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
0.03/0.24	% Computer   : n004.star.cs.uiowa.edu
0.03/0.24	% Model      : x86_64 x86_64
0.03/0.24	% CPU        : Intel(R) Xeon(R) CPU E5-2609 0 @ 2.40GHz
0.03/0.24	% Memory     : 32218.625MB
0.03/0.24	% OS         : Linux 3.10.0-693.2.2.el7.x86_64
0.03/0.24	% CPULimit   : 300
0.03/0.24	% DateTime   : Sat Jul 14 05:51:55 CDT 2018
0.03/0.24	% CPUTime    : 
27.30/27.56	% SZS status Theorem
27.30/27.56	
27.43/27.67	% SZS output start Proof
27.43/27.67	Take the following subset of the input axioms:
27.53/27.79	  fof(and_1, axiom,
27.53/27.79	      and_1 <=> ![X, Y]: is_a_theorem(implies(and(X, Y), X))).
27.53/27.79	  fof(and_3, axiom,
27.53/27.79	      ![X, Y]: is_a_theorem(implies(X, implies(Y, and(X, Y))))
27.53/27.79	        <=> and_3).
27.53/27.79	  fof(axiom_4, axiom,
27.53/27.79	      ![X]:
27.53/27.79	        is_a_theorem(implies(necessarily(X), necessarily(necessarily(X))))
27.53/27.79	        <=> axiom_4).
27.53/27.79	  fof(axiom_5, axiom,
27.53/27.79	      ![X]: is_a_theorem(implies(possibly(X), necessarily(possibly(X))))
27.53/27.79	        <=> axiom_5).
27.53/27.79	  fof(axiom_B, axiom,
27.53/27.79	      axiom_B
27.53/27.79	        <=> ![X]: is_a_theorem(implies(X, necessarily(possibly(X))))).
27.53/27.79	  fof(axiom_M, axiom,
27.53/27.79	      axiom_M <=> ![X]: is_a_theorem(implies(necessarily(X), X))).
27.53/27.79	  fof(axiom_m1, axiom,
27.53/27.79	      ![X, Y]: is_a_theorem(strict_implies(and(X, Y), and(Y, X)))
27.53/27.79	        <=> axiom_m1).
27.53/27.79	  fof(cn3, axiom,
27.53/27.79	      ![P]: is_a_theorem(implies(implies(not(P), P), P)) <=> cn3).
27.53/27.79	  fof(hilbert_and_1, axiom, and_1).
27.53/27.79	  fof(hilbert_and_3, axiom, and_3).
27.53/27.79	  fof(hilbert_implies_1, axiom, implies_1).
27.53/27.79	  fof(hilbert_implies_2, axiom, implies_2).
27.53/27.79	  fof(hilbert_modus_ponens, axiom, modus_ponens).
27.53/27.79	  fof(hilbert_modus_tollens, axiom, modus_tollens).
27.53/27.79	  fof(hilbert_op_equiv, axiom, op_equiv).
27.53/27.79	  fof(hilbert_op_implies_and, axiom, op_implies_and).
27.53/27.79	  fof(hilbert_op_or, axiom, op_or).
27.53/27.79	  fof(hilbert_or_1, axiom, or_1).
27.53/27.79	  fof(implies_1, axiom,
27.53/27.79	      ![X, Y]: is_a_theorem(implies(X, implies(Y, X))) <=> implies_1).
27.53/27.79	  fof(implies_2, axiom,
27.53/27.79	      implies_2
27.53/27.79	        <=> ![X, Y]:
27.53/27.79	              is_a_theorem(implies(implies(X, implies(X, Y)), implies(X, Y)))).
27.53/27.79	  fof(km4b_axiom_4, axiom, axiom_4).
27.53/27.79	  fof(km4b_axiom_B, axiom, axiom_B).
27.53/27.79	  fof(km4b_axiom_M, axiom, axiom_M).
27.53/27.79	  fof(km4b_op_possibly, axiom, op_possibly).
27.53/27.79	  fof(kn1, axiom, ![P]: is_a_theorem(implies(P, and(P, P))) <=> kn1).
27.53/27.79	  fof(modus_ponens, axiom,
27.53/27.79	      modus_ponens
27.53/27.79	        <=> ![X, Y]:
27.53/27.79	              ((is_a_theorem(X) & is_a_theorem(implies(X, Y)))
27.53/27.79	                 => is_a_theorem(Y))).
27.53/27.79	  fof(modus_tollens, axiom,
27.53/27.79	      modus_tollens
27.53/27.79	        <=> ![X, Y]:
27.53/27.79	              is_a_theorem(implies(implies(not(Y), not(X)), implies(X, Y)))).
27.53/27.79	  fof(op_and, axiom,
27.53/27.79	      ![X, Y]: and(X, Y)=not(or(not(X), not(Y))) <= op_and).
27.53/27.79	  fof(op_equiv, axiom,
27.53/27.79	      op_equiv
27.53/27.79	        => ![X, Y]: equiv(X, Y)=and(implies(X, Y), implies(Y, X))).
27.53/27.79	  fof(op_implies_and, axiom,
27.53/27.79	      ![X, Y]: not(and(X, not(Y)))=implies(X, Y) <= op_implies_and).
27.53/27.79	  fof(op_or, axiom,
27.53/27.79	      ![X, Y]: or(X, Y)=not(and(not(X), not(Y))) <= op_or).
27.53/27.79	  fof(op_possibly, axiom,
27.53/27.79	      ![X]: not(necessarily(not(X)))=possibly(X) <= op_possibly).
27.53/27.79	  fof(op_strict_implies, axiom,
27.53/27.79	      op_strict_implies
27.53/27.79	        => ![X, Y]: necessarily(implies(X, Y))=strict_implies(X, Y)).
27.53/27.79	  fof(or_1, axiom,
27.53/27.79	      or_1 <=> ![X, Y]: is_a_theorem(implies(X, or(X, Y)))).
27.53/27.79	  fof(r1, axiom, ![P]: is_a_theorem(implies(or(P, P), P)) <=> r1).
27.53/27.79	  fof(r3, axiom,
27.53/27.79	      r3 <=> ![P, Q]: is_a_theorem(implies(or(P, Q), or(Q, P)))).
27.53/27.79	  fof(s1_0_axiom_m1, conjecture, axiom_m1).
27.53/27.79	  fof(s1_0_op_strict_implies, axiom, op_strict_implies).
27.53/27.79	  fof(substitution_of_equivalents, axiom,
27.53/27.79	      substitution_of_equivalents
27.53/27.79	        <=> ![X, Y]: (is_a_theorem(equiv(X, Y)) => Y=X)).
27.53/27.79	  fof(substitution_of_equivalents, axiom,
27.53/27.79	      substitution_of_equivalents).
27.53/27.79	
27.53/27.79	Now clausify the problem and encode Horn clauses using encoding 3 of
27.53/27.79	http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
27.53/27.79	We repeatedly replace C & s=t => u=v by the two clauses:
27.53/27.79	  $$fresh(y, y, x1...xn) = u
27.53/27.79	  C => $$fresh(s, t, x1...xn) = v
27.53/27.79	where $$fresh is a fresh function symbol and x1..xn are the free
27.53/27.79	variables of u and v.
27.53/27.79	A predicate p(X) is encoded as p(X)=$$true (this is sound, because the
27.53/27.79	input problem has no model of domain size 1).
27.53/27.79	
27.53/27.79	The encoding turns the above axioms into the following unit equations and goals:
27.53/27.79	
27.53/27.79	Axiom 6 (and_1_1): $$fresh107(X, X, Y, Z) = $$true.
27.53/27.79	Axiom 10 (and_3_1): $$fresh103(X, X, Y, Z) = $$true.
27.53/27.79	Axiom 12 (axiom_4_1): $$fresh101(X, X, Y) = $$true.
27.53/27.79	Axiom 16 (axiom_B_1): $$fresh97(X, X, Y) = $$true.
27.53/27.79	Axiom 20 (axiom_M_1): $$fresh93(X, X, Y) = $$true.
27.53/27.79	Axiom 21 (axiom_m1): $$fresh92(X, X) = $$true.
27.53/27.79	Axiom 62 (implies_1_1): $$fresh51(X, X, Y, Z) = $$true.
27.53/27.79	Axiom 64 (implies_2_1): $$fresh49(X, X, Y, Z) = $$true.
27.53/27.79	Axiom 73 (modus_ponens_2): $$fresh40(X, X, Y, Z) = is_a_theorem(Z).
27.53/27.79	Axiom 74 (modus_ponens_2): $$fresh115(X, X, Y) = $$true.
27.53/27.79	Axiom 75 (modus_ponens_2): $$fresh116(X, X, Y, Z) = $$fresh115(is_a_theorem(Y), $$true, Z).
27.53/27.79	Axiom 82 (modus_tollens_1): $$fresh35(X, X, Y, Z) = $$true.
27.53/27.79	Axiom 87 (op_equiv): $$fresh30(X, X, Y, Z) = equiv(Y, Z).
27.53/27.79	Axiom 88 (op_implies_and): $$fresh29(X, X, Y, Z) = implies(Y, Z).
27.53/27.79	Axiom 91 (op_or): $$fresh26(X, X, Y, Z) = or(Y, Z).
27.53/27.79	Axiom 92 (op_possibly): $$fresh25(X, X, Y) = possibly(Y).
27.53/27.79	Axiom 94 (op_strict_implies): $$fresh23(X, X, Y, Z) = strict_implies(Y, Z).
27.53/27.79	Axiom 96 (or_1_1): $$fresh21(X, X, Y, Z) = $$true.
27.53/27.79	Axiom 112 (substitution_of_equivalents_2): $$fresh3(X, X, Y, Z) = Y.
27.53/27.79	Axiom 113 (substitution_of_equivalents_2): $$fresh4(X, X, Y, Z) = Z.
27.53/27.79	Axiom 121 (substitution_of_equivalents_2): $$fresh4(substitution_of_equivalents, $$true, X, Y) = $$fresh3(is_a_theorem(equiv(X, Y)), $$true, X, Y).
27.53/27.79	Axiom 127 (kn1_1): $$fresh45(kn1, $$true, X) = is_a_theorem(implies(X, and(X, X))).
27.53/27.79	Axiom 135 (or_1_1): $$fresh21(or_1, $$true, X, Y) = is_a_theorem(implies(X, or(X, Y))).
27.53/27.79	Axiom 140 (modus_ponens_2): $$fresh116(modus_ponens, $$true, X, Y) = $$fresh40(is_a_theorem(implies(X, Y)), $$true, X, Y).
