0.00/0.03	% 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
0.02/0.23	% Computer   : n164.star.cs.uiowa.edu
0.02/0.23	% Model      : x86_64 x86_64
0.02/0.23	% CPU        : Intel(R) Xeon(R) CPU E5-2609 0 @ 2.40GHz
0.02/0.23	% Memory     : 32218.625MB
0.02/0.23	% OS         : Linux 3.10.0-693.2.2.el7.x86_64
0.02/0.23	% CPULimit   : 300
0.02/0.23	% DateTime   : Sat Jul 14 06:08:54 CDT 2018
0.02/0.23	% CPUTime    : 
92.42/92.65	% SZS status Theorem
92.42/92.65	
92.42/92.67	% SZS output start Proof
92.42/92.67	Take the following subset of the input axioms:
105.53/105.74	  fof(and_1, axiom,
105.53/105.74	      and_1 <=> ![X, Y]: is_a_theorem(implies(and(X, Y), X))).
105.53/105.74	  fof(and_2, axiom,
105.53/105.74	      ![X, Y]: is_a_theorem(implies(and(X, Y), Y)) <=> and_2).
105.53/105.74	  fof(and_3, axiom,
105.53/105.74	      ![X, Y]: is_a_theorem(implies(X, implies(Y, and(X, Y))))
105.53/105.74	        <=> and_3).
105.53/105.74	  fof(cn2, axiom,
105.53/105.74	      cn2 <=> ![P, Q]: is_a_theorem(implies(P, implies(not(P), Q)))).
105.53/105.74	  fof(equivalence_3, axiom,
105.53/105.74	      ![X, Y]:
105.53/105.74	        is_a_theorem(implies(implies(X, Y),
105.53/105.74	                             implies(implies(Y, X), equiv(X, Y))))
105.53/105.74	        <=> equivalence_3).
105.53/105.74	  fof(hilbert_op_implies_and, axiom, op_implies_and).
105.53/105.74	  fof(hilbert_op_or, axiom, op_or).
105.53/105.74	  fof(hilbert_or_3, conjecture, or_3).
105.53/105.74	  fof(kn1, axiom, ![P]: is_a_theorem(implies(P, and(P, P))) <=> kn1).
105.53/105.74	  fof(modus_ponens, axiom,
105.53/105.74	      modus_ponens
105.53/105.74	        <=> ![X, Y]:
105.53/105.74	              ((is_a_theorem(X) & is_a_theorem(implies(X, Y)))
105.53/105.74	                 => is_a_theorem(Y))).
105.53/105.74	  fof(op_and, axiom,
105.53/105.74	      ![X, Y]: and(X, Y)=not(or(not(X), not(Y))) <= op_and).
105.53/105.74	  fof(op_equiv, axiom,
105.53/105.74	      op_equiv
105.53/105.74	        => ![X, Y]: equiv(X, Y)=and(implies(X, Y), implies(Y, X))).
105.53/105.74	  fof(op_implies_and, axiom,
105.53/105.74	      ![X, Y]: not(and(X, not(Y)))=implies(X, Y) <= op_implies_and).
105.53/105.74	  fof(op_implies_or, axiom,
105.53/105.74	      ![X, Y]: or(not(X), Y)=implies(X, Y) <= op_implies_or).
105.53/105.74	  fof(op_or, axiom,
105.53/105.74	      ![X, Y]: or(X, Y)=not(and(not(X), not(Y))) <= op_or).
105.53/105.74	  fof(or_1, axiom,
105.53/105.74	      or_1 <=> ![X, Y]: is_a_theorem(implies(X, or(X, Y)))).
105.53/105.74	  fof(or_2, axiom,
105.53/105.74	      ![X, Y]: is_a_theorem(implies(Y, or(X, Y))) <=> or_2).
105.53/105.74	  fof(or_3, axiom,
105.53/105.74	      or_3
105.53/105.74	        <=> ![X, Y, Z]:
105.53/105.74	              is_a_theorem(implies(implies(X, Z),
105.53/105.74	                                   implies(implies(Y, Z), implies(or(X, Y), Z))))).
105.53/105.74	  fof(principia_modus_ponens, axiom, modus_ponens).
105.53/105.74	  fof(principia_op_and, axiom, op_and).
105.53/105.74	  fof(principia_op_equiv, axiom, op_equiv).
105.53/105.74	  fof(principia_op_implies_or, axiom, op_implies_or).
105.53/105.74	  fof(principia_r1, axiom, r1).
105.53/105.74	  fof(principia_r2, axiom, r2).
105.53/105.74	  fof(principia_r3, axiom, r3).
105.53/105.74	  fof(principia_r4, axiom, r4).
105.53/105.74	  fof(principia_r5, axiom, r5).
