TSTP Solution File: LCL537+1 by Twee---2.4.2
View Problem
- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : LCL537+1 : TPTP v8.1.2. Released v3.3.0.
% Transfm : none
% Format : tptp:raw
% Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% Computer : n002.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 : Thu Aug 31 08:19:21 EDT 2023
% Result : Theorem 85.22s 11.44s
% Output : Proof 88.29s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.12/0.13 % Problem : LCL537+1 : TPTP v8.1.2. Released v3.3.0.
% 0.12/0.15 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.16/0.39 % Computer : n002.cluster.edu
% 0.16/0.39 % Model : x86_64 x86_64
% 0.16/0.39 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.16/0.39 % Memory : 8042.1875MB
% 0.16/0.39 % OS : Linux 3.10.0-693.el7.x86_64
% 0.16/0.39 % CPULimit : 300
% 0.16/0.39 % WCLimit : 300
% 0.16/0.39 % DateTime : Fri Aug 25 07:49:32 EDT 2023
% 0.16/0.39 % CPUTime :
% 85.22/11.44 Command-line arguments: --no-flatten-goal
% 85.22/11.44
% 85.22/11.44 % SZS status Theorem
% 85.22/11.44
% 86.28/11.56 % SZS output start Proof
% 86.28/11.56 Take the following subset of the input axioms:
% 86.28/11.56 fof(and_1, axiom, and_1 <=> ![X, Y]: is_a_theorem(implies(and(X, Y), X))).
% 86.28/11.56 fof(and_3, axiom, and_3 <=> ![X2, Y2]: is_a_theorem(implies(X2, implies(Y2, and(X2, Y2))))).
% 86.28/11.56 fof(axiom_4, axiom, axiom_4 <=> ![X2]: is_a_theorem(implies(necessarily(X2), necessarily(necessarily(X2))))).
% 86.28/11.56 fof(axiom_5, axiom, axiom_5 <=> ![X2]: is_a_theorem(implies(possibly(X2), necessarily(possibly(X2))))).
% 86.28/11.56 fof(axiom_B, axiom, axiom_B <=> ![X2]: is_a_theorem(implies(X2, necessarily(possibly(X2))))).
% 86.28/11.56 fof(axiom_M, axiom, axiom_M <=> ![X2]: is_a_theorem(implies(necessarily(X2), X2))).
% 86.28/11.56 fof(hilbert_and_1, axiom, and_1).
% 86.28/11.56 fof(hilbert_and_3, axiom, and_3).
% 86.28/11.56 fof(hilbert_implies_1, axiom, implies_1).
% 86.28/11.56 fof(hilbert_implies_2, axiom, implies_2).
% 86.28/11.56 fof(hilbert_modus_ponens, axiom, modus_ponens).
% 86.28/11.56 fof(hilbert_modus_tollens, axiom, modus_tollens).
% 86.28/11.56 fof(hilbert_op_equiv, axiom, op_equiv).
% 86.28/11.56 fof(hilbert_op_implies_and, axiom, op_implies_and).
% 86.28/11.56 fof(hilbert_op_or, axiom, op_or).
% 86.28/11.56 fof(hilbert_or_1, axiom, or_1).
% 86.28/11.56 fof(implies_1, axiom, implies_1 <=> ![X2, Y2]: is_a_theorem(implies(X2, implies(Y2, X2)))).
% 86.28/11.56 fof(implies_2, axiom, implies_2 <=> ![X2, Y2]: is_a_theorem(implies(implies(X2, implies(X2, Y2)), implies(X2, Y2)))).
% 86.28/11.56 fof(km4b_axiom_4, axiom, axiom_4).
% 86.28/11.56 fof(km4b_axiom_B, axiom, axiom_B).
% 86.28/11.56 fof(km4b_axiom_M, axiom, axiom_M).
% 86.28/11.56 fof(km4b_op_possibly, axiom, op_possibly).
% 86.28/11.56 fof(km5_axiom_5, conjecture, axiom_5).
% 86.28/11.56 fof(kn1, axiom, kn1 <=> ![P]: is_a_theorem(implies(P, and(P, P)))).
% 86.28/11.56 fof(kn2, axiom, kn2 <=> ![Q, P2]: is_a_theorem(implies(and(P2, Q), P2))).
% 86.28/11.57 fof(modus_ponens, axiom, modus_ponens <=> ![X2, Y2]: ((is_a_theorem(X2) & is_a_theorem(implies(X2, Y2))) => is_a_theorem(Y2))).
% 86.28/11.57 fof(modus_tollens, axiom, modus_tollens <=> ![X2, Y2]: is_a_theorem(implies(implies(not(Y2), not(X2)), implies(X2, Y2)))).
% 86.28/11.57 fof(op_equiv, axiom, op_equiv => ![X2, Y2]: equiv(X2, Y2)=and(implies(X2, Y2), implies(Y2, X2))).
% 86.28/11.57 fof(op_implies_and, axiom, op_implies_and => ![X2, Y2]: implies(X2, Y2)=not(and(X2, not(Y2)))).
% 86.28/11.57 fof(op_or, axiom, op_or => ![X2, Y2]: or(X2, Y2)=not(and(not(X2), not(Y2)))).
% 86.28/11.57 fof(op_possibly, axiom, op_possibly => ![X2]: possibly(X2)=not(necessarily(not(X2)))).
% 86.28/11.57 fof(or_1, axiom, or_1 <=> ![X2, Y2]: is_a_theorem(implies(X2, or(X2, Y2)))).
% 86.28/11.57 fof(substitution_of_equivalents, axiom, substitution_of_equivalents <=> ![X2, Y2]: (is_a_theorem(equiv(X2, Y2)) => X2=Y2)).
% 86.28/11.57 fof(substitution_of_equivalents, axiom, substitution_of_equivalents).
% 86.28/11.57
% 86.28/11.57 Now clausify the problem and encode Horn clauses using encoding 3 of
% 86.28/11.57 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 86.28/11.57 We repeatedly replace C & s=t => u=v by the two clauses:
% 86.28/11.57 fresh(y, y, x1...xn) = u
% 86.28/11.57 C => fresh(s, t, x1...xn) = v
% 86.28/11.57 where fresh is a fresh function symbol and x1..xn are the free
% 86.28/11.57 variables of u and v.
% 86.28/11.57 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 86.28/11.57 input problem has no model of domain size 1).
% 86.28/11.57
% 86.28/11.57 The encoding turns the above axioms into the following unit equations and goals:
% 86.28/11.57
% 86.28/11.57 Axiom 1 (hilbert_op_or): op_or = true.
% 86.28/11.57 Axiom 2 (hilbert_op_implies_and): op_implies_and = true.
% 86.28/11.57 Axiom 3 (hilbert_op_equiv): op_equiv = true.
% 86.28/11.57 Axiom 4 (km4b_op_possibly): op_possibly = true.
% 86.28/11.57 Axiom 5 (hilbert_modus_ponens): modus_ponens = true.
% 86.28/11.57 Axiom 6 (substitution_of_equivalents): substitution_of_equivalents = true.
% 86.28/11.57 Axiom 7 (hilbert_modus_tollens): modus_tollens = true.
% 86.28/11.57 Axiom 8 (hilbert_implies_1): implies_1 = true.
% 86.28/11.57 Axiom 9 (hilbert_implies_2): implies_2 = true.
% 86.28/11.57 Axiom 10 (hilbert_and_1): and_1 = true.
% 86.28/11.57 Axiom 11 (hilbert_and_3): and_3 = true.
% 86.28/11.57 Axiom 12 (hilbert_or_1): or_1 = true.
% 86.28/11.57 Axiom 13 (km4b_axiom_M): axiom_M = true.
% 86.28/11.57 Axiom 14 (km4b_axiom_4): axiom_4 = true.
% 86.28/11.57 Axiom 15 (km4b_axiom_B): axiom_B = true.
% 86.28/11.57 Axiom 16 (axiom_5): fresh100(X, X) = true.
% 86.28/11.57 Axiom 17 (kn1): fresh46(X, X) = true.
% 86.28/11.57 Axiom 18 (kn2): fresh44(X, X) = true.
% 86.28/11.57 Axiom 19 (modus_ponens_2): fresh116(X, X, Y) = true.
% 86.28/11.57 Axiom 20 (axiom_4_1): fresh101(X, X, Y) = true.
% 86.28/11.57 Axiom 21 (axiom_B_1): fresh97(X, X, Y) = true.
% 86.28/11.57 Axiom 22 (axiom_M_1): fresh93(X, X, Y) = true.
% 86.28/11.57 Axiom 23 (modus_ponens_2): fresh40(X, X, Y) = is_a_theorem(Y).
% 86.28/11.57 Axiom 24 (op_possibly): fresh25(X, X, Y) = possibly(Y).
% 86.28/11.57 Axiom 25 (op_possibly): fresh25(op_possibly, true, X) = not(necessarily(not(X))).
% 86.28/11.57 Axiom 26 (modus_ponens_2): fresh115(X, X, Y, Z) = fresh116(modus_ponens, true, Z).
% 86.28/11.57 Axiom 27 (and_1_1): fresh107(X, X, Y, Z) = true.
% 86.28/11.57 Axiom 28 (and_3_1): fresh103(X, X, Y, Z) = true.
% 86.28/11.57 Axiom 29 (implies_1_1): fresh51(X, X, Y, Z) = true.
% 86.28/11.57 Axiom 30 (implies_2_1): fresh49(X, X, Y, Z) = true.
% 86.28/11.57 Axiom 31 (modus_tollens_1): fresh35(X, X, Y, Z) = true.
% 86.28/11.57 Axiom 32 (op_equiv): fresh30(X, X, Y, Z) = equiv(Y, Z).
% 87.79/11.57 Axiom 33 (op_implies_and): fresh29(X, X, Y, Z) = implies(Y, Z).
% 87.79/11.57 Axiom 34 (op_or): fresh26(X, X, Y, Z) = or(Y, Z).
% 87.79/11.57 Axiom 35 (or_1_1): fresh21(X, X, Y, Z) = true.
% 87.79/11.57 Axiom 36 (substitution_of_equivalents_2): fresh4(X, X, Y, Z) = Y.
% 87.79/11.57 Axiom 37 (substitution_of_equivalents_2): fresh3(X, X, Y, Z) = Z.
% 87.79/11.57 Axiom 38 (op_implies_and): fresh29(op_implies_and, true, X, Y) = not(and(X, not(Y))).
% 87.79/11.57 Axiom 39 (axiom_M_1): fresh93(axiom_M, true, X) = is_a_theorem(implies(necessarily(X), X)).
% 87.79/11.57 Axiom 40 (op_or): fresh26(op_or, true, X, Y) = not(and(not(X), not(Y))).
% 87.79/11.57 Axiom 41 (or_1_1): fresh21(or_1, true, X, Y) = is_a_theorem(implies(X, or(X, Y))).
% 87.79/11.57 Axiom 42 (axiom_B_1): fresh97(axiom_B, true, X) = is_a_theorem(implies(X, necessarily(possibly(X)))).
% 87.79/11.57 Axiom 43 (kn1_1): fresh45(kn1, true, X) = is_a_theorem(implies(X, and(X, X))).
% 87.79/11.57 Axiom 44 (implies_1_1): fresh51(implies_1, true, X, Y) = is_a_theorem(implies(X, implies(Y, X))).
% 87.79/11.57 Axiom 45 (and_1_1): fresh107(and_1, true, X, Y) = is_a_theorem(implies(and(X, Y), X)).
% 87.79/11.57 Axiom 46 (op_equiv): fresh30(op_equiv, true, X, Y) = and(implies(X, Y), implies(Y, X)).
