0.00/0.04 % Problem : theBenchmark.p : TPTP v0.0.0. Released v0.0.0. 0.00/0.04 % Command : twee %s --tstp --casc --quiet --conditional-encoding if --smaller --drop-non-horn 0.03/0.23 % Computer : n171.star.cs.uiowa.edu 0.03/0.23 % Model : x86_64 x86_64 0.03/0.23 % CPU : Intel(R) Xeon(R) CPU E5-2609 0 @ 2.40GHz 0.03/0.23 % Memory : 32218.625MB 0.03/0.23 % OS : Linux 3.10.0-693.2.2.el7.x86_64 0.03/0.23 % CPULimit : 300 0.03/0.23 % DateTime : Sat Jul 14 05:44:40 CDT 2018 0.03/0.23 % CPUTime : 9.62/9.86 % SZS status Theorem 9.62/9.86 9.73/9.91 % SZS output start Proof 9.73/9.91 Take the following subset of the input axioms: 13.33/13.55 fof(and_1, axiom, 13.33/13.55 and_1 <=> ![X, Y]: is_a_theorem(implies(and(X, Y), X))). 13.33/13.55 fof(and_2, axiom, 13.33/13.55 ![X, Y]: is_a_theorem(implies(and(X, Y), Y)) <=> and_2). 13.33/13.55 fof(and_3, axiom, 13.33/13.55 ![X, Y]: is_a_theorem(implies(X, implies(Y, and(X, Y)))) 13.33/13.55 <=> and_3). 13.33/13.55 fof(axiom_5, axiom, 13.33/13.55 ![X]: is_a_theorem(implies(possibly(X), necessarily(possibly(X)))) 13.33/13.55 <=> axiom_5). 13.33/13.55 fof(axiom_B, axiom, 13.33/13.55 axiom_B 13.33/13.55 <=> ![X]: is_a_theorem(implies(X, necessarily(possibly(X))))). 13.33/13.55 fof(axiom_M, axiom, 13.33/13.55 axiom_M <=> ![X]: is_a_theorem(implies(necessarily(X), X))). 13.33/13.55 fof(cn3, axiom, 13.33/13.55 ![P]: is_a_theorem(implies(implies(not(P), P), P)) <=> cn3). 13.33/13.55 fof(hilbert_and_1, axiom, and_1). 13.33/13.55 fof(hilbert_and_2, axiom, and_2). 13.33/13.55 fof(hilbert_and_3, axiom, and_3). 13.33/13.55 fof(hilbert_implies_1, axiom, implies_1). 13.33/13.55 fof(hilbert_implies_2, axiom, implies_2). 13.33/13.55 fof(hilbert_modus_ponens, axiom, modus_ponens). 13.33/13.55 fof(hilbert_modus_tollens, axiom, modus_tollens). 13.33/13.55 fof(hilbert_op_equiv, axiom, op_equiv). 13.33/13.55 fof(hilbert_op_implies_and, axiom, op_implies_and). 13.33/13.55 fof(hilbert_op_or, axiom, op_or). 13.33/13.55 fof(hilbert_or_1, axiom, or_1). 13.33/13.55 fof(hilbert_or_2, axiom, or_2). 13.33/13.55 fof(implies_1, axiom, 13.33/13.55 ![X, Y]: is_a_theorem(implies(X, implies(Y, X))) <=> implies_1). 13.33/13.55 fof(implies_2, axiom, 13.33/13.55 implies_2 13.33/13.55 <=> ![X, Y]: 13.33/13.55 is_a_theorem(implies(implies(X, implies(X, Y)), implies(X, Y)))). 13.33/13.55 fof(km4b_axiom_B, conjecture, axiom_B). 13.33/13.55 fof(km5_axiom_5, axiom, axiom_5). 13.33/13.55 fof(km5_axiom_M, axiom, axiom_M). 13.33/13.55 fof(km5_necessitation, axiom, necessitation). 13.33/13.55 fof(km5_op_possibly, axiom, op_possibly). 13.33/13.55 fof(kn1, axiom, ![P]: is_a_theorem(implies(P, and(P, P))) <=> kn1). 13.33/13.55 fof(modus_ponens, axiom, 13.33/13.55 modus_ponens 13.33/13.55 <=> ![X, Y]: 13.33/13.55 ((is_a_theorem(X) & is_a_theorem(implies(X, Y))) 13.33/13.55 => is_a_theorem(Y))). 13.33/13.55 fof(modus_tollens, axiom, 13.33/13.55 modus_tollens 13.33/13.55 <=> ![X, Y]: 13.33/13.55 is_a_theorem(implies(implies(not(Y), not(X)), implies(X, Y)))). 13.33/13.55 fof(necessitation, axiom, 13.33/13.55 necessitation 13.33/13.55 <=> ![X]: (is_a_theorem(necessarily(X)) <= is_a_theorem(X))). 13.33/13.55 fof(op_and, axiom, 13.33/13.55 ![X, Y]: and(X, Y)=not(or(not(X), not(Y))) <= op_and). 13.33/13.55 fof(op_equiv, axiom, 13.33/13.55 op_equiv 13.33/13.55 => ![X, Y]: equiv(X, Y)=and(implies(X, Y), implies(Y, X))). 13.33/13.55 fof(op_implies_and, axiom, 13.33/13.55 ![X, Y]: not(and(X, not(Y)))=implies(X, Y) <= op_implies_and). 13.33/13.55 fof(op_or, axiom, 13.33/13.55 ![X, Y]: or(X, Y)=not(and(not(X), not(Y))) <= op_or). 13.33/13.55 fof(op_possibly, axiom, 13.33/13.55 ![X]: not(necessarily(not(X)))=possibly(X) <= op_possibly). 13.33/13.55 fof(or_1, axiom, 13.33/13.55 or_1 <=> ![X, Y]: is_a_theorem(implies(X, or(X, Y)))). 13.33/13.55 fof(or_2, axiom, 13.33/13.55 ![X, Y]: is_a_theorem(implies(Y, or(X, Y))) <=> or_2). 13.33/13.55 fof(r1, axiom, ![P]: is_a_theorem(implies(or(P, P), P)) <=> r1). 13.33/13.55 fof(r3, axiom, 13.33/13.55 r3 <=> ![P, Q]: is_a_theorem(implies(or(P, Q), or(Q, P)))). 13.33/13.55 fof(substitution_of_equivalents, axiom, 13.33/13.55 substitution_of_equivalents 13.33/13.55 <=> ![X, Y]: (is_a_theorem(equiv(X, Y)) => Y=X)). 13.33/13.55 fof(substitution_of_equivalents, axiom, 13.33/13.55 substitution_of_equivalents). 13.33/13.55 13.33/13.55 Now clausify the problem and encode Horn clauses using encoding 3 of 13.33/13.55 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf. 13.33/13.55 We repeatedly replace C & s=t => u=v by the two clauses: 13.33/13.55 $$fresh(y, y, x1...xn) = u 13.33/13.55 C => $$fresh(s, t, x1...xn) = v 13.33/13.55 where $$fresh is a fresh function symbol and x1..xn are the free 13.33/13.55 variables of u and v. 13.33/13.55 A predicate p(X) is encoded as p(X)=$$true (this is sound, because the 13.33/13.55 input problem has no model of domain size 1). 13.33/13.55 13.33/13.55 The encoding turns the above axioms into the following unit equations and goals: 13.33/13.55 13.33/13.55 Axiom 6 (and_1_1): $$fresh107(X, X, Y, Z) = $$true. 13.33/13.55 Axiom 8 (and_2_1): $$fresh105(X, X, Y, Z) = $$true. 13.33/13.55 Axiom 10 (and_3_1): $$fresh103(X, X, Y, Z) = $$true. 13.33/13.55 Axiom 14 (axiom_5_1): $$fresh99(X, X, Y) = $$true. 13.33/13.55 Axiom 15 (axiom_B): $$fresh98(X, X) = $$true. 13.33/13.55 Axiom 20 (axiom_M_1): $$fresh93(X, X, Y) = $$true. 13.33/13.55 Axiom 62 (implies_1_1): $$fresh51(X, X, Y, Z) = $$true. 13.33/13.55 Axiom 64 (implies_2_1): $$fresh49(X, X, Y, Z) = $$true. 13.33/13.55 Axiom 73 (modus_ponens_2): $$fresh40(X, X, Y, Z) = is_a_theorem(Z). 13.33/13.55 Axiom 74 (modus_ponens_2): $$fresh115(X, X, Y) = $$true. 13.33/13.55 Axiom 75 (modus_ponens_2): $$fresh116(X, X, Y, Z) = $$fresh115(is_a_theorem(Y), $$true, Z). 13.33/13.55 Axiom 82 (modus_tollens_1): $$fresh35(X, X, Y, Z) = $$true. 13.33/13.55 Axiom 83 (necessitation_1): $$fresh34(X, X, Y) = is_a_theorem(necessarily(Y)). 13.33/13.55 Axiom 84 (necessitation_1): $$fresh33(X, X, Y) = $$true. 13.33/13.55 Axiom 87 (op_equiv): $$fresh30(X, X, Y, Z) = equiv(Y, Z). 13.33/13.55 Axiom 88 (op_implies_and): $$fresh29(X, X, Y, Z) = implies(Y, Z). 13.33/13.55 Axiom 91 (op_or): $$fresh26(X, X, Y, Z) = or(Y, Z). 13.33/13.55 Axiom 92 (op_possibly): $$fresh25(X, X, Y) = possibly(Y). 13.33/13.55 Axiom 96 (or_1_1): $$fresh21(X, X, Y, Z) = $$true. 13.33/13.55 Axiom 98 (or_2_1): $$fresh19(X, X, Y, Z) = $$true. 13.33/13.55 Axiom 112 (substitution_of_equivalents_2): $$fresh3(X, X, Y, Z) = Y. 13.33/13.55 Axiom 113 (substitution_of_equivalents_2): $$fresh4(X, X, Y, Z) = Z. 13.33/13.55 Axiom 117 (and_2_1): $$fresh105(and_2, $$true, X, Y) = is_a_theorem(implies(and(X, Y), Y)). 13.33/13.55 Axiom 121 (substitution_of_equivalents_2): $$fresh4(substitution_of_equivalents, $$true, X, Y) = $$fresh3(is_a_theorem(equiv(X, Y)), $$true, X, Y). 13.33/13.55 Axiom 127 (kn1_1): $$fresh45(kn1, $$true, X) = is_a_theorem(implies(X, and(X, X))). 13.33/13.55 Axiom 135 (or_1_1): $$fresh21(or_1, $$true, X, Y) = is_a_theorem(implies(X, or(X, Y))). 13.33/13.55 Axiom 137 (or_2_1): $$fresh19(or_2, $$true, X, Y) = is_a_theorem(implies(Y, or(X, Y))). 13.33/13.55 Axiom 140 (modus_ponens_2): $$fresh116(modus_ponens, $$true, X, Y) = $$fresh40(is_a_theorem(implies(X, Y)), $$true, X, Y). 13.33/13.55 Axiom 143 (r1_1): $$fresh15(r1, $$true, X) = is_a_theorem(implies(or(X, X), X)). 13.33/13.55 Axiom 145 (and_3_1): $$fresh103(and_3, $$true, X, Y) = is_a_theorem(implies(X, implies(Y, and(X, Y)))). 13.33/13.55 Axiom 149 (and_1_1): $$fresh107(and_1, $$true, X, Y) = is_a_theorem(implies(and(X, Y), X)). 13.33/13.55 Axiom 151 (cn3_1): $$fresh59(cn3, $$true, X) = is_a_theorem(implies(implies(not(X), X), X)). 13.33/13.55 Axiom 153 (r3_1): $$fresh11(r3, $$true, X, Y) = is_a_theorem(implies(or(X, Y), or(Y, X))). 13.33/13.55 Axiom 159 (implies_1_1): $$fresh51(implies_1, $$true, X, Y) = is_a_theorem(implies(X, implies(Y, X))). 13.33/13.55 Axiom 161 (modus_tollens_1): $$fresh35(modus_tollens, $$true, X, Y) = is_a_theorem(implies(implies(not(Y), not(X)), implies(X, Y))). 13.33/13.55 Axiom 167 (implies_2_1): $$fresh49(implies_2, $$true, X, Y) = is_a_theorem(implies(implies(X, implies(X, Y)), implies(X, Y))). 13.33/13.55 Axiom 169 (op_or): $$fresh26(op_or, $$true, X, Y) = not(and(not(X), not(Y))). 13.33/13.55 Axiom 170 (op_and): $$fresh31(op_and, $$true, X, Y) = not(or(not(X), not(Y))). 13.33/13.55 Axiom 171 (op_implies_and): $$fresh29(op_implies_and, $$true, X, Y) = not(and(X, not(Y))). 13.33/13.55 Axiom 173 (op_equiv): $$fresh30(op_equiv, $$true, X, Y) = and(implies(X, Y), implies(Y, X)). 13.33/13.55 Axiom 175 (hilbert_and_3): and_3 = $$true. 13.33/13.55 Axiom 176 (hilbert_op_or): op_or = $$true. 13.33/13.55 Axiom 177 (hilbert_and_2): and_2 = $$true. 13.33/13.55 Axiom 178 (hilbert_and_1): and_1 = $$true. 13.33/13.55 Axiom 179 (substitution_of_equivalents): substitution_of_equivalents = $$true. 13.33/13.55 Axiom 180 (hilbert_implies_2): implies_2 = $$true. 13.33/13.55 Axiom 183 (hilbert_op_implies_and): op_implies_and = $$true. 13.33/13.55 Axiom 184 (hilbert_or_1): or_1 = $$true. 13.33/13.55 Axiom 186 (hilbert_or_2): or_2 = $$true. 13.33/13.55 Axiom 187 (hilbert_implies_1): implies_1 = $$true. 13.33/13.55 Axiom 188 (hilbert_modus_tollens): modus_tollens = $$true. 13.33/13.55 Axiom 189 (hilbert_op_equiv): op_equiv = $$true. 13.33/13.55 Axiom 190 (hilbert_modus_ponens): modus_ponens = $$true. 13.33/13.55 Axiom 216 (axiom_5_1): $$fresh99(axiom_5, $$true, X) = is_a_theorem(implies(possibly(X), necessarily(possibly(X)))). 13.33/13.55 Axiom 222 (axiom_M_1): $$fresh93(axiom_M, $$true, X) = is_a_theorem(implies(necessarily(X), X)). 13.33/13.55 Axiom 227 (necessitation_1): $$fresh34(necessitation, $$true, X) = $$fresh33(is_a_theorem(X), $$true, X). 13.33/13.55 Axiom 230 (axiom_B_1): $$fresh97(axiom_B, $$true, X) = is_a_theorem(implies(X, necessarily(possibly(X)))). 13.33/13.55 Axiom 231 (axiom_B): $$fresh98(is_a_theorem(implies(sK7_axiom_B_X, necessarily(possibly(sK7_axiom_B_X)))), $$true) = axiom_B. 13.33/13.55 Axiom 241 (op_possibly): $$fresh25(op_possibly, $$true, X) = not(necessarily(not(X))). 13.33/13.55 Axiom 242 (km5_necessitation): necessitation = $$true. 13.33/13.55 Axiom 243 (km5_axiom_M): axiom_M = $$true. 13.33/13.55 Axiom 244 (km5_axiom_5): axiom_5 = $$true. 13.33/13.55 Axiom 245 (km5_op_possibly): op_possibly = $$true. 13.33/13.55 13.33/13.55 Lemma 246: $$fresh116(X, X, Y, Z) = $$fresh116(?, ?, Y, Z). 13.33/13.55 Proof: 13.33/13.55 $$fresh116(X, X, Y, Z) 13.33/13.55 = { by axiom 75 (modus_ponens_2) } 13.33/13.55 $$fresh115(is_a_theorem(Y), $$true, Z) 13.33/13.55 = { by axiom 75 (modus_ponens_2) } 13.33/13.55 $$fresh116(?, ?, Y, Z) 13.33/13.55 13.33/13.55 Lemma 247: not(necessarily(not(X))) = possibly(X). 13.33/13.55 Proof: 13.33/13.55 not(necessarily(not(X))) 13.33/13.55 = { by axiom 241 (op_possibly) } 13.33/13.55 $$fresh25(op_possibly, $$true, X) 13.33/13.55 = { by axiom 245 (km5_op_possibly) } 13.33/13.55 $$fresh25($$true, $$true, X) 13.33/13.55 = { by axiom 92 (op_possibly) } 13.33/13.55 possibly(X) 13.33/13.55 13.33/13.55 Lemma 248: not(and(X, not(Y))) = implies(X, Y). 13.33/13.55 Proof: 13.33/13.55 not(and(X, not(Y))) 13.33/13.55 = { by axiom 171 (op_implies_and) } 13.33/13.55 $$fresh29(op_implies_and, $$true, X, Y) 13.33/13.55 = { by axiom 183 (hilbert_op_implies_and) } 13.33/13.55 $$fresh29($$true, $$true, X, Y) 13.33/13.55 = { by axiom 88 (op_implies_and) } 13.33/13.55 implies(X, Y) 13.33/13.55 13.33/13.55 Lemma 249: implies(not(X), Y) = or(X, Y). 13.33/13.55 Proof: 13.33/13.55 implies(not(X), Y) 13.33/13.55 = { by lemma 248 } 13.33/13.55 not(and(not(X), not(Y))) 13.33/13.55 = { by axiom 169 (op_or) } 13.33/13.55 $$fresh26(op_or, $$true, X, Y) 13.33/13.55 = { by axiom 176 (hilbert_op_or) } 13.33/13.55 $$fresh26($$true, $$true, X, Y) 13.33/13.55 = { by axiom 91 (op_or) } 13.33/13.55 or(X, Y) 13.33/13.55 13.33/13.55 Lemma 250: $$fresh15(r1, $$true, X) = $$fresh59(cn3, $$true, X). 13.33/13.55 Proof: 13.33/13.55 $$fresh15(r1, $$true, X) 13.33/13.55 = { by axiom 143 (r1_1) } 13.33/13.55 is_a_theorem(implies(or(X, X), X)) 13.33/13.55 = { by lemma 249 } 13.33/13.55 is_a_theorem(implies(implies(not(X), X), X)) 13.33/13.55 = { by axiom 151 (cn3_1) } 13.33/13.55 $$fresh59(cn3, $$true, X) 13.33/13.55 13.33/13.55 Lemma 251: not(and(X, possibly(Y))) = implies(X, necessarily(not(Y))). 13.33/13.55 Proof: 13.33/13.55 not(and(X, possibly(Y))) 13.33/13.55 = { by lemma 247 } 13.33/13.55 not(and(X, not(necessarily(not(Y))))) 13.33/13.55 = { by lemma 248 } 13.33/13.55 implies(X, necessarily(not(Y))) 13.33/13.55 13.33/13.55 Lemma 252: not(and(X, implies(Y, Z))) = implies(X, and(Y, not(Z))). 13.33/13.55 Proof: 13.33/13.55 not(and(X, implies(Y, Z))) 13.33/13.55 = { by lemma 248 } 13.33/13.55 not(and(X, not(and(Y, not(Z))))) 13.33/13.55 = { by lemma 248 } 13.33/13.55 implies(X, and(Y, not(Z))) 13.33/13.55 13.33/13.55 Lemma 253: $$fresh116(?, ?, implies(necessarily(X), X), Y) = $$true. 13.33/13.55 Proof: 13.33/13.55 $$fresh116(?, ?, implies(necessarily(X), X), Y) 13.33/13.55 = { by axiom 75 (modus_ponens_2) } 13.33/13.55 $$fresh115(is_a_theorem(implies(necessarily(X), X)), $$true, Y) 13.33/13.55 = { by axiom 222 (axiom_M_1) } 13.33/13.55 $$fresh115($$fresh93(axiom_M, $$true, X), $$true, Y) 13.33/13.55 = { by axiom 243 (km5_axiom_M) } 13.33/13.55 $$fresh115($$fresh93($$true, $$true, X), $$true, Y) 13.33/13.55 = { by axiom 20 (axiom_M_1) } 13.33/13.55 $$fresh115($$true, $$true, Y) 13.33/13.55 = { by axiom 74 (modus_ponens_2) } 13.33/13.55 $$true 13.33/13.55 13.33/13.55 Lemma 254: is_a_theorem(implies(or(X, not(Y)), implies(Y, X))) = $$true. 13.33/13.55 Proof: 13.33/13.55 is_a_theorem(implies(or(X, not(Y)), implies(Y, X))) 13.33/13.55 = { by lemma 249 } 13.33/13.55 is_a_theorem(implies(implies(not(X), not(Y)), implies(Y, X))) 13.33/13.55 = { by axiom 161 (modus_tollens_1) } 13.33/13.55 $$fresh35(modus_tollens, $$true, Y, X) 13.33/13.55 = { by axiom 188 (hilbert_modus_tollens) } 13.33/13.55 $$fresh35($$true, $$true, Y, X) 13.33/13.55 = { by axiom 82 (modus_tollens_1) } 13.33/13.55 $$true 13.33/13.55 13.33/13.55 Lemma 255: $$fresh116(?, ?, implies(and(X, Y), Y), Z) = $$true. 13.33/13.55 Proof: 13.33/13.55 $$fresh116(?, ?, implies(and(X, Y), Y), Z) 13.33/13.55 = { by axiom 75 (modus_ponens_2) } 13.33/13.55 $$fresh115(is_a_theorem(implies(and(X, Y), Y)), $$true, Z) 13.33/13.55 = { by axiom 117 (and_2_1) } 13.33/13.55 $$fresh115($$fresh105(and_2, $$true, X, Y), $$true, Z) 13.33/13.55 = { by axiom 177 (hilbert_and_2) } 13.33/13.55 $$fresh115($$fresh105($$true, $$true, X, Y), $$true, Z) 13.33/13.55 = { by axiom 8 (and_2_1) } 13.33/13.55 $$fresh115($$true, $$true, Z) 13.33/13.55 = { by axiom 74 (modus_ponens_2) } 13.33/13.55 $$true 13.33/13.55 13.33/13.55 Lemma 256: $$fresh116(?, ?, implies(X, or(X, Y)), Z) = $$true. 13.33/13.55 Proof: 13.33/13.55 $$fresh116(?, ?, implies(X, or(X, Y)), Z) 13.33/13.55 = { by axiom 75 (modus_ponens_2) } 13.33/13.55 $$fresh115(is_a_theorem(implies(X, or(X, Y))), $$true, Z) 13.33/13.55 = { by axiom 135 (or_1_1) } 13.33/13.55 $$fresh115($$fresh21(or_1, $$true, X, Y), $$true, Z) 13.33/13.55 = { by axiom 184 (hilbert_or_1) } 13.33/13.55 $$fresh115($$fresh21($$true, $$true, X, Y), $$true, Z) 13.33/13.55 = { by axiom 96 (or_1_1) } 13.33/13.55 $$fresh115($$true, $$true, Z) 13.33/13.55 = { by axiom 74 (modus_ponens_2) } 13.33/13.55 $$true 13.33/13.55 13.33/13.55 Lemma 257: $$fresh116(?, ?, implies(X, implies(Y, X)), Z) = $$true. 13.33/13.55 Proof: 13.33/13.55 $$fresh116(?, ?, implies(X, implies(Y, X)), Z) 13.33/13.55 = { by axiom 75 (modus_ponens_2) } 13.33/13.55 $$fresh115(is_a_theorem(implies(X, implies(Y, X))), $$true, Z) 13.33/13.55 = { by axiom 159 (implies_1_1) } 13.33/13.55 $$fresh115($$fresh51(implies_1, $$true, X, Y), $$true, Z) 13.33/13.55 = { by axiom 187 (hilbert_implies_1) } 13.33/13.55 $$fresh115($$fresh51($$true, $$true, X, Y), $$true, Z) 13.33/13.55 = { by axiom 62 (implies_1_1) } 13.33/13.55 $$fresh115($$true, $$true, Z) 13.33/13.55 = { by axiom 74 (modus_ponens_2) } 13.33/13.56 $$true 13.33/13.56 13.33/13.56 Lemma 258: $$fresh116(?, ?, X, implies(Y, X)) = is_a_theorem(implies(Y, X)). 