27.53/27.79	Axiom 143 (r1_1): $$fresh15(r1, $$true, X) = is_a_theorem(implies(or(X, X), X)).
27.53/27.79	Axiom 145 (and_3_1): $$fresh103(and_3, $$true, X, Y) = is_a_theorem(implies(X, implies(Y, and(X, Y)))).
27.53/27.79	Axiom 149 (and_1_1): $$fresh107(and_1, $$true, X, Y) = is_a_theorem(implies(and(X, Y), X)).
27.53/27.79	Axiom 151 (cn3_1): $$fresh59(cn3, $$true, X) = is_a_theorem(implies(implies(not(X), X), X)).
27.53/27.79	Axiom 153 (r3_1): $$fresh11(r3, $$true, X, Y) = is_a_theorem(implies(or(X, Y), or(Y, X))).
27.53/27.79	Axiom 159 (implies_1_1): $$fresh51(implies_1, $$true, X, Y) = is_a_theorem(implies(X, implies(Y, X))).
27.53/27.79	Axiom 161 (modus_tollens_1): $$fresh35(modus_tollens, $$true, X, Y) = is_a_theorem(implies(implies(not(Y), not(X)), implies(X, Y))).
27.53/27.79	Axiom 167 (implies_2_1): $$fresh49(implies_2, $$true, X, Y) = is_a_theorem(implies(implies(X, implies(X, Y)), implies(X, Y))).
27.53/27.79	Axiom 169 (op_or): $$fresh26(op_or, $$true, X, Y) = not(and(not(X), not(Y))).
27.53/27.79	Axiom 170 (op_and): $$fresh31(op_and, $$true, X, Y) = not(or(not(X), not(Y))).
27.53/27.79	Axiom 171 (op_implies_and): $$fresh29(op_implies_and, $$true, X, Y) = not(and(X, not(Y))).
27.53/27.79	Axiom 173 (op_equiv): $$fresh30(op_equiv, $$true, X, Y) = and(implies(X, Y), implies(Y, X)).
27.53/27.79	Axiom 175 (hilbert_and_3): and_3 = $$true.
27.53/27.79	Axiom 176 (hilbert_op_or): op_or = $$true.
27.53/27.79	Axiom 178 (hilbert_and_1): and_1 = $$true.
27.53/27.79	Axiom 179 (substitution_of_equivalents): substitution_of_equivalents = $$true.
27.53/27.79	Axiom 180 (hilbert_implies_2): implies_2 = $$true.
27.53/27.79	Axiom 183 (hilbert_op_implies_and): op_implies_and = $$true.
27.53/27.79	Axiom 184 (hilbert_or_1): or_1 = $$true.
27.53/27.79	Axiom 187 (hilbert_implies_1): implies_1 = $$true.
27.53/27.79	Axiom 188 (hilbert_modus_tollens): modus_tollens = $$true.
27.53/27.79	Axiom 189 (hilbert_op_equiv): op_equiv = $$true.
27.53/27.79	Axiom 190 (hilbert_modus_ponens): modus_ponens = $$true.
27.53/27.79	Axiom 210 (axiom_4_1): $$fresh101(axiom_4, $$true, X) = is_a_theorem(implies(necessarily(X), necessarily(necessarily(X)))).
27.53/27.79	Axiom 216 (axiom_5_1): $$fresh99(axiom_5, $$true, X) = is_a_theorem(implies(possibly(X), necessarily(possibly(X)))).
27.53/27.79	Axiom 218 (axiom_m1_1): $$fresh89(axiom_m1, $$true, X, Y) = is_a_theorem(strict_implies(and(X, Y), and(Y, X))).
27.53/27.79	Axiom 219 (axiom_m1): $$fresh92(is_a_theorem(strict_implies(and(sK15_axiom_m1_X, sK14_axiom_m1_Y), and(sK14_axiom_m1_Y, sK15_axiom_m1_X))), $$true) = axiom_m1.
27.53/27.79	Axiom 222 (axiom_M_1): $$fresh93(axiom_M, $$true, X) = is_a_theorem(implies(necessarily(X), X)).
27.53/27.79	Axiom 230 (axiom_B_1): $$fresh97(axiom_B, $$true, X) = is_a_theorem(implies(X, necessarily(possibly(X)))).
27.53/27.79	Axiom 240 (op_strict_implies): $$fresh23(op_strict_implies, $$true, X, Y) = necessarily(implies(X, Y)).
27.53/27.79	Axiom 241 (op_possibly): $$fresh25(op_possibly, $$true, X) = not(necessarily(not(X))).
27.53/27.79	Axiom 242 (km4b_axiom_M): axiom_M = $$true.
27.53/27.79	Axiom 244 (km4b_axiom_4): axiom_4 = $$true.
27.53/27.79	Axiom 246 (km4b_op_possibly): op_possibly = $$true.
27.53/27.79	Axiom 247 (km4b_axiom_B): axiom_B = $$true.
27.53/27.79	Axiom 250 (s1_0_op_strict_implies): op_strict_implies = $$true.
27.53/27.79	
27.53/27.79	Lemma 251: $$fresh116(X, X, Y, Z) = $$fresh116(?, ?, Y, Z).
27.53/27.79	Proof:
27.53/27.79	  $$fresh116(X, X, Y, Z)
27.53/27.79	= { by axiom 75 (modus_ponens_2) }
27.53/27.79	  $$fresh115(is_a_theorem(Y), $$true, Z)
27.53/27.79	= { by axiom 75 (modus_ponens_2) }
27.53/27.79	  $$fresh116(?, ?, Y, Z)
27.53/27.79	
27.53/27.79	Lemma 252: not(necessarily(not(X))) = possibly(X).
27.53/27.79	Proof:
27.53/27.79	  not(necessarily(not(X)))
27.53/27.79	= { by axiom 241 (op_possibly) }
27.53/27.79	  $$fresh25(op_possibly, $$true, X)
27.53/27.79	= { by axiom 246 (km4b_op_possibly) }
27.53/27.79	  $$fresh25($$true, $$true, X)
27.53/27.79	= { by axiom 92 (op_possibly) }
27.53/27.79	  possibly(X)
27.53/27.79	
27.53/27.79	Lemma 253: not(and(X, not(Y))) = implies(X, Y).
27.53/27.79	Proof:
27.53/27.79	  not(and(X, not(Y)))
27.53/27.79	= { by axiom 171 (op_implies_and) }
27.53/27.79	  $$fresh29(op_implies_and, $$true, X, Y)
27.53/27.79	= { by axiom 183 (hilbert_op_implies_and) }
27.53/27.79	  $$fresh29($$true, $$true, X, Y)
27.53/27.79	= { by axiom 88 (op_implies_and) }
27.53/27.79	  implies(X, Y)
27.53/27.79	
27.53/27.79	Lemma 254: implies(not(X), Y) = or(X, Y).
27.53/27.79	Proof:
27.53/27.79	  implies(not(X), Y)
27.53/27.79	= { by lemma 253 }
27.53/27.79	  not(and(not(X), not(Y)))
27.53/27.79	= { by axiom 169 (op_or) }
27.53/27.79	  $$fresh26(op_or, $$true, X, Y)
27.53/27.79	= { by axiom 176 (hilbert_op_or) }
27.53/27.79	  $$fresh26($$true, $$true, X, Y)
27.53/27.79	= { by axiom 91 (op_or) }
27.53/27.79	  or(X, Y)
27.53/27.79	
27.53/27.79	Lemma 255: $$fresh15(r1, $$true, X) = $$fresh59(cn3, $$true, X).
27.53/27.79	Proof:
27.53/27.79	  $$fresh15(r1, $$true, X)
27.53/27.79	= { by axiom 143 (r1_1) }
27.53/27.79	  is_a_theorem(implies(or(X, X), X))
27.53/27.79	= { by lemma 254 }
27.53/27.79	  is_a_theorem(implies(implies(not(X), X), X))
27.53/27.79	= { by axiom 151 (cn3_1) }
27.53/27.79	  $$fresh59(cn3, $$true, X)
27.53/27.79	
27.53/27.79	Lemma 256: possibly(and(X, not(Y))) = not(strict_implies(X, Y)).
27.53/27.79	Proof:
27.53/27.79	  possibly(and(X, not(Y)))
27.53/27.79	= { by lemma 252 }
27.53/27.79	  not(necessarily(not(and(X, not(Y)))))
27.53/27.79	= { by lemma 253 }
27.53/27.79	  not(necessarily(implies(X, Y)))
27.53/27.79	= { by axiom 240 (op_strict_implies) }
27.53/27.79	  not($$fresh23(op_strict_implies, $$true, X, Y))
27.53/27.79	= { by axiom 250 (s1_0_op_strict_implies) }
27.53/27.79	  not($$fresh23($$true, $$true, X, Y))
27.53/27.79	= { by axiom 94 (op_strict_implies) }
27.53/27.79	  not(strict_implies(X, Y))
27.53/27.79	
27.53/27.79	Lemma 257: $$fresh116(?, ?, implies(necessarily(X), X), Y) = $$true.
27.53/27.79	Proof:
27.53/27.79	  $$fresh116(?, ?, implies(necessarily(X), X), Y)
27.53/27.79	= { by axiom 75 (modus_ponens_2) }
27.53/27.79	  $$fresh115(is_a_theorem(implies(necessarily(X), X)), $$true, Y)
27.53/27.79	= { by axiom 222 (axiom_M_1) }
27.53/27.79	  $$fresh115($$fresh93(axiom_M, $$true, X), $$true, Y)
27.53/27.79	= { by axiom 242 (km4b_axiom_M) }
27.53/27.79	  $$fresh115($$fresh93($$true, $$true, X), $$true, Y)
27.53/27.79	= { by axiom 20 (axiom_M_1) }
27.53/27.79	  $$fresh115($$true, $$true, Y)
27.53/27.79	= { by axiom 74 (modus_ponens_2) }
27.53/27.79	  $$true
27.53/27.79	
27.53/27.79	Lemma 258: is_a_theorem(implies(or(X, not(Y)), implies(Y, X))) = $$true.
27.53/27.79	Proof:
27.53/27.79	  is_a_theorem(implies(or(X, not(Y)), implies(Y, X)))
27.53/27.79	= { by lemma 254 }
27.53/27.79	  is_a_theorem(implies(implies(not(X), not(Y)), implies(Y, X)))
27.53/27.79	= { by axiom 161 (modus_tollens_1) }
27.53/27.79	  $$fresh35(modus_tollens, $$true, Y, X)
27.53/27.79	= { by axiom 188 (hilbert_modus_tollens) }
27.53/27.79	  $$fresh35($$true, $$true, Y, X)
27.53/27.79	= { by axiom 82 (modus_tollens_1) }
27.53/27.79	  $$true
27.53/27.79	
27.53/27.79	Lemma 259: $$fresh116(?, ?, implies(X, or(X, Y)), Z) = $$true.