% 87.79/11.57 Axiom 47 (axiom_5_1): fresh99(axiom_5, true, X) = is_a_theorem(implies(possibly(X), necessarily(possibly(X)))).
% 87.79/11.57 Axiom 48 (axiom_4_1): fresh101(axiom_4, true, X) = is_a_theorem(implies(necessarily(X), necessarily(necessarily(X)))).
% 87.79/11.57 Axiom 49 (modus_ponens_2): fresh115(is_a_theorem(implies(X, Y)), true, X, Y) = fresh40(is_a_theorem(X), true, Y).
% 87.79/11.57 Axiom 50 (kn1): fresh46(is_a_theorem(implies(p14, and(p14, p14))), true) = kn1.
% 87.79/11.57 Axiom 51 (kn2): fresh44(is_a_theorem(implies(and(p13, q11), p13)), true) = kn2.
% 87.79/11.57 Axiom 52 (substitution_of_equivalents_2): fresh4(substitution_of_equivalents, true, X, Y) = fresh3(is_a_theorem(equiv(X, Y)), true, X, Y).
% 87.79/11.57 Axiom 53 (and_3_1): fresh103(and_3, true, X, Y) = is_a_theorem(implies(X, implies(Y, and(X, Y)))).
% 87.79/11.57 Axiom 54 (axiom_5): fresh100(is_a_theorem(implies(possibly(x12), necessarily(possibly(x12)))), true) = axiom_5.
% 87.79/11.57 Axiom 55 (implies_2_1): fresh49(implies_2, true, X, Y) = is_a_theorem(implies(implies(X, implies(X, Y)), implies(X, Y))).
% 87.79/11.57 Axiom 56 (modus_tollens_1): fresh35(modus_tollens, true, X, Y) = is_a_theorem(implies(implies(not(Y), not(X)), implies(X, Y))).
% 87.79/11.57
% 87.79/11.57 Lemma 57: modus_ponens = op_or.
% 87.79/11.57 Proof:
% 87.79/11.57 modus_ponens
% 87.79/11.57 = { by axiom 5 (hilbert_modus_ponens) }
% 87.79/11.57 true
% 87.79/11.57 = { by axiom 1 (hilbert_op_or) R->L }
% 87.79/11.57 op_or
% 87.79/11.57
% 87.79/11.57 Lemma 58: and_1 = op_or.
% 87.79/11.57 Proof:
% 87.79/11.57 and_1
% 87.79/11.57 = { by axiom 10 (hilbert_and_1) }
% 87.79/11.57 true
% 87.79/11.57 = { by axiom 1 (hilbert_op_or) R->L }
% 87.79/11.57 op_or
% 87.79/11.57
% 87.79/11.57 Lemma 59: fresh107(X, X, Y, Z) = op_or.
% 87.79/11.57 Proof:
% 87.79/11.57 fresh107(X, X, Y, Z)
% 87.79/11.57 = { by axiom 27 (and_1_1) }
% 87.79/11.57 true
% 87.79/11.57 = { by axiom 1 (hilbert_op_or) R->L }
% 87.79/11.57 op_or
% 87.79/11.57
% 87.79/11.57 Lemma 60: op_or = kn2.
% 87.79/11.57 Proof:
% 87.79/11.57 op_or
% 87.79/11.57 = { by axiom 1 (hilbert_op_or) }
% 87.79/11.57 true
% 87.79/11.57 = { by axiom 18 (kn2) R->L }
% 87.79/11.57 fresh44(op_or, op_or)
% 87.79/11.57 = { by axiom 1 (hilbert_op_or) }
% 87.79/11.57 fresh44(op_or, true)
% 87.79/11.57 = { by lemma 59 R->L }
% 87.79/11.57 fresh44(fresh107(op_or, op_or, p13, q11), true)
% 87.79/11.57 = { by axiom 1 (hilbert_op_or) }
% 87.79/11.57 fresh44(fresh107(op_or, true, p13, q11), true)
% 87.79/11.57 = { by lemma 58 R->L }
% 87.79/11.57 fresh44(fresh107(and_1, true, p13, q11), true)
% 87.79/11.57 = { by axiom 45 (and_1_1) }
% 87.79/11.57 fresh44(is_a_theorem(implies(and(p13, q11), p13)), true)
% 87.79/11.57 = { by axiom 51 (kn2) }
% 87.79/11.57 kn2
% 87.79/11.57
% 87.79/11.57 Lemma 61: is_a_theorem(implies(X, and(X, X))) = fresh45(kn1, op_or, X).
% 87.79/11.57 Proof:
% 87.79/11.57 is_a_theorem(implies(X, and(X, X)))
% 87.79/11.57 = { by axiom 43 (kn1_1) R->L }
% 87.79/11.57 fresh45(kn1, true, X)
% 87.79/11.57 = { by axiom 1 (hilbert_op_or) R->L }
% 87.79/11.57 fresh45(kn1, op_or, X)
% 87.79/11.57
% 87.79/11.57 Lemma 62: is_a_theorem(implies(X, implies(Y, and(X, Y)))) = kn2.
% 87.79/11.57 Proof:
% 87.79/11.57 is_a_theorem(implies(X, implies(Y, and(X, Y))))
% 87.79/11.57 = { by axiom 53 (and_3_1) R->L }
% 87.79/11.57 fresh103(and_3, true, X, Y)
% 87.79/11.57 = { by axiom 11 (hilbert_and_3) }
% 87.79/11.57 fresh103(true, true, X, Y)
% 87.79/11.57 = { by axiom 1 (hilbert_op_or) R->L }
% 87.79/11.57 fresh103(op_or, true, X, Y)
% 87.79/11.57 = { by lemma 60 }
% 87.79/11.57 fresh103(kn2, true, X, Y)
% 87.79/11.57 = { by axiom 1 (hilbert_op_or) R->L }
% 87.79/11.57 fresh103(kn2, op_or, X, Y)
% 87.79/11.57 = { by lemma 60 }
% 87.79/11.57 fresh103(kn2, kn2, X, Y)
% 87.79/11.57 = { by axiom 28 (and_3_1) }
% 87.79/11.57 true
% 87.79/11.57 = { by axiom 1 (hilbert_op_or) R->L }
% 87.79/11.57 op_or
% 87.79/11.57 = { by lemma 60 }
% 87.79/11.57 kn2
% 87.79/11.57
% 87.79/11.57 Lemma 63: fresh115(is_a_theorem(implies(X, Y)), op_or, X, Y) = fresh40(is_a_theorem(X), op_or, Y).
% 87.79/11.57 Proof:
% 87.79/11.57 fresh115(is_a_theorem(implies(X, Y)), op_or, X, Y)
% 87.79/11.57 = { by axiom 1 (hilbert_op_or) }
% 87.79/11.57 fresh115(is_a_theorem(implies(X, Y)), true, X, Y)
% 87.79/11.57 = { by axiom 49 (modus_ponens_2) }
% 87.79/11.57 fresh40(is_a_theorem(X), true, Y)
% 87.79/11.57 = { by axiom 1 (hilbert_op_or) R->L }
% 87.79/11.57 fresh40(is_a_theorem(X), op_or, Y)
% 87.79/11.57
% 87.79/11.57 Lemma 64: fresh116(X, X, Y) = op_or.
% 87.79/11.57 Proof:
% 87.79/11.57 fresh116(X, X, Y)
% 87.79/11.57 = { by axiom 19 (modus_ponens_2) }
% 87.79/11.57 true
% 87.79/11.57 = { by axiom 1 (hilbert_op_or) R->L }
% 87.79/11.57 op_or
% 87.79/11.57
% 87.79/11.57 Lemma 65: fresh115(X, X, Y, Z) = op_or.
% 87.79/11.57 Proof:
% 87.79/11.57 fresh115(X, X, Y, Z)
% 87.79/11.57 = { by axiom 26 (modus_ponens_2) }
% 87.79/11.57 fresh116(modus_ponens, true, Z)
% 87.79/11.57 = { by lemma 57 }
% 87.79/11.57 fresh116(op_or, true, Z)
% 87.79/11.57 = { by axiom 1 (hilbert_op_or) R->L }
% 87.79/11.57 fresh116(op_or, op_or, Z)
% 87.79/11.57 = { by lemma 64 }
% 87.79/11.57 op_or
% 87.79/11.58
% 87.79/11.58 Lemma 66: fresh45(kn1, kn2, X) = kn2.
% 87.79/11.58 Proof:
% 87.79/11.58 fresh45(kn1, kn2, X)
% 87.79/11.58 = { by lemma 60 R->L }
% 87.79/11.58 fresh45(kn1, op_or, X)
% 87.79/11.58 = { by lemma 61 R->L }
% 87.79/11.58 is_a_theorem(implies(X, and(X, X)))
% 87.79/11.58 = { by axiom 23 (modus_ponens_2) R->L }
% 87.79/11.58 fresh40(kn2, kn2, implies(X, and(X, X)))
% 87.79/11.58 = { by lemma 62 R->L }
% 87.79/11.58 fresh40(is_a_theorem(implies(X, implies(X, and(X, X)))), kn2, implies(X, and(X, X)))
% 87.79/11.58 = { by lemma 60 R->L }
% 87.79/11.58 fresh40(is_a_theorem(implies(X, implies(X, and(X, X)))), op_or, implies(X, and(X, X)))
% 87.79/11.58 = { by lemma 63 R->L }
% 87.79/11.58 fresh115(is_a_theorem(implies(implies(X, implies(X, and(X, X))), implies(X, and(X, X)))), op_or, implies(X, implies(X, and(X, X))), implies(X, and(X, X)))
% 87.79/11.58 = { by axiom 55 (implies_2_1) R->L }
% 87.79/11.58 fresh115(fresh49(implies_2, true, X, and(X, X)), op_or, implies(X, implies(X, and(X, X))), implies(X, and(X, X)))
% 87.79/11.58 = { by axiom 9 (hilbert_implies_2) }
% 87.79/11.58 fresh115(fresh49(true, true, X, and(X, X)), op_or, implies(X, implies(X, and(X, X))), implies(X, and(X, X)))
% 87.79/11.58 = { by axiom 1 (hilbert_op_or) R->L }
% 87.79/11.58 fresh115(fresh49(op_or, true, X, and(X, X)), op_or, implies(X, implies(X, and(X, X))), implies(X, and(X, X)))
% 87.79/11.58 = { by lemma 60 }
% 87.79/11.58 fresh115(fresh49(kn2, true, X, and(X, X)), op_or, implies(X, implies(X, and(X, X))), implies(X, and(X, X)))
% 87.79/11.58 = { by axiom 1 (hilbert_op_or) R->L }
% 87.79/11.58 fresh115(fresh49(kn2, op_or, X, and(X, X)), op_or, implies(X, implies(X, and(X, X))), implies(X, and(X, X)))
% 87.79/11.58 = { by lemma 60 }
% 87.79/11.58 fresh115(fresh49(kn2, kn2, X, and(X, X)), op_or, implies(X, implies(X, and(X, X))), implies(X, and(X, X)))
% 87.79/11.58 = { by axiom 30 (implies_2_1) }
% 87.79/11.58 fresh115(true, op_or, implies(X, implies(X, and(X, X))), implies(X, and(X, X)))
% 87.79/11.58 = { by axiom 1 (hilbert_op_or) R->L }
% 87.79/11.58 fresh115(op_or, op_or, implies(X, implies(X, and(X, X))), implies(X, and(X, X)))
% 87.79/11.58 = { by lemma 60 }
% 87.79/11.58 fresh115(kn2, op_or, implies(X, implies(X, and(X, X))), implies(X, and(X, X)))
% 87.79/11.58 = { by lemma 60 }
% 87.79/11.58 fresh115(kn2, kn2, implies(X, implies(X, and(X, X))), implies(X, and(X, X)))
% 87.79/11.58 = { by lemma 65 }
% 87.79/11.58 op_or
% 87.79/11.58 = { by lemma 60 }
% 87.79/11.58 kn2
% 87.79/11.58
% 87.79/11.58 Lemma 67: kn2 = kn1.