13.33/13.56 Proof: 13.33/13.56 $$fresh116(?, ?, X, implies(Y, X)) 13.33/13.56 = { by lemma 246 } 13.33/13.56 $$fresh116($$true, $$true, X, implies(Y, X)) 13.33/13.56 = { by axiom 190 (hilbert_modus_ponens) } 13.33/13.56 $$fresh116(modus_ponens, $$true, X, implies(Y, X)) 13.33/13.56 = { by axiom 140 (modus_ponens_2) } 13.33/13.56 $$fresh40(is_a_theorem(implies(X, implies(Y, X))), $$true, X, implies(Y, X)) 13.33/13.56 = { by axiom 159 (implies_1_1) } 13.33/13.56 $$fresh40($$fresh51(implies_1, $$true, X, Y), $$true, X, implies(Y, X)) 13.33/13.56 = { by axiom 187 (hilbert_implies_1) } 13.33/13.56 $$fresh40($$fresh51($$true, $$true, X, Y), $$true, X, implies(Y, X)) 13.33/13.56 = { by axiom 62 (implies_1_1) } 13.33/13.56 $$fresh40($$true, $$true, X, implies(Y, X)) 13.33/13.56 = { by axiom 73 (modus_ponens_2) } 13.33/13.56 is_a_theorem(implies(Y, X)) 13.33/13.56 13.33/13.56 Lemma 259: $$fresh116(?, ?, or(X, not(Y)), implies(Y, X)) = is_a_theorem(implies(Y, X)). 13.33/13.56 Proof: 13.33/13.56 $$fresh116(?, ?, or(X, not(Y)), implies(Y, X)) 13.33/13.56 = { by lemma 246 } 13.33/13.56 $$fresh116($$true, $$true, or(X, not(Y)), implies(Y, X)) 13.33/13.56 = { by axiom 190 (hilbert_modus_ponens) } 13.33/13.56 $$fresh116(modus_ponens, $$true, or(X, not(Y)), implies(Y, X)) 13.33/13.56 = { by axiom 140 (modus_ponens_2) } 13.33/13.56 $$fresh40(is_a_theorem(implies(or(X, not(Y)), implies(Y, X))), $$true, or(X, not(Y)), implies(Y, X)) 13.33/13.56 = { by lemma 254 } 13.33/13.56 $$fresh40($$true, $$true, or(X, not(Y)), implies(Y, X)) 13.33/13.56 = { by axiom 73 (modus_ponens_2) } 13.33/13.56 is_a_theorem(implies(Y, X)) 13.33/13.56 13.33/13.56 Lemma 260: is_a_theorem(implies(X, X)) = $$true. 13.33/13.56 Proof: 13.33/13.56 is_a_theorem(implies(X, X)) 13.33/13.56 = { by lemma 259 } 13.33/13.56 $$fresh116(?, ?, or(X, not(X)), implies(X, X)) 13.33/13.56 = { by axiom 75 (modus_ponens_2) } 13.33/13.56 $$fresh115(is_a_theorem(or(X, not(X))), $$true, implies(X, X)) 13.33/13.56 = { by axiom 73 (modus_ponens_2) } 13.33/13.56 $$fresh115($$fresh40($$true, $$true, or(X, or(X, not(X))), or(X, not(X))), $$true, implies(X, X)) 13.33/13.56 = { by axiom 64 (implies_2_1) } 13.33/13.56 $$fresh115($$fresh40($$fresh49($$true, $$true, not(X), not(X)), $$true, or(X, or(X, not(X))), or(X, not(X))), $$true, implies(X, X)) 13.33/13.56 = { by axiom 180 (hilbert_implies_2) } 13.33/13.56 $$fresh115($$fresh40($$fresh49(implies_2, $$true, not(X), not(X)), $$true, or(X, or(X, not(X))), or(X, not(X))), $$true, implies(X, X)) 13.33/13.56 = { by axiom 167 (implies_2_1) } 13.33/13.56 $$fresh115($$fresh40(is_a_theorem(implies(implies(not(X), implies(not(X), not(X))), implies(not(X), not(X)))), $$true, or(X, or(X, not(X))), or(X, not(X))), $$true, implies(X, X)) 13.33/13.56 = { by lemma 249 } 13.33/13.56 $$fresh115($$fresh40(is_a_theorem(implies(or(X, implies(not(X), not(X))), implies(not(X), not(X)))), $$true, or(X, or(X, not(X))), or(X, not(X))), $$true, implies(X, X)) 13.33/13.56 = { by lemma 249 } 13.33/13.56 $$fresh115($$fresh40(is_a_theorem(implies(or(X, or(X, not(X))), implies(not(X), not(X)))), $$true, or(X, or(X, not(X))), or(X, not(X))), $$true, implies(X, X)) 13.33/13.56 = { by lemma 249 } 13.33/13.56 $$fresh115($$fresh40(is_a_theorem(implies(or(X, or(X, not(X))), or(X, not(X)))), $$true, or(X, or(X, not(X))), or(X, not(X))), $$true, implies(X, X)) 13.33/13.56 = { by axiom 140 (modus_ponens_2) } 13.33/13.56 $$fresh115($$fresh116(modus_ponens, $$true, or(X, or(X, not(X))), or(X, not(X))), $$true, implies(X, X)) 13.33/13.56 = { by axiom 190 (hilbert_modus_ponens) } 13.33/13.56 $$fresh115($$fresh116($$true, $$true, or(X, or(X, not(X))), or(X, not(X))), $$true, implies(X, X)) 13.33/13.56 = { by lemma 246 } 13.33/13.56 $$fresh115($$fresh116(?, ?, or(X, or(X, not(X))), or(X, not(X))), $$true, implies(X, X)) 13.33/13.56 = { by lemma 249 } 13.33/13.56 $$fresh115($$fresh116(?, ?, implies(not(X), or(X, not(X))), or(X, not(X))), $$true, implies(X, X)) 13.33/13.56 = { by axiom 75 (modus_ponens_2) } 13.33/13.56 $$fresh115($$fresh115(is_a_theorem(implies(not(X), or(X, not(X)))), $$true, or(X, not(X))), $$true, implies(X, X)) 13.33/13.56 = { by axiom 137 (or_2_1) } 13.33/13.56 $$fresh115($$fresh115($$fresh19(or_2, $$true, X, not(X)), $$true, or(X, not(X))), $$true, implies(X, X)) 13.33/13.56 = { by axiom 186 (hilbert_or_2) } 13.33/13.56 $$fresh115($$fresh115($$fresh19($$true, $$true, X, not(X)), $$true, or(X, not(X))), $$true, implies(X, X)) 13.33/13.56 = { by axiom 98 (or_2_1) } 13.33/13.56 $$fresh115($$fresh115($$true, $$true, or(X, not(X))), $$true, implies(X, X)) 13.33/13.56 = { by axiom 74 (modus_ponens_2) } 13.33/13.56 $$fresh115($$true, $$true, implies(X, X)) 13.33/13.56 = { by axiom 74 (modus_ponens_2) } 13.33/13.56 $$true 13.33/13.56 13.33/13.56 Lemma 261: $$fresh3(is_a_theorem(equiv(X, Y)), $$true, X, Y) = Y. 13.33/13.56 Proof: 13.33/13.56 $$fresh3(is_a_theorem(equiv(X, Y)), $$true, X, Y) 13.33/13.56 = { by axiom 121 (substitution_of_equivalents_2) } 13.33/13.56 $$fresh4(substitution_of_equivalents, $$true, X, Y) 13.33/13.56 = { by axiom 179 (substitution_of_equivalents) } 13.33/13.56 $$fresh4($$true, $$true, X, Y) 13.33/13.56 = { by axiom 113 (substitution_of_equivalents_2) } 13.33/13.56 Y 13.33/13.56 13.33/13.56 Lemma 262: $$fresh40(is_a_theorem(implies(X, Y)), $$true, X, Y) = $$fresh116(?, ?, X, Y). 13.33/13.56 Proof: 13.33/13.56 $$fresh40(is_a_theorem(implies(X, Y)), $$true, X, Y) 13.33/13.56 = { by axiom 140 (modus_ponens_2) } 13.33/13.56 $$fresh116(modus_ponens, $$true, X, Y) 13.33/13.56 = { by axiom 190 (hilbert_modus_ponens) } 13.33/13.56 $$fresh116($$true, $$true, X, Y) 13.33/13.56 = { by lemma 246 } 13.33/13.56 $$fresh116(?, ?, X, Y) 13.33/13.56 13.33/13.56 Lemma 263: and(implies(X, Y), implies(Y, X)) = equiv(X, Y). 13.33/13.56 Proof: 13.33/13.56 and(implies(X, Y), implies(Y, X)) 13.33/13.56 = { by axiom 173 (op_equiv) } 13.33/13.56 $$fresh30(op_equiv, $$true, X, Y) 13.33/13.56 = { by axiom 189 (hilbert_op_equiv) } 13.33/13.56 $$fresh30($$true, $$true, X, Y) 13.33/13.56 = { by axiom 87 (op_equiv) } 13.33/13.56 equiv(X, Y) 13.33/13.56 13.33/13.56 Lemma 264: $$fresh116(?, ?, X, implies(Y, and(X, Y))) = is_a_theorem(implies(Y, and(X, Y))). 13.33/13.56 Proof: 13.33/13.56 $$fresh116(?, ?, X, implies(Y, and(X, Y))) 13.33/13.56 = { by lemma 262 } 13.33/13.56 $$fresh40(is_a_theorem(implies(X, implies(Y, and(X, Y)))), $$true, X, implies(Y, and(X, Y))) 13.33/13.56 = { by axiom 145 (and_3_1) } 13.33/13.56 $$fresh40($$fresh103(and_3, $$true, X, Y), $$true, X, implies(Y, and(X, Y))) 13.33/13.56 = { by axiom 175 (hilbert_and_3) } 13.33/13.56 $$fresh40($$fresh103($$true, $$true, X, Y), $$true, X, implies(Y, and(X, Y))) 13.33/13.56 = { by axiom 10 (and_3_1) } 13.33/13.56 $$fresh40($$true, $$true, X, implies(Y, and(X, Y))) 13.33/13.56 = { by axiom 73 (modus_ponens_2) } 13.33/13.56 is_a_theorem(implies(Y, and(X, Y))) 13.33/13.56 13.33/13.56 Lemma 265: is_a_theorem(implies(X, and(implies(necessarily(Y), Y), X))) = $$true. 13.33/13.56 Proof: 13.33/13.56 is_a_theorem(implies(X, and(implies(necessarily(Y), Y), X))) 13.33/13.56 = { by lemma 264 } 13.33/13.56 $$fresh116(?, ?, implies(necessarily(Y), Y), implies(X, and(implies(necessarily(Y), Y), X))) 13.33/13.56 = { by lemma 253 } 13.33/13.56 $$true 13.33/13.56 13.33/13.56 Lemma 266: is_a_theorem(implies(X, and(implies(and(Y, Z), Z), X))) = $$true. 13.33/13.56 Proof: 13.33/13.56 is_a_theorem(implies(X, and(implies(and(Y, Z), Z), X))) 13.33/13.56 = { by lemma 264 } 13.33/13.56 $$fresh116(?, ?, implies(and(Y, Z), Z), implies(X, and(implies(and(Y, Z), Z), X))) 13.33/13.56 = { by lemma 255 } 13.33/13.58 $$true 13.33/13.58 13.33/13.58 Lemma 267: and(X, X) = X. 13.33/13.58 Proof: 13.33/13.58 and(X, X) 13.33/13.58 = { by lemma 261 } 13.33/13.58 $$fresh3(is_a_theorem(equiv(X, and(X, X))), $$true, X, and(X, X)) 13.33/13.58 = { by axiom 73 (modus_ponens_2) } 13.33/13.58 $$fresh3($$fresh40($$true, $$true, implies(and(X, X), X), equiv(X, and(X, X))), $$true, X, and(X, X)) 13.33/13.58 = { by axiom 74 (modus_ponens_2) } 13.33/13.58 $$fresh3($$fresh40($$fresh115($$true, $$true, implies(implies(and(X, X), X), and(implies(X, and(X, X)), implies(and(X, X), X)))), $$true, implies(and(X, X), X), equiv(X, and(X, X))), $$true, X, and(X, X)) 13.33/13.58 = { by axiom 74 (modus_ponens_2) } 13.33/13.58 $$fresh3($$fresh40($$fresh115($$fresh115($$true, $$true, implies(X, and(X, X))), $$true, implies(implies(and(X, X), X), and(implies(X, and(X, X)), implies(and(X, X), X)))), $$true, implies(and(X, X), X), equiv(X, and(X, X))), $$true, X, and(X, X)) 13.33/13.58 = { by axiom 10 (and_3_1) } 13.33/13.58 $$fresh3($$fresh40($$fresh115($$fresh115($$fresh103($$true, $$true, X, X), $$true, implies(X, and(X, X))), $$true, implies(implies(and(X, X), X), and(implies(X, and(X, X)), implies(and(X, X), X)))), $$true, implies(and(X, X), X), equiv(X, and(X, X))), $$true, X, and(X, X)) 13.33/13.58 = { by axiom 175 (hilbert_and_3) } 13.33/13.58 $$fresh3($$fresh40($$fresh115($$fresh115($$fresh103(and_3, $$true, X, X), $$true, implies(X, and(X, X))), $$true, implies(implies(and(X, X), X), and(implies(X, and(X, X)), implies(and(X, X), X)))), $$true, implies(and(X, X), X), equiv(X, and(X, X))), $$true, X, and(X, X)) 13.33/13.58 = { by axiom 145 (and_3_1) } 13.33/13.58 $$fresh3($$fresh40($$fresh115($$fresh115(is_a_theorem(implies(X, implies(X, and(X, X)))), $$true, implies(X, and(X, X))), $$true, implies(implies(and(X, X), X), and(implies(X, and(X, X)), implies(and(X, X), X)))), $$true, implies(and(X, X), X), equiv(X, and(X, X))), $$true, X, and(X, X)) 13.33/13.58 = { by axiom 75 (modus_ponens_2) } 13.33/13.58 $$fresh3($$fresh40($$fresh115($$fresh116(?, ?, implies(X, implies(X, and(X, X))), implies(X, and(X, X))), $$true, implies(implies(and(X, X), X), and(implies(X, and(X, X)), implies(and(X, X), X)))), $$true, implies(and(X, X), X), equiv(X, and(X, X))), $$true, X, and(X, X)) 13.33/13.58 = { by lemma 262 } 13.33/13.58 $$fresh3($$fresh40($$fresh115($$fresh40(is_a_theorem(implies(implies(X, implies(X, and(X, X))), implies(X, and(X, X)))), $$true, implies(X, implies(X, and(X, X))), implies(X, and(X, X))), $$true, implies(implies(and(X, X), X), and(implies(X, and(X, X)), implies(and(X, X), X)))), $$true, implies(and(X, X), X), equiv(X, and(X, X))), $$true, X, and(X, X)) 13.33/13.58 = { by axiom 167 (implies_2_1) } 13.33/13.58 $$fresh3($$fresh40($$fresh115($$fresh40($$fresh49(implies_2, $$true, X, and(X, X)), $$true, implies(X, implies(X, and(X, X))), implies(X, and(X, X))), $$true, implies(implies(and(X, X), X), and(implies(X, and(X, X)), implies(and(X, X), X)))), $$true, implies(and(X, X), X), equiv(X, and(X, X))), $$true, X, and(X, X)) 13.33/13.58 = { by axiom 180 (hilbert_implies_2) } 13.33/13.58 $$fresh3($$fresh40($$fresh115($$fresh40($$fresh49($$true, $$true, X, and(X, X)), $$true, implies(X, implies(X, and(X, X))), implies(X, and(X, X))), $$true, implies(implies(and(X, X), X), and(implies(X, and(X, X)), implies(and(X, X), X)))), $$true, implies(and(X, X), X), equiv(X, and(X, X))), $$true, X, and(X, X)) 13.33/13.58 = { by axiom 64 (implies_2_1) } 13.33/13.58 $$fresh3($$fresh40($$fresh115($$fresh40($$true, $$true, implies(X, implies(X, and(X, X))), implies(X, and(X, X))), $$true, implies(implies(and(X, X), X), and(implies(X, and(X, X)), implies(and(X, X), X)))), $$true, implies(and(X, X), X), equiv(X, and(X, X))), $$true, X, and(X, X)) 13.33/13.58 = { by axiom 73 (modus_ponens_2) } 13.33/13.58 $$fresh3($$fresh40($$fresh115(is_a_theorem(implies(X, and(X, X))), $$true, implies(implies(and(X, X), X), and(implies(X, and(X, X)), implies(and(X, X), X)))), $$true, implies(and(X, X), X), equiv(X, and(X, X))), $$true, X, and(X, X)) 13.33/13.58 = { by axiom 75 (modus_ponens_2) } 13.33/13.58 $$fresh3($$fresh40($$fresh116(?, ?, implies(X, and(X, X)), implies(implies(and(X, X), X), and(implies(X, and(X, X)), implies(and(X, X), X)))), $$true, implies(and(X, X), X), equiv(X, and(X, X))), $$true, X, and(X, X)) 13.33/13.58 = { by lemma 264 } 13.33/13.58 $$fresh3($$fresh40(is_a_theorem(implies(implies(and(X, X), X), and(implies(X, and(X, X)), implies(and(X, X), X)))), $$true, implies(and(X, X), X), equiv(X, and(X, X))), $$true, X, and(X, X)) 13.33/13.58 = { by lemma 263 } 13.33/13.58 $$fresh3($$fresh40(is_a_theorem(implies(implies(and(X, X), X), equiv(X, and(X, X)))), $$true, implies(and(X, X), X), equiv(X, and(X, X))), $$true, X, and(X, X)) 13.33/13.58 = { by lemma 262 } 13.33/13.58 $$fresh3($$fresh116(?, ?, implies(and(X, X), X), equiv(X, and(X, X))), $$true, X, and(X, X)) 13.33/13.58 = { by axiom 75 (modus_ponens_2) } 13.33/13.58 $$fresh3($$fresh115(is_a_theorem(implies(and(X, X), X)), $$true, equiv(X, and(X, X))), $$true, X, and(X, X)) 13.33/13.58 = { by axiom 149 (and_1_1) } 13.33/13.58 $$fresh3($$fresh115($$fresh107(and_1, $$true, X, X), $$true, equiv(X, and(X, X))), $$true, X, and(X, X)) 13.33/13.58 = { by axiom 178 (hilbert_and_1) } 13.33/13.58 $$fresh3($$fresh115($$fresh107($$true, $$true, X, X), $$true, equiv(X, and(X, X))), $$true, X, and(X, X)) 13.33/13.58 = { by axiom 6 (and_1_1) } 13.33/13.58 $$fresh3($$fresh115($$true, $$true, equiv(X, and(X, X))), $$true, X, and(X, X)) 13.33/13.58 = { by axiom 74 (modus_ponens_2) } 13.33/13.58 $$fresh3($$true, $$true, X, and(X, X)) 13.33/13.58 = { by axiom 112 (substitution_of_equivalents_2) } 13.33/13.59 X 13.33/13.59 13.33/13.59 Lemma 268: or(X, X) = not(not(X)). 