27.53/27.79	Proof:
27.53/27.79	  $$fresh116(?, ?, implies(X, or(X, Y)), Z)
27.53/27.79	= { by axiom 75 (modus_ponens_2) }
27.53/27.79	  $$fresh115(is_a_theorem(implies(X, or(X, Y))), $$true, Z)
27.53/27.79	= { by axiom 135 (or_1_1) }
27.53/27.79	  $$fresh115($$fresh21(or_1, $$true, X, Y), $$true, Z)
27.53/27.79	= { by axiom 184 (hilbert_or_1) }
27.53/27.79	  $$fresh115($$fresh21($$true, $$true, X, Y), $$true, Z)
27.53/27.79	= { by axiom 96 (or_1_1) }
27.53/27.79	  $$fresh115($$true, $$true, Z)
27.53/27.79	= { by axiom 74 (modus_ponens_2) }
27.53/27.79	  $$true
27.53/27.79	
27.53/27.79	Lemma 260: $$fresh3(is_a_theorem(equiv(X, Y)), $$true, X, Y) = Y.
27.53/27.79	Proof:
27.53/27.79	  $$fresh3(is_a_theorem(equiv(X, Y)), $$true, X, Y)
27.53/27.79	= { by axiom 121 (substitution_of_equivalents_2) }
27.53/27.79	  $$fresh4(substitution_of_equivalents, $$true, X, Y)
27.53/27.79	= { by axiom 179 (substitution_of_equivalents) }
27.53/27.79	  $$fresh4($$true, $$true, X, Y)
27.53/27.79	= { by axiom 113 (substitution_of_equivalents_2) }
27.53/27.79	  Y
27.53/27.79	
27.53/27.79	Lemma 261: $$fresh40(is_a_theorem(implies(X, Y)), $$true, X, Y) = $$fresh116(?, ?, X, Y).
27.53/27.79	Proof:
27.53/27.79	  $$fresh40(is_a_theorem(implies(X, Y)), $$true, X, Y)
27.53/27.79	= { by axiom 140 (modus_ponens_2) }
27.53/27.79	  $$fresh116(modus_ponens, $$true, X, Y)
27.53/27.79	= { by axiom 190 (hilbert_modus_ponens) }
27.53/27.79	  $$fresh116($$true, $$true, X, Y)
27.53/27.79	= { by lemma 251 }
27.53/27.79	  $$fresh116(?, ?, X, Y)
27.53/27.79	
27.53/27.79	Lemma 262: is_a_theorem(implies(X, implies(Y, and(X, Y)))) = $$true.
27.53/27.79	Proof:
27.53/27.79	  is_a_theorem(implies(X, implies(Y, and(X, Y))))
27.53/27.79	= { by axiom 145 (and_3_1) }
27.53/27.79	  $$fresh103(and_3, $$true, X, Y)
27.53/27.79	= { by axiom 175 (hilbert_and_3) }
27.53/27.79	  $$fresh103($$true, $$true, X, Y)
27.53/27.79	= { by axiom 10 (and_3_1) }
27.53/27.79	  $$true
27.53/27.79	
27.53/27.79	Lemma 263: and(implies(X, Y), implies(Y, X)) = equiv(X, Y).
27.53/27.79	Proof:
27.53/27.79	  and(implies(X, Y), implies(Y, X))
27.53/27.79	= { by axiom 173 (op_equiv) }
27.53/27.79	  $$fresh30(op_equiv, $$true, X, Y)
27.53/27.79	= { by axiom 189 (hilbert_op_equiv) }
27.53/27.79	  $$fresh30($$true, $$true, X, Y)
27.53/27.79	= { by axiom 87 (op_equiv) }
27.53/27.79	  equiv(X, Y)
27.53/27.79	
27.53/27.79	Lemma 264: $$fresh116(?, ?, implies(X, implies(X, Y)), implies(X, Y)) = is_a_theorem(implies(X, Y)).
27.53/27.79	Proof:
27.53/27.79	  $$fresh116(?, ?, implies(X, implies(X, Y)), implies(X, Y))
27.53/27.79	= { by lemma 261 }
27.53/27.79	  $$fresh40(is_a_theorem(implies(implies(X, implies(X, Y)), implies(X, Y))), $$true, implies(X, implies(X, Y)), implies(X, Y))
27.53/27.79	= { by axiom 167 (implies_2_1) }
27.53/27.79	  $$fresh40($$fresh49(implies_2, $$true, X, Y), $$true, implies(X, implies(X, Y)), implies(X, Y))
27.53/27.79	= { by axiom 180 (hilbert_implies_2) }
27.53/27.79	  $$fresh40($$fresh49($$true, $$true, X, Y), $$true, implies(X, implies(X, Y)), implies(X, Y))
27.53/27.79	= { by axiom 64 (implies_2_1) }
27.53/27.79	  $$fresh40($$true, $$true, implies(X, implies(X, Y)), implies(X, Y))
27.53/27.79	= { by axiom 73 (modus_ponens_2) }
27.53/27.79	  is_a_theorem(implies(X, Y))
27.53/27.79	
27.53/27.79	Lemma 265: is_a_theorem(implies(X, X)) = $$true.
27.53/27.79	Proof:
27.53/27.79	  is_a_theorem(implies(X, X))
27.53/27.79	= { by lemma 264 }
27.53/27.79	  $$fresh116(?, ?, implies(X, implies(X, X)), implies(X, X))
27.53/27.79	= { by axiom 75 (modus_ponens_2) }
27.53/27.79	  $$fresh115(is_a_theorem(implies(X, implies(X, X))), $$true, implies(X, X))
27.53/27.79	= { by axiom 159 (implies_1_1) }
27.53/27.79	  $$fresh115($$fresh51(implies_1, $$true, X, X), $$true, implies(X, X))
27.53/27.79	= { by axiom 187 (hilbert_implies_1) }
27.53/27.79	  $$fresh115($$fresh51($$true, $$true, X, X), $$true, implies(X, X))
27.53/27.79	= { by axiom 62 (implies_1_1) }
27.53/27.79	  $$fresh115($$true, $$true, implies(X, X))
27.53/27.79	= { by axiom 74 (modus_ponens_2) }
27.53/27.79	  $$true
27.53/27.79	
27.53/27.79	Lemma 266: $$fresh116(?, ?, X, implies(Y, and(X, Y))) = is_a_theorem(implies(Y, and(X, Y))).
27.53/27.79	Proof:
27.53/27.79	  $$fresh116(?, ?, X, implies(Y, and(X, Y)))
27.53/27.79	= { by lemma 261 }
27.53/27.79	  $$fresh40(is_a_theorem(implies(X, implies(Y, and(X, Y)))), $$true, X, implies(Y, and(X, Y)))
27.53/27.79	= { by lemma 262 }
27.53/27.79	  $$fresh40($$true, $$true, X, implies(Y, and(X, Y)))
27.53/27.79	= { by axiom 73 (modus_ponens_2) }
27.53/27.79	  is_a_theorem(implies(Y, and(X, Y)))
27.53/27.79	
27.53/27.79	Lemma 267: is_a_theorem(implies(X, and(implies(necessarily(Y), Y), X))) = $$true.
27.53/27.79	Proof:
27.53/27.79	  is_a_theorem(implies(X, and(implies(necessarily(Y), Y), X)))
27.53/27.79	= { by lemma 266 }
27.53/27.79	  $$fresh116(?, ?, implies(necessarily(Y), Y), implies(X, and(implies(necessarily(Y), Y), X)))
27.53/27.79	= { by lemma 257 }
27.61/27.80	  $$true
27.61/27.80	
27.61/27.80	Lemma 268: or(X, X) = not(not(X)).