% 87.79/11.58 Proof:
% 87.79/11.58 kn2
% 87.79/11.58 = { by lemma 60 R->L }
% 87.79/11.58 op_or
% 87.79/11.58 = { by axiom 1 (hilbert_op_or) }
% 87.79/11.58 true
% 87.79/11.58 = { by axiom 17 (kn1) R->L }
% 87.79/11.58 fresh46(kn2, kn2)
% 87.79/11.58 = { by lemma 66 R->L }
% 87.79/11.58 fresh46(fresh45(kn1, kn2, p14), kn2)
% 87.79/11.58 = { by lemma 60 R->L }
% 87.79/11.58 fresh46(fresh45(kn1, kn2, p14), op_or)
% 87.79/11.58 = { by lemma 60 R->L }
% 87.79/11.58 fresh46(fresh45(kn1, op_or, p14), op_or)
% 87.79/11.58 = { by axiom 1 (hilbert_op_or) }
% 87.79/11.58 fresh46(fresh45(kn1, op_or, p14), true)
% 87.79/11.58 = { by lemma 61 R->L }
% 87.79/11.58 fresh46(is_a_theorem(implies(p14, and(p14, p14))), true)
% 87.79/11.58 = { by axiom 50 (kn1) }
% 87.79/11.58 kn1
% 87.79/11.58
% 87.79/11.58 Lemma 68: fresh115(X, X, Y, Z) = kn1.
% 87.79/11.58 Proof:
% 87.79/11.58 fresh115(X, X, Y, Z)
% 87.79/11.58 = { by axiom 26 (modus_ponens_2) }
% 87.79/11.58 fresh116(modus_ponens, true, Z)
% 87.79/11.58 = { by lemma 57 }
% 87.79/11.58 fresh116(op_or, true, Z)
% 87.79/11.58 = { by axiom 1 (hilbert_op_or) R->L }
% 87.79/11.58 fresh116(op_or, op_or, Z)
% 87.79/11.58 = { by lemma 64 }
% 87.79/11.58 op_or
% 87.79/11.58 = { by lemma 60 }
% 87.79/11.58 kn2
% 87.79/11.58 = { by lemma 67 }
% 87.79/11.58 kn1
% 87.79/11.58
% 87.79/11.58 Lemma 69: fresh40(is_a_theorem(X), kn1, implies(Y, and(X, Y))) = kn1.
% 87.79/11.58 Proof:
% 87.79/11.58 fresh40(is_a_theorem(X), kn1, implies(Y, and(X, Y)))
% 87.79/11.58 = { by lemma 67 R->L }
% 87.79/11.58 fresh40(is_a_theorem(X), kn2, implies(Y, and(X, Y)))
% 87.79/11.58 = { by lemma 60 R->L }
% 87.79/11.58 fresh40(is_a_theorem(X), op_or, implies(Y, and(X, Y)))
% 87.79/11.58 = { by lemma 63 R->L }
% 87.79/11.58 fresh115(is_a_theorem(implies(X, implies(Y, and(X, Y)))), op_or, X, implies(Y, and(X, Y)))
% 87.79/11.58 = { by lemma 62 }
% 87.79/11.58 fresh115(kn2, op_or, X, implies(Y, and(X, Y)))
% 87.79/11.58 = { by lemma 60 }
% 87.79/11.58 fresh115(kn2, kn2, X, implies(Y, and(X, Y)))
% 87.79/11.58 = { by lemma 65 }
% 87.79/11.58 op_or
% 87.79/11.58 = { by lemma 60 }
% 87.79/11.58 kn2
% 87.79/11.58 = { by lemma 67 }
% 87.79/11.58 kn1
% 87.79/11.58
% 87.79/11.58 Lemma 70: and(implies(X, Y), implies(Y, X)) = equiv(X, Y).
% 87.79/11.58 Proof:
% 87.79/11.58 and(implies(X, Y), implies(Y, X))
% 87.79/11.58 = { by axiom 46 (op_equiv) R->L }
% 87.79/11.58 fresh30(op_equiv, true, X, Y)
% 87.79/11.58 = { by axiom 3 (hilbert_op_equiv) }
% 87.79/11.58 fresh30(true, true, X, Y)
% 87.79/11.58 = { by axiom 1 (hilbert_op_or) R->L }
% 87.79/11.58 fresh30(op_or, true, X, Y)
% 87.79/11.58 = { by axiom 1 (hilbert_op_or) R->L }
% 87.79/11.58 fresh30(op_or, op_or, X, Y)
% 87.79/11.58 = { by axiom 32 (op_equiv) }
% 87.79/11.58 equiv(X, Y)
% 87.79/11.58
% 87.79/11.58 Lemma 71: fresh115(is_a_theorem(implies(X, Y)), kn1, X, Y) = fresh40(is_a_theorem(X), kn1, Y).
% 87.79/11.58 Proof:
% 87.79/11.58 fresh115(is_a_theorem(implies(X, Y)), kn1, X, Y)
% 87.79/11.58 = { by lemma 67 R->L }
% 87.79/11.58 fresh115(is_a_theorem(implies(X, Y)), kn2, X, Y)
% 87.79/11.58 = { by lemma 60 R->L }
% 87.79/11.58 fresh115(is_a_theorem(implies(X, Y)), op_or, X, Y)
% 87.79/11.58 = { by lemma 63 }
% 87.79/11.58 fresh40(is_a_theorem(X), op_or, Y)
% 87.79/11.58 = { by lemma 60 }
% 87.79/11.58 fresh40(is_a_theorem(X), kn2, Y)
% 87.79/11.58 = { by lemma 67 }
% 87.79/11.58 fresh40(is_a_theorem(X), kn1, Y)
% 87.79/11.58
% 87.79/11.58 Lemma 72: fresh3(is_a_theorem(equiv(X, Y)), kn1, X, Y) = X.
% 87.79/11.58 Proof:
% 87.79/11.58 fresh3(is_a_theorem(equiv(X, Y)), kn1, X, Y)
% 87.79/11.58 = { by lemma 67 R->L }
% 87.79/11.58 fresh3(is_a_theorem(equiv(X, Y)), kn2, X, Y)
% 87.79/11.58 = { by lemma 60 R->L }
% 87.79/11.58 fresh3(is_a_theorem(equiv(X, Y)), op_or, X, Y)
% 87.79/11.58 = { by axiom 1 (hilbert_op_or) }
% 87.79/11.58 fresh3(is_a_theorem(equiv(X, Y)), true, X, Y)
% 87.79/11.58 = { by axiom 52 (substitution_of_equivalents_2) R->L }
% 87.79/11.58 fresh4(substitution_of_equivalents, true, X, Y)
% 87.79/11.58 = { by axiom 6 (substitution_of_equivalents) }
% 87.79/11.58 fresh4(true, true, X, Y)
% 87.79/11.58 = { by axiom 1 (hilbert_op_or) R->L }
% 87.79/11.58 fresh4(op_or, true, X, Y)
% 87.79/11.58 = { by lemma 60 }
% 87.79/11.58 fresh4(kn2, true, X, Y)
% 87.79/11.58 = { by axiom 1 (hilbert_op_or) R->L }
% 87.79/11.58 fresh4(kn2, op_or, X, Y)
% 87.79/11.58 = { by lemma 60 }
% 87.79/11.58 fresh4(kn2, kn2, X, Y)
% 87.79/11.58 = { by axiom 36 (substitution_of_equivalents_2) }
% 87.79/11.58 X
% 87.79/11.58
% 87.79/11.58 Lemma 73: and(X, X) = X.
% 87.79/11.58 Proof:
% 87.79/11.58 and(X, X)
% 87.79/11.58 = { by axiom 37 (substitution_of_equivalents_2) R->L }
% 87.79/11.58 fresh3(kn1, kn1, X, and(X, X))
% 87.79/11.58 = { by lemma 68 R->L }
% 87.79/11.58 fresh3(fresh115(kn1, kn1, implies(and(X, X), X), equiv(X, and(X, X))), kn1, X, and(X, X))
% 87.79/11.58 = { by lemma 69 R->L }
% 87.79/11.58 fresh3(fresh115(fresh40(is_a_theorem(implies(X, and(X, X))), kn1, implies(implies(and(X, X), X), and(implies(X, and(X, X)), implies(and(X, X), X)))), kn1, implies(and(X, X), X), equiv(X, and(X, X))), kn1, X, and(X, X))
% 87.79/11.58 = { by axiom 43 (kn1_1) R->L }
% 87.79/11.58 fresh3(fresh115(fresh40(fresh45(kn1, true, X), kn1, implies(implies(and(X, X), X), and(implies(X, and(X, X)), implies(and(X, X), X)))), kn1, implies(and(X, X), X), equiv(X, and(X, X))), kn1, X, and(X, X))
% 87.79/11.58 = { by axiom 1 (hilbert_op_or) R->L }
% 87.79/11.58 fresh3(fresh115(fresh40(fresh45(kn1, op_or, X), kn1, implies(implies(and(X, X), X), and(implies(X, and(X, X)), implies(and(X, X), X)))), kn1, implies(and(X, X), X), equiv(X, and(X, X))), kn1, X, and(X, X))
% 87.79/11.58 = { by lemma 60 }
% 87.79/11.59 fresh3(fresh115(fresh40(fresh45(kn1, kn2, X), kn1, implies(implies(and(X, X), X), and(implies(X, and(X, X)), implies(and(X, X), X)))), kn1, implies(and(X, X), X), equiv(X, and(X, X))), kn1, X, and(X, X))
% 87.79/11.59 = { by lemma 66 }
% 87.79/11.59 fresh3(fresh115(fresh40(kn2, kn1, implies(implies(and(X, X), X), and(implies(X, and(X, X)), implies(and(X, X), X)))), kn1, implies(and(X, X), X), equiv(X, and(X, X))), kn1, X, and(X, X))
% 87.79/11.59 = { by lemma 67 }
% 87.79/11.59 fresh3(fresh115(fresh40(kn1, kn1, implies(implies(and(X, X), X), and(implies(X, and(X, X)), implies(and(X, X), X)))), kn1, implies(and(X, X), X), equiv(X, and(X, X))), kn1, X, and(X, X))
% 87.79/11.59 = { by axiom 23 (modus_ponens_2) }
% 87.79/11.59 fresh3(fresh115(is_a_theorem(implies(implies(and(X, X), X), and(implies(X, and(X, X)), implies(and(X, X), X)))), kn1, implies(and(X, X), X), equiv(X, and(X, X))), kn1, X, and(X, X))
% 87.79/11.59 = { by lemma 70 }
% 87.79/11.59 fresh3(fresh115(is_a_theorem(implies(implies(and(X, X), X), equiv(X, and(X, X)))), kn1, implies(and(X, X), X), equiv(X, and(X, X))), kn1, X, and(X, X))
% 87.79/11.59 = { by lemma 71 }
% 87.79/11.59 fresh3(fresh40(is_a_theorem(implies(and(X, X), X)), kn1, equiv(X, and(X, X))), kn1, X, and(X, X))
% 87.79/11.59 = { by axiom 45 (and_1_1) R->L }
% 87.79/11.59 fresh3(fresh40(fresh107(and_1, true, X, X), kn1, equiv(X, and(X, X))), kn1, X, and(X, X))
% 87.79/11.59 = { by lemma 58 }
% 87.79/11.59 fresh3(fresh40(fresh107(op_or, true, X, X), kn1, equiv(X, and(X, X))), kn1, X, and(X, X))
% 87.79/11.59 = { by axiom 1 (hilbert_op_or) R->L }
% 87.79/11.59 fresh3(fresh40(fresh107(op_or, op_or, X, X), kn1, equiv(X, and(X, X))), kn1, X, and(X, X))
% 87.79/11.59 = { by lemma 59 }
% 87.79/11.59 fresh3(fresh40(op_or, kn1, equiv(X, and(X, X))), kn1, X, and(X, X))
% 87.79/11.59 = { by lemma 60 }
% 87.79/11.59 fresh3(fresh40(kn2, kn1, equiv(X, and(X, X))), kn1, X, and(X, X))
% 87.79/11.59 = { by lemma 67 }
% 87.79/11.59 fresh3(fresh40(kn1, kn1, equiv(X, and(X, X))), kn1, X, and(X, X))
% 87.79/11.59 = { by axiom 23 (modus_ponens_2) }
% 87.79/11.59 fresh3(is_a_theorem(equiv(X, and(X, X))), kn1, X, and(X, X))
% 87.79/11.59 = { by lemma 72 }
% 87.79/11.59 X
% 87.79/11.59
% 87.79/11.59 Lemma 74: not(and(X, not(Y))) = implies(X, Y).