13.33/13.59 Proof: 13.33/13.59 or(X, X) 13.33/13.59 = { by lemma 249 } 13.33/13.59 implies(not(X), X) 13.33/13.59 = { by lemma 248 } 13.33/13.59 not(and(not(X), not(X))) 13.33/13.59 = { by lemma 267 } 13.33/13.59 not(not(X)) 13.33/13.59 13.33/13.59 Lemma 269: is_a_theorem(implies(implies(or(X, Y), X), equiv(X, or(X, Y)))) = $$true. 13.33/13.59 Proof: 13.33/13.59 is_a_theorem(implies(implies(or(X, Y), X), equiv(X, or(X, Y)))) 13.33/13.59 = { by lemma 263 } 13.33/13.59 is_a_theorem(implies(implies(or(X, Y), X), and(implies(X, or(X, Y)), implies(or(X, Y), X)))) 13.33/13.59 = { by lemma 264 } 13.33/13.59 $$fresh116(?, ?, implies(X, or(X, Y)), implies(implies(or(X, Y), X), and(implies(X, or(X, Y)), implies(or(X, Y), X)))) 13.33/13.59 = { by lemma 256 } 13.33/13.59 $$true 13.33/13.59 13.33/13.59 Lemma 270: necessarily(possibly(X)) = possibly(X). 13.33/13.59 Proof: 13.33/13.59 necessarily(possibly(X)) 13.33/13.59 = { by lemma 261 } 13.33/13.59 $$fresh3(is_a_theorem(equiv(possibly(X), necessarily(possibly(X)))), $$true, possibly(X), necessarily(possibly(X))) 13.33/13.59 = { by axiom 73 (modus_ponens_2) } 13.33/13.59 $$fresh3($$fresh40($$true, $$true, implies(necessarily(possibly(X)), possibly(X)), equiv(possibly(X), necessarily(possibly(X)))), $$true, possibly(X), necessarily(possibly(X))) 13.33/13.59 = { by axiom 74 (modus_ponens_2) } 13.33/13.59 $$fresh3($$fresh40($$fresh115($$true, $$true, implies(implies(necessarily(possibly(X)), possibly(X)), and(implies(possibly(X), necessarily(possibly(X))), implies(necessarily(possibly(X)), possibly(X))))), $$true, implies(necessarily(possibly(X)), possibly(X)), equiv(possibly(X), necessarily(possibly(X)))), $$true, possibly(X), necessarily(possibly(X))) 13.33/13.59 = { by axiom 14 (axiom_5_1) } 13.33/13.59 $$fresh3($$fresh40($$fresh115($$fresh99($$true, $$true, X), $$true, implies(implies(necessarily(possibly(X)), possibly(X)), and(implies(possibly(X), necessarily(possibly(X))), implies(necessarily(possibly(X)), possibly(X))))), $$true, implies(necessarily(possibly(X)), possibly(X)), equiv(possibly(X), necessarily(possibly(X)))), $$true, possibly(X), necessarily(possibly(X))) 13.33/13.59 = { by axiom 244 (km5_axiom_5) } 13.33/13.59 $$fresh3($$fresh40($$fresh115($$fresh99(axiom_5, $$true, X), $$true, implies(implies(necessarily(possibly(X)), possibly(X)), and(implies(possibly(X), necessarily(possibly(X))), implies(necessarily(possibly(X)), possibly(X))))), $$true, implies(necessarily(possibly(X)), possibly(X)), equiv(possibly(X), necessarily(possibly(X)))), $$true, possibly(X), necessarily(possibly(X))) 13.33/13.59 = { by axiom 216 (axiom_5_1) } 13.33/13.59 $$fresh3($$fresh40($$fresh115(is_a_theorem(implies(possibly(X), necessarily(possibly(X)))), $$true, implies(implies(necessarily(possibly(X)), possibly(X)), and(implies(possibly(X), necessarily(possibly(X))), implies(necessarily(possibly(X)), possibly(X))))), $$true, implies(necessarily(possibly(X)), possibly(X)), equiv(possibly(X), necessarily(possibly(X)))), $$true, possibly(X), necessarily(possibly(X))) 13.33/13.59 = { by axiom 75 (modus_ponens_2) } 13.33/13.59 $$fresh3($$fresh40($$fresh116(?, ?, implies(possibly(X), necessarily(possibly(X))), implies(implies(necessarily(possibly(X)), possibly(X)), and(implies(possibly(X), necessarily(possibly(X))), implies(necessarily(possibly(X)), possibly(X))))), $$true, implies(necessarily(possibly(X)), possibly(X)), equiv(possibly(X), necessarily(possibly(X)))), $$true, possibly(X), necessarily(possibly(X))) 13.33/13.59 = { by lemma 264 } 13.33/13.59 $$fresh3($$fresh40(is_a_theorem(implies(implies(necessarily(possibly(X)), possibly(X)), and(implies(possibly(X), necessarily(possibly(X))), implies(necessarily(possibly(X)), possibly(X))))), $$true, implies(necessarily(possibly(X)), possibly(X)), equiv(possibly(X), necessarily(possibly(X)))), $$true, possibly(X), necessarily(possibly(X))) 13.33/13.59 = { by lemma 263 } 13.33/13.59 $$fresh3($$fresh40(is_a_theorem(implies(implies(necessarily(possibly(X)), possibly(X)), equiv(possibly(X), necessarily(possibly(X))))), $$true, implies(necessarily(possibly(X)), possibly(X)), equiv(possibly(X), necessarily(possibly(X)))), $$true, possibly(X), necessarily(possibly(X))) 13.33/13.59 = { by lemma 262 } 13.33/13.59 $$fresh3($$fresh116(?, ?, implies(necessarily(possibly(X)), possibly(X)), equiv(possibly(X), necessarily(possibly(X)))), $$true, possibly(X), necessarily(possibly(X))) 13.33/13.59 = { by lemma 253 } 13.33/13.59 $$fresh3($$true, $$true, possibly(X), necessarily(possibly(X))) 13.33/13.59 = { by axiom 112 (substitution_of_equivalents_2) } 13.33/13.59 possibly(X) 13.33/13.59 13.33/13.59 Lemma 271: implies(implies(X, Y), and(X, not(Y))) = not(implies(X, Y)). 13.33/13.59 Proof: 13.33/13.59 implies(implies(X, Y), and(X, not(Y))) 13.33/13.59 = { by lemma 252 } 13.33/13.59 not(and(implies(X, Y), implies(X, Y))) 13.33/13.59 = { by lemma 267 } 13.33/13.59 not(implies(X, Y)) 13.33/13.59 13.33/13.59 Lemma 272: $$fresh116(?, ?, implies(X, necessarily(X)), equiv(necessarily(X), X)) = is_a_theorem(equiv(necessarily(X), X)). 13.33/13.59 Proof: 13.33/13.59 $$fresh116(?, ?, implies(X, necessarily(X)), equiv(necessarily(X), X)) 13.33/13.59 = { by lemma 262 } 13.33/13.59 $$fresh40(is_a_theorem(implies(implies(X, necessarily(X)), equiv(necessarily(X), X))), $$true, implies(X, necessarily(X)), equiv(necessarily(X), X)) 13.33/13.59 = { by lemma 263 } 13.33/13.59 $$fresh40(is_a_theorem(implies(implies(X, necessarily(X)), and(implies(necessarily(X), X), implies(X, necessarily(X))))), $$true, implies(X, necessarily(X)), equiv(necessarily(X), X)) 13.33/13.59 = { by lemma 265 } 13.33/13.59 $$fresh40($$true, $$true, implies(X, necessarily(X)), equiv(necessarily(X), X)) 13.33/13.59 = { by axiom 73 (modus_ponens_2) } 13.33/13.60 is_a_theorem(equiv(necessarily(X), X)) 13.33/13.60 13.33/13.60 Lemma 273: necessarily(implies(necessarily(X), X)) = implies(necessarily(X), X). 13.33/13.60 Proof: 13.33/13.60 necessarily(implies(necessarily(X), X)) 13.33/13.60 = { by axiom 112 (substitution_of_equivalents_2) } 13.33/13.60 $$fresh3($$true, $$true, necessarily(implies(necessarily(X), X)), implies(necessarily(X), X)) 13.33/13.60 = { by axiom 74 (modus_ponens_2) } 13.33/13.60 $$fresh3($$fresh115($$true, $$true, equiv(necessarily(implies(necessarily(X), X)), implies(necessarily(X), X))), $$true, necessarily(implies(necessarily(X), X)), implies(necessarily(X), X)) 13.33/13.60 = { by axiom 74 (modus_ponens_2) } 13.33/13.60 $$fresh3($$fresh115($$fresh115($$true, $$true, implies(implies(necessarily(X), X), necessarily(implies(necessarily(X), X)))), $$true, equiv(necessarily(implies(necessarily(X), X)), implies(necessarily(X), X))), $$true, necessarily(implies(necessarily(X), X)), implies(necessarily(X), X)) 13.33/13.60 = { by axiom 84 (necessitation_1) } 13.33/13.60 $$fresh3($$fresh115($$fresh115($$fresh33($$true, $$true, implies(necessarily(X), X)), $$true, implies(implies(necessarily(X), X), necessarily(implies(necessarily(X), X)))), $$true, equiv(necessarily(implies(necessarily(X), X)), implies(necessarily(X), X))), $$true, necessarily(implies(necessarily(X), X)), implies(necessarily(X), X)) 13.33/13.60 = { by axiom 20 (axiom_M_1) } 13.33/13.60 $$fresh3($$fresh115($$fresh115($$fresh33($$fresh93($$true, $$true, X), $$true, implies(necessarily(X), X)), $$true, implies(implies(necessarily(X), X), necessarily(implies(necessarily(X), X)))), $$true, equiv(necessarily(implies(necessarily(X), X)), implies(necessarily(X), X))), $$true, necessarily(implies(necessarily(X), X)), implies(necessarily(X), X)) 13.33/13.60 = { by axiom 243 (km5_axiom_M) } 13.33/13.60 $$fresh3($$fresh115($$fresh115($$fresh33($$fresh93(axiom_M, $$true, X), $$true, implies(necessarily(X), X)), $$true, implies(implies(necessarily(X), X), necessarily(implies(necessarily(X), X)))), $$true, equiv(necessarily(implies(necessarily(X), X)), implies(necessarily(X), X))), $$true, necessarily(implies(necessarily(X), X)), implies(necessarily(X), X)) 13.33/13.60 = { by axiom 222 (axiom_M_1) } 13.33/13.60 $$fresh3($$fresh115($$fresh115($$fresh33(is_a_theorem(implies(necessarily(X), X)), $$true, implies(necessarily(X), X)), $$true, implies(implies(necessarily(X), X), necessarily(implies(necessarily(X), X)))), $$true, equiv(necessarily(implies(necessarily(X), X)), implies(necessarily(X), X))), $$true, necessarily(implies(necessarily(X), X)), implies(necessarily(X), X)) 13.33/13.60 = { by axiom 227 (necessitation_1) } 13.33/13.60 $$fresh3($$fresh115($$fresh115($$fresh34(necessitation, $$true, implies(necessarily(X), X)), $$true, implies(implies(necessarily(X), X), necessarily(implies(necessarily(X), X)))), $$true, equiv(necessarily(implies(necessarily(X), X)), implies(necessarily(X), X))), $$true, necessarily(implies(necessarily(X), X)), implies(necessarily(X), X)) 13.33/13.60 = { by axiom 242 (km5_necessitation) } 13.33/13.60 $$fresh3($$fresh115($$fresh115($$fresh34($$true, $$true, implies(necessarily(X), X)), $$true, implies(implies(necessarily(X), X), necessarily(implies(necessarily(X), X)))), $$true, equiv(necessarily(implies(necessarily(X), X)), implies(necessarily(X), X))), $$true, necessarily(implies(necessarily(X), X)), implies(necessarily(X), X)) 13.33/13.60 = { by axiom 83 (necessitation_1) } 13.33/13.60 $$fresh3($$fresh115($$fresh115(is_a_theorem(necessarily(implies(necessarily(X), X))), $$true, implies(implies(necessarily(X), X), necessarily(implies(necessarily(X), X)))), $$true, equiv(necessarily(implies(necessarily(X), X)), implies(necessarily(X), X))), $$true, necessarily(implies(necessarily(X), X)), implies(necessarily(X), X)) 13.33/13.60 = { by axiom 75 (modus_ponens_2) } 13.33/13.60 $$fresh3($$fresh115($$fresh116(?, ?, necessarily(implies(necessarily(X), X)), implies(implies(necessarily(X), X), necessarily(implies(necessarily(X), X)))), $$true, equiv(necessarily(implies(necessarily(X), X)), implies(necessarily(X), X))), $$true, necessarily(implies(necessarily(X), X)), implies(necessarily(X), X)) 13.33/13.60 = { by lemma 258 } 13.33/13.60 $$fresh3($$fresh115(is_a_theorem(implies(implies(necessarily(X), X), necessarily(implies(necessarily(X), X)))), $$true, equiv(necessarily(implies(necessarily(X), X)), implies(necessarily(X), X))), $$true, necessarily(implies(necessarily(X), X)), implies(necessarily(X), X)) 13.33/13.60 = { by axiom 75 (modus_ponens_2) } 13.42/13.60 $$fresh3($$fresh116(?, ?, implies(implies(necessarily(X), X), necessarily(implies(necessarily(X), X))), equiv(necessarily(implies(necessarily(X), X)), implies(necessarily(X), X))), $$true, necessarily(implies(necessarily(X), X)), implies(necessarily(X), X)) 13.42/13.60 = { by lemma 272 } 13.42/13.60 $$fresh3(is_a_theorem(equiv(necessarily(implies(necessarily(X), X)), implies(necessarily(X), X))), $$true, necessarily(implies(necessarily(X), X)), implies(necessarily(X), X)) 13.42/13.60 = { by lemma 261 } 13.42/13.60 implies(necessarily(X), X) 13.42/13.60 13.42/13.60 Lemma 274: implies(X, implies(X, Y)) = implies(X, Y). 13.42/13.60 Proof: 13.42/13.60 implies(X, implies(X, Y)) 13.42/13.60 = { by axiom 112 (substitution_of_equivalents_2) } 13.42/13.60 $$fresh3($$true, $$true, implies(X, implies(X, Y)), implies(X, Y)) 13.42/13.60 = { by lemma 257 } 13.42/13.60 $$fresh3($$fresh116(?, ?, implies(implies(X, Y), implies(X, implies(X, Y))), equiv(implies(X, implies(X, Y)), implies(X, Y))), $$true, implies(X, implies(X, Y)), implies(X, Y)) 13.42/13.60 = { by lemma 262 } 13.42/13.60 $$fresh3($$fresh40(is_a_theorem(implies(implies(implies(X, Y), implies(X, implies(X, Y))), equiv(implies(X, implies(X, Y)), implies(X, Y)))), $$true, implies(implies(X, Y), implies(X, implies(X, Y))), equiv(implies(X, implies(X, Y)), implies(X, Y))), $$true, implies(X, implies(X, Y)), implies(X, Y)) 13.42/13.60 = { by lemma 263 } 13.42/13.60 $$fresh3($$fresh40(is_a_theorem(implies(implies(implies(X, Y), implies(X, implies(X, Y))), and(implies(implies(X, implies(X, Y)), implies(X, Y)), implies(implies(X, Y), implies(X, implies(X, Y)))))), $$true, implies(implies(X, Y), implies(X, implies(X, Y))), equiv(implies(X, implies(X, Y)), implies(X, Y))), $$true, implies(X, implies(X, Y)), implies(X, Y)) 13.42/13.60 = { by lemma 264 } 13.42/13.60 $$fresh3($$fresh40($$fresh116(?, ?, implies(implies(X, implies(X, Y)), implies(X, Y)), implies(implies(implies(X, Y), implies(X, implies(X, Y))), and(implies(implies(X, implies(X, Y)), implies(X, Y)), implies(implies(X, Y), implies(X, implies(X, Y)))))), $$true, implies(implies(X, Y), implies(X, implies(X, Y))), equiv(implies(X, implies(X, Y)), implies(X, Y))), $$true, implies(X, implies(X, Y)), implies(X, Y)) 13.42/13.60 = { by axiom 75 (modus_ponens_2) } 13.42/13.60 $$fresh3($$fresh40($$fresh115(is_a_theorem(implies(implies(X, implies(X, Y)), implies(X, Y))), $$true, implies(implies(implies(X, Y), implies(X, implies(X, Y))), and(implies(implies(X, implies(X, Y)), implies(X, Y)), implies(implies(X, Y), implies(X, implies(X, Y)))))), $$true, implies(implies(X, Y), implies(X, implies(X, Y))), equiv(implies(X, implies(X, Y)), implies(X, Y))), $$true, implies(X, implies(X, Y)), implies(X, Y)) 13.42/13.60 = { by axiom 167 (implies_2_1) } 13.42/13.60 $$fresh3($$fresh40($$fresh115($$fresh49(implies_2, $$true, X, Y), $$true, implies(implies(implies(X, Y), implies(X, implies(X, Y))), and(implies(implies(X, implies(X, Y)), implies(X, Y)), implies(implies(X, Y), implies(X, implies(X, Y)))))), $$true, implies(implies(X, Y), implies(X, implies(X, Y))), equiv(implies(X, implies(X, Y)), implies(X, Y))), $$true, implies(X, implies(X, Y)), implies(X, Y)) 13.42/13.60 = { by axiom 180 (hilbert_implies_2) } 13.42/13.60 $$fresh3($$fresh40($$fresh115($$fresh49($$true, $$true, X, Y), $$true, implies(implies(implies(X, Y), implies(X, implies(X, Y))), and(implies(implies(X, implies(X, Y)), implies(X, Y)), implies(implies(X, Y), implies(X, implies(X, Y)))))), $$true, implies(implies(X, Y), implies(X, implies(X, Y))), equiv(implies(X, implies(X, Y)), implies(X, Y))), $$true, implies(X, implies(X, Y)), implies(X, Y)) 13.42/13.60 = { by axiom 64 (implies_2_1) } 13.42/13.60 $$fresh3($$fresh40($$fresh115($$true, $$true, implies(implies(implies(X, Y), implies(X, implies(X, Y))), and(implies(implies(X, implies(X, Y)), implies(X, Y)), implies(implies(X, Y), implies(X, implies(X, Y)))))), $$true, implies(implies(X, Y), implies(X, implies(X, Y))), equiv(implies(X, implies(X, Y)), implies(X, Y))), $$true, implies(X, implies(X, Y)), implies(X, Y)) 13.42/13.60 = { by axiom 74 (modus_ponens_2) } 13.42/13.60 $$fresh3($$fresh40($$true, $$true, implies(implies(X, Y), implies(X, implies(X, Y))), equiv(implies(X, implies(X, Y)), implies(X, Y))), $$true, implies(X, implies(X, Y)), implies(X, Y)) 13.42/13.60 = { by axiom 73 (modus_ponens_2) } 13.42/13.60 $$fresh3(is_a_theorem(equiv(implies(X, implies(X, Y)), implies(X, Y))), $$true, implies(X, implies(X, Y)), implies(X, Y)) 13.42/13.60 = { by lemma 261 } 13.42/13.60 implies(X, Y) 13.42/13.60 13.42/13.60 Lemma 275: not(equiv(implies(X, Y), not(implies(X, Y)))) = or(implies(X, Y), not(implies(X, Y))). 