27.61/27.80	Proof:
27.61/27.80	  or(X, X)
27.61/27.80	= { by lemma 254 }
27.61/27.80	  implies(not(X), X)
27.61/27.80	= { by lemma 253 }
27.61/27.80	  not(and(not(X), not(X)))
27.61/27.80	= { by lemma 260 }
27.61/27.80	  not($$fresh3(is_a_theorem(equiv(not(X), and(not(X), not(X)))), $$true, not(X), and(not(X), not(X))))
27.61/27.80	= { by axiom 73 (modus_ponens_2) }
27.61/27.80	  not($$fresh3($$fresh40($$true, $$true, implies(and(not(X), not(X)), not(X)), equiv(not(X), and(not(X), not(X)))), $$true, not(X), and(not(X), not(X))))
27.61/27.80	= { by axiom 74 (modus_ponens_2) }
27.61/27.80	  not($$fresh3($$fresh40($$fresh115($$true, $$true, implies(implies(and(not(X), not(X)), not(X)), and(implies(not(X), and(not(X), not(X))), implies(and(not(X), not(X)), not(X))))), $$true, implies(and(not(X), not(X)), not(X)), equiv(not(X), and(not(X), not(X)))), $$true, not(X), and(not(X), not(X))))
27.61/27.80	= { by axiom 74 (modus_ponens_2) }
27.61/27.80	  not($$fresh3($$fresh40($$fresh115($$fresh115($$true, $$true, implies(not(X), and(not(X), not(X)))), $$true, implies(implies(and(not(X), not(X)), not(X)), and(implies(not(X), and(not(X), not(X))), implies(and(not(X), not(X)), not(X))))), $$true, implies(and(not(X), not(X)), not(X)), equiv(not(X), and(not(X), not(X)))), $$true, not(X), and(not(X), not(X))))
27.61/27.80	= { by lemma 262 }
27.61/27.80	  not($$fresh3($$fresh40($$fresh115($$fresh115(is_a_theorem(implies(not(X), implies(not(X), and(not(X), not(X))))), $$true, implies(not(X), and(not(X), not(X)))), $$true, implies(implies(and(not(X), not(X)), not(X)), and(implies(not(X), and(not(X), not(X))), implies(and(not(X), not(X)), not(X))))), $$true, implies(and(not(X), not(X)), not(X)), equiv(not(X), and(not(X), not(X)))), $$true, not(X), and(not(X), not(X))))
27.61/27.80	= { by axiom 75 (modus_ponens_2) }
27.61/27.80	  not($$fresh3($$fresh40($$fresh115($$fresh116(?, ?, implies(not(X), implies(not(X), and(not(X), not(X)))), implies(not(X), and(not(X), not(X)))), $$true, implies(implies(and(not(X), not(X)), not(X)), and(implies(not(X), and(not(X), not(X))), implies(and(not(X), not(X)), not(X))))), $$true, implies(and(not(X), not(X)), not(X)), equiv(not(X), and(not(X), not(X)))), $$true, not(X), and(not(X), not(X))))
27.61/27.80	= { by lemma 264 }
27.61/27.81	  not($$fresh3($$fresh40($$fresh115(is_a_theorem(implies(not(X), and(not(X), not(X)))), $$true, implies(implies(and(not(X), not(X)), not(X)), and(implies(not(X), and(not(X), not(X))), implies(and(not(X), not(X)), not(X))))), $$true, implies(and(not(X), not(X)), not(X)), equiv(not(X), and(not(X), not(X)))), $$true, not(X), and(not(X), not(X))))
27.61/27.81	= { by axiom 75 (modus_ponens_2) }
27.61/27.81	  not($$fresh3($$fresh40($$fresh116(?, ?, implies(not(X), and(not(X), not(X))), implies(implies(and(not(X), not(X)), not(X)), and(implies(not(X), and(not(X), not(X))), implies(and(not(X), not(X)), not(X))))), $$true, implies(and(not(X), not(X)), not(X)), equiv(not(X), and(not(X), not(X)))), $$true, not(X), and(not(X), not(X))))
27.61/27.81	= { by lemma 266 }
27.61/27.81	  not($$fresh3($$fresh40(is_a_theorem(implies(implies(and(not(X), not(X)), not(X)), and(implies(not(X), and(not(X), not(X))), implies(and(not(X), not(X)), not(X))))), $$true, implies(and(not(X), not(X)), not(X)), equiv(not(X), and(not(X), not(X)))), $$true, not(X), and(not(X), not(X))))
27.61/27.81	= { by lemma 263 }
27.61/27.81	  not($$fresh3($$fresh40(is_a_theorem(implies(implies(and(not(X), not(X)), not(X)), equiv(not(X), and(not(X), not(X))))), $$true, implies(and(not(X), not(X)), not(X)), equiv(not(X), and(not(X), not(X)))), $$true, not(X), and(not(X), not(X))))
27.61/27.81	= { by lemma 261 }
27.61/27.81	  not($$fresh3($$fresh116(?, ?, implies(and(not(X), not(X)), not(X)), equiv(not(X), and(not(X), not(X)))), $$true, not(X), and(not(X), not(X))))
27.61/27.81	= { by axiom 75 (modus_ponens_2) }
27.61/27.81	  not($$fresh3($$fresh115(is_a_theorem(implies(and(not(X), not(X)), not(X))), $$true, equiv(not(X), and(not(X), not(X)))), $$true, not(X), and(not(X), not(X))))
27.61/27.81	= { by axiom 149 (and_1_1) }
27.61/27.81	  not($$fresh3($$fresh115($$fresh107(and_1, $$true, not(X), not(X)), $$true, equiv(not(X), and(not(X), not(X)))), $$true, not(X), and(not(X), not(X))))
27.61/27.81	= { by axiom 178 (hilbert_and_1) }
27.61/27.81	  not($$fresh3($$fresh115($$fresh107($$true, $$true, not(X), not(X)), $$true, equiv(not(X), and(not(X), not(X)))), $$true, not(X), and(not(X), not(X))))
27.61/27.81	= { by axiom 6 (and_1_1) }
27.61/27.81	  not($$fresh3($$fresh115($$true, $$true, equiv(not(X), and(not(X), not(X)))), $$true, not(X), and(not(X), not(X))))
27.61/27.81	= { by axiom 74 (modus_ponens_2) }
27.61/27.81	  not($$fresh3($$true, $$true, not(X), and(not(X), not(X))))
27.61/27.81	= { by axiom 112 (substitution_of_equivalents_2) }
27.61/27.82	  not(not(X))
27.61/27.82	
27.61/27.82	Lemma 269: necessarily(not(implies(X, X))) = not(implies(X, X)).
27.61/27.82	Proof:
27.61/27.82	  necessarily(not(implies(X, X)))
27.61/27.82	= { by axiom 112 (substitution_of_equivalents_2) }
27.61/27.82	  $$fresh3($$true, $$true, necessarily(not(implies(X, X))), not(implies(X, X)))
27.61/27.82	= { by axiom 74 (modus_ponens_2) }
27.61/27.82	  $$fresh3($$fresh115($$true, $$true, and(implies(necessarily(not(implies(X, X))), not(implies(X, X))), or(implies(X, X), necessarily(not(implies(X, X)))))), $$true, necessarily(not(implies(X, X))), not(implies(X, X)))
27.61/27.82	= { by axiom 74 (modus_ponens_2) }
27.61/27.82	  $$fresh3($$fresh115($$fresh115($$true, $$true, or(implies(X, X), necessarily(not(implies(X, X))))), $$true, and(implies(necessarily(not(implies(X, X))), not(implies(X, X))), or(implies(X, X), necessarily(not(implies(X, X)))))), $$true, necessarily(not(implies(X, X))), not(implies(X, X)))
27.61/27.82	= { by lemma 265 }
27.61/27.82	  $$fresh3($$fresh115($$fresh115(is_a_theorem(implies(X, X)), $$true, or(implies(X, X), necessarily(not(implies(X, X))))), $$true, and(implies(necessarily(not(implies(X, X))), not(implies(X, X))), or(implies(X, X), necessarily(not(implies(X, X)))))), $$true, necessarily(not(implies(X, X))), not(implies(X, X)))
27.61/27.82	= { by axiom 75 (modus_ponens_2) }
27.61/27.82	  $$fresh3($$fresh115($$fresh116(?, ?, implies(X, X), or(implies(X, X), necessarily(not(implies(X, X))))), $$true, and(implies(necessarily(not(implies(X, X))), not(implies(X, X))), or(implies(X, X), necessarily(not(implies(X, X)))))), $$true, necessarily(not(implies(X, X))), not(implies(X, X)))
27.61/27.82	= { by lemma 251 }
27.61/27.82	  $$fresh3($$fresh115($$fresh116($$true, $$true, implies(X, X), or(implies(X, X), necessarily(not(implies(X, X))))), $$true, and(implies(necessarily(not(implies(X, X))), not(implies(X, X))), or(implies(X, X), necessarily(not(implies(X, X)))))), $$true, necessarily(not(implies(X, X))), not(implies(X, X)))
27.61/27.82	= { by axiom 190 (hilbert_modus_ponens) }
27.61/27.82	  $$fresh3($$fresh115($$fresh116(modus_ponens, $$true, implies(X, X), or(implies(X, X), necessarily(not(implies(X, X))))), $$true, and(implies(necessarily(not(implies(X, X))), not(implies(X, X))), or(implies(X, X), necessarily(not(implies(X, X)))))), $$true, necessarily(not(implies(X, X))), not(implies(X, X)))
27.61/27.82	= { by axiom 140 (modus_ponens_2) }
27.61/27.82	  $$fresh3($$fresh115($$fresh40(is_a_theorem(implies(implies(X, X), or(implies(X, X), necessarily(not(implies(X, X)))))), $$true, implies(X, X), or(implies(X, X), necessarily(not(implies(X, X))))), $$true, and(implies(necessarily(not(implies(X, X))), not(implies(X, X))), or(implies(X, X), necessarily(not(implies(X, X)))))), $$true, necessarily(not(implies(X, X))), not(implies(X, X)))
27.61/27.82	= { by axiom 135 (or_1_1) }
27.61/27.82	  $$fresh3($$fresh115($$fresh40($$fresh21(or_1, $$true, implies(X, X), necessarily(not(implies(X, X)))), $$true, implies(X, X), or(implies(X, X), necessarily(not(implies(X, X))))), $$true, and(implies(necessarily(not(implies(X, X))), not(implies(X, X))), or(implies(X, X), necessarily(not(implies(X, X)))))), $$true, necessarily(not(implies(X, X))), not(implies(X, X)))
27.61/27.82	= { by axiom 184 (hilbert_or_1) }
27.61/27.82	  $$fresh3($$fresh115($$fresh40($$fresh21($$true, $$true, implies(X, X), necessarily(not(implies(X, X)))), $$true, implies(X, X), or(implies(X, X), necessarily(not(implies(X, X))))), $$true, and(implies(necessarily(not(implies(X, X))), not(implies(X, X))), or(implies(X, X), necessarily(not(implies(X, X)))))), $$true, necessarily(not(implies(X, X))), not(implies(X, X)))
27.61/27.82	= { by axiom 96 (or_1_1) }
27.61/27.82	  $$fresh3($$fresh115($$fresh40($$true, $$true, implies(X, X), or(implies(X, X), necessarily(not(implies(X, X))))), $$true, and(implies(necessarily(not(implies(X, X))), not(implies(X, X))), or(implies(X, X), necessarily(not(implies(X, X)))))), $$true, necessarily(not(implies(X, X))), not(implies(X, X)))
27.61/27.82	= { by axiom 73 (modus_ponens_2) }
27.61/27.82	  $$fresh3($$fresh115(is_a_theorem(or(implies(X, X), necessarily(not(implies(X, X))))), $$true, and(implies(necessarily(not(implies(X, X))), not(implies(X, X))), or(implies(X, X), necessarily(not(implies(X, X)))))), $$true, necessarily(not(implies(X, X))), not(implies(X, X)))
27.61/27.82	= { by axiom 75 (modus_ponens_2) }
27.61/27.82	  $$fresh3($$fresh116(?, ?, or(implies(X, X), necessarily(not(implies(X, X)))), and(implies(necessarily(not(implies(X, X))), not(implies(X, X))), or(implies(X, X), necessarily(not(implies(X, X)))))), $$true, necessarily(not(implies(X, X))), not(implies(X, X)))
27.61/27.82	= { by lemma 261 }
27.61/27.82	  $$fresh3($$fresh40(is_a_theorem(implies(or(implies(X, X), necessarily(not(implies(X, X)))), and(implies(necessarily(not(implies(X, X))), not(implies(X, X))), or(implies(X, X), necessarily(not(implies(X, X))))))), $$true, or(implies(X, X), necessarily(not(implies(X, X)))), and(implies(necessarily(not(implies(X, X))), not(implies(X, X))), or(implies(X, X), necessarily(not(implies(X, X)))))), $$true, necessarily(not(implies(X, X))), not(implies(X, X)))
27.61/27.82	= { by lemma 267 }
27.61/27.82	  $$fresh3($$fresh40($$true, $$true, or(implies(X, X), necessarily(not(implies(X, X)))), and(implies(necessarily(not(implies(X, X))), not(implies(X, X))), or(implies(X, X), necessarily(not(implies(X, X)))))), $$true, necessarily(not(implies(X, X))), not(implies(X, X)))
27.61/27.82	= { by axiom 73 (modus_ponens_2) }
27.61/27.82	  $$fresh3(is_a_theorem(and(implies(necessarily(not(implies(X, X))), not(implies(X, X))), or(implies(X, X), necessarily(not(implies(X, X)))))), $$true, necessarily(not(implies(X, X))), not(implies(X, X)))
27.61/27.82	= { by lemma 254 }
27.61/27.82	  $$fresh3(is_a_theorem(and(implies(necessarily(not(implies(X, X))), not(implies(X, X))), implies(not(implies(X, X)), necessarily(not(implies(X, X)))))), $$true, necessarily(not(implies(X, X))), not(implies(X, X)))
27.61/27.82	= { by axiom 173 (op_equiv) }
27.61/27.82	  $$fresh3(is_a_theorem($$fresh30(op_equiv, $$true, necessarily(not(implies(X, X))), not(implies(X, X)))), $$true, necessarily(not(implies(X, X))), not(implies(X, X)))
27.61/27.82	= { by axiom 189 (hilbert_op_equiv) }
27.61/27.82	  $$fresh3(is_a_theorem($$fresh30($$true, $$true, necessarily(not(implies(X, X))), not(implies(X, X)))), $$true, necessarily(not(implies(X, X))), not(implies(X, X)))
27.61/27.82	= { by axiom 87 (op_equiv) }
27.61/27.82	  $$fresh3(is_a_theorem(equiv(necessarily(not(implies(X, X))), not(implies(X, X)))), $$true, necessarily(not(implies(X, X))), not(implies(X, X)))
27.61/27.82	= { by lemma 260 }
27.61/27.83	  not(implies(X, X))
27.61/27.83	
27.61/27.83	Lemma 270: not(not(X)) = X.