% 87.79/11.59 Proof:
% 87.79/11.59 not(and(X, not(Y)))
% 87.79/11.59 = { by axiom 38 (op_implies_and) R->L }
% 87.79/11.59 fresh29(op_implies_and, true, X, Y)
% 87.79/11.59 = { by axiom 2 (hilbert_op_implies_and) }
% 87.79/11.59 fresh29(true, true, X, Y)
% 87.79/11.59 = { by axiom 1 (hilbert_op_or) R->L }
% 87.79/11.59 fresh29(op_or, true, X, Y)
% 87.79/11.59 = { by axiom 1 (hilbert_op_or) R->L }
% 87.79/11.59 fresh29(op_or, op_or, X, Y)
% 87.79/11.59 = { by axiom 33 (op_implies_and) }
% 87.79/11.59 implies(X, Y)
% 87.79/11.59
% 87.79/11.59 Lemma 75: implies(not(X), Y) = or(X, Y).
% 87.79/11.59 Proof:
% 87.79/11.59 implies(not(X), Y)
% 87.79/11.59 = { by lemma 74 R->L }
% 87.79/11.59 not(and(not(X), not(Y)))
% 87.79/11.59 = { by axiom 40 (op_or) R->L }
% 87.79/11.59 fresh26(op_or, true, X, Y)
% 87.79/11.59 = { by axiom 1 (hilbert_op_or) R->L }
% 87.79/11.59 fresh26(op_or, op_or, X, Y)
% 87.79/11.59 = { by axiom 34 (op_or) }
% 87.79/11.59 or(X, Y)
% 87.79/11.59
% 87.79/11.59 Lemma 76: implies(not(X), X) = not(not(X)).
% 87.79/11.59 Proof:
% 87.79/11.59 implies(not(X), X)
% 87.79/11.59 = { by lemma 74 R->L }
% 87.79/11.59 not(and(not(X), not(X)))
% 87.79/11.59 = { by lemma 73 }
% 87.79/11.59 not(not(X))
% 87.79/11.59
% 87.79/11.59 Lemma 77: or(X, X) = not(not(X)).
% 87.79/11.59 Proof:
% 87.79/11.59 or(X, X)
% 87.79/11.59 = { by lemma 75 R->L }
% 87.79/11.59 implies(not(X), X)
% 87.79/11.59 = { by lemma 76 }
% 87.79/11.59 not(not(X))
% 87.79/11.59
% 87.79/11.59 Lemma 78: not(necessarily(not(X))) = possibly(X).
% 87.79/11.59 Proof:
% 87.79/11.59 not(necessarily(not(X)))
% 87.79/11.59 = { by axiom 25 (op_possibly) R->L }
% 87.79/11.59 fresh25(op_possibly, true, X)
% 87.79/11.59 = { by axiom 4 (km4b_op_possibly) }
% 87.79/11.59 fresh25(true, true, X)
% 87.79/11.59 = { by axiom 1 (hilbert_op_or) R->L }
% 87.79/11.59 fresh25(op_or, true, X)
% 87.79/11.59 = { by axiom 1 (hilbert_op_or) R->L }
% 87.79/11.59 fresh25(op_or, op_or, X)
% 87.79/11.59 = { by axiom 24 (op_possibly) }
% 87.79/11.59 possibly(X)
% 87.79/11.59
% 87.79/11.59 Lemma 79: possibly(possibly(X)) = possibly(X).
% 87.79/11.59 Proof:
% 87.79/11.59 possibly(possibly(X))
% 87.79/11.59 = { by lemma 78 R->L }
% 87.79/11.59 possibly(not(necessarily(not(X))))
% 87.79/11.59 = { by lemma 78 R->L }
% 87.79/11.59 not(necessarily(not(not(necessarily(not(X))))))
% 87.79/11.59 = { by axiom 37 (substitution_of_equivalents_2) R->L }
% 87.79/11.59 not(necessarily(fresh3(kn1, kn1, necessarily(not(X)), not(not(necessarily(not(X)))))))
% 87.79/11.59 = { by lemma 68 R->L }
% 87.79/11.59 not(necessarily(fresh3(fresh115(kn1, kn1, implies(or(necessarily(not(X)), necessarily(not(X))), necessarily(not(X))), equiv(necessarily(not(X)), or(necessarily(not(X)), necessarily(not(X))))), kn1, necessarily(not(X)), not(not(necessarily(not(X)))))))
% 87.79/11.59 = { by lemma 67 R->L }
% 87.79/11.59 not(necessarily(fresh3(fresh115(kn2, kn1, implies(or(necessarily(not(X)), necessarily(not(X))), necessarily(not(X))), equiv(necessarily(not(X)), or(necessarily(not(X)), necessarily(not(X))))), kn1, necessarily(not(X)), not(not(necessarily(not(X)))))))
% 87.79/11.59 = { by lemma 60 R->L }
% 87.79/11.59 not(necessarily(fresh3(fresh115(op_or, kn1, implies(or(necessarily(not(X)), necessarily(not(X))), necessarily(not(X))), equiv(necessarily(not(X)), or(necessarily(not(X)), necessarily(not(X))))), kn1, necessarily(not(X)), not(not(necessarily(not(X)))))))
% 87.79/11.60 = { by lemma 65 R->L }
% 87.79/11.60 not(necessarily(fresh3(fresh115(fresh115(kn2, kn2, implies(necessarily(not(X)), or(necessarily(not(X)), necessarily(not(X)))), implies(implies(or(necessarily(not(X)), necessarily(not(X))), necessarily(not(X))), and(implies(necessarily(not(X)), or(necessarily(not(X)), necessarily(not(X)))), implies(or(necessarily(not(X)), necessarily(not(X))), necessarily(not(X)))))), kn1, implies(or(necessarily(not(X)), necessarily(not(X))), necessarily(not(X))), equiv(necessarily(not(X)), or(necessarily(not(X)), necessarily(not(X))))), kn1, necessarily(not(X)), not(not(necessarily(not(X)))))))
% 87.79/11.60 = { by lemma 60 R->L }
% 87.79/11.60 not(necessarily(fresh3(fresh115(fresh115(kn2, op_or, implies(necessarily(not(X)), or(necessarily(not(X)), necessarily(not(X)))), implies(implies(or(necessarily(not(X)), necessarily(not(X))), necessarily(not(X))), and(implies(necessarily(not(X)), or(necessarily(not(X)), necessarily(not(X)))), implies(or(necessarily(not(X)), necessarily(not(X))), necessarily(not(X)))))), kn1, implies(or(necessarily(not(X)), necessarily(not(X))), necessarily(not(X))), equiv(necessarily(not(X)), or(necessarily(not(X)), necessarily(not(X))))), kn1, necessarily(not(X)), not(not(necessarily(not(X)))))))
% 87.79/11.60 = { by lemma 62 R->L }
% 87.79/11.60 not(necessarily(fresh3(fresh115(fresh115(is_a_theorem(implies(implies(necessarily(not(X)), or(necessarily(not(X)), necessarily(not(X)))), implies(implies(or(necessarily(not(X)), necessarily(not(X))), necessarily(not(X))), and(implies(necessarily(not(X)), or(necessarily(not(X)), necessarily(not(X)))), implies(or(necessarily(not(X)), necessarily(not(X))), necessarily(not(X))))))), op_or, implies(necessarily(not(X)), or(necessarily(not(X)), necessarily(not(X)))), implies(implies(or(necessarily(not(X)), necessarily(not(X))), necessarily(not(X))), and(implies(necessarily(not(X)), or(necessarily(not(X)), necessarily(not(X)))), implies(or(necessarily(not(X)), necessarily(not(X))), necessarily(not(X)))))), kn1, implies(or(necessarily(not(X)), necessarily(not(X))), necessarily(not(X))), equiv(necessarily(not(X)), or(necessarily(not(X)), necessarily(not(X))))), kn1, necessarily(not(X)), not(not(necessarily(not(X)))))))
% 87.79/11.60 = { by lemma 63 }
% 87.79/11.60 not(necessarily(fresh3(fresh115(fresh40(is_a_theorem(implies(necessarily(not(X)), or(necessarily(not(X)), necessarily(not(X))))), op_or, implies(implies(or(necessarily(not(X)), necessarily(not(X))), necessarily(not(X))), and(implies(necessarily(not(X)), or(necessarily(not(X)), necessarily(not(X)))), implies(or(necessarily(not(X)), necessarily(not(X))), necessarily(not(X)))))), kn1, implies(or(necessarily(not(X)), necessarily(not(X))), necessarily(not(X))), equiv(necessarily(not(X)), or(necessarily(not(X)), necessarily(not(X))))), kn1, necessarily(not(X)), not(not(necessarily(not(X)))))))
% 87.79/11.60 = { by lemma 60 }
% 87.79/11.60 not(necessarily(fresh3(fresh115(fresh40(is_a_theorem(implies(necessarily(not(X)), or(necessarily(not(X)), necessarily(not(X))))), kn2, implies(implies(or(necessarily(not(X)), necessarily(not(X))), necessarily(not(X))), and(implies(necessarily(not(X)), or(necessarily(not(X)), necessarily(not(X)))), implies(or(necessarily(not(X)), necessarily(not(X))), necessarily(not(X)))))), kn1, implies(or(necessarily(not(X)), necessarily(not(X))), necessarily(not(X))), equiv(necessarily(not(X)), or(necessarily(not(X)), necessarily(not(X))))), kn1, necessarily(not(X)), not(not(necessarily(not(X)))))))
% 87.79/11.60 = { by axiom 41 (or_1_1) R->L }
% 87.79/11.60 not(necessarily(fresh3(fresh115(fresh40(fresh21(or_1, true, necessarily(not(X)), necessarily(not(X))), kn2, implies(implies(or(necessarily(not(X)), necessarily(not(X))), necessarily(not(X))), and(implies(necessarily(not(X)), or(necessarily(not(X)), necessarily(not(X)))), implies(or(necessarily(not(X)), necessarily(not(X))), necessarily(not(X)))))), kn1, implies(or(necessarily(not(X)), necessarily(not(X))), necessarily(not(X))), equiv(necessarily(not(X)), or(necessarily(not(X)), necessarily(not(X))))), kn1, necessarily(not(X)), not(not(necessarily(not(X)))))))
% 87.79/11.60 = { by axiom 12 (hilbert_or_1) }
% 87.79/11.60 not(necessarily(fresh3(fresh115(fresh40(fresh21(true, true, necessarily(not(X)), necessarily(not(X))), kn2, implies(implies(or(necessarily(not(X)), necessarily(not(X))), necessarily(not(X))), and(implies(necessarily(not(X)), or(necessarily(not(X)), necessarily(not(X)))), implies(or(necessarily(not(X)), necessarily(not(X))), necessarily(not(X)))))), kn1, implies(or(necessarily(not(X)), necessarily(not(X))), necessarily(not(X))), equiv(necessarily(not(X)), or(necessarily(not(X)), necessarily(not(X))))), kn1, necessarily(not(X)), not(not(necessarily(not(X)))))))
% 87.79/11.60 = { by axiom 1 (hilbert_op_or) R->L }