13.42/13.60 Proof: 13.42/13.60 not(equiv(implies(X, Y), not(implies(X, Y)))) 13.42/13.60 = { by lemma 271 } 13.42/13.60 not(equiv(implies(X, Y), implies(implies(X, Y), and(X, not(Y))))) 13.42/13.60 = { by axiom 87 (op_equiv) } 13.42/13.60 not($$fresh30($$true, $$true, implies(X, Y), implies(implies(X, Y), and(X, not(Y))))) 13.42/13.60 = { by axiom 189 (hilbert_op_equiv) } 13.42/13.60 not($$fresh30(op_equiv, $$true, implies(X, Y), implies(implies(X, Y), and(X, not(Y))))) 13.42/13.60 = { by axiom 173 (op_equiv) } 13.42/13.60 not(and(implies(implies(X, Y), implies(implies(X, Y), and(X, not(Y)))), implies(implies(implies(X, Y), and(X, not(Y))), implies(X, Y)))) 13.42/13.60 = { by lemma 252 } 13.42/13.60 implies(implies(implies(X, Y), implies(implies(X, Y), and(X, not(Y)))), and(implies(implies(X, Y), and(X, not(Y))), not(implies(X, Y)))) 13.42/13.60 = { by lemma 274 } 13.42/13.60 implies(implies(implies(X, Y), and(X, not(Y))), and(implies(implies(X, Y), and(X, not(Y))), not(implies(X, Y)))) 13.42/13.60 = { by lemma 271 } 13.42/13.60 implies(not(implies(X, Y)), and(implies(implies(X, Y), and(X, not(Y))), not(implies(X, Y)))) 13.42/13.60 = { by lemma 249 } 13.42/13.60 or(implies(X, Y), and(implies(implies(X, Y), and(X, not(Y))), not(implies(X, Y)))) 13.42/13.60 = { by lemma 271 } 13.42/13.60 or(implies(X, Y), and(not(implies(X, Y)), not(implies(X, Y)))) 13.42/13.60 = { by lemma 249 } 13.42/13.60 implies(not(implies(X, Y)), and(not(implies(X, Y)), not(implies(X, Y)))) 13.42/13.60 = { by lemma 248 } 13.42/13.60 not(and(not(implies(X, Y)), not(and(not(implies(X, Y)), not(implies(X, Y)))))) 13.42/13.60 = { by axiom 169 (op_or) } 13.42/13.60 not(and(not(implies(X, Y)), $$fresh26(op_or, $$true, implies(X, Y), implies(X, Y)))) 13.42/13.60 = { by axiom 176 (hilbert_op_or) } 13.42/13.60 not(and(not(implies(X, Y)), $$fresh26($$true, $$true, implies(X, Y), implies(X, Y)))) 13.42/13.60 = { by axiom 91 (op_or) } 13.42/13.60 not(and(not(implies(X, Y)), or(implies(X, Y), implies(X, Y)))) 13.42/13.60 = { by lemma 268 } 13.42/13.60 not(and(not(implies(X, Y)), not(not(implies(X, Y))))) 13.42/13.60 = { by lemma 248 } 13.42/13.60 implies(not(implies(X, Y)), not(implies(X, Y))) 13.42/13.60 = { by lemma 249 } 13.42/13.61 or(implies(X, Y), not(implies(X, Y))) 13.42/13.61 13.42/13.61 Lemma 276: is_a_theorem(equiv(necessarily(implies(and(X, Y), Y)), implies(and(X, Y), Y))) = $$true. 13.42/13.61 Proof: 13.42/13.61 is_a_theorem(equiv(necessarily(implies(and(X, Y), Y)), implies(and(X, Y), Y))) 13.42/13.61 = { by lemma 272 } 13.42/13.61 $$fresh116(?, ?, implies(implies(and(X, Y), Y), necessarily(implies(and(X, Y), Y))), equiv(necessarily(implies(and(X, Y), Y)), implies(and(X, Y), Y))) 13.42/13.61 = { by axiom 75 (modus_ponens_2) } 13.42/13.61 $$fresh115(is_a_theorem(implies(implies(and(X, Y), Y), necessarily(implies(and(X, Y), Y)))), $$true, equiv(necessarily(implies(and(X, Y), Y)), implies(and(X, Y), Y))) 13.42/13.61 = { by lemma 258 } 13.42/13.61 $$fresh115($$fresh116(?, ?, necessarily(implies(and(X, Y), Y)), implies(implies(and(X, Y), Y), necessarily(implies(and(X, Y), Y)))), $$true, equiv(necessarily(implies(and(X, Y), Y)), implies(and(X, Y), Y))) 13.42/13.61 = { by axiom 75 (modus_ponens_2) } 13.42/13.61 $$fresh115($$fresh115(is_a_theorem(necessarily(implies(and(X, Y), Y))), $$true, implies(implies(and(X, Y), Y), necessarily(implies(and(X, Y), Y)))), $$true, equiv(necessarily(implies(and(X, Y), Y)), implies(and(X, Y), Y))) 13.42/13.61 = { by axiom 83 (necessitation_1) } 13.42/13.61 $$fresh115($$fresh115($$fresh34($$true, $$true, implies(and(X, Y), Y)), $$true, implies(implies(and(X, Y), Y), necessarily(implies(and(X, Y), Y)))), $$true, equiv(necessarily(implies(and(X, Y), Y)), implies(and(X, Y), Y))) 13.42/13.61 = { by axiom 242 (km5_necessitation) } 13.42/13.61 $$fresh115($$fresh115($$fresh34(necessitation, $$true, implies(and(X, Y), Y)), $$true, implies(implies(and(X, Y), Y), necessarily(implies(and(X, Y), Y)))), $$true, equiv(necessarily(implies(and(X, Y), Y)), implies(and(X, Y), Y))) 13.42/13.61 = { by axiom 227 (necessitation_1) } 13.42/13.61 $$fresh115($$fresh115($$fresh33(is_a_theorem(implies(and(X, Y), Y)), $$true, implies(and(X, Y), Y)), $$true, implies(implies(and(X, Y), Y), necessarily(implies(and(X, Y), Y)))), $$true, equiv(necessarily(implies(and(X, Y), Y)), implies(and(X, Y), Y))) 13.42/13.61 = { by axiom 117 (and_2_1) } 13.42/13.61 $$fresh115($$fresh115($$fresh33($$fresh105(and_2, $$true, X, Y), $$true, implies(and(X, Y), Y)), $$true, implies(implies(and(X, Y), Y), necessarily(implies(and(X, Y), Y)))), $$true, equiv(necessarily(implies(and(X, Y), Y)), implies(and(X, Y), Y))) 13.42/13.61 = { by axiom 177 (hilbert_and_2) } 13.42/13.61 $$fresh115($$fresh115($$fresh33($$fresh105($$true, $$true, X, Y), $$true, implies(and(X, Y), Y)), $$true, implies(implies(and(X, Y), Y), necessarily(implies(and(X, Y), Y)))), $$true, equiv(necessarily(implies(and(X, Y), Y)), implies(and(X, Y), Y))) 13.42/13.61 = { by axiom 8 (and_2_1) } 13.42/13.61 $$fresh115($$fresh115($$fresh33($$true, $$true, implies(and(X, Y), Y)), $$true, implies(implies(and(X, Y), Y), necessarily(implies(and(X, Y), Y)))), $$true, equiv(necessarily(implies(and(X, Y), Y)), implies(and(X, Y), Y))) 13.42/13.61 = { by axiom 84 (necessitation_1) } 13.42/13.61 $$fresh115($$fresh115($$true, $$true, implies(implies(and(X, Y), Y), necessarily(implies(and(X, Y), Y)))), $$true, equiv(necessarily(implies(and(X, Y), Y)), implies(and(X, Y), Y))) 13.42/13.61 = { by axiom 74 (modus_ponens_2) } 13.42/13.61 $$fresh115($$true, $$true, equiv(necessarily(implies(and(X, Y), Y)), implies(and(X, Y), Y))) 13.42/13.61 = { by axiom 74 (modus_ponens_2) } 13.42/13.61 $$true 13.42/13.61 13.42/13.61 Lemma 277: necessarily(implies(X, X)) = implies(X, X). 13.42/13.61 Proof: 13.42/13.61 necessarily(implies(X, X)) 13.42/13.61 = { by axiom 112 (substitution_of_equivalents_2) } 13.42/13.61 $$fresh3($$true, $$true, necessarily(implies(X, X)), implies(X, X)) 13.42/13.61 = { by lemma 276 } 13.42/13.61 $$fresh3(is_a_theorem(equiv(necessarily(implies(and(X, X), X)), implies(and(X, X), X))), $$true, necessarily(implies(X, X)), implies(X, X)) 13.42/13.61 = { by lemma 267 } 13.42/13.61 $$fresh3(is_a_theorem(equiv(necessarily(implies(X, X)), implies(and(X, X), X))), $$true, necessarily(implies(X, X)), implies(X, X)) 13.42/13.61 = { by lemma 267 } 13.42/13.61 $$fresh3(is_a_theorem(equiv(necessarily(implies(X, X)), implies(X, X))), $$true, necessarily(implies(X, X)), implies(X, X)) 13.42/13.61 = { by lemma 261 } 13.42/13.62 implies(X, X) 13.42/13.62 13.42/13.62 Lemma 278: implies(implies(X, X), implies(X, X)) = implies(X, X). 13.42/13.62 Proof: 13.42/13.62 implies(implies(X, X), implies(X, X)) 13.42/13.62 = { by lemma 261 } 13.42/13.62 $$fresh3(is_a_theorem(equiv(implies(X, X), implies(implies(X, X), implies(X, X)))), $$true, implies(X, X), implies(implies(X, X), implies(X, X))) 13.42/13.62 = { by lemma 263 } 13.42/13.62 $$fresh3(is_a_theorem(and(implies(implies(X, X), implies(implies(X, X), implies(X, X))), implies(implies(implies(X, X), implies(X, X)), implies(X, X)))), $$true, implies(X, X), implies(implies(X, X), implies(X, X))) 13.42/13.62 = { by lemma 274 } 13.42/13.62 $$fresh3(is_a_theorem(and(implies(implies(X, X), implies(X, X)), implies(implies(implies(X, X), implies(X, X)), implies(X, X)))), $$true, implies(X, X), implies(implies(X, X), implies(X, X))) 13.42/13.62 = { by lemma 267 } 13.42/13.62 $$fresh3(is_a_theorem(and(implies(and(implies(X, X), implies(X, X)), implies(X, X)), implies(implies(implies(X, X), implies(X, X)), implies(X, X)))), $$true, implies(X, X), implies(implies(X, X), implies(X, X))) 13.42/13.62 = { by axiom 73 (modus_ponens_2) } 13.42/13.62 $$fresh3($$fresh40($$true, $$true, implies(implies(implies(X, X), implies(X, X)), implies(X, X)), and(implies(and(implies(X, X), implies(X, X)), implies(X, X)), implies(implies(implies(X, X), implies(X, X)), implies(X, X)))), $$true, implies(X, X), implies(implies(X, X), implies(X, X))) 13.42/13.62 = { by lemma 266 } 13.42/13.62 $$fresh3($$fresh40(is_a_theorem(implies(implies(implies(implies(X, X), implies(X, X)), implies(X, X)), and(implies(and(implies(X, X), implies(X, X)), implies(X, X)), implies(implies(implies(X, X), implies(X, X)), implies(X, X))))), $$true, implies(implies(implies(X, X), implies(X, X)), implies(X, X)), and(implies(and(implies(X, X), implies(X, X)), implies(X, X)), implies(implies(implies(X, X), implies(X, X)), implies(X, X)))), $$true, implies(X, X), implies(implies(X, X), implies(X, X))) 13.42/13.62 = { by lemma 262 } 13.42/13.62 $$fresh3($$fresh116(?, ?, implies(implies(implies(X, X), implies(X, X)), implies(X, X)), and(implies(and(implies(X, X), implies(X, X)), implies(X, X)), implies(implies(implies(X, X), implies(X, X)), implies(X, X)))), $$true, implies(X, X), implies(implies(X, X), implies(X, X))) 13.42/13.62 = { by lemma 267 } 13.42/13.62 $$fresh3($$fresh116(?, ?, implies(implies(implies(X, X), implies(X, X)), implies(and(X, X), X)), and(implies(and(implies(X, X), implies(X, X)), implies(X, X)), implies(implies(implies(X, X), implies(X, X)), implies(X, X)))), $$true, implies(X, X), implies(implies(X, X), implies(X, X))) 13.42/13.62 = { by lemma 267 } 13.42/13.62 $$fresh3($$fresh116(?, ?, implies(implies(implies(X, X), implies(X, X)), implies(and(X, X), X)), and(implies(implies(X, X), implies(X, X)), implies(implies(implies(X, X), implies(X, X)), implies(X, X)))), $$true, implies(X, X), implies(implies(X, X), implies(X, X))) 13.42/13.62 = { by axiom 75 (modus_ponens_2) } 13.42/13.62 $$fresh3($$fresh115(is_a_theorem(implies(implies(implies(X, X), implies(X, X)), implies(and(X, X), X))), $$true, and(implies(implies(X, X), implies(X, X)), implies(implies(implies(X, X), implies(X, X)), implies(X, X)))), $$true, implies(X, X), implies(implies(X, X), implies(X, X))) 13.42/13.62 = { by lemma 258 } 13.42/13.62 $$fresh3($$fresh115($$fresh116(?, ?, implies(and(X, X), X), implies(implies(implies(X, X), implies(X, X)), implies(and(X, X), X))), $$true, and(implies(implies(X, X), implies(X, X)), implies(implies(implies(X, X), implies(X, X)), implies(X, X)))), $$true, implies(X, X), implies(implies(X, X), implies(X, X))) 13.42/13.62 = { by lemma 255 } 13.42/13.62 $$fresh3($$fresh115($$true, $$true, and(implies(implies(X, X), implies(X, X)), implies(implies(implies(X, X), implies(X, X)), implies(X, X)))), $$true, implies(X, X), implies(implies(X, X), implies(X, X))) 13.42/13.62 = { by axiom 74 (modus_ponens_2) } 13.42/13.62 $$fresh3($$true, $$true, implies(X, X), implies(implies(X, X), implies(X, X))) 13.42/13.62 = { by axiom 112 (substitution_of_equivalents_2) } 13.42/13.62 implies(X, X) 13.42/13.62 13.42/13.62 Lemma 279: not(not(X)) = X. 13.42/13.62 Proof: 13.42/13.62 not(not(X)) 13.42/13.62 = { by lemma 261 } 13.42/13.62 $$fresh3(is_a_theorem(equiv(X, not(not(X)))), $$true, X, not(not(X))) 13.42/13.62 = { by axiom 73 (modus_ponens_2) } 13.42/13.62 $$fresh3($$fresh40($$true, $$true, or(not(X), X), equiv(X, not(not(X)))), $$true, X, not(not(X))) 13.42/13.62 = { by lemma 269 } 13.42/13.62 $$fresh3($$fresh40(is_a_theorem(implies(implies(or(X, X), X), equiv(X, or(X, X)))), $$true, or(not(X), X), equiv(X, not(not(X)))), $$true, X, not(not(X))) 13.42/13.62 = { by lemma 268 } 13.42/13.62 $$fresh3($$fresh40(is_a_theorem(implies(implies(not(not(X)), X), equiv(X, or(X, X)))), $$true, or(not(X), X), equiv(X, not(not(X)))), $$true, X, not(not(X))) 13.42/13.62 = { by lemma 249 } 13.42/13.62 $$fresh3($$fresh40(is_a_theorem(implies(or(not(X), X), equiv(X, or(X, X)))), $$true, or(not(X), X), equiv(X, not(not(X)))), $$true, X, not(not(X))) 13.42/13.62 = { by lemma 268 } 13.42/13.62 $$fresh3($$fresh40(is_a_theorem(implies(or(not(X), X), equiv(X, not(not(X))))), $$true, or(not(X), X), equiv(X, not(not(X)))), $$true, X, not(not(X))) 13.42/13.62 = { by lemma 262 } 13.42/13.62 $$fresh3($$fresh116(?, ?, or(not(X), X), equiv(X, not(not(X)))), $$true, X, not(not(X))) 13.42/13.62 = { by lemma 249 } 13.42/13.62 $$fresh3($$fresh116(?, ?, implies(not(not(X)), X), equiv(X, not(not(X)))), $$true, X, not(not(X))) 13.42/13.62 = { by lemma 268 } 13.42/13.62 $$fresh3($$fresh116(?, ?, implies(or(X, X), X), equiv(X, not(not(X)))), $$true, X, not(not(X))) 13.42/13.62 = { by axiom 75 (modus_ponens_2) } 13.42/13.62 $$fresh3($$fresh115(is_a_theorem(implies(or(X, X), X)), $$true, equiv(X, not(not(X)))), $$true, X, not(not(X))) 13.42/13.62 = { by lemma 268 } 13.42/13.62 $$fresh3($$fresh115(is_a_theorem(implies(not(not(X)), X)), $$true, equiv(X, not(not(X)))), $$true, X, not(not(X))) 13.42/13.62 = { by lemma 259 } 13.42/13.62 $$fresh3($$fresh115($$fresh116(?, ?, or(X, not(not(not(X)))), implies(not(not(X)), X)), $$true, equiv(X, not(not(X)))), $$true, X, not(not(X))) 13.42/13.62 = { by lemma 268 } 13.42/13.62 $$fresh3($$fresh115($$fresh116(?, ?, or(X, or(not(X), not(X))), implies(not(not(X)), X)), $$true, equiv(X, not(not(X)))), $$true, X, not(not(X))) 13.42/13.62 = { by lemma 249 } 13.42/13.62 $$fresh3($$fresh115($$fresh116(?, ?, implies(not(X), or(not(X), not(X))), implies(not(not(X)), X)), $$true, equiv(X, not(not(X)))), $$true, X, not(not(X))) 13.42/13.62 = { by lemma 256 } 13.42/13.62 $$fresh3($$fresh115($$true, $$true, equiv(X, not(not(X)))), $$true, X, not(not(X))) 13.42/13.62 = { by axiom 74 (modus_ponens_2) } 13.42/13.62 $$fresh3($$true, $$true, X, not(not(X))) 13.42/13.62 = { by axiom 112 (substitution_of_equivalents_2) } 13.42/13.62 X 13.42/13.62 13.42/13.62 Lemma 280: not(possibly(X)) = necessarily(not(X)). 13.42/13.62 Proof: 13.42/13.62 not(possibly(X)) 13.42/13.62 = { by lemma 247 } 13.42/13.62 not(not(necessarily(not(X)))) 13.42/13.62 = { by lemma 279 } 13.42/13.62 necessarily(not(X)) 13.42/13.62 13.42/13.62 Lemma 281: not(necessarily(X)) = possibly(not(X)). 13.42/13.62 Proof: 13.42/13.62 not(necessarily(X)) 13.42/13.62 = { by lemma 279 } 13.42/13.62 not(necessarily(not(not(X)))) 13.42/13.62 = { by lemma 247 } 13.42/13.62 possibly(not(X)) 13.42/13.62 13.42/13.62 Lemma 282: possibly(necessarily(X)) = necessarily(X). 13.42/13.62 Proof: 13.42/13.62 possibly(necessarily(X)) 13.42/13.62 = { by lemma 247 } 13.42/13.62 not(necessarily(not(necessarily(X)))) 13.42/13.62 = { by lemma 281 } 13.42/13.62 not(necessarily(possibly(not(X)))) 13.42/13.62 = { by lemma 270 } 13.42/13.62 not(possibly(not(X))) 13.42/13.62 = { by lemma 280 } 13.42/13.62 necessarily(not(not(X))) 13.42/13.62 = { by lemma 279 } 13.42/13.62 necessarily(X) 13.42/13.62 13.42/13.62 Lemma 283: or(not(X), Y) = implies(X, Y). 13.42/13.62 Proof: 13.42/13.62 or(not(X), Y) 13.42/13.62 = { by lemma 249 } 13.42/13.62 implies(not(not(X)), Y) 13.42/13.62 = { by lemma 279 } 13.42/13.62 implies(X, Y) 13.42/13.62 13.42/13.62 Lemma 284: not(implies(X, Y)) = and(X, not(Y)). 13.42/13.62 Proof: 13.42/13.62 not(implies(X, Y)) 13.42/13.62 = { by lemma 248 } 13.42/13.62 not(not(and(X, not(Y)))) 13.42/13.62 = { by lemma 279 } 13.42/13.62 and(X, not(Y)) 13.42/13.62 13.42/13.62 Lemma 285: not(and(X, Y)) = implies(X, not(Y)). 