27.61/27.83	Proof:
27.61/27.83	  not(not(X))
27.61/27.83	= { by lemma 260 }
27.61/27.83	  $$fresh3(is_a_theorem(equiv(X, not(not(X)))), $$true, X, not(not(X)))
27.61/27.83	= { by axiom 73 (modus_ponens_2) }
27.61/27.83	  $$fresh3($$fresh40($$true, $$true, or(not(X), X), equiv(X, not(not(X)))), $$true, X, not(not(X)))
27.61/27.83	= { by lemma 259 }
27.61/27.83	  $$fresh3($$fresh40($$fresh116(?, ?, implies(X, or(X, X)), implies(implies(or(X, X), X), and(implies(X, or(X, X)), implies(or(X, X), X)))), $$true, or(not(X), X), equiv(X, not(not(X)))), $$true, X, not(not(X)))
27.61/27.83	= { by lemma 266 }
27.61/27.83	  $$fresh3($$fresh40(is_a_theorem(implies(implies(or(X, X), X), and(implies(X, or(X, X)), implies(or(X, X), X)))), $$true, or(not(X), X), equiv(X, not(not(X)))), $$true, X, not(not(X)))
27.61/27.83	= { by lemma 268 }
27.61/27.83	  $$fresh3($$fresh40(is_a_theorem(implies(implies(not(not(X)), X), and(implies(X, or(X, X)), implies(or(X, X), X)))), $$true, or(not(X), X), equiv(X, not(not(X)))), $$true, X, not(not(X)))
27.61/27.83	= { by lemma 254 }
27.61/27.83	  $$fresh3($$fresh40(is_a_theorem(implies(or(not(X), X), and(implies(X, or(X, X)), implies(or(X, X), X)))), $$true, or(not(X), X), equiv(X, not(not(X)))), $$true, X, not(not(X)))
27.61/27.83	= { by lemma 263 }
27.61/27.83	  $$fresh3($$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)))
27.61/27.83	= { by lemma 268 }
27.61/27.83	  $$fresh3($$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)))
27.61/27.83	= { by lemma 261 }
27.61/27.83	  $$fresh3($$fresh116(?, ?, or(not(X), X), equiv(X, not(not(X)))), $$true, X, not(not(X)))
27.61/27.83	= { by lemma 254 }
27.61/27.83	  $$fresh3($$fresh116(?, ?, implies(not(not(X)), X), equiv(X, not(not(X)))), $$true, X, not(not(X)))
27.61/27.83	= { by lemma 268 }
27.61/27.83	  $$fresh3($$fresh116(?, ?, implies(or(X, X), X), equiv(X, not(not(X)))), $$true, X, not(not(X)))
27.61/27.83	= { by axiom 75 (modus_ponens_2) }
27.61/27.83	  $$fresh3($$fresh115(is_a_theorem(implies(or(X, X), X)), $$true, equiv(X, not(not(X)))), $$true, X, not(not(X)))
27.61/27.83	= { by lemma 268 }
27.61/27.83	  $$fresh3($$fresh115(is_a_theorem(implies(not(not(X)), X)), $$true, equiv(X, not(not(X)))), $$true, X, not(not(X)))
27.61/27.83	= { by axiom 73 (modus_ponens_2) }
27.61/27.83	  $$fresh3($$fresh115($$fresh40($$true, $$true, or(X, not(not(not(X)))), implies(not(not(X)), X)), $$true, equiv(X, not(not(X)))), $$true, X, not(not(X)))
27.61/27.83	= { by lemma 258 }
27.61/27.83	  $$fresh3($$fresh115($$fresh40(is_a_theorem(implies(or(X, not(not(not(X)))), implies(not(not(X)), X))), $$true, or(X, not(not(not(X)))), implies(not(not(X)), X)), $$true, equiv(X, not(not(X)))), $$true, X, not(not(X)))
27.61/27.83	= { by lemma 261 }
27.61/27.83	  $$fresh3($$fresh115($$fresh116(?, ?, or(X, not(not(not(X)))), implies(not(not(X)), X)), $$true, equiv(X, not(not(X)))), $$true, X, not(not(X)))
27.61/27.83	= { by lemma 268 }
27.61/27.83	  $$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)))
27.61/27.83	= { by lemma 254 }
27.61/27.83	  $$fresh3($$fresh115($$fresh116(?, ?, implies(not(X), or(not(X), not(X))), implies(not(not(X)), X)), $$true, equiv(X, not(not(X)))), $$true, X, not(not(X)))
27.61/27.83	= { by lemma 259 }
27.61/27.83	  $$fresh3($$fresh115($$true, $$true, equiv(X, not(not(X)))), $$true, X, not(not(X)))
27.61/27.83	= { by axiom 74 (modus_ponens_2) }
27.61/27.83	  $$fresh3($$true, $$true, X, not(not(X)))
27.61/27.83	= { by axiom 112 (substitution_of_equivalents_2) }
27.61/27.83	  X
27.61/27.83	
27.61/27.83	Lemma 271: not(necessarily(X)) = possibly(not(X)).
27.61/27.83	Proof:
27.61/27.83	  not(necessarily(X))
27.61/27.83	= { by lemma 270 }
27.61/27.83	  not(necessarily(not(not(X))))
27.61/27.83	= { by lemma 252 }
27.61/27.83	  possibly(not(X))
27.61/27.83	
27.61/27.83	Lemma 272: or(not(X), Y) = implies(X, Y).
27.61/27.83	Proof:
27.61/27.83	  or(not(X), Y)
27.61/27.83	= { by lemma 254 }
27.61/27.83	  implies(not(not(X)), Y)
27.61/27.83	= { by lemma 270 }
27.61/27.83	  implies(X, Y)
27.61/27.83	
27.61/27.83	Lemma 273: not(implies(X, Y)) = and(X, not(Y)).
27.61/27.83	Proof:
27.61/27.83	  not(implies(X, Y))
27.61/27.83	= { by lemma 253 }
27.61/27.83	  not(not(and(X, not(Y))))
27.61/27.83	= { by lemma 270 }
27.61/27.83	  and(X, not(Y))
27.61/27.83	
27.61/27.83	Lemma 274: $$fresh11(r3, $$true, X, Y) = $$true.
27.61/27.83	Proof:
27.61/27.83	  $$fresh11(r3, $$true, X, Y)
27.61/27.83	= { by axiom 153 (r3_1) }
27.61/27.83	  is_a_theorem(implies(or(X, Y), or(Y, X)))
27.61/27.83	= { by lemma 270 }
27.61/27.83	  is_a_theorem(implies(or(X, not(not(Y))), or(Y, X)))
27.61/27.83	= { by lemma 254 }
27.61/27.83	  is_a_theorem(implies(or(X, not(not(Y))), implies(not(Y), X)))
27.61/27.83	= { by lemma 258 }
27.71/27.91	  $$true
27.71/27.91	
27.71/27.91	Lemma 275: necessarily(possibly(X)) = possibly(X).