% 87.79/11.60 not(necessarily(fresh3(fresh115(fresh40(fresh21(op_or, true, necessarily(not(X)), necessarily(not(X))), kn2, implies(implies(or(necessarily(not(X)), necessarily(not(X))), necessarily(not(X))), and(implies(necessarily(not(X)), or(necessarily(not(X)), necessarily(not(X)))), implies(or(necessarily(not(X)), necessarily(not(X))), necessarily(not(X)))))), kn1, implies(or(necessarily(not(X)), necessarily(not(X))), necessarily(not(X))), equiv(necessarily(not(X)), or(necessarily(not(X)), necessarily(not(X))))), kn1, necessarily(not(X)), not(not(necessarily(not(X)))))))
% 87.79/11.60 = { by axiom 1 (hilbert_op_or) R->L }
% 87.79/11.60 not(necessarily(fresh3(fresh115(fresh40(fresh21(op_or, op_or, necessarily(not(X)), necessarily(not(X))), kn2, implies(implies(or(necessarily(not(X)), necessarily(not(X))), necessarily(not(X))), and(implies(necessarily(not(X)), or(necessarily(not(X)), necessarily(not(X)))), implies(or(necessarily(not(X)), necessarily(not(X))), necessarily(not(X)))))), kn1, implies(or(necessarily(not(X)), necessarily(not(X))), necessarily(not(X))), equiv(necessarily(not(X)), or(necessarily(not(X)), necessarily(not(X))))), kn1, necessarily(not(X)), not(not(necessarily(not(X)))))))
% 87.79/11.60 = { by axiom 35 (or_1_1) }
% 87.79/11.60 not(necessarily(fresh3(fresh115(fresh40(true, kn2, implies(implies(or(necessarily(not(X)), necessarily(not(X))), necessarily(not(X))), and(implies(necessarily(not(X)), or(necessarily(not(X)), necessarily(not(X)))), implies(or(necessarily(not(X)), necessarily(not(X))), necessarily(not(X)))))), kn1, implies(or(necessarily(not(X)), necessarily(not(X))), necessarily(not(X))), equiv(necessarily(not(X)), or(necessarily(not(X)), necessarily(not(X))))), kn1, necessarily(not(X)), not(not(necessarily(not(X)))))))
% 87.79/11.60 = { by axiom 1 (hilbert_op_or) R->L }
% 87.79/11.60 not(necessarily(fresh3(fresh115(fresh40(op_or, kn2, implies(implies(or(necessarily(not(X)), necessarily(not(X))), necessarily(not(X))), and(implies(necessarily(not(X)), or(necessarily(not(X)), necessarily(not(X)))), implies(or(necessarily(not(X)), necessarily(not(X))), necessarily(not(X)))))), kn1, implies(or(necessarily(not(X)), necessarily(not(X))), necessarily(not(X))), equiv(necessarily(not(X)), or(necessarily(not(X)), necessarily(not(X))))), kn1, necessarily(not(X)), not(not(necessarily(not(X)))))))
% 87.79/11.60 = { by lemma 60 }
% 87.79/11.60 not(necessarily(fresh3(fresh115(fresh40(kn2, kn2, implies(implies(or(necessarily(not(X)), necessarily(not(X))), necessarily(not(X))), and(implies(necessarily(not(X)), or(necessarily(not(X)), necessarily(not(X)))), implies(or(necessarily(not(X)), necessarily(not(X))), necessarily(not(X)))))), kn1, implies(or(necessarily(not(X)), necessarily(not(X))), necessarily(not(X))), equiv(necessarily(not(X)), or(necessarily(not(X)), necessarily(not(X))))), kn1, necessarily(not(X)), not(not(necessarily(not(X)))))))
% 87.79/11.60 = { by axiom 23 (modus_ponens_2) }
% 87.79/11.60 not(necessarily(fresh3(fresh115(is_a_theorem(implies(implies(or(necessarily(not(X)), necessarily(not(X))), necessarily(not(X))), and(implies(necessarily(not(X)), or(necessarily(not(X)), necessarily(not(X)))), implies(or(necessarily(not(X)), necessarily(not(X))), necessarily(not(X)))))), kn1, implies(or(necessarily(not(X)), necessarily(not(X))), necessarily(not(X))), equiv(necessarily(not(X)), or(necessarily(not(X)), necessarily(not(X))))), kn1, necessarily(not(X)), not(not(necessarily(not(X)))))))
% 87.79/11.60 = { by lemma 70 }
% 87.79/11.60 not(necessarily(fresh3(fresh115(is_a_theorem(implies(implies(or(necessarily(not(X)), necessarily(not(X))), necessarily(not(X))), equiv(necessarily(not(X)), or(necessarily(not(X)), necessarily(not(X)))))), kn1, implies(or(necessarily(not(X)), necessarily(not(X))), necessarily(not(X))), equiv(necessarily(not(X)), or(necessarily(not(X)), necessarily(not(X))))), kn1, necessarily(not(X)), not(not(necessarily(not(X)))))))
% 87.79/11.60 = { by lemma 71 }
% 87.79/11.60 not(necessarily(fresh3(fresh40(is_a_theorem(implies(or(necessarily(not(X)), necessarily(not(X))), necessarily(not(X)))), kn1, equiv(necessarily(not(X)), or(necessarily(not(X)), necessarily(not(X))))), kn1, necessarily(not(X)), not(not(necessarily(not(X)))))))
% 87.79/11.60 = { by lemma 77 }
% 87.79/11.60 not(necessarily(fresh3(fresh40(is_a_theorem(implies(not(not(necessarily(not(X)))), necessarily(not(X)))), kn1, equiv(necessarily(not(X)), or(necessarily(not(X)), necessarily(not(X))))), kn1, necessarily(not(X)), not(not(necessarily(not(X)))))))
% 87.79/11.60 = { by axiom 23 (modus_ponens_2) R->L }
% 87.79/11.61 not(necessarily(fresh3(fresh40(fresh40(kn1, kn1, implies(not(not(necessarily(not(X)))), necessarily(not(X)))), kn1, equiv(necessarily(not(X)), or(necessarily(not(X)), necessarily(not(X))))), kn1, necessarily(not(X)), not(not(necessarily(not(X)))))))
% 87.79/11.61 = { by lemma 67 R->L }
% 87.79/11.61 not(necessarily(fresh3(fresh40(fresh40(kn2, kn1, implies(not(not(necessarily(not(X)))), necessarily(not(X)))), kn1, equiv(necessarily(not(X)), or(necessarily(not(X)), necessarily(not(X))))), kn1, necessarily(not(X)), not(not(necessarily(not(X)))))))
% 87.79/11.61 = { by lemma 60 R->L }
% 87.79/11.61 not(necessarily(fresh3(fresh40(fresh40(op_or, kn1, implies(not(not(necessarily(not(X)))), necessarily(not(X)))), kn1, equiv(necessarily(not(X)), or(necessarily(not(X)), necessarily(not(X))))), kn1, necessarily(not(X)), not(not(necessarily(not(X)))))))
% 87.79/11.61 = { by axiom 1 (hilbert_op_or) }
% 87.79/11.61 not(necessarily(fresh3(fresh40(fresh40(true, kn1, implies(not(not(necessarily(not(X)))), necessarily(not(X)))), kn1, equiv(necessarily(not(X)), or(necessarily(not(X)), necessarily(not(X))))), kn1, necessarily(not(X)), not(not(necessarily(not(X)))))))
% 87.79/11.61 = { by axiom 29 (implies_1_1) R->L }
% 87.79/11.61 not(necessarily(fresh3(fresh40(fresh40(fresh51(op_or, op_or, not(necessarily(not(X))), not(not(necessarily(not(X))))), kn1, implies(not(not(necessarily(not(X)))), necessarily(not(X)))), kn1, equiv(necessarily(not(X)), or(necessarily(not(X)), necessarily(not(X))))), kn1, necessarily(not(X)), not(not(necessarily(not(X)))))))
% 87.79/11.61 = { by axiom 1 (hilbert_op_or) }
% 87.79/11.61 not(necessarily(fresh3(fresh40(fresh40(fresh51(op_or, true, not(necessarily(not(X))), not(not(necessarily(not(X))))), kn1, implies(not(not(necessarily(not(X)))), necessarily(not(X)))), kn1, equiv(necessarily(not(X)), or(necessarily(not(X)), necessarily(not(X))))), kn1, necessarily(not(X)), not(not(necessarily(not(X)))))))
% 87.79/11.61 = { by axiom 1 (hilbert_op_or) }
% 87.79/11.61 not(necessarily(fresh3(fresh40(fresh40(fresh51(true, true, not(necessarily(not(X))), not(not(necessarily(not(X))))), kn1, implies(not(not(necessarily(not(X)))), necessarily(not(X)))), kn1, equiv(necessarily(not(X)), or(necessarily(not(X)), necessarily(not(X))))), kn1, necessarily(not(X)), not(not(necessarily(not(X)))))))
% 87.79/11.61 = { by axiom 8 (hilbert_implies_1) R->L }
% 87.79/11.61 not(necessarily(fresh3(fresh40(fresh40(fresh51(implies_1, true, not(necessarily(not(X))), not(not(necessarily(not(X))))), kn1, implies(not(not(necessarily(not(X)))), necessarily(not(X)))), kn1, equiv(necessarily(not(X)), or(necessarily(not(X)), necessarily(not(X))))), kn1, necessarily(not(X)), not(not(necessarily(not(X)))))))
% 87.79/11.61 = { by axiom 44 (implies_1_1) }
% 87.79/11.61 not(necessarily(fresh3(fresh40(fresh40(is_a_theorem(implies(not(necessarily(not(X))), implies(not(not(necessarily(not(X)))), not(necessarily(not(X)))))), kn1, implies(not(not(necessarily(not(X)))), necessarily(not(X)))), kn1, equiv(necessarily(not(X)), or(necessarily(not(X)), necessarily(not(X))))), kn1, necessarily(not(X)), not(not(necessarily(not(X)))))))
% 87.79/11.61 = { by lemma 75 }
% 87.79/11.61 not(necessarily(fresh3(fresh40(fresh40(is_a_theorem(or(necessarily(not(X)), implies(not(not(necessarily(not(X)))), not(necessarily(not(X)))))), kn1, implies(not(not(necessarily(not(X)))), necessarily(not(X)))), kn1, equiv(necessarily(not(X)), or(necessarily(not(X)), necessarily(not(X))))), kn1, necessarily(not(X)), not(not(necessarily(not(X)))))))
% 87.79/11.61 = { by lemma 76 }