13.42/13.62 Proof: 13.42/13.62 not(and(X, Y)) 13.42/13.62 = { by lemma 279 } 13.42/13.62 not(and(X, not(not(Y)))) 13.42/13.62 = { by lemma 248 } 13.42/13.62 implies(X, not(Y)) 13.42/13.62 13.42/13.62 Lemma 286: implies(necessarily(not(X)), Y) = or(possibly(X), Y). 13.42/13.62 Proof: 13.42/13.62 implies(necessarily(not(X)), Y) 13.42/13.62 = { by lemma 280 } 13.42/13.62 implies(not(possibly(X)), Y) 13.42/13.62 = { by lemma 249 } 13.42/13.62 or(possibly(X), Y) 13.42/13.62 13.42/13.62 Lemma 287: not(or(X, not(Y))) = and(not(X), Y). 13.42/13.62 Proof: 13.42/13.62 not(or(X, not(Y))) 13.42/13.62 = { by lemma 279 } 13.42/13.62 not(or(not(not(X)), not(Y))) 13.42/13.62 = { by lemma 283 } 13.42/13.62 not(implies(not(X), not(Y))) 13.42/13.62 = { by lemma 284 } 13.42/13.62 and(not(X), not(not(Y))) 13.42/13.62 = { by lemma 279 } 13.42/13.63 and(not(X), Y) 13.42/13.63 13.42/13.63 Lemma 288: $$fresh116(?, ?, implies(X, and(Y, X)), equiv(and(Y, X), X)) = is_a_theorem(equiv(and(Y, X), X)). 13.42/13.63 Proof: 13.42/13.63 $$fresh116(?, ?, implies(X, and(Y, X)), equiv(and(Y, X), X)) 13.42/13.63 = { by lemma 262 } 13.42/13.63 $$fresh40(is_a_theorem(implies(implies(X, and(Y, X)), equiv(and(Y, X), X))), $$true, implies(X, and(Y, X)), equiv(and(Y, X), X)) 13.42/13.63 = { by lemma 263 } 13.42/13.63 $$fresh40(is_a_theorem(implies(implies(X, and(Y, X)), and(implies(and(Y, X), X), implies(X, and(Y, X))))), $$true, implies(X, and(Y, X)), equiv(and(Y, X), X)) 13.42/13.63 = { by lemma 266 } 13.42/13.63 $$fresh40($$true, $$true, implies(X, and(Y, X)), equiv(and(Y, X), X)) 13.42/13.63 = { by axiom 73 (modus_ponens_2) } 13.42/13.63 is_a_theorem(equiv(and(Y, X), X)) 13.42/13.63 13.42/13.63 Lemma 289: and(implies(necessarily(Y), Y), X) = X. 13.42/13.63 Proof: 13.42/13.63 and(implies(necessarily(Y), Y), X) 13.42/13.63 = { by axiom 112 (substitution_of_equivalents_2) } 13.42/13.63 $$fresh3($$true, $$true, and(implies(necessarily(Y), Y), X), X) 13.42/13.63 = { by axiom 74 (modus_ponens_2) } 13.42/13.63 $$fresh3($$fresh115($$true, $$true, equiv(and(implies(necessarily(Y), Y), X), X)), $$true, and(implies(necessarily(Y), Y), X), X) 13.42/13.63 = { by lemma 265 } 13.42/13.63 $$fresh3($$fresh115(is_a_theorem(implies(X, and(implies(necessarily(Y), Y), X))), $$true, equiv(and(implies(necessarily(Y), Y), X), X)), $$true, and(implies(necessarily(Y), Y), X), X) 13.42/13.63 = { by axiom 75 (modus_ponens_2) } 13.42/13.63 $$fresh3($$fresh116(?, ?, implies(X, and(implies(necessarily(Y), Y), X)), equiv(and(implies(necessarily(Y), Y), X), X)), $$true, and(implies(necessarily(Y), Y), X), X) 13.42/13.63 = { by lemma 288 } 13.42/13.63 $$fresh3(is_a_theorem(equiv(and(implies(necessarily(Y), Y), X), X)), $$true, and(implies(necessarily(Y), Y), X), X) 13.42/13.63 = { by lemma 261 } 13.42/13.63 X 13.42/13.63 13.42/13.63 Lemma 290: and(implies(Y, Y), X) = X. 13.42/13.63 Proof: 13.42/13.63 and(implies(Y, Y), X) 13.42/13.63 = { by lemma 278 } 13.42/13.63 and(implies(implies(Y, Y), implies(Y, Y)), X) 13.42/13.63 = { by lemma 277 } 13.42/13.63 and(implies(necessarily(implies(Y, Y)), implies(Y, Y)), X) 13.42/13.63 = { by lemma 289 } 13.42/13.63 X 13.42/13.63 13.42/13.63 Lemma 291: implies(implies(necessarily(X), X), Y) = Y. 13.42/13.63 Proof: 13.42/13.63 implies(implies(necessarily(X), X), Y) 13.42/13.63 = { by lemma 248 } 13.42/13.63 not(and(implies(necessarily(X), X), not(Y))) 13.42/13.63 = { by lemma 289 } 13.42/13.63 not(not(Y)) 13.42/13.63 = { by lemma 279 } 13.42/13.63 Y 13.42/13.63 13.42/13.63 Lemma 292: implies(implies(Y, Y), X) = X. 13.42/13.63 Proof: 13.42/13.63 implies(implies(Y, Y), X) 13.42/13.63 = { by lemma 278 } 13.42/13.63 implies(implies(implies(Y, Y), implies(Y, Y)), X) 13.42/13.63 = { by lemma 277 } 13.42/13.63 implies(implies(necessarily(implies(Y, Y)), implies(Y, Y)), X) 13.42/13.63 = { by lemma 291 } 13.42/13.63 X 13.42/13.63 13.42/13.63 Lemma 293: implies(and(Y, not(Z)), W) = or(implies(Y, Z), W). 13.42/13.63 Proof: 13.42/13.63 implies(and(Y, not(Z)), W) 13.42/13.63 = { by lemma 291 } 13.42/13.63 implies(implies(implies(necessarily(?), ?), and(Y, not(Z))), W) 13.42/13.63 = { by lemma 248 } 13.42/13.63 not(and(implies(implies(necessarily(?), ?), and(Y, not(Z))), not(W))) 13.42/13.63 = { by lemma 248 } 13.42/13.63 not(and(not(and(implies(necessarily(?), ?), not(and(Y, not(Z))))), not(W))) 13.42/13.63 = { by axiom 169 (op_or) } 13.42/13.63 $$fresh26(op_or, $$true, and(implies(necessarily(?), ?), not(and(Y, not(Z)))), W) 13.42/13.63 = { by axiom 176 (hilbert_op_or) } 13.42/13.63 $$fresh26($$true, $$true, and(implies(necessarily(?), ?), not(and(Y, not(Z)))), W) 13.42/13.63 = { by axiom 91 (op_or) } 13.42/13.63 or(and(implies(necessarily(?), ?), not(and(Y, not(Z)))), W) 13.42/13.63 = { by lemma 248 } 13.42/13.63 or(and(implies(necessarily(?), ?), implies(Y, Z)), W) 13.42/13.63 = { by lemma 289 } 13.42/13.63 or(implies(Y, Z), W) 13.42/13.63 13.42/13.63 Lemma 294: possibly(and(necessarily(not(X)), X)) = and(necessarily(not(X)), X). 13.42/13.63 Proof: 13.42/13.63 possibly(and(necessarily(not(X)), X)) 13.42/13.63 = { by lemma 280 } 13.42/13.63 possibly(and(not(possibly(X)), X)) 13.42/13.63 = { by lemma 287 } 13.42/13.63 possibly(not(or(possibly(X), not(X)))) 13.42/13.63 = { by lemma 289 } 13.42/13.63 possibly(not(or(and(implies(necessarily(?), ?), possibly(X)), not(X)))) 13.42/13.63 = { by lemma 281 } 13.42/13.63 not(necessarily(or(and(implies(necessarily(?), ?), possibly(X)), not(X)))) 13.42/13.63 = { by axiom 91 (op_or) } 13.42/13.63 not(necessarily($$fresh26($$true, $$true, and(implies(necessarily(?), ?), possibly(X)), not(X)))) 13.42/13.63 = { by axiom 176 (hilbert_op_or) } 13.42/13.63 not(necessarily($$fresh26(op_or, $$true, and(implies(necessarily(?), ?), possibly(X)), not(X)))) 13.42/13.63 = { by axiom 169 (op_or) } 13.42/13.63 not(necessarily(not(and(not(and(implies(necessarily(?), ?), possibly(X))), not(not(X)))))) 13.42/13.63 = { by lemma 247 } 13.42/13.63 possibly(and(not(and(implies(necessarily(?), ?), possibly(X))), not(not(X)))) 13.42/13.63 = { by lemma 251 } 13.42/13.63 possibly(and(implies(implies(necessarily(?), ?), necessarily(not(X))), not(not(X)))) 13.42/13.63 = { by lemma 291 } 13.42/13.63 possibly(and(necessarily(not(X)), not(not(X)))) 13.42/13.63 = { by lemma 247 } 13.42/13.63 not(necessarily(not(and(necessarily(not(X)), not(not(X)))))) 13.42/13.63 = { by lemma 248 } 13.42/13.63 not(necessarily(implies(necessarily(not(X)), not(X)))) 13.42/13.63 = { by lemma 273 } 13.42/13.63 not(implies(necessarily(not(X)), not(X))) 13.42/13.63 = { by lemma 284 } 13.42/13.63 and(necessarily(not(X)), not(not(X))) 13.42/13.63 = { by lemma 279 } 13.42/13.63 and(necessarily(not(X)), X) 13.42/13.63 13.42/13.63 Lemma 295: equiv(implies(X, X), implies(Y, Y)) = implies(X, X). 13.42/13.63 Proof: 13.42/13.63 equiv(implies(X, X), implies(Y, Y)) 13.42/13.63 = { by lemma 263 } 13.42/13.63 and(implies(implies(X, X), implies(Y, Y)), implies(implies(Y, Y), implies(X, X))) 13.42/13.63 = { by lemma 292 } 13.42/13.63 and(implies(Y, Y), implies(implies(Y, Y), implies(X, X))) 13.42/13.63 = { by lemma 290 } 13.42/13.63 implies(implies(Y, Y), implies(X, X)) 13.42/13.63 = { by lemma 292 } 13.42/13.63 implies(X, X) 13.42/13.63 13.42/13.63 Lemma 296: implies(X, X) = implies(?, ?). 13.42/13.63 Proof: 13.42/13.63 implies(X, X) 13.42/13.63 = { by lemma 261 } 13.42/13.63 $$fresh3(is_a_theorem(equiv(implies(Y, Y), implies(X, X))), $$true, implies(Y, Y), implies(X, X)) 13.42/13.63 = { by lemma 295 } 13.42/13.63 $$fresh3(is_a_theorem(implies(Y, Y)), $$true, implies(Y, Y), implies(X, X)) 13.42/13.63 = { by lemma 260 } 13.42/13.63 $$fresh3($$true, $$true, implies(Y, Y), implies(X, X)) 13.42/13.63 = { by axiom 112 (substitution_of_equivalents_2) } 13.42/13.63 implies(Y, Y) 13.42/13.63 = { by axiom 112 (substitution_of_equivalents_2) } 13.42/13.63 $$fresh3($$true, $$true, implies(Y, Y), implies(?, ?)) 13.42/13.63 = { by lemma 260 } 13.42/13.63 $$fresh3(is_a_theorem(implies(Y, Y)), $$true, implies(Y, Y), implies(?, ?)) 13.42/13.63 = { by lemma 295 } 13.42/13.63 $$fresh3(is_a_theorem(equiv(implies(Y, Y), implies(?, ?))), $$true, implies(Y, Y), implies(?, ?)) 13.42/13.63 = { by lemma 261 } 13.42/13.63 implies(?, ?) 13.42/13.63 13.42/13.63 Lemma 297: $$fresh116(?, ?, implies(or(X, Y), X), equiv(X, or(X, Y))) = is_a_theorem(equiv(X, or(X, Y))). 13.42/13.63 Proof: 13.42/13.63 $$fresh116(?, ?, implies(or(X, Y), X), equiv(X, or(X, Y))) 13.42/13.63 = { by lemma 262 } 13.42/13.63 $$fresh40(is_a_theorem(implies(implies(or(X, Y), X), equiv(X, or(X, Y)))), $$true, implies(or(X, Y), X), equiv(X, or(X, Y))) 13.42/13.63 = { by lemma 269 } 13.42/13.63 $$fresh40($$true, $$true, implies(or(X, Y), X), equiv(X, or(X, Y))) 13.42/13.63 = { by axiom 73 (modus_ponens_2) } 13.42/13.63 is_a_theorem(equiv(X, or(X, Y))) 13.42/13.63 13.42/13.63 Lemma 298: or(implies(X, or(X, Y)), Z) = implies(X, or(X, Y)). 13.42/13.63 Proof: 13.42/13.63 or(implies(X, or(X, Y)), Z) 13.42/13.63 = { by lemma 261 } 13.42/13.63 $$fresh3(is_a_theorem(equiv(implies(X, or(X, Y)), or(implies(X, or(X, Y)), Z))), $$true, implies(X, or(X, Y)), or(implies(X, or(X, Y)), Z)) 13.42/13.63 = { by lemma 297 } 13.42/13.63 $$fresh3($$fresh116(?, ?, implies(or(implies(X, or(X, Y)), Z), implies(X, or(X, Y))), equiv(implies(X, or(X, Y)), or(implies(X, or(X, Y)), Z))), $$true, implies(X, or(X, Y)), or(implies(X, or(X, Y)), Z)) 13.42/13.63 = { by axiom 75 (modus_ponens_2) } 13.42/13.63 $$fresh3($$fresh115(is_a_theorem(implies(or(implies(X, or(X, Y)), Z), implies(X, or(X, Y)))), $$true, equiv(implies(X, or(X, Y)), or(implies(X, or(X, Y)), Z))), $$true, implies(X, or(X, Y)), or(implies(X, or(X, Y)), Z)) 13.42/13.63 = { by lemma 258 } 13.42/13.63 $$fresh3($$fresh115($$fresh116(?, ?, implies(X, or(X, Y)), implies(or(implies(X, or(X, Y)), Z), implies(X, or(X, Y)))), $$true, equiv(implies(X, or(X, Y)), or(implies(X, or(X, Y)), Z))), $$true, implies(X, or(X, Y)), or(implies(X, or(X, Y)), Z)) 13.42/13.63 = { by lemma 256 } 13.42/13.63 $$fresh3($$fresh115($$true, $$true, equiv(implies(X, or(X, Y)), or(implies(X, or(X, Y)), Z))), $$true, implies(X, or(X, Y)), or(implies(X, or(X, Y)), Z)) 13.42/13.63 = { by axiom 74 (modus_ponens_2) } 13.42/13.63 $$fresh3($$true, $$true, implies(X, or(X, Y)), or(implies(X, or(X, Y)), Z)) 13.42/13.63 = { by axiom 112 (substitution_of_equivalents_2) } 16.93/17.17 implies(X, or(X, Y)) 16.93/17.17 16.93/17.17 Goal 1 (km4b_axiom_B): axiom_B = $$true. 16.93/17.17 Proof: 16.93/17.17 axiom_B 16.93/17.17 = { by axiom 231 (axiom_B) } 16.93/17.17 $$fresh98(is_a_theorem(implies(sK7_axiom_B_X, necessarily(possibly(sK7_axiom_B_X)))), $$true) 16.93/17.17 = { by lemma 270 } 16.93/17.17 $$fresh98(is_a_theorem(implies(sK7_axiom_B_X, possibly(sK7_axiom_B_X))), $$true) 16.93/17.17 = { by lemma 283 } 16.93/17.17 $$fresh98(is_a_theorem(or(not(sK7_axiom_B_X), possibly(sK7_axiom_B_X))), $$true) 16.93/17.17 = { by lemma 292 } 16.93/17.17 $$fresh98(is_a_theorem(implies(implies(?, ?), or(not(sK7_axiom_B_X), possibly(sK7_axiom_B_X)))), $$true) 16.93/17.17 = { by lemma 248 } 16.93/17.17 $$fresh98(is_a_theorem(implies(not(and(?, not(?))), or(not(sK7_axiom_B_X), possibly(sK7_axiom_B_X)))), $$true) 16.93/17.17 = { by lemma 284 } 16.93/17.17 $$fresh98(is_a_theorem(implies(not(not(implies(?, ?))), or(not(sK7_axiom_B_X), possibly(sK7_axiom_B_X)))), $$true) 16.93/17.17 = { by lemma 296 } 16.93/17.17 $$fresh98(is_a_theorem(implies(not(not(implies(implies(necessarily(or(implies(not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))), and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))), not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))))), or(implies(not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))), and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))), not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))))), implies(necessarily(or(implies(not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))), and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))), not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))))), or(implies(not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))), and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))), not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))))))))), or(not(sK7_axiom_B_X), possibly(sK7_axiom_B_X)))), $$true) 16.93/17.17 = { by lemma 291 } 16.93/17.17 $$fresh98(is_a_theorem(implies(not(not(implies(necessarily(or(implies(not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))), and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))), not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))))), or(implies(not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))), and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))), not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))))))), or(not(sK7_axiom_B_X), possibly(sK7_axiom_B_X)))), $$true) 16.93/17.17 = { by lemma 273 } 16.93/17.17 $$fresh98(is_a_theorem(implies(not(not(necessarily(implies(necessarily(or(implies(not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))), and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))), not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))))), or(implies(not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))), and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))), not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))))))))), or(not(sK7_axiom_B_X), possibly(sK7_axiom_B_X)))), $$true) 16.93/17.17 = { by lemma 293 } 16.93/17.18 $$fresh98(is_a_theorem(implies(not(not(necessarily(implies(necessarily(implies(and(not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))), not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))))), not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))))), or(implies(not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))), and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))), not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))))))))), or(not(sK7_axiom_B_X), possibly(sK7_axiom_B_X)))), $$true) 16.93/17.18 = { by axiom 112 (substitution_of_equivalents_2) } 16.93/17.18 $$fresh98(is_a_theorem(implies(not(not(necessarily(implies($$fresh3($$true, $$true, necessarily(implies(and(not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))), not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))))), not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))))), implies(and(not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))), not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))))), not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))))), or(implies(not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))), and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))), not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))))))))), or(not(sK7_axiom_B_X), possibly(sK7_axiom_B_X)))), $$true) 16.93/17.18 = { by lemma 276 } 16.93/17.18 $$fresh98(is_a_theorem(implies(not(not(necessarily(implies($$fresh3(is_a_theorem(equiv(necessarily(implies(and(not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))), not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))))), not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))))), implies(and(not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))), not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))))), not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))))))), $$true, necessarily(implies(and(not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))), not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))))), not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))))), implies(and(not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))), not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))))), not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))))), or(implies(not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))), and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))), not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))))))))), or(not(sK7_axiom_B_X), possibly(sK7_axiom_B_X)))), $$true) 16.93/17.18 = { by lemma 261 } 16.93/17.18 $$fresh98(is_a_theorem(implies(not(not(necessarily(implies(implies(and(not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))), not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))))), not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))))), or(implies(not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))), and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))), not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))))))))), or(not(sK7_axiom_B_X), possibly(sK7_axiom_B_X)))), $$true) 16.93/17.18 = { by lemma 293 } 16.93/17.18 $$fresh98(is_a_theorem(implies(not(not(necessarily(implies(or(implies(not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))), and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))), not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))))), or(implies(not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))), and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))), not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))))))))), or(not(sK7_axiom_B_X), possibly(sK7_axiom_B_X)))), $$true) 16.93/17.18 = { by lemma 277 } 16.93/17.18 $$fresh98(is_a_theorem(implies(not(not(implies(or(implies(not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))), and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))), not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))))), or(implies(not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))), and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))), not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))))))), or(not(sK7_axiom_B_X), possibly(sK7_axiom_B_X)))), $$true) 16.93/17.18 = { by lemma 293 } 16.93/17.18 $$fresh98(is_a_theorem(implies(not(not(implies(implies(and(not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))), not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))))), not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))))), or(implies(not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))), and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))), not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))))))), or(not(sK7_axiom_B_X), possibly(sK7_axiom_B_X)))), $$true) 16.93/17.18 = { by lemma 248 } 16.93/17.18 $$fresh98(is_a_theorem(implies(not(not(not(and(implies(and(not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))), not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))))), not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))))), not(or(implies(not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))), and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))), not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))))))))), or(not(sK7_axiom_B_X), possibly(sK7_axiom_B_X)))), $$true) 16.93/17.18 = { by axiom 112 (substitution_of_equivalents_2) } 16.93/17.18 $$fresh98(is_a_theorem(implies(not(not(not($$fresh3($$true, $$true, and(implies(and(not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))), not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))))), not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))))), not(or(implies(not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))), and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))), not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))))))), not(or(implies(not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))), and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))), not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))))))))), or(not(sK7_axiom_B_X), possibly(sK7_axiom_B_X)))), $$true) 16.93/17.18 = { by axiom 74 (modus_ponens_2) } 16.93/17.18 $$fresh98(is_a_theorem(implies(not(not(not($$fresh3($$fresh115($$true, $$true, equiv(and(implies(and(not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))), not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))))), not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))))), not(or(implies(not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))), and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))), not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))))))), not(or(implies(not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))), and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))), not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))))))), $$true, and(implies(and(not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))), not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))))), not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))))), not(or(implies(not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))), and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))), not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))))))), not(or(implies(not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))), and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))), not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))))))))), or(not(sK7_axiom_B_X), possibly(sK7_axiom_B_X)))), $$true) 16.93/17.18 = { by lemma 266 } 16.93/17.18 $$fresh98(is_a_theorem(implies(not(not(not($$fresh3($$fresh115(is_a_theorem(implies(not(or(implies(not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))), and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))), not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))))), and(implies(and(not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))), not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))))), not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))))), not(or(implies(not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))), and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))), not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))))))))), $$true, equiv(and(implies(and(not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))), not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))))), not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))))), not(or(implies(not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))), and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))), not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))))))), not(or(implies(not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))), and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))), not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))))))), $$true, and(implies(and(not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))), not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))))), not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))))), not(or(implies(not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))), and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))), not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))))))), not(or(implies(not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))), and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))), not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))))))))), or(not(sK7_axiom_B_X), possibly(sK7_axiom_B_X)))), $$true) 16.93/17.18 = { by axiom 75 (modus_ponens_2) } 16.93/17.19 $$fresh98(is_a_theorem(implies(not(not(not($$fresh3($$fresh116(?, ?, implies(not(or(implies(not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))), and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))), not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))))), and(implies(and(not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))), not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))))), not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))))), not(or(implies(not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))), and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))), not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))))))), equiv(and(implies(and(not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))), not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))))), not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))))), not(or(implies(not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))), and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))), not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))))))), not(or(implies(not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))), and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))), not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))))))), $$true, and(implies(and(not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))), not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))))), not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))))), not(or(implies(not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))), and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))), not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))))))), not(or(implies(not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))), and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))), not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))))))))), or(not(sK7_axiom_B_X), possibly(sK7_axiom_B_X)))), $$true) 16.93/17.19 = { by lemma 288 } 16.93/17.19 $$fresh98(is_a_theorem(implies(not(not(not($$fresh3(is_a_theorem(equiv(and(implies(and(not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))), not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))))), not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))))), not(or(implies(not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))), and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))), not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))))))), not(or(implies(not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))), and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))), not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))))))), $$true, and(implies(and(not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))), not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))))), not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))))), not(or(implies(not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))), and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))), not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))))))), not(or(implies(not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))), and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))), not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))))))))), or(not(sK7_axiom_B_X), possibly(sK7_axiom_B_X)))), $$true) 16.93/17.19 = { by lemma 261 } 16.93/17.19 $$fresh98(is_a_theorem(implies(not(not(not(not(or(implies(not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))), and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))), not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))))))))), or(not(sK7_axiom_B_X), possibly(sK7_axiom_B_X)))), $$true) 16.93/17.19 = { by lemma 279 } 16.93/17.19 $$fresh98(is_a_theorem(implies(not(not(or(implies(not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))), and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))), not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))))))), or(not(sK7_axiom_B_X), possibly(sK7_axiom_B_X)))), $$true) 16.93/17.19 = { by lemma 287 } 16.93/17.19 $$fresh98(is_a_theorem(implies(not(and(not(implies(not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))), and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))))), and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))))), or(not(sK7_axiom_B_X), possibly(sK7_axiom_B_X)))), $$true) 16.93/17.19 = { by lemma 284 } 16.93/17.19 $$fresh98(is_a_theorem(implies(not(and(and(not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))), not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))))), and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))))), or(not(sK7_axiom_B_X), possibly(sK7_axiom_B_X)))), $$true) 16.93/17.19 = { by lemma 267 } 16.93/17.19 $$fresh98(is_a_theorem(implies(not(and(not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))), and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))))), or(not(sK7_axiom_B_X), possibly(sK7_axiom_B_X)))), $$true) 16.93/17.19 = { by lemma 279 } 16.93/17.19 $$fresh98(is_a_theorem(implies(not(and(not(not(not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))))), and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))))), or(not(sK7_axiom_B_X), possibly(sK7_axiom_B_X)))), $$true) 16.93/17.19 = { by lemma 268 } 16.93/17.19 $$fresh98(is_a_theorem(implies(not(and(or(not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))), not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))))), and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))))), or(not(sK7_axiom_B_X), possibly(sK7_axiom_B_X)))), $$true) 16.93/17.19 = { by lemma 249 } 16.93/17.19 $$fresh98(is_a_theorem(implies(not(and(implies(not(not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))))), not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))))), and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))))), or(not(sK7_axiom_B_X), possibly(sK7_axiom_B_X)))), $$true) 16.93/17.19 = { by lemma 279 } 16.93/17.19 $$fresh98(is_a_theorem(implies(not(and(implies(not(not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))))), not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))))), not(not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))))))), or(not(sK7_axiom_B_X), possibly(sK7_axiom_B_X)))), $$true) 16.93/17.19 = { by lemma 268 } 16.93/17.19 $$fresh98(is_a_theorem(implies(not(and(implies(not(not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))))), not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))))), or(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))))), or(not(sK7_axiom_B_X), possibly(sK7_axiom_B_X)))), $$true) 