105.53/105.74	  fof(r1, axiom, ![P]: is_a_theorem(implies(or(P, P), P)) <=> r1).
105.53/105.74	  fof(r2, axiom, r2 <=> ![P, Q]: is_a_theorem(implies(Q, or(P, Q)))).
105.53/105.74	  fof(r3, axiom,
105.53/105.74	      r3 <=> ![P, Q]: is_a_theorem(implies(or(P, Q), or(Q, P)))).
105.53/105.74	  fof(r4, axiom,
105.53/105.74	      r4
105.53/105.74	        <=> ![P, Q, R]:
105.53/105.74	              is_a_theorem(implies(or(P, or(Q, R)), or(Q, or(P, R))))).
105.53/105.74	  fof(r5, axiom,
105.53/105.74	      r5
105.53/105.74	        <=> ![P, Q, R]:
105.53/105.74	              is_a_theorem(implies(implies(Q, R), implies(or(P, Q), or(P, R))))).
105.53/105.74	  fof(substitution_of_equivalents, axiom,
105.53/105.74	      substitution_of_equivalents
105.53/105.74	        <=> ![X, Y]: (is_a_theorem(equiv(X, Y)) => Y=X)).
105.53/105.74	  fof(substitution_of_equivalents, axiom,
105.53/105.74	      substitution_of_equivalents).
105.53/105.74	
105.53/105.74	Now clausify the problem and encode Horn clauses using encoding 3 of
105.53/105.74	http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
105.53/105.74	We repeatedly replace C & s=t => u=v by the two clauses:
105.53/105.74	  $$fresh(y, y, x1...xn) = u
105.53/105.74	  C => $$fresh(s, t, x1...xn) = v
105.53/105.74	where $$fresh is a fresh function symbol and x1..xn are the free
105.53/105.74	variables of u and v.
105.53/105.74	A predicate p(X) is encoded as p(X)=$$true (this is sound, because the
105.53/105.74	input problem has no model of domain size 1).
105.53/105.74	
105.53/105.74	The encoding turns the above axioms into the following unit equations and goals:
105.53/105.74	
105.53/105.74	Axiom 31 (modus_ponens_2): $$fresh28(X, X, Y, Z) = is_a_theorem(Z).
105.53/105.74	Axiom 32 (modus_ponens_2): $$fresh59(X, X, Y) = $$true.
105.53/105.74	Axiom 33 (modus_ponens_2): $$fresh60(X, X, Y, Z) = $$fresh59(is_a_theorem(Y), $$true, Z).
105.53/105.74	Axiom 37 (op_and): $$fresh24(X, X, Y, Z) = and(Y, Z).
105.53/105.74	Axiom 38 (op_equiv): $$fresh23(X, X, Y, Z) = equiv(Y, Z).
105.53/105.74	Axiom 39 (op_implies_and): $$fresh22(X, X, Y, Z) = implies(Y, Z).
105.53/105.74	Axiom 40 (op_implies_or): $$fresh21(X, X, Y, Z) = implies(Y, Z).
105.53/105.74	Axiom 41 (op_or): $$fresh20(X, X, Y, Z) = or(Y, Z).
105.53/105.74	Axiom 46 (or_3): $$fresh15(X, X) = $$true.
105.53/105.74	Axiom 49 (r1_1): $$fresh12(X, X, Y) = $$true.
105.53/105.74	Axiom 51 (r2_1): $$fresh10(X, X, Y, Z) = $$true.
105.53/105.74	Axiom 53 (r3_1): $$fresh8(X, X, Y, Z) = $$true.
105.53/105.74	Axiom 55 (r4_1): $$fresh6(X, X, Y, Z, W) = $$true.
105.53/105.74	Axiom 57 (r5_1): $$fresh4(X, X, Y, Z, W) = $$true.
105.53/105.74	Axiom 59 (substitution_of_equivalents_2): $$fresh(X, X, Y, Z) = Y.
105.53/105.74	Axiom 60 (substitution_of_equivalents_2): $$fresh2(X, X, Y, Z) = Z.
105.53/105.74	Axiom 61 (and_2_1): $$fresh55(and_2, $$true, X, Y) = is_a_theorem(implies(and(X, Y), Y)).
105.53/105.74	Axiom 65 (substitution_of_equivalents_2): $$fresh2(substitution_of_equivalents, $$true, X, Y) = $$fresh(is_a_theorem(equiv(X, Y)), $$true, X, Y).
105.53/105.74	Axiom 69 (r5_1): $$fresh4(r5, $$true, X, Y, Z) = is_a_theorem(implies(implies(Y, Z), implies(or(X, Y), or(X, Z)))).