27.71/27.91	Proof:
27.71/27.91	  necessarily(possibly(X))
27.71/27.91	= { by axiom 112 (substitution_of_equivalents_2) }
27.71/27.91	  $$fresh3($$true, $$true, necessarily(possibly(X)), possibly(X))
27.71/27.91	= { by axiom 74 (modus_ponens_2) }
27.71/27.91	  $$fresh3($$fresh115($$true, $$true, equiv(necessarily(possibly(X)), possibly(X))), $$true, necessarily(possibly(X)), possibly(X))
27.71/27.91	= { by axiom 16 (axiom_B_1) }
27.71/27.91	  $$fresh3($$fresh115($$fresh97($$true, $$true, possibly(X)), $$true, equiv(necessarily(possibly(X)), possibly(X))), $$true, necessarily(possibly(X)), possibly(X))
27.71/27.91	= { by axiom 247 (km4b_axiom_B) }
27.71/27.91	  $$fresh3($$fresh115($$fresh97(axiom_B, $$true, possibly(X)), $$true, equiv(necessarily(possibly(X)), possibly(X))), $$true, necessarily(possibly(X)), possibly(X))
27.71/27.91	= { by axiom 230 (axiom_B_1) }
27.71/27.91	  $$fresh3($$fresh115(is_a_theorem(implies(possibly(X), necessarily(possibly(possibly(X))))), $$true, equiv(necessarily(possibly(X)), possibly(X))), $$true, necessarily(possibly(X)), possibly(X))
27.71/27.91	= { by lemma 270 }
27.71/27.91	  $$fresh3($$fresh115(is_a_theorem(implies(possibly(X), necessarily(possibly(possibly(not(not(X))))))), $$true, equiv(necessarily(possibly(X)), possibly(X))), $$true, necessarily(possibly(X)), possibly(X))
27.71/27.91	= { by lemma 271 }
27.71/27.91	  $$fresh3($$fresh115(is_a_theorem(implies(possibly(X), necessarily(possibly(not(necessarily(not(X))))))), $$true, equiv(necessarily(possibly(X)), possibly(X))), $$true, necessarily(possibly(X)), possibly(X))
27.71/27.91	= { by lemma 271 }
27.71/27.91	  $$fresh3($$fresh115(is_a_theorem(implies(possibly(X), necessarily(not(necessarily(necessarily(not(X))))))), $$true, equiv(necessarily(possibly(X)), possibly(X))), $$true, necessarily(possibly(X)), possibly(X))
27.71/27.91	= { by lemma 260 }
27.71/27.91	  $$fresh3($$fresh115(is_a_theorem(implies(possibly(X), necessarily(not($$fresh3(is_a_theorem(equiv(necessarily(not(X)), necessarily(necessarily(not(X))))), $$true, necessarily(not(X)), necessarily(necessarily(not(X)))))))), $$true, equiv(necessarily(possibly(X)), possibly(X))), $$true, necessarily(possibly(X)), possibly(X))
27.71/27.91	= { by axiom 73 (modus_ponens_2) }
27.71/27.91	  $$fresh3($$fresh115(is_a_theorem(implies(possibly(X), necessarily(not($$fresh3($$fresh40($$true, $$true, implies(necessarily(necessarily(not(X))), necessarily(not(X))), equiv(necessarily(not(X)), necessarily(necessarily(not(X))))), $$true, necessarily(not(X)), necessarily(necessarily(not(X)))))))), $$true, equiv(necessarily(possibly(X)), possibly(X))), $$true, necessarily(possibly(X)), possibly(X))
27.71/27.91	= { by axiom 74 (modus_ponens_2) }
27.71/27.91	  $$fresh3($$fresh115(is_a_theorem(implies(possibly(X), necessarily(not($$fresh3($$fresh40($$fresh115($$true, $$true, implies(implies(necessarily(necessarily(not(X))), necessarily(not(X))), and(implies(necessarily(not(X)), necessarily(necessarily(not(X)))), implies(necessarily(necessarily(not(X))), necessarily(not(X)))))), $$true, implies(necessarily(necessarily(not(X))), necessarily(not(X))), equiv(necessarily(not(X)), necessarily(necessarily(not(X))))), $$true, necessarily(not(X)), necessarily(necessarily(not(X)))))))), $$true, equiv(necessarily(possibly(X)), possibly(X))), $$true, necessarily(possibly(X)), possibly(X))
27.71/27.91	= { by axiom 12 (axiom_4_1) }
27.71/27.91	  $$fresh3($$fresh115(is_a_theorem(implies(possibly(X), necessarily(not($$fresh3($$fresh40($$fresh115($$fresh101($$true, $$true, not(X)), $$true, implies(implies(necessarily(necessarily(not(X))), necessarily(not(X))), and(implies(necessarily(not(X)), necessarily(necessarily(not(X)))), implies(necessarily(necessarily(not(X))), necessarily(not(X)))))), $$true, implies(necessarily(necessarily(not(X))), necessarily(not(X))), equiv(necessarily(not(X)), necessarily(necessarily(not(X))))), $$true, necessarily(not(X)), necessarily(necessarily(not(X)))))))), $$true, equiv(necessarily(possibly(X)), possibly(X))), $$true, necessarily(possibly(X)), possibly(X))
27.71/27.91	= { by axiom 244 (km4b_axiom_4) }
27.71/27.91	  $$fresh3($$fresh115(is_a_theorem(implies(possibly(X), necessarily(not($$fresh3($$fresh40($$fresh115($$fresh101(axiom_4, $$true, not(X)), $$true, implies(implies(necessarily(necessarily(not(X))), necessarily(not(X))), and(implies(necessarily(not(X)), necessarily(necessarily(not(X)))), implies(necessarily(necessarily(not(X))), necessarily(not(X)))))), $$true, implies(necessarily(necessarily(not(X))), necessarily(not(X))), equiv(necessarily(not(X)), necessarily(necessarily(not(X))))), $$true, necessarily(not(X)), necessarily(necessarily(not(X)))))))), $$true, equiv(necessarily(possibly(X)), possibly(X))), $$true, necessarily(possibly(X)), possibly(X))
27.71/27.91	= { by axiom 210 (axiom_4_1) }
27.71/27.91	  $$fresh3($$fresh115(is_a_theorem(implies(possibly(X), necessarily(not($$fresh3($$fresh40($$fresh115(is_a_theorem(implies(necessarily(not(X)), necessarily(necessarily(not(X))))), $$true, implies(implies(necessarily(necessarily(not(X))), necessarily(not(X))), and(implies(necessarily(not(X)), necessarily(necessarily(not(X)))), implies(necessarily(necessarily(not(X))), necessarily(not(X)))))), $$true, implies(necessarily(necessarily(not(X))), necessarily(not(X))), equiv(necessarily(not(X)), necessarily(necessarily(not(X))))), $$true, necessarily(not(X)), necessarily(necessarily(not(X)))))))), $$true, equiv(necessarily(possibly(X)), possibly(X))), $$true, necessarily(possibly(X)), possibly(X))
27.71/27.91	= { by axiom 75 (modus_ponens_2) }
27.71/27.91	  $$fresh3($$fresh115(is_a_theorem(implies(possibly(X), necessarily(not($$fresh3($$fresh40($$fresh116(?, ?, implies(necessarily(not(X)), necessarily(necessarily(not(X)))), implies(implies(necessarily(necessarily(not(X))), necessarily(not(X))), and(implies(necessarily(not(X)), necessarily(necessarily(not(X)))), implies(necessarily(necessarily(not(X))), necessarily(not(X)))))), $$true, implies(necessarily(necessarily(not(X))), necessarily(not(X))), equiv(necessarily(not(X)), necessarily(necessarily(not(X))))), $$true, necessarily(not(X)), necessarily(necessarily(not(X)))))))), $$true, equiv(necessarily(possibly(X)), possibly(X))), $$true, necessarily(possibly(X)), possibly(X))
27.71/27.91	= { by lemma 266 }
27.71/27.91	  $$fresh3($$fresh115(is_a_theorem(implies(possibly(X), necessarily(not($$fresh3($$fresh40(is_a_theorem(implies(implies(necessarily(necessarily(not(X))), necessarily(not(X))), and(implies(necessarily(not(X)), necessarily(necessarily(not(X)))), implies(necessarily(necessarily(not(X))), necessarily(not(X)))))), $$true, implies(necessarily(necessarily(not(X))), necessarily(not(X))), equiv(necessarily(not(X)), necessarily(necessarily(not(X))))), $$true, necessarily(not(X)), necessarily(necessarily(not(X)))))))), $$true, equiv(necessarily(possibly(X)), possibly(X))), $$true, necessarily(possibly(X)), possibly(X))
27.71/27.91	= { by lemma 263 }
27.71/27.91	  $$fresh3($$fresh115(is_a_theorem(implies(possibly(X), necessarily(not($$fresh3($$fresh40(is_a_theorem(implies(implies(necessarily(necessarily(not(X))), necessarily(not(X))), equiv(necessarily(not(X)), necessarily(necessarily(not(X)))))), $$true, implies(necessarily(necessarily(not(X))), necessarily(not(X))), equiv(necessarily(not(X)), necessarily(necessarily(not(X))))), $$true, necessarily(not(X)), necessarily(necessarily(not(X)))))))), $$true, equiv(necessarily(possibly(X)), possibly(X))), $$true, necessarily(possibly(X)), possibly(X))
27.71/27.91	= { by lemma 261 }
27.71/27.91	  $$fresh3($$fresh115(is_a_theorem(implies(possibly(X), necessarily(not($$fresh3($$fresh116(?, ?, implies(necessarily(necessarily(not(X))), necessarily(not(X))), equiv(necessarily(not(X)), necessarily(necessarily(not(X))))), $$true, necessarily(not(X)), necessarily(necessarily(not(X)))))))), $$true, equiv(necessarily(possibly(X)), possibly(X))), $$true, necessarily(possibly(X)), possibly(X))
27.71/27.91	= { by lemma 257 }
27.71/27.91	  $$fresh3($$fresh115(is_a_theorem(implies(possibly(X), necessarily(not($$fresh3($$true, $$true, necessarily(not(X)), necessarily(necessarily(not(X)))))))), $$true, equiv(necessarily(possibly(X)), possibly(X))), $$true, necessarily(possibly(X)), possibly(X))
27.71/27.91	= { by axiom 112 (substitution_of_equivalents_2) }
27.71/27.91	  $$fresh3($$fresh115(is_a_theorem(implies(possibly(X), necessarily(not(necessarily(not(X)))))), $$true, equiv(necessarily(possibly(X)), possibly(X))), $$true, necessarily(possibly(X)), possibly(X))
27.71/27.91	= { by lemma 271 }
27.71/27.91	  $$fresh3($$fresh115(is_a_theorem(implies(possibly(X), necessarily(possibly(not(not(X)))))), $$true, equiv(necessarily(possibly(X)), possibly(X))), $$true, necessarily(possibly(X)), possibly(X))
27.71/27.91	= { by lemma 270 }
27.71/27.91	  $$fresh3($$fresh115(is_a_theorem(implies(possibly(X), necessarily(possibly(X)))), $$true, equiv(necessarily(possibly(X)), possibly(X))), $$true, necessarily(possibly(X)), possibly(X))
27.71/27.91	= { by axiom 75 (modus_ponens_2) }
27.71/27.91	  $$fresh3($$fresh116(?, ?, implies(possibly(X), necessarily(possibly(X))), equiv(necessarily(possibly(X)), possibly(X))), $$true, necessarily(possibly(X)), possibly(X))
27.71/27.91	= { by lemma 261 }
27.71/27.91	  $$fresh3($$fresh40(is_a_theorem(implies(implies(possibly(X), necessarily(possibly(X))), equiv(necessarily(possibly(X)), possibly(X)))), $$true, implies(possibly(X), necessarily(possibly(X))), equiv(necessarily(possibly(X)), possibly(X))), $$true, necessarily(possibly(X)), possibly(X))
27.71/27.91	= { by lemma 263 }
27.71/27.91	  $$fresh3($$fresh40(is_a_theorem(implies(implies(possibly(X), necessarily(possibly(X))), and(implies(necessarily(possibly(X)), possibly(X)), implies(possibly(X), necessarily(possibly(X)))))), $$true, implies(possibly(X), necessarily(possibly(X))), equiv(necessarily(possibly(X)), possibly(X))), $$true, necessarily(possibly(X)), possibly(X))
27.71/27.91	= { by lemma 267 }
27.71/27.91	  $$fresh3($$fresh40($$true, $$true, implies(possibly(X), necessarily(possibly(X))), equiv(necessarily(possibly(X)), possibly(X))), $$true, necessarily(possibly(X)), possibly(X))
27.71/27.91	= { by axiom 73 (modus_ponens_2) }
27.71/27.91	  $$fresh3(is_a_theorem(equiv(necessarily(possibly(X)), possibly(X))), $$true, necessarily(possibly(X)), possibly(X))
27.71/27.91	= { by lemma 260 }
27.82/28.02	  possibly(X)
27.82/28.02	
27.82/28.02	Goal 1 (s1_0_axiom_m1): axiom_m1 = $$true.