% 87.79/11.61 not(necessarily(fresh3(fresh40(fresh40(is_a_theorem(or(necessarily(not(X)), not(not(not(necessarily(not(X))))))), kn1, implies(not(not(necessarily(not(X)))), necessarily(not(X)))), kn1, equiv(necessarily(not(X)), or(necessarily(not(X)), necessarily(not(X))))), kn1, necessarily(not(X)), not(not(necessarily(not(X)))))))
% 87.79/11.61 = { by lemma 67 R->L }
% 87.79/11.61 not(necessarily(fresh3(fresh40(fresh40(is_a_theorem(or(necessarily(not(X)), not(not(not(necessarily(not(X))))))), kn2, implies(not(not(necessarily(not(X)))), necessarily(not(X)))), kn1, equiv(necessarily(not(X)), or(necessarily(not(X)), necessarily(not(X))))), kn1, necessarily(not(X)), not(not(necessarily(not(X)))))))
% 87.79/11.61 = { by lemma 60 R->L }
% 87.79/11.61 not(necessarily(fresh3(fresh40(fresh40(is_a_theorem(or(necessarily(not(X)), not(not(not(necessarily(not(X))))))), op_or, implies(not(not(necessarily(not(X)))), necessarily(not(X)))), kn1, equiv(necessarily(not(X)), or(necessarily(not(X)), necessarily(not(X))))), kn1, necessarily(not(X)), not(not(necessarily(not(X)))))))
% 87.79/11.61 = { by lemma 63 R->L }
% 87.79/11.61 not(necessarily(fresh3(fresh40(fresh115(is_a_theorem(implies(or(necessarily(not(X)), not(not(not(necessarily(not(X)))))), implies(not(not(necessarily(not(X)))), necessarily(not(X))))), op_or, or(necessarily(not(X)), not(not(not(necessarily(not(X)))))), implies(not(not(necessarily(not(X)))), necessarily(not(X)))), kn1, equiv(necessarily(not(X)), or(necessarily(not(X)), necessarily(not(X))))), kn1, necessarily(not(X)), not(not(necessarily(not(X)))))))
% 87.79/11.61 = { by lemma 75 R->L }
% 87.79/11.61 not(necessarily(fresh3(fresh40(fresh115(is_a_theorem(implies(implies(not(necessarily(not(X))), not(not(not(necessarily(not(X)))))), implies(not(not(necessarily(not(X)))), necessarily(not(X))))), op_or, or(necessarily(not(X)), not(not(not(necessarily(not(X)))))), implies(not(not(necessarily(not(X)))), necessarily(not(X)))), kn1, equiv(necessarily(not(X)), or(necessarily(not(X)), necessarily(not(X))))), kn1, necessarily(not(X)), not(not(necessarily(not(X)))))))
% 87.79/11.61 = { by axiom 56 (modus_tollens_1) R->L }
% 87.79/11.61 not(necessarily(fresh3(fresh40(fresh115(fresh35(modus_tollens, true, not(not(necessarily(not(X)))), necessarily(not(X))), op_or, or(necessarily(not(X)), not(not(not(necessarily(not(X)))))), implies(not(not(necessarily(not(X)))), necessarily(not(X)))), kn1, equiv(necessarily(not(X)), or(necessarily(not(X)), necessarily(not(X))))), kn1, necessarily(not(X)), not(not(necessarily(not(X)))))))
% 87.79/11.61 = { by axiom 7 (hilbert_modus_tollens) }
% 87.79/11.61 not(necessarily(fresh3(fresh40(fresh115(fresh35(true, true, not(not(necessarily(not(X)))), necessarily(not(X))), op_or, or(necessarily(not(X)), not(not(not(necessarily(not(X)))))), implies(not(not(necessarily(not(X)))), necessarily(not(X)))), kn1, equiv(necessarily(not(X)), or(necessarily(not(X)), necessarily(not(X))))), kn1, necessarily(not(X)), not(not(necessarily(not(X)))))))
% 87.79/11.61 = { by axiom 1 (hilbert_op_or) R->L }
% 87.79/11.61 not(necessarily(fresh3(fresh40(fresh115(fresh35(op_or, true, not(not(necessarily(not(X)))), necessarily(not(X))), op_or, or(necessarily(not(X)), not(not(not(necessarily(not(X)))))), implies(not(not(necessarily(not(X)))), necessarily(not(X)))), kn1, equiv(necessarily(not(X)), or(necessarily(not(X)), necessarily(not(X))))), kn1, necessarily(not(X)), not(not(necessarily(not(X)))))))
% 87.79/11.61 = { by lemma 60 }
% 87.79/11.61 not(necessarily(fresh3(fresh40(fresh115(fresh35(kn2, true, not(not(necessarily(not(X)))), necessarily(not(X))), op_or, or(necessarily(not(X)), not(not(not(necessarily(not(X)))))), implies(not(not(necessarily(not(X)))), necessarily(not(X)))), kn1, equiv(necessarily(not(X)), or(necessarily(not(X)), necessarily(not(X))))), kn1, necessarily(not(X)), not(not(necessarily(not(X)))))))
% 87.79/11.61 = { by axiom 1 (hilbert_op_or) R->L }
% 87.79/11.61 not(necessarily(fresh3(fresh40(fresh115(fresh35(kn2, op_or, not(not(necessarily(not(X)))), necessarily(not(X))), op_or, or(necessarily(not(X)), not(not(not(necessarily(not(X)))))), implies(not(not(necessarily(not(X)))), necessarily(not(X)))), kn1, equiv(necessarily(not(X)), or(necessarily(not(X)), necessarily(not(X))))), kn1, necessarily(not(X)), not(not(necessarily(not(X)))))))
% 87.79/11.61 = { by lemma 60 }
% 87.79/11.61 not(necessarily(fresh3(fresh40(fresh115(fresh35(kn2, kn2, not(not(necessarily(not(X)))), necessarily(not(X))), op_or, or(necessarily(not(X)), not(not(not(necessarily(not(X)))))), implies(not(not(necessarily(not(X)))), necessarily(not(X)))), kn1, equiv(necessarily(not(X)), or(necessarily(not(X)), necessarily(not(X))))), kn1, necessarily(not(X)), not(not(necessarily(not(X)))))))
% 87.79/11.61 = { by axiom 31 (modus_tollens_1) }
% 87.79/11.61 not(necessarily(fresh3(fresh40(fresh115(true, op_or, or(necessarily(not(X)), not(not(not(necessarily(not(X)))))), implies(not(not(necessarily(not(X)))), necessarily(not(X)))), kn1, equiv(necessarily(not(X)), or(necessarily(not(X)), necessarily(not(X))))), kn1, necessarily(not(X)), not(not(necessarily(not(X)))))))
% 87.79/11.61 = { by axiom 1 (hilbert_op_or) R->L }
% 87.79/11.61 not(necessarily(fresh3(fresh40(fresh115(op_or, op_or, or(necessarily(not(X)), not(not(not(necessarily(not(X)))))), implies(not(not(necessarily(not(X)))), necessarily(not(X)))), kn1, equiv(necessarily(not(X)), or(necessarily(not(X)), necessarily(not(X))))), kn1, necessarily(not(X)), not(not(necessarily(not(X)))))))
% 87.79/11.61 = { by lemma 60 }
% 87.79/11.62 not(necessarily(fresh3(fresh40(fresh115(kn2, op_or, or(necessarily(not(X)), not(not(not(necessarily(not(X)))))), implies(not(not(necessarily(not(X)))), necessarily(not(X)))), kn1, equiv(necessarily(not(X)), or(necessarily(not(X)), necessarily(not(X))))), kn1, necessarily(not(X)), not(not(necessarily(not(X)))))))
% 87.79/11.62 = { by lemma 67 }
% 87.79/11.62 not(necessarily(fresh3(fresh40(fresh115(kn1, op_or, or(necessarily(not(X)), not(not(not(necessarily(not(X)))))), implies(not(not(necessarily(not(X)))), necessarily(not(X)))), kn1, equiv(necessarily(not(X)), or(necessarily(not(X)), necessarily(not(X))))), kn1, necessarily(not(X)), not(not(necessarily(not(X)))))))
% 87.79/11.62 = { by lemma 60 }
% 87.79/11.62 not(necessarily(fresh3(fresh40(fresh115(kn1, kn2, or(necessarily(not(X)), not(not(not(necessarily(not(X)))))), implies(not(not(necessarily(not(X)))), necessarily(not(X)))), kn1, equiv(necessarily(not(X)), or(necessarily(not(X)), necessarily(not(X))))), kn1, necessarily(not(X)), not(not(necessarily(not(X)))))))
% 87.79/11.62 = { by lemma 67 }
% 87.79/11.62 not(necessarily(fresh3(fresh40(fresh115(kn1, kn1, or(necessarily(not(X)), not(not(not(necessarily(not(X)))))), implies(not(not(necessarily(not(X)))), necessarily(not(X)))), kn1, equiv(necessarily(not(X)), or(necessarily(not(X)), necessarily(not(X))))), kn1, necessarily(not(X)), not(not(necessarily(not(X)))))))
% 87.79/11.62 = { by lemma 65 }
% 87.79/11.62 not(necessarily(fresh3(fresh40(op_or, kn1, equiv(necessarily(not(X)), or(necessarily(not(X)), necessarily(not(X))))), kn1, necessarily(not(X)), not(not(necessarily(not(X)))))))
% 87.79/11.62 = { by lemma 60 }
% 87.79/11.62 not(necessarily(fresh3(fresh40(kn2, kn1, equiv(necessarily(not(X)), or(necessarily(not(X)), necessarily(not(X))))), kn1, necessarily(not(X)), not(not(necessarily(not(X)))))))
% 87.79/11.62 = { by lemma 67 }
% 87.79/11.62 not(necessarily(fresh3(fresh40(kn1, kn1, equiv(necessarily(not(X)), or(necessarily(not(X)), necessarily(not(X))))), kn1, necessarily(not(X)), not(not(necessarily(not(X)))))))
% 87.79/11.62 = { by axiom 23 (modus_ponens_2) }
% 87.79/11.62 not(necessarily(fresh3(is_a_theorem(equiv(necessarily(not(X)), or(necessarily(not(X)), necessarily(not(X))))), kn1, necessarily(not(X)), not(not(necessarily(not(X)))))))
% 87.79/11.62 = { by lemma 77 }
% 87.79/11.62 not(necessarily(fresh3(is_a_theorem(equiv(necessarily(not(X)), not(not(necessarily(not(X)))))), kn1, necessarily(not(X)), not(not(necessarily(not(X)))))))
% 87.79/11.62 = { by lemma 72 }
% 87.79/11.62 not(necessarily(necessarily(not(X))))
% 87.79/11.62 = { by axiom 37 (substitution_of_equivalents_2) R->L }
% 87.79/11.62 not(fresh3(kn1, kn1, necessarily(not(X)), necessarily(necessarily(not(X)))))
% 87.79/11.62 = { by lemma 68 R->L }