16.93/17.19 = { by lemma 249 } 16.93/17.19 $$fresh98(is_a_theorem(implies(not(and(implies(not(not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))))), not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))))), implies(not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))), and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))))), or(not(sK7_axiom_B_X), possibly(sK7_axiom_B_X)))), $$true) 16.93/17.19 = { by lemma 274 } 16.93/17.19 $$fresh98(is_a_theorem(implies(not(and(implies(not(not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))))), not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))))), implies(not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))), implies(not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))), and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))))))), or(not(sK7_axiom_B_X), possibly(sK7_axiom_B_X)))), $$true) 16.93/17.19 = { by lemma 249 } 16.93/17.19 $$fresh98(is_a_theorem(implies(not(and(implies(not(not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))))), not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))))), or(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), implies(not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))), and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))))))), or(not(sK7_axiom_B_X), possibly(sK7_axiom_B_X)))), $$true) 16.93/17.19 = { by lemma 249 } 16.93/17.19 $$fresh98(is_a_theorem(implies(not(and(implies(not(not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))))), not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))))), or(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), or(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))))))), or(not(sK7_axiom_B_X), possibly(sK7_axiom_B_X)))), $$true) 16.93/17.19 = { by lemma 268 } 16.93/17.19 $$fresh98(is_a_theorem(implies(not(and(implies(not(not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))))), not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))))), or(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), not(not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))))))), or(not(sK7_axiom_B_X), possibly(sK7_axiom_B_X)))), $$true) 16.93/17.19 = { by lemma 249 } 16.93/17.19 $$fresh98(is_a_theorem(implies(not(and(implies(not(not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))))), not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))))), implies(not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))), not(not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))))))), or(not(sK7_axiom_B_X), possibly(sK7_axiom_B_X)))), $$true) 16.93/17.19 = { by axiom 173 (op_equiv) } 16.93/17.19 $$fresh98(is_a_theorem(implies(not($$fresh30(op_equiv, $$true, not(not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))))), not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))))), or(not(sK7_axiom_B_X), possibly(sK7_axiom_B_X)))), $$true) 16.93/17.19 = { by axiom 189 (hilbert_op_equiv) } 16.93/17.19 $$fresh98(is_a_theorem(implies(not($$fresh30($$true, $$true, not(not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))))), not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))))), or(not(sK7_axiom_B_X), possibly(sK7_axiom_B_X)))), $$true) 16.93/17.19 = { by axiom 87 (op_equiv) } 16.93/17.19 $$fresh98(is_a_theorem(implies(not(equiv(not(not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))))), not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))))), or(not(sK7_axiom_B_X), possibly(sK7_axiom_B_X)))), $$true) 16.93/17.19 = { by lemma 279 } 16.93/17.19 $$fresh98(is_a_theorem(implies(not(equiv(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))))), or(not(sK7_axiom_B_X), possibly(sK7_axiom_B_X)))), $$true) 16.93/17.19 = { by axiom 87 (op_equiv) } 16.93/17.19 $$fresh98(is_a_theorem(implies(not($$fresh30($$true, $$true, and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))))), or(not(sK7_axiom_B_X), possibly(sK7_axiom_B_X)))), $$true) 16.93/17.19 = { by axiom 189 (hilbert_op_equiv) } 16.93/17.19 $$fresh98(is_a_theorem(implies(not($$fresh30(op_equiv, $$true, and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))))), or(not(sK7_axiom_B_X), possibly(sK7_axiom_B_X)))), $$true) 16.93/17.19 = { by axiom 173 (op_equiv) } 16.93/17.19 $$fresh98(is_a_theorem(implies(not(and(implies(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))))), implies(not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))), and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))))), or(not(sK7_axiom_B_X), possibly(sK7_axiom_B_X)))), $$true) 16.93/17.19 = { by lemma 279 } 16.93/17.19 $$fresh98(is_a_theorem(implies(not(and(implies(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), not(not(possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))))), not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))))), implies(not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))), and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))))), or(not(sK7_axiom_B_X), possibly(sK7_axiom_B_X)))), $$true) 16.93/17.19 = { by lemma 293 } 16.93/17.19 $$fresh98(is_a_theorem(implies(not(and(or(implies(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), not(possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))), not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))))), implies(not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))), and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))))), or(not(sK7_axiom_B_X), possibly(sK7_axiom_B_X)))), $$true) 16.93/17.19 = { by lemma 267 } 16.93/17.19 $$fresh98(is_a_theorem(implies(not(and(or(implies(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), not(and(possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))))), not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))))), implies(not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))), and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))))), or(not(sK7_axiom_B_X), possibly(sK7_axiom_B_X)))), $$true) 16.93/17.19 = { by lemma 251 } 16.93/17.19 $$fresh98(is_a_theorem(implies(not(and(or(implies(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), implies(possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)), necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))))), not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))))), implies(not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))), and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))))), or(not(sK7_axiom_B_X), possibly(sK7_axiom_B_X)))), $$true) 16.93/17.19 = { by lemma 261 } 16.93/17.19 $$fresh98(is_a_theorem(implies(not(and($$fresh3(is_a_theorem(equiv(implies(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), implies(possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)), necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))))), or(implies(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), implies(possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)), necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))))), not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))))))), $$true, implies(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), implies(possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)), necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))))), or(implies(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), implies(possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)), necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))))), not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))))), implies(not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))), and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))))), or(not(sK7_axiom_B_X), possibly(sK7_axiom_B_X)))), $$true) 16.93/17.19 = { by lemma 297 } 16.93/17.19 $$fresh98(is_a_theorem(implies(not(and($$fresh3($$fresh116(?, ?, implies(or(implies(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), implies(possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)), necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))))), not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))))), implies(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), implies(possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)), necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))))), equiv(implies(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), implies(possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)), necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))))), or(implies(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), implies(possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)), necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))))), not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))))))), $$true, implies(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), implies(possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)), necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))))), or(implies(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), implies(possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)), necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))))), not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))))), implies(not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))), and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))))), or(not(sK7_axiom_B_X), possibly(sK7_axiom_B_X)))), $$true) 16.93/17.19 = { by axiom 75 (modus_ponens_2) } 16.93/17.19 $$fresh98(is_a_theorem(implies(not(and($$fresh3($$fresh115(is_a_theorem(implies(or(implies(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), implies(possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)), necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))))), not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))))), implies(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), implies(possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)), necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))))))), $$true, equiv(implies(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), implies(possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)), necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))))), or(implies(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), implies(possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)), necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))))), not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))))))), $$true, implies(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), implies(possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)), necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))))), or(implies(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), implies(possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)), necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))))), not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))))), implies(not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))), and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))))), or(not(sK7_axiom_B_X), possibly(sK7_axiom_B_X)))), $$true) 16.93/17.19 = { by lemma 258 } 16.93/17.19 $$fresh98(is_a_theorem(implies(not(and($$fresh3($$fresh115($$fresh116(?, ?, implies(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), implies(possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)), necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))))), implies(or(implies(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), implies(possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)), necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))))), not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))))), implies(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), implies(possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)), necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))))))), $$true, equiv(implies(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), implies(possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)), necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))))), or(implies(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), implies(possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)), necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))))), not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))))))), $$true, implies(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), implies(possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)), necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))))), or(implies(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), implies(possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)), necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))))), not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))))), implies(not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))), and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))))), or(not(sK7_axiom_B_X), possibly(sK7_axiom_B_X)))), $$true) 16.93/17.19 = { by lemma 257 } 16.93/17.19 $$fresh98(is_a_theorem(implies(not(and($$fresh3($$fresh115($$true, $$true, equiv(implies(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), implies(possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)), necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))))), or(implies(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), implies(possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)), necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))))), not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))))))), $$true, implies(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), implies(possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)), necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))))), or(implies(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), implies(possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)), necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))))), not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))))), implies(not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))), and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))))), or(not(sK7_axiom_B_X), possibly(sK7_axiom_B_X)))), $$true) 16.93/17.19 = { by axiom 74 (modus_ponens_2) } 16.93/17.19 $$fresh98(is_a_theorem(implies(not(and($$fresh3($$true, $$true, implies(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), implies(possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)), necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))))), or(implies(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), implies(possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)), necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))))), not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))))), implies(not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))), and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))))), or(not(sK7_axiom_B_X), possibly(sK7_axiom_B_X)))), $$true) 