27.82/28.02	Proof:
27.82/28.02	  axiom_m1
27.82/28.02	= { by axiom 219 (axiom_m1) }
27.82/28.02	  $$fresh92(is_a_theorem(strict_implies(and(sK15_axiom_m1_X, sK14_axiom_m1_Y), and(sK14_axiom_m1_Y, sK15_axiom_m1_X))), $$true)
27.82/28.02	= { by lemma 270 }
27.82/28.02	  $$fresh92(is_a_theorem(strict_implies(and(sK15_axiom_m1_X, not(not(sK14_axiom_m1_Y))), and(sK14_axiom_m1_Y, sK15_axiom_m1_X))), $$true)
27.82/28.02	= { by lemma 273 }
27.82/28.02	  $$fresh92(is_a_theorem(strict_implies(not(implies(sK15_axiom_m1_X, not(sK14_axiom_m1_Y))), and(sK14_axiom_m1_Y, sK15_axiom_m1_X))), $$true)
27.82/28.02	= { by lemma 272 }
27.82/28.02	  $$fresh92(is_a_theorem(strict_implies(not(or(not(sK15_axiom_m1_X), not(sK14_axiom_m1_Y))), and(sK14_axiom_m1_Y, sK15_axiom_m1_X))), $$true)
27.82/28.02	= { by axiom 112 (substitution_of_equivalents_2) }
27.82/28.02	  $$fresh92(is_a_theorem(strict_implies(not($$fresh3($$true, $$true, or(not(sK15_axiom_m1_X), not(sK14_axiom_m1_Y)), or(not(sK14_axiom_m1_Y), not(sK15_axiom_m1_X)))), and(sK14_axiom_m1_Y, sK15_axiom_m1_X))), $$true)
27.82/28.02	= { by axiom 74 (modus_ponens_2) }
27.82/28.02	  $$fresh92(is_a_theorem(strict_implies(not($$fresh3($$fresh115($$true, $$true, equiv(or(not(sK15_axiom_m1_X), not(sK14_axiom_m1_Y)), or(not(sK14_axiom_m1_Y), not(sK15_axiom_m1_X)))), $$true, or(not(sK15_axiom_m1_X), not(sK14_axiom_m1_Y)), or(not(sK14_axiom_m1_Y), not(sK15_axiom_m1_X)))), and(sK14_axiom_m1_Y, sK15_axiom_m1_X))), $$true)
27.82/28.02	= { by lemma 274 }
27.82/28.02	  $$fresh92(is_a_theorem(strict_implies(not($$fresh3($$fresh115($$fresh11(r3, $$true, not(sK14_axiom_m1_Y), not(sK15_axiom_m1_X)), $$true, equiv(or(not(sK15_axiom_m1_X), not(sK14_axiom_m1_Y)), or(not(sK14_axiom_m1_Y), not(sK15_axiom_m1_X)))), $$true, or(not(sK15_axiom_m1_X), not(sK14_axiom_m1_Y)), or(not(sK14_axiom_m1_Y), not(sK15_axiom_m1_X)))), and(sK14_axiom_m1_Y, sK15_axiom_m1_X))), $$true)
27.82/28.02	= { by axiom 153 (r3_1) }
27.82/28.02	  $$fresh92(is_a_theorem(strict_implies(not($$fresh3($$fresh115(is_a_theorem(implies(or(not(sK14_axiom_m1_Y), not(sK15_axiom_m1_X)), or(not(sK15_axiom_m1_X), not(sK14_axiom_m1_Y)))), $$true, equiv(or(not(sK15_axiom_m1_X), not(sK14_axiom_m1_Y)), or(not(sK14_axiom_m1_Y), not(sK15_axiom_m1_X)))), $$true, or(not(sK15_axiom_m1_X), not(sK14_axiom_m1_Y)), or(not(sK14_axiom_m1_Y), not(sK15_axiom_m1_X)))), and(sK14_axiom_m1_Y, sK15_axiom_m1_X))), $$true)
27.82/28.02	= { by axiom 75 (modus_ponens_2) }
27.82/28.02	  $$fresh92(is_a_theorem(strict_implies(not($$fresh3($$fresh116(?, ?, implies(or(not(sK14_axiom_m1_Y), not(sK15_axiom_m1_X)), or(not(sK15_axiom_m1_X), not(sK14_axiom_m1_Y))), equiv(or(not(sK15_axiom_m1_X), not(sK14_axiom_m1_Y)), or(not(sK14_axiom_m1_Y), not(sK15_axiom_m1_X)))), $$true, or(not(sK15_axiom_m1_X), not(sK14_axiom_m1_Y)), or(not(sK14_axiom_m1_Y), not(sK15_axiom_m1_X)))), and(sK14_axiom_m1_Y, sK15_axiom_m1_X))), $$true)
27.82/28.02	= { by lemma 261 }
27.82/28.02	  $$fresh92(is_a_theorem(strict_implies(not($$fresh3($$fresh40(is_a_theorem(implies(implies(or(not(sK14_axiom_m1_Y), not(sK15_axiom_m1_X)), or(not(sK15_axiom_m1_X), not(sK14_axiom_m1_Y))), equiv(or(not(sK15_axiom_m1_X), not(sK14_axiom_m1_Y)), or(not(sK14_axiom_m1_Y), not(sK15_axiom_m1_X))))), $$true, implies(or(not(sK14_axiom_m1_Y), not(sK15_axiom_m1_X)), or(not(sK15_axiom_m1_X), not(sK14_axiom_m1_Y))), equiv(or(not(sK15_axiom_m1_X), not(sK14_axiom_m1_Y)), or(not(sK14_axiom_m1_Y), not(sK15_axiom_m1_X)))), $$true, or(not(sK15_axiom_m1_X), not(sK14_axiom_m1_Y)), or(not(sK14_axiom_m1_Y), not(sK15_axiom_m1_X)))), and(sK14_axiom_m1_Y, sK15_axiom_m1_X))), $$true)
27.82/28.02	= { by lemma 263 }
27.82/28.02	  $$fresh92(is_a_theorem(strict_implies(not($$fresh3($$fresh40(is_a_theorem(implies(implies(or(not(sK14_axiom_m1_Y), not(sK15_axiom_m1_X)), or(not(sK15_axiom_m1_X), not(sK14_axiom_m1_Y))), and(implies(or(not(sK15_axiom_m1_X), not(sK14_axiom_m1_Y)), or(not(sK14_axiom_m1_Y), not(sK15_axiom_m1_X))), implies(or(not(sK14_axiom_m1_Y), not(sK15_axiom_m1_X)), or(not(sK15_axiom_m1_X), not(sK14_axiom_m1_Y)))))), $$true, implies(or(not(sK14_axiom_m1_Y), not(sK15_axiom_m1_X)), or(not(sK15_axiom_m1_X), not(sK14_axiom_m1_Y))), equiv(or(not(sK15_axiom_m1_X), not(sK14_axiom_m1_Y)), or(not(sK14_axiom_m1_Y), not(sK15_axiom_m1_X)))), $$true, or(not(sK15_axiom_m1_X), not(sK14_axiom_m1_Y)), or(not(sK14_axiom_m1_Y), not(sK15_axiom_m1_X)))), and(sK14_axiom_m1_Y, sK15_axiom_m1_X))), $$true)
27.82/28.02	= { by lemma 266 }
27.82/28.02	  $$fresh92(is_a_theorem(strict_implies(not($$fresh3($$fresh40($$fresh116(?, ?, implies(or(not(sK15_axiom_m1_X), not(sK14_axiom_m1_Y)), or(not(sK14_axiom_m1_Y), not(sK15_axiom_m1_X))), implies(implies(or(not(sK14_axiom_m1_Y), not(sK15_axiom_m1_X)), or(not(sK15_axiom_m1_X), not(sK14_axiom_m1_Y))), and(implies(or(not(sK15_axiom_m1_X), not(sK14_axiom_m1_Y)), or(not(sK14_axiom_m1_Y), not(sK15_axiom_m1_X))), implies(or(not(sK14_axiom_m1_Y), not(sK15_axiom_m1_X)), or(not(sK15_axiom_m1_X), not(sK14_axiom_m1_Y)))))), $$true, implies(or(not(sK14_axiom_m1_Y), not(sK15_axiom_m1_X)), or(not(sK15_axiom_m1_X), not(sK14_axiom_m1_Y))), equiv(or(not(sK15_axiom_m1_X), not(sK14_axiom_m1_Y)), or(not(sK14_axiom_m1_Y), not(sK15_axiom_m1_X)))), $$true, or(not(sK15_axiom_m1_X), not(sK14_axiom_m1_Y)), or(not(sK14_axiom_m1_Y), not(sK15_axiom_m1_X)))), and(sK14_axiom_m1_Y, sK15_axiom_m1_X))), $$true)
27.82/28.02	= { by axiom 75 (modus_ponens_2) }
27.82/28.02	  $$fresh92(is_a_theorem(strict_implies(not($$fresh3($$fresh40($$fresh115(is_a_theorem(implies(or(not(sK15_axiom_m1_X), not(sK14_axiom_m1_Y)), or(not(sK14_axiom_m1_Y), not(sK15_axiom_m1_X)))), $$true, implies(implies(or(not(sK14_axiom_m1_Y), not(sK15_axiom_m1_X)), or(not(sK15_axiom_m1_X), not(sK14_axiom_m1_Y))), and(implies(or(not(sK15_axiom_m1_X), not(sK14_axiom_m1_Y)), or(not(sK14_axiom_m1_Y), not(sK15_axiom_m1_X))), implies(or(not(sK14_axiom_m1_Y), not(sK15_axiom_m1_X)), or(not(sK15_axiom_m1_X), not(sK14_axiom_m1_Y)))))), $$true, implies(or(not(sK14_axiom_m1_Y), not(sK15_axiom_m1_X)), or(not(sK15_axiom_m1_X), not(sK14_axiom_m1_Y))), equiv(or(not(sK15_axiom_m1_X), not(sK14_axiom_m1_Y)), or(not(sK14_axiom_m1_Y), not(sK15_axiom_m1_X)))), $$true, or(not(sK15_axiom_m1_X), not(sK14_axiom_m1_Y)), or(not(sK14_axiom_m1_Y), not(sK15_axiom_m1_X)))), and(sK14_axiom_m1_Y, sK15_axiom_m1_X))), $$true)
27.82/28.02	= { by axiom 153 (r3_1) }