% 87.79/11.62 not(fresh3(fresh115(kn1, kn1, implies(necessarily(necessarily(not(X))), necessarily(not(X))), equiv(necessarily(not(X)), necessarily(necessarily(not(X))))), kn1, necessarily(not(X)), necessarily(necessarily(not(X)))))
% 87.79/11.62 = { by lemma 69 R->L }
% 87.79/11.62 not(fresh3(fresh115(fresh40(is_a_theorem(implies(necessarily(not(X)), necessarily(necessarily(not(X))))), kn1, implies(implies(necessarily(necessarily(not(X))), necessarily(not(X))), and(implies(necessarily(not(X)), necessarily(necessarily(not(X)))), implies(necessarily(necessarily(not(X))), necessarily(not(X)))))), kn1, implies(necessarily(necessarily(not(X))), necessarily(not(X))), equiv(necessarily(not(X)), necessarily(necessarily(not(X))))), kn1, necessarily(not(X)), necessarily(necessarily(not(X)))))
% 87.79/11.62 = { by axiom 48 (axiom_4_1) R->L }
% 87.79/11.62 not(fresh3(fresh115(fresh40(fresh101(axiom_4, true, not(X)), kn1, implies(implies(necessarily(necessarily(not(X))), necessarily(not(X))), and(implies(necessarily(not(X)), necessarily(necessarily(not(X)))), implies(necessarily(necessarily(not(X))), necessarily(not(X)))))), kn1, implies(necessarily(necessarily(not(X))), necessarily(not(X))), equiv(necessarily(not(X)), necessarily(necessarily(not(X))))), kn1, necessarily(not(X)), necessarily(necessarily(not(X)))))
% 87.79/11.62 = { by axiom 14 (km4b_axiom_4) }
% 87.79/11.62 not(fresh3(fresh115(fresh40(fresh101(true, true, not(X)), kn1, implies(implies(necessarily(necessarily(not(X))), necessarily(not(X))), and(implies(necessarily(not(X)), necessarily(necessarily(not(X)))), implies(necessarily(necessarily(not(X))), necessarily(not(X)))))), kn1, implies(necessarily(necessarily(not(X))), necessarily(not(X))), equiv(necessarily(not(X)), necessarily(necessarily(not(X))))), kn1, necessarily(not(X)), necessarily(necessarily(not(X)))))
% 87.79/11.62 = { by axiom 1 (hilbert_op_or) R->L }
% 87.79/11.62 not(fresh3(fresh115(fresh40(fresh101(op_or, true, not(X)), kn1, implies(implies(necessarily(necessarily(not(X))), necessarily(not(X))), and(implies(necessarily(not(X)), necessarily(necessarily(not(X)))), implies(necessarily(necessarily(not(X))), necessarily(not(X)))))), kn1, implies(necessarily(necessarily(not(X))), necessarily(not(X))), equiv(necessarily(not(X)), necessarily(necessarily(not(X))))), kn1, necessarily(not(X)), necessarily(necessarily(not(X)))))
% 87.79/11.62 = { by axiom 1 (hilbert_op_or) R->L }
% 87.79/11.62 not(fresh3(fresh115(fresh40(fresh101(op_or, op_or, not(X)), kn1, implies(implies(necessarily(necessarily(not(X))), necessarily(not(X))), and(implies(necessarily(not(X)), necessarily(necessarily(not(X)))), implies(necessarily(necessarily(not(X))), necessarily(not(X)))))), kn1, implies(necessarily(necessarily(not(X))), necessarily(not(X))), equiv(necessarily(not(X)), necessarily(necessarily(not(X))))), kn1, necessarily(not(X)), necessarily(necessarily(not(X)))))
% 87.79/11.62 = { by axiom 20 (axiom_4_1) }
% 87.79/11.62 not(fresh3(fresh115(fresh40(true, kn1, implies(implies(necessarily(necessarily(not(X))), necessarily(not(X))), and(implies(necessarily(not(X)), necessarily(necessarily(not(X)))), implies(necessarily(necessarily(not(X))), necessarily(not(X)))))), kn1, implies(necessarily(necessarily(not(X))), necessarily(not(X))), equiv(necessarily(not(X)), necessarily(necessarily(not(X))))), kn1, necessarily(not(X)), necessarily(necessarily(not(X)))))
% 87.79/11.62 = { by axiom 1 (hilbert_op_or) R->L }
% 87.79/11.62 not(fresh3(fresh115(fresh40(op_or, kn1, implies(implies(necessarily(necessarily(not(X))), necessarily(not(X))), and(implies(necessarily(not(X)), necessarily(necessarily(not(X)))), implies(necessarily(necessarily(not(X))), necessarily(not(X)))))), kn1, implies(necessarily(necessarily(not(X))), necessarily(not(X))), equiv(necessarily(not(X)), necessarily(necessarily(not(X))))), kn1, necessarily(not(X)), necessarily(necessarily(not(X)))))
% 87.79/11.62 = { by lemma 60 }
% 87.79/11.62 not(fresh3(fresh115(fresh40(kn2, kn1, implies(implies(necessarily(necessarily(not(X))), necessarily(not(X))), and(implies(necessarily(not(X)), necessarily(necessarily(not(X)))), implies(necessarily(necessarily(not(X))), necessarily(not(X)))))), kn1, implies(necessarily(necessarily(not(X))), necessarily(not(X))), equiv(necessarily(not(X)), necessarily(necessarily(not(X))))), kn1, necessarily(not(X)), necessarily(necessarily(not(X)))))
% 87.79/11.62 = { by lemma 67 }
% 87.79/11.62 not(fresh3(fresh115(fresh40(kn1, kn1, implies(implies(necessarily(necessarily(not(X))), necessarily(not(X))), and(implies(necessarily(not(X)), necessarily(necessarily(not(X)))), implies(necessarily(necessarily(not(X))), necessarily(not(X)))))), kn1, implies(necessarily(necessarily(not(X))), necessarily(not(X))), equiv(necessarily(not(X)), necessarily(necessarily(not(X))))), kn1, necessarily(not(X)), necessarily(necessarily(not(X)))))
% 87.79/11.62 = { by axiom 23 (modus_ponens_2) }
% 87.79/11.62 not(fresh3(fresh115(is_a_theorem(implies(implies(necessarily(necessarily(not(X))), necessarily(not(X))), and(implies(necessarily(not(X)), necessarily(necessarily(not(X)))), implies(necessarily(necessarily(not(X))), necessarily(not(X)))))), kn1, implies(necessarily(necessarily(not(X))), necessarily(not(X))), equiv(necessarily(not(X)), necessarily(necessarily(not(X))))), kn1, necessarily(not(X)), necessarily(necessarily(not(X)))))
% 87.79/11.62 = { by lemma 70 }
% 87.79/11.62 not(fresh3(fresh115(is_a_theorem(implies(implies(necessarily(necessarily(not(X))), necessarily(not(X))), equiv(necessarily(not(X)), necessarily(necessarily(not(X)))))), kn1, implies(necessarily(necessarily(not(X))), necessarily(not(X))), equiv(necessarily(not(X)), necessarily(necessarily(not(X))))), kn1, necessarily(not(X)), necessarily(necessarily(not(X)))))
% 87.79/11.62 = { by lemma 71 }
% 87.79/11.62 not(fresh3(fresh40(is_a_theorem(implies(necessarily(necessarily(not(X))), necessarily(not(X)))), kn1, equiv(necessarily(not(X)), necessarily(necessarily(not(X))))), kn1, necessarily(not(X)), necessarily(necessarily(not(X)))))
% 87.79/11.62 = { by axiom 39 (axiom_M_1) R->L }
% 87.79/11.62 not(fresh3(fresh40(fresh93(axiom_M, true, necessarily(not(X))), kn1, equiv(necessarily(not(X)), necessarily(necessarily(not(X))))), kn1, necessarily(not(X)), necessarily(necessarily(not(X)))))
% 87.79/11.62 = { by axiom 13 (km4b_axiom_M) }
% 87.79/11.62 not(fresh3(fresh40(fresh93(true, true, necessarily(not(X))), kn1, equiv(necessarily(not(X)), necessarily(necessarily(not(X))))), kn1, necessarily(not(X)), necessarily(necessarily(not(X)))))
% 87.79/11.62 = { by axiom 1 (hilbert_op_or) R->L }
% 87.79/11.62 not(fresh3(fresh40(fresh93(op_or, true, necessarily(not(X))), kn1, equiv(necessarily(not(X)), necessarily(necessarily(not(X))))), kn1, necessarily(not(X)), necessarily(necessarily(not(X)))))
% 87.79/11.62 = { by axiom 1 (hilbert_op_or) R->L }
% 87.79/11.62 not(fresh3(fresh40(fresh93(op_or, op_or, necessarily(not(X))), kn1, equiv(necessarily(not(X)), necessarily(necessarily(not(X))))), kn1, necessarily(not(X)), necessarily(necessarily(not(X)))))
% 87.79/11.62 = { by axiom 22 (axiom_M_1) }
% 87.79/11.62 not(fresh3(fresh40(true, kn1, equiv(necessarily(not(X)), necessarily(necessarily(not(X))))), kn1, necessarily(not(X)), necessarily(necessarily(not(X)))))
% 87.79/11.62 = { by axiom 1 (hilbert_op_or) R->L }
% 87.79/11.62 not(fresh3(fresh40(op_or, kn1, equiv(necessarily(not(X)), necessarily(necessarily(not(X))))), kn1, necessarily(not(X)), necessarily(necessarily(not(X)))))
% 87.79/11.62 = { by lemma 60 }
% 87.79/11.62 not(fresh3(fresh40(kn2, kn1, equiv(necessarily(not(X)), necessarily(necessarily(not(X))))), kn1, necessarily(not(X)), necessarily(necessarily(not(X)))))
% 87.79/11.62 = { by lemma 67 }
% 87.79/11.62 not(fresh3(fresh40(kn1, kn1, equiv(necessarily(not(X)), necessarily(necessarily(not(X))))), kn1, necessarily(not(X)), necessarily(necessarily(not(X)))))
% 87.79/11.62 = { by axiom 23 (modus_ponens_2) }
% 87.79/11.62 not(fresh3(is_a_theorem(equiv(necessarily(not(X)), necessarily(necessarily(not(X))))), kn1, necessarily(not(X)), necessarily(necessarily(not(X)))))
% 87.79/11.62 = { by lemma 72 }
% 87.79/11.62 not(necessarily(not(X)))
% 87.79/11.62 = { by lemma 78 }
% 87.79/11.63 possibly(X)
% 87.79/11.63
% 87.79/11.63 Goal 1 (km5_axiom_5): axiom_5 = true.