16.93/17.19 = { by axiom 112 (substitution_of_equivalents_2) } 16.93/17.19 $$fresh98(is_a_theorem(implies(not(and(implies(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), implies(possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)), necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))))), implies(not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))), and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))))), or(not(sK7_axiom_B_X), possibly(sK7_axiom_B_X)))), $$true) 16.93/17.19 = { by lemma 286 } 16.93/17.19 $$fresh98(is_a_theorem(implies(not(and(or(possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)), implies(possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)), necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))))), implies(not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))), and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))))), or(not(sK7_axiom_B_X), possibly(sK7_axiom_B_X)))), $$true) 16.93/17.19 = { by lemma 283 } 16.93/17.19 $$fresh98(is_a_theorem(implies(not(and(or(possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)), or(not(possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))))), implies(not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))), and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))))), or(not(sK7_axiom_B_X), possibly(sK7_axiom_B_X)))), $$true) 16.93/17.19 = { by lemma 249 } 17.02/17.19 $$fresh98(is_a_theorem(implies(not(and(implies(not(possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), or(not(possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))))), implies(not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))), and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))))), or(not(sK7_axiom_B_X), possibly(sK7_axiom_B_X)))), $$true) 17.02/17.19 = { by lemma 298 } 17.02/17.19 $$fresh98(is_a_theorem(implies(not(and(or(implies(not(possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), or(not(possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))))), not(implies(not(possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), or(not(possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))))))), implies(not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))), and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))))), or(not(sK7_axiom_B_X), possibly(sK7_axiom_B_X)))), $$true) 17.02/17.19 = { by lemma 275 } 17.02/17.19 $$fresh98(is_a_theorem(implies(not(and(not(equiv(implies(not(possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), or(not(possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))))), not(implies(not(possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), or(not(possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))))))), implies(not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))), and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))))), or(not(sK7_axiom_B_X), possibly(sK7_axiom_B_X)))), $$true) 17.02/17.19 = { by lemma 298 } 17.02/17.20 $$fresh98(is_a_theorem(implies(not(and(not(equiv(or(implies(not(possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), or(not(possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))))), ?), not(implies(not(possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), or(not(possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))))))), implies(not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))), and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))))), or(not(sK7_axiom_B_X), possibly(sK7_axiom_B_X)))), $$true) 17.02/17.20 = { by lemma 249 } 17.02/17.20 $$fresh98(is_a_theorem(implies(not(and(not(equiv(implies(not(implies(not(possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), or(not(possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))))), ?), not(implies(not(possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), or(not(possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))))))), implies(not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))), and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))))), or(not(sK7_axiom_B_X), possibly(sK7_axiom_B_X)))), $$true) 17.02/17.20 = { by lemma 298 } 17.02/17.20 $$fresh98(is_a_theorem(implies(not(and(not(equiv(implies(not(implies(not(possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), or(not(possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))))), ?), not(or(implies(not(possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), or(not(possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))))), ?)))), implies(not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))), and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))))), or(not(sK7_axiom_B_X), possibly(sK7_axiom_B_X)))), $$true) 17.02/17.20 = { by lemma 249 } 17.02/17.20 $$fresh98(is_a_theorem(implies(not(and(not(equiv(implies(not(implies(not(possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), or(not(possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))))), ?), not(implies(not(implies(not(possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), or(not(possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))))), ?)))), implies(not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))), and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))))), or(not(sK7_axiom_B_X), possibly(sK7_axiom_B_X)))), $$true) 17.02/17.20 = { by lemma 275 } 17.02/17.20 $$fresh98(is_a_theorem(implies(not(and(or(implies(not(implies(not(possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), or(not(possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))))), ?), not(implies(not(implies(not(possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), or(not(possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))))), ?))), implies(not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))), and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))))), or(not(sK7_axiom_B_X), possibly(sK7_axiom_B_X)))), $$true) 17.02/17.20 = { by lemma 249 } 17.02/17.20 $$fresh98(is_a_theorem(implies(not(and(or(or(implies(not(possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), or(not(possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))))), ?), not(implies(not(implies(not(possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), or(not(possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))))), ?))), implies(not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))), and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))))), or(not(sK7_axiom_B_X), possibly(sK7_axiom_B_X)))), $$true) 17.02/17.20 = { by lemma 249 } 17.02/17.20 $$fresh98(is_a_theorem(implies(not(and(or(or(implies(not(possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), or(not(possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))))), ?), not(or(implies(not(possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), or(not(possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))))), ?))), implies(not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))), and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))))), or(not(sK7_axiom_B_X), possibly(sK7_axiom_B_X)))), $$true) 17.02/17.20 = { by lemma 249 } 17.02/17.20 $$fresh98(is_a_theorem(implies(not(and(implies(not(or(implies(not(possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), or(not(possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))))), ?)), not(or(implies(not(possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), or(not(possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))))), ?))), implies(not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))), and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))))), or(not(sK7_axiom_B_X), possibly(sK7_axiom_B_X)))), $$true) 17.02/17.20 = { by lemma 296 } 17.02/17.20 $$fresh98(is_a_theorem(implies(not(and(implies(?, ?), implies(not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))), and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))))), or(not(sK7_axiom_B_X), possibly(sK7_axiom_B_X)))), $$true) 17.02/17.20 = { by lemma 249 } 17.02/17.20 $$fresh98(is_a_theorem(implies(not(and(implies(?, ?), or(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))))), or(not(sK7_axiom_B_X), possibly(sK7_axiom_B_X)))), $$true) 17.02/17.20 = { by lemma 268 } 17.02/17.20 $$fresh98(is_a_theorem(implies(not(and(implies(?, ?), not(not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))))))), or(not(sK7_axiom_B_X), possibly(sK7_axiom_B_X)))), $$true) 17.02/17.20 = { by lemma 290 } 17.02/17.20 $$fresh98(is_a_theorem(implies(not(not(not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))))), or(not(sK7_axiom_B_X), possibly(sK7_axiom_B_X)))), $$true) 17.02/17.20 = { by lemma 279 } 17.02/17.20 $$fresh98(is_a_theorem(implies(not(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))), or(not(sK7_axiom_B_X), possibly(sK7_axiom_B_X)))), $$true) 17.02/17.20 = { by lemma 251 } 17.02/17.20 $$fresh98(is_a_theorem(implies(implies(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))), or(not(sK7_axiom_B_X), possibly(sK7_axiom_B_X)))), $$true) 17.02/17.20 = { by lemma 286 } 17.02/17.20 $$fresh98(is_a_theorem(implies(or(possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)), necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))), or(not(sK7_axiom_B_X), possibly(sK7_axiom_B_X)))), $$true) 17.02/17.20 = { by lemma 294 } 17.02/17.20 $$fresh98(is_a_theorem(implies(or(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X), necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))), or(not(sK7_axiom_B_X), possibly(sK7_axiom_B_X)))), $$true) 17.02/17.20 = { by lemma 289 } 17.02/17.20 $$fresh98(is_a_theorem(implies(or(and(implies(necessarily(not(sK7_axiom_B_X)), not(sK7_axiom_B_X)), and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)), necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))), or(not(sK7_axiom_B_X), possibly(sK7_axiom_B_X)))), $$true) 17.02/17.20 = { by lemma 273 } 17.02/17.20 $$fresh98(is_a_theorem(implies(or(and(necessarily(implies(necessarily(not(sK7_axiom_B_X)), not(sK7_axiom_B_X))), and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)), necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))), or(not(sK7_axiom_B_X), possibly(sK7_axiom_B_X)))), $$true) 17.02/17.20 = { by lemma 285 } 17.02/17.20 $$fresh98(is_a_theorem(implies(or(and(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)), necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))), or(not(sK7_axiom_B_X), possibly(sK7_axiom_B_X)))), $$true) 17.02/17.20 = { by lemma 282 } 17.02/17.20 $$fresh98(is_a_theorem(implies(or(and(possibly(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))), and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)), necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))), or(not(sK7_axiom_B_X), possibly(sK7_axiom_B_X)))), $$true) 17.02/17.20 = { by lemma 270 } 17.02/17.20 $$fresh98(is_a_theorem(implies(or(and(necessarily(possibly(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))))), and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)), necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))), or(not(sK7_axiom_B_X), possibly(sK7_axiom_B_X)))), $$true) 17.02/17.20 = { by lemma 282 } 17.02/17.20 $$fresh98(is_a_theorem(implies(or(and(necessarily(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))), and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)), necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))), or(not(sK7_axiom_B_X), possibly(sK7_axiom_B_X)))), $$true) 17.02/17.20 = { by lemma 294 } 17.02/17.20 $$fresh98(is_a_theorem(implies(or(and(necessarily(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))), or(not(sK7_axiom_B_X), possibly(sK7_axiom_B_X)))), $$true) 17.02/17.20 = { by axiom 91 (op_or) } 17.02/17.20 $$fresh98(is_a_theorem(implies($$fresh26($$true, $$true, and(necessarily(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))), or(not(sK7_axiom_B_X), possibly(sK7_axiom_B_X)))), $$true) 17.02/17.20 = { by axiom 176 (hilbert_op_or) } 17.02/17.20 $$fresh98(is_a_theorem(implies($$fresh26(op_or, $$true, and(necessarily(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))), or(not(sK7_axiom_B_X), possibly(sK7_axiom_B_X)))), $$true) 17.02/17.20 = { by axiom 169 (op_or) } 17.02/17.20 $$fresh98(is_a_theorem(implies(not(and(not(and(necessarily(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))), possibly(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))), not(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))))), or(not(sK7_axiom_B_X), possibly(sK7_axiom_B_X)))), $$true) 17.02/17.20 = { by lemma 251 } 17.02/17.20 $$fresh98(is_a_theorem(implies(not(and(implies(necessarily(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))), necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))), not(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))))), or(not(sK7_axiom_B_X), possibly(sK7_axiom_B_X)))), $$true) 17.02/17.20 = { by lemma 248 } 17.02/17.20 $$fresh98(is_a_theorem(implies(implies(implies(necessarily(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))), necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))), necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X)))), or(not(sK7_axiom_B_X), possibly(sK7_axiom_B_X)))), $$true) 17.02/17.20 = { by lemma 291 } 17.02/17.20 $$fresh98(is_a_theorem(implies(necessarily(not(and(necessarily(not(sK7_axiom_B_X)), sK7_axiom_B_X))), or(not(sK7_axiom_B_X), possibly(sK7_axiom_B_X)))), $$true) 17.02/17.20 = { by lemma 285 } 17.02/17.20 $$fresh98(is_a_theorem(implies(necessarily(implies(necessarily(not(sK7_axiom_B_X)), not(sK7_axiom_B_X))), or(not(sK7_axiom_B_X), possibly(sK7_axiom_B_X)))), $$true) 17.02/17.20 = { by lemma 273 } 17.02/17.20 $$fresh98(is_a_theorem(implies(implies(necessarily(not(sK7_axiom_B_X)), not(sK7_axiom_B_X)), or(not(sK7_axiom_B_X), possibly(sK7_axiom_B_X)))), $$true) 17.02/17.20 = { by lemma 286 } 17.02/17.20 $$fresh98(is_a_theorem(implies(or(possibly(sK7_axiom_B_X), not(sK7_axiom_B_X)), or(not(sK7_axiom_B_X), possibly(sK7_axiom_B_X)))), $$true) 17.02/17.20 = { by lemma 279 } 17.02/17.20 $$fresh98(is_a_theorem(implies(or(possibly(sK7_axiom_B_X), not(not(not(sK7_axiom_B_X)))), or(not(sK7_axiom_B_X), possibly(sK7_axiom_B_X)))), $$true) 17.02/17.20 = { by lemma 249 } 17.02/17.20 $$fresh98(is_a_theorem(implies(or(possibly(sK7_axiom_B_X), not(not(not(sK7_axiom_B_X)))), implies(not(not(sK7_axiom_B_X)), possibly(sK7_axiom_B_X)))), $$true) 17.02/17.20 = { by lemma 254 } 17.02/17.20 $$fresh98($$true, $$true) 17.02/17.20 = { by axiom 15 (axiom_B) } 17.02/17.20 $$true 17.02/17.20 % SZS output end Proof 17.02/17.20 17.02/17.20 RESULT: Theorem (the conjecture is true). 17.02/17.20 EOF