27.82/28.02	  $$fresh92(is_a_theorem(strict_implies(not($$fresh3($$fresh40($$fresh115($$fresh11(r3, $$true, not(sK15_axiom_m1_X), not(sK14_axiom_m1_Y)), $$true, implies(implies(or(not(sK14_axiom_m1_Y), not(sK15_axiom_m1_X)), or(not(sK15_axiom_m1_X), not(sK14_axiom_m1_Y))), and(implies(or(not(sK15_axiom_m1_X), not(sK14_axiom_m1_Y)), or(not(sK14_axiom_m1_Y), not(sK15_axiom_m1_X))), implies(or(not(sK14_axiom_m1_Y), not(sK15_axiom_m1_X)), or(not(sK15_axiom_m1_X), not(sK14_axiom_m1_Y)))))), $$true, implies(or(not(sK14_axiom_m1_Y), not(sK15_axiom_m1_X)), or(not(sK15_axiom_m1_X), not(sK14_axiom_m1_Y))), equiv(or(not(sK15_axiom_m1_X), not(sK14_axiom_m1_Y)), or(not(sK14_axiom_m1_Y), not(sK15_axiom_m1_X)))), $$true, or(not(sK15_axiom_m1_X), not(sK14_axiom_m1_Y)), or(not(sK14_axiom_m1_Y), not(sK15_axiom_m1_X)))), and(sK14_axiom_m1_Y, sK15_axiom_m1_X))), $$true)
27.82/28.02	= { by lemma 274 }
27.82/28.02	  $$fresh92(is_a_theorem(strict_implies(not($$fresh3($$fresh40($$fresh115($$true, $$true, implies(implies(or(not(sK14_axiom_m1_Y), not(sK15_axiom_m1_X)), or(not(sK15_axiom_m1_X), not(sK14_axiom_m1_Y))), and(implies(or(not(sK15_axiom_m1_X), not(sK14_axiom_m1_Y)), or(not(sK14_axiom_m1_Y), not(sK15_axiom_m1_X))), implies(or(not(sK14_axiom_m1_Y), not(sK15_axiom_m1_X)), or(not(sK15_axiom_m1_X), not(sK14_axiom_m1_Y)))))), $$true, implies(or(not(sK14_axiom_m1_Y), not(sK15_axiom_m1_X)), or(not(sK15_axiom_m1_X), not(sK14_axiom_m1_Y))), equiv(or(not(sK15_axiom_m1_X), not(sK14_axiom_m1_Y)), or(not(sK14_axiom_m1_Y), not(sK15_axiom_m1_X)))), $$true, or(not(sK15_axiom_m1_X), not(sK14_axiom_m1_Y)), or(not(sK14_axiom_m1_Y), not(sK15_axiom_m1_X)))), and(sK14_axiom_m1_Y, sK15_axiom_m1_X))), $$true)
27.82/28.02	= { by axiom 74 (modus_ponens_2) }
27.82/28.02	  $$fresh92(is_a_theorem(strict_implies(not($$fresh3($$fresh40($$true, $$true, implies(or(not(sK14_axiom_m1_Y), not(sK15_axiom_m1_X)), or(not(sK15_axiom_m1_X), not(sK14_axiom_m1_Y))), equiv(or(not(sK15_axiom_m1_X), not(sK14_axiom_m1_Y)), or(not(sK14_axiom_m1_Y), not(sK15_axiom_m1_X)))), $$true, or(not(sK15_axiom_m1_X), not(sK14_axiom_m1_Y)), or(not(sK14_axiom_m1_Y), not(sK15_axiom_m1_X)))), and(sK14_axiom_m1_Y, sK15_axiom_m1_X))), $$true)
27.82/28.02	= { by axiom 73 (modus_ponens_2) }
27.82/28.02	  $$fresh92(is_a_theorem(strict_implies(not($$fresh3(is_a_theorem(equiv(or(not(sK15_axiom_m1_X), not(sK14_axiom_m1_Y)), or(not(sK14_axiom_m1_Y), not(sK15_axiom_m1_X)))), $$true, or(not(sK15_axiom_m1_X), not(sK14_axiom_m1_Y)), or(not(sK14_axiom_m1_Y), not(sK15_axiom_m1_X)))), and(sK14_axiom_m1_Y, sK15_axiom_m1_X))), $$true)
27.82/28.02	= { by lemma 260 }
27.82/28.02	  $$fresh92(is_a_theorem(strict_implies(not(or(not(sK14_axiom_m1_Y), not(sK15_axiom_m1_X))), and(sK14_axiom_m1_Y, sK15_axiom_m1_X))), $$true)
27.82/28.02	= { by lemma 270 }
27.82/28.02	  $$fresh92(is_a_theorem(strict_implies(not(or(not(not(not(sK14_axiom_m1_Y))), not(sK15_axiom_m1_X))), and(sK14_axiom_m1_Y, sK15_axiom_m1_X))), $$true)
27.82/28.02	= { by lemma 272 }
27.82/28.02	  $$fresh92(is_a_theorem(strict_implies(not(implies(not(not(sK14_axiom_m1_Y)), not(sK15_axiom_m1_X))), and(sK14_axiom_m1_Y, sK15_axiom_m1_X))), $$true)
27.82/28.02	= { by lemma 273 }
27.82/28.02	  $$fresh92(is_a_theorem(strict_implies(and(not(not(sK14_axiom_m1_Y)), not(not(sK15_axiom_m1_X))), and(sK14_axiom_m1_Y, sK15_axiom_m1_X))), $$true)
27.82/28.02	= { by lemma 270 }
27.82/28.02	  $$fresh92(is_a_theorem(strict_implies(and(sK14_axiom_m1_Y, not(not(sK15_axiom_m1_X))), and(sK14_axiom_m1_Y, sK15_axiom_m1_X))), $$true)
27.82/28.02	= { by lemma 270 }
27.82/28.02	  $$fresh92(is_a_theorem(strict_implies(and(sK14_axiom_m1_Y, sK15_axiom_m1_X), and(sK14_axiom_m1_Y, sK15_axiom_m1_X))), $$true)
27.82/28.02	= { by lemma 270 }
27.82/28.02	  $$fresh92(is_a_theorem(not(not(strict_implies(and(sK14_axiom_m1_Y, sK15_axiom_m1_X), and(sK14_axiom_m1_Y, sK15_axiom_m1_X))))), $$true)
27.82/28.02	= { by lemma 256 }
27.82/28.02	  $$fresh92(is_a_theorem(not(possibly(and(and(sK14_axiom_m1_Y, sK15_axiom_m1_X), not(and(sK14_axiom_m1_Y, sK15_axiom_m1_X)))))), $$true)
27.82/28.02	= { by lemma 275 }
27.82/28.02	  $$fresh92(is_a_theorem(not(necessarily(possibly(and(and(sK14_axiom_m1_Y, sK15_axiom_m1_X), not(and(sK14_axiom_m1_Y, sK15_axiom_m1_X))))))), $$true)
27.82/28.02	= { by lemma 273 }
27.82/28.02	  $$fresh92(is_a_theorem(not(necessarily(possibly(not(implies(and(sK14_axiom_m1_Y, sK15_axiom_m1_X), and(sK14_axiom_m1_Y, sK15_axiom_m1_X))))))), $$true)
27.82/28.02	= { by lemma 269 }
27.82/28.02	  $$fresh92(is_a_theorem(not(necessarily(possibly(necessarily(not(implies(and(sK14_axiom_m1_Y, sK15_axiom_m1_X), and(sK14_axiom_m1_Y, sK15_axiom_m1_X)))))))), $$true)
27.82/28.02	= { by lemma 252 }
27.82/28.02	  $$fresh92(is_a_theorem(not(necessarily(not(necessarily(not(necessarily(not(implies(and(sK14_axiom_m1_Y, sK15_axiom_m1_X), and(sK14_axiom_m1_Y, sK15_axiom_m1_X)))))))))), $$true)
27.82/28.02	= { by lemma 252 }
27.82/28.02	  $$fresh92(is_a_theorem(not(necessarily(not(necessarily(possibly(implies(and(sK14_axiom_m1_Y, sK15_axiom_m1_X), and(sK14_axiom_m1_Y, sK15_axiom_m1_X)))))))), $$true)
27.82/28.02	= { by lemma 275 }
27.82/28.02	  $$fresh92(is_a_theorem(not(necessarily(not(possibly(implies(and(sK14_axiom_m1_Y, sK15_axiom_m1_X), and(sK14_axiom_m1_Y, sK15_axiom_m1_X))))))), $$true)
27.82/28.02	= { by lemma 252 }
27.82/28.02	  $$fresh92(is_a_theorem(not(necessarily(not(not(necessarily(not(implies(and(sK14_axiom_m1_Y, sK15_axiom_m1_X), and(sK14_axiom_m1_Y, sK15_axiom_m1_X))))))))), $$true)
27.82/28.02	= { by lemma 270 }
27.82/28.02	  $$fresh92(is_a_theorem(not(necessarily(necessarily(not(implies(and(sK14_axiom_m1_Y, sK15_axiom_m1_X), and(sK14_axiom_m1_Y, sK15_axiom_m1_X))))))), $$true)
27.82/28.02	= { by lemma 269 }
27.82/28.02	  $$fresh92(is_a_theorem(not(necessarily(not(implies(and(sK14_axiom_m1_Y, sK15_axiom_m1_X), and(sK14_axiom_m1_Y, sK15_axiom_m1_X)))))), $$true)
27.82/28.02	= { by lemma 269 }
27.82/28.02	  $$fresh92(is_a_theorem(not(not(implies(and(sK14_axiom_m1_Y, sK15_axiom_m1_X), and(sK14_axiom_m1_Y, sK15_axiom_m1_X))))), $$true)
27.82/28.02	= { by lemma 270 }
27.82/28.02	  $$fresh92(is_a_theorem(implies(and(sK14_axiom_m1_Y, sK15_axiom_m1_X), and(sK14_axiom_m1_Y, sK15_axiom_m1_X))), $$true)
27.82/28.02	= { by lemma 265 }
27.82/28.02	  $$fresh92($$true, $$true)
27.82/28.02	= { by axiom 21 (axiom_m1) }
27.82/28.02	  $$true
27.82/28.02	% SZS output end Proof
27.82/28.02	
27.82/28.02	RESULT: Theorem (the conjecture is true).
27.82/28.04	EOF