% 87.79/11.63 Proof:
% 87.79/11.63 axiom_5
% 87.79/11.63 = { by axiom 54 (axiom_5) R->L }
% 87.79/11.63 fresh100(is_a_theorem(implies(possibly(x12), necessarily(possibly(x12)))), true)
% 87.79/11.63 = { by axiom 47 (axiom_5_1) R->L }
% 87.79/11.63 fresh100(fresh99(axiom_5, true, x12), true)
% 87.79/11.63 = { by axiom 1 (hilbert_op_or) R->L }
% 87.79/11.63 fresh100(fresh99(axiom_5, op_or, x12), true)
% 87.79/11.63 = { by lemma 60 }
% 87.79/11.63 fresh100(fresh99(axiom_5, kn2, x12), true)
% 87.79/11.63 = { by axiom 1 (hilbert_op_or) R->L }
% 87.79/11.63 fresh100(fresh99(axiom_5, kn2, x12), op_or)
% 87.79/11.63 = { by lemma 60 }
% 87.79/11.63 fresh100(fresh99(axiom_5, kn2, x12), kn2)
% 87.79/11.63 = { by lemma 67 }
% 87.79/11.63 fresh100(fresh99(axiom_5, kn1, x12), kn2)
% 87.79/11.63 = { by lemma 67 }
% 87.79/11.63 fresh100(fresh99(axiom_5, kn1, x12), kn1)
% 87.79/11.63 = { by lemma 67 R->L }
% 87.79/11.63 fresh100(fresh99(axiom_5, kn2, x12), kn1)
% 87.79/11.63 = { by lemma 60 R->L }
% 87.79/11.63 fresh100(fresh99(axiom_5, op_or, x12), kn1)
% 87.79/11.63 = { by axiom 1 (hilbert_op_or) }
% 87.79/11.63 fresh100(fresh99(axiom_5, true, x12), kn1)
% 87.79/11.63 = { by axiom 47 (axiom_5_1) }
% 87.79/11.63 fresh100(is_a_theorem(implies(possibly(x12), necessarily(possibly(x12)))), kn1)
% 87.79/11.63 = { by lemma 73 R->L }
% 87.79/11.63 fresh100(is_a_theorem(and(implies(possibly(x12), necessarily(possibly(x12))), implies(possibly(x12), necessarily(possibly(x12))))), kn1)
% 87.79/11.63 = { by lemma 79 R->L }
% 87.79/11.63 fresh100(is_a_theorem(and(implies(possibly(x12), necessarily(possibly(x12))), implies(possibly(x12), necessarily(possibly(possibly(x12)))))), kn1)
% 87.79/11.63 = { by lemma 79 R->L }
% 87.79/11.63 fresh100(is_a_theorem(and(implies(possibly(x12), necessarily(possibly(possibly(x12)))), implies(possibly(x12), necessarily(possibly(possibly(x12)))))), kn1)
% 87.79/11.63 = { by axiom 23 (modus_ponens_2) R->L }
% 87.79/11.63 fresh100(fresh40(kn1, kn1, and(implies(possibly(x12), necessarily(possibly(possibly(x12)))), implies(possibly(x12), necessarily(possibly(possibly(x12)))))), kn1)
% 87.79/11.63 = { by lemma 67 R->L }
% 87.79/11.63 fresh100(fresh40(kn2, kn1, and(implies(possibly(x12), necessarily(possibly(possibly(x12)))), implies(possibly(x12), necessarily(possibly(possibly(x12)))))), kn1)
% 87.79/11.63 = { by lemma 60 R->L }
% 87.79/11.63 fresh100(fresh40(op_or, kn1, and(implies(possibly(x12), necessarily(possibly(possibly(x12)))), implies(possibly(x12), necessarily(possibly(possibly(x12)))))), kn1)
% 87.79/11.63 = { by axiom 1 (hilbert_op_or) }
% 87.79/11.63 fresh100(fresh40(true, kn1, and(implies(possibly(x12), necessarily(possibly(possibly(x12)))), implies(possibly(x12), necessarily(possibly(possibly(x12)))))), kn1)
% 87.79/11.63 = { by axiom 21 (axiom_B_1) R->L }
% 87.79/11.63 fresh100(fresh40(fresh97(op_or, op_or, possibly(x12)), kn1, and(implies(possibly(x12), necessarily(possibly(possibly(x12)))), implies(possibly(x12), necessarily(possibly(possibly(x12)))))), kn1)
% 87.79/11.63 = { by axiom 1 (hilbert_op_or) }
% 87.79/11.63 fresh100(fresh40(fresh97(op_or, true, possibly(x12)), kn1, and(implies(possibly(x12), necessarily(possibly(possibly(x12)))), implies(possibly(x12), necessarily(possibly(possibly(x12)))))), kn1)
% 87.79/11.63 = { by axiom 1 (hilbert_op_or) }
% 87.79/11.63 fresh100(fresh40(fresh97(true, true, possibly(x12)), kn1, and(implies(possibly(x12), necessarily(possibly(possibly(x12)))), implies(possibly(x12), necessarily(possibly(possibly(x12)))))), kn1)
% 87.79/11.63 = { by axiom 15 (km4b_axiom_B) R->L }
% 87.79/11.63 fresh100(fresh40(fresh97(axiom_B, true, possibly(x12)), kn1, and(implies(possibly(x12), necessarily(possibly(possibly(x12)))), implies(possibly(x12), necessarily(possibly(possibly(x12)))))), kn1)
% 87.79/11.63 = { by axiom 42 (axiom_B_1) }
% 87.79/11.63 fresh100(fresh40(is_a_theorem(implies(possibly(x12), necessarily(possibly(possibly(x12))))), kn1, and(implies(possibly(x12), necessarily(possibly(possibly(x12)))), implies(possibly(x12), necessarily(possibly(possibly(x12)))))), kn1)
% 87.79/11.63 = { by lemma 74 R->L }
% 87.79/11.63 fresh100(fresh40(is_a_theorem(implies(possibly(x12), necessarily(possibly(possibly(x12))))), kn1, and(not(and(possibly(x12), not(necessarily(possibly(possibly(x12)))))), implies(possibly(x12), necessarily(possibly(possibly(x12)))))), kn1)
% 87.79/11.63 = { by lemma 74 R->L }
% 87.79/11.63 fresh100(fresh40(is_a_theorem(implies(possibly(x12), necessarily(possibly(possibly(x12))))), kn1, and(not(and(possibly(x12), not(necessarily(possibly(possibly(x12)))))), not(and(possibly(x12), not(necessarily(possibly(possibly(x12)))))))), kn1)
% 87.79/11.63 = { by lemma 74 R->L }
% 87.79/11.63 fresh100(fresh40(is_a_theorem(not(and(possibly(x12), not(necessarily(possibly(possibly(x12))))))), kn1, and(not(and(possibly(x12), not(necessarily(possibly(possibly(x12)))))), not(and(possibly(x12), not(necessarily(possibly(possibly(x12)))))))), kn1)
% 87.79/11.63 = { by lemma 67 R->L }
% 87.79/11.63 fresh100(fresh40(is_a_theorem(not(and(possibly(x12), not(necessarily(possibly(possibly(x12))))))), kn2, and(not(and(possibly(x12), not(necessarily(possibly(possibly(x12)))))), not(and(possibly(x12), not(necessarily(possibly(possibly(x12)))))))), kn1)
% 87.79/11.63 = { by lemma 60 R->L }
% 87.79/11.63 fresh100(fresh40(is_a_theorem(not(and(possibly(x12), not(necessarily(possibly(possibly(x12))))))), op_or, and(not(and(possibly(x12), not(necessarily(possibly(possibly(x12)))))), not(and(possibly(x12), not(necessarily(possibly(possibly(x12)))))))), kn1)
% 88.29/11.63 = { by lemma 63 R->L }
% 88.29/11.63 fresh100(fresh115(is_a_theorem(implies(not(and(possibly(x12), not(necessarily(possibly(possibly(x12)))))), and(not(and(possibly(x12), not(necessarily(possibly(possibly(x12)))))), not(and(possibly(x12), not(necessarily(possibly(possibly(x12))))))))), op_or, not(and(possibly(x12), not(necessarily(possibly(possibly(x12)))))), and(not(and(possibly(x12), not(necessarily(possibly(possibly(x12)))))), not(and(possibly(x12), not(necessarily(possibly(possibly(x12)))))))), kn1)
% 88.29/11.63 = { by lemma 75 }
% 88.29/11.63 fresh100(fresh115(is_a_theorem(or(and(possibly(x12), not(necessarily(possibly(possibly(x12))))), and(not(and(possibly(x12), not(necessarily(possibly(possibly(x12)))))), not(and(possibly(x12), not(necessarily(possibly(possibly(x12))))))))), op_or, not(and(possibly(x12), not(necessarily(possibly(possibly(x12)))))), and(not(and(possibly(x12), not(necessarily(possibly(possibly(x12)))))), not(and(possibly(x12), not(necessarily(possibly(possibly(x12)))))))), kn1)
% 88.29/11.63 = { by lemma 60 }
% 88.29/11.63 fresh100(fresh115(is_a_theorem(or(and(possibly(x12), not(necessarily(possibly(possibly(x12))))), and(not(and(possibly(x12), not(necessarily(possibly(possibly(x12)))))), not(and(possibly(x12), not(necessarily(possibly(possibly(x12))))))))), kn2, not(and(possibly(x12), not(necessarily(possibly(possibly(x12)))))), and(not(and(possibly(x12), not(necessarily(possibly(possibly(x12)))))), not(and(possibly(x12), not(necessarily(possibly(possibly(x12)))))))), kn1)
% 88.29/11.63 = { by lemma 67 }
% 88.29/11.63 fresh100(fresh115(is_a_theorem(or(and(possibly(x12), not(necessarily(possibly(possibly(x12))))), and(not(and(possibly(x12), not(necessarily(possibly(possibly(x12)))))), not(and(possibly(x12), not(necessarily(possibly(possibly(x12))))))))), kn1, not(and(possibly(x12), not(necessarily(possibly(possibly(x12)))))), and(not(and(possibly(x12), not(necessarily(possibly(possibly(x12)))))), not(and(possibly(x12), not(necessarily(possibly(possibly(x12)))))))), kn1)
% 88.29/11.63 = { by lemma 75 R->L }
% 88.29/11.63 fresh100(fresh115(is_a_theorem(implies(not(and(possibly(x12), not(necessarily(possibly(possibly(x12)))))), and(not(and(possibly(x12), not(necessarily(possibly(possibly(x12)))))), not(and(possibly(x12), not(necessarily(possibly(possibly(x12))))))))), kn1, not(and(possibly(x12), not(necessarily(possibly(possibly(x12)))))), and(not(and(possibly(x12), not(necessarily(possibly(possibly(x12)))))), not(and(possibly(x12), not(necessarily(possibly(possibly(x12)))))))), kn1)
% 88.29/11.63 = { by lemma 61 }
% 88.29/11.63 fresh100(fresh115(fresh45(kn1, op_or, not(and(possibly(x12), not(necessarily(possibly(possibly(x12))))))), kn1, not(and(possibly(x12), not(necessarily(possibly(possibly(x12)))))), and(not(and(possibly(x12), not(necessarily(possibly(possibly(x12)))))), not(and(possibly(x12), not(necessarily(possibly(possibly(x12)))))))), kn1)
% 88.29/11.63 = { by lemma 60 }
% 88.29/11.63 fresh100(fresh115(fresh45(kn1, kn2, not(and(possibly(x12), not(necessarily(possibly(possibly(x12))))))), kn1, not(and(possibly(x12), not(necessarily(possibly(possibly(x12)))))), and(not(and(possibly(x12), not(necessarily(possibly(possibly(x12)))))), not(and(possibly(x12), not(necessarily(possibly(possibly(x12)))))))), kn1)
% 88.29/11.63 = { by lemma 66 }
% 88.29/11.63 fresh100(fresh115(kn2, kn1, not(and(possibly(x12), not(necessarily(possibly(possibly(x12)))))), and(not(and(possibly(x12), not(necessarily(possibly(possibly(x12)))))), not(and(possibly(x12), not(necessarily(possibly(possibly(x12)))))))), kn1)
% 88.29/11.63 = { by lemma 67 }
% 88.29/11.63 fresh100(fresh115(kn1, kn1, not(and(possibly(x12), not(necessarily(possibly(possibly(x12)))))), and(not(and(possibly(x12), not(necessarily(possibly(possibly(x12)))))), not(and(possibly(x12), not(necessarily(possibly(possibly(x12)))))))), kn1)
% 88.29/11.63 = { by lemma 65 }
% 88.29/11.63 fresh100(op_or, kn1)
% 88.29/11.63 = { by lemma 60 }
% 88.29/11.63 fresh100(kn2, kn1)
% 88.29/11.63 = { by lemma 67 }
% 88.29/11.63 fresh100(kn1, kn1)
% 88.29/11.63 = { by axiom 16 (axiom_5) }
% 88.29/11.63 true
% 88.29/11.63 % SZS output end Proof
% 88.29/11.63
% 88.29/11.63 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------