105.53/105.74	Axiom 71 (kn1_1): $$fresh33(kn1, $$true, X) = is_a_theorem(implies(X, and(X, X))).
105.53/105.74	Axiom 75 (r4_1): $$fresh6(r4, $$true, X, Y, Z) = is_a_theorem(implies(or(X, or(Y, Z)), or(Y, or(X, Z)))).
105.53/105.74	Axiom 77 (cn2_1): $$fresh49(cn2, $$true, X, Y) = is_a_theorem(implies(X, implies(not(X), Y))).
105.53/105.74	Axiom 79 (or_1_1): $$fresh18(or_1, $$true, X, Y) = is_a_theorem(implies(X, or(X, Y))).
105.53/105.74	Axiom 81 (or_2_1): $$fresh16(or_2, $$true, X, Y) = is_a_theorem(implies(Y, or(X, Y))).
105.53/105.74	Axiom 84 (modus_ponens_2): $$fresh60(modus_ponens, $$true, X, Y) = $$fresh28(is_a_theorem(implies(X, Y)), $$true, X, Y).
105.53/105.74	Axiom 87 (r1_1): $$fresh12(r1, $$true, X) = is_a_theorem(implies(or(X, X), X)).
105.53/105.74	Axiom 89 (and_3_1): $$fresh53(and_3, $$true, X, Y) = is_a_theorem(implies(X, implies(Y, and(X, Y)))).
105.53/105.74	Axiom 91 (r2_1): $$fresh10(r2, $$true, X, Y) = is_a_theorem(implies(Y, or(X, Y))).
105.53/105.74	Axiom 93 (and_1_1): $$fresh57(and_1, $$true, X, Y) = is_a_theorem(implies(and(X, Y), X)).
105.53/105.74	Axiom 97 (r3_1): $$fresh8(r3, $$true, X, Y) = is_a_theorem(implies(or(X, Y), or(Y, X))).
105.53/105.74	Axiom 101 (or_3_1): $$fresh14(or_3, $$true, X, Y, Z) = is_a_theorem(implies(implies(X, Z), implies(implies(Y, Z), implies(or(X, Y), Z)))).
105.53/105.74	Axiom 102 (or_3): $$fresh15(is_a_theorem(implies(implies(sK13_or_3_X, sK11_or_3_Z), implies(implies(sK12_or_3_Y, sK11_or_3_Z), implies(or(sK13_or_3_X, sK12_or_3_Y), sK11_or_3_Z)))), $$true) = or_3.
105.53/105.74	Axiom 109 (equivalence_3_1): $$fresh41(equivalence_3, $$true, X, Y) = is_a_theorem(implies(implies(X, Y), implies(implies(Y, X), equiv(X, Y)))).
105.53/105.74	Axiom 113 (op_or): $$fresh20(op_or, $$true, X, Y) = not(and(not(X), not(Y))).
105.53/105.74	Axiom 114 (op_and): $$fresh24(op_and, $$true, X, Y) = not(or(not(X), not(Y))).
105.53/105.74	Axiom 115 (op_implies_and): $$fresh22(op_implies_and, $$true, X, Y) = not(and(X, not(Y))).
105.53/105.74	Axiom 116 (op_implies_or): $$fresh21(op_implies_or, $$true, X, Y) = or(not(X), Y).
105.53/105.74	Axiom 117 (op_equiv): $$fresh23(op_equiv, $$true, X, Y) = and(implies(X, Y), implies(Y, X)).
105.53/105.74	Axiom 118 (principia_r4): r4 = $$true.
105.53/105.74	Axiom 119 (principia_op_equiv): op_equiv = $$true.
105.53/105.74	Axiom 120 (principia_r2): r2 = $$true.
105.53/105.74	Axiom 121 (principia_r1): r1 = $$true.
105.53/105.74	Axiom 122 (principia_r3): r3 = $$true.
105.53/105.74	Axiom 123 (principia_modus_ponens): modus_ponens = $$true.
105.53/105.74	Axiom 124 (principia_r5): r5 = $$true.
105.53/105.74	Axiom 125 (principia_op_implies_or): op_implies_or = $$true.
105.53/105.74	Axiom 126 (principia_op_and): op_and = $$true.
105.53/105.74	Axiom 127 (substitution_of_equivalents): substitution_of_equivalents = $$true.
105.53/105.74	Axiom 129 (hilbert_op_implies_and): op_implies_and = $$true.
105.53/105.74	Axiom 130 (hilbert_op_or): op_or = $$true.
105.53/105.74	
105.53/105.74	Lemma 131: $$fresh60(X, X, Y, Z) = $$fresh60(?, ?, Y, Z).
105.53/105.74	Proof:
105.53/105.74	  $$fresh60(X, X, Y, Z)
105.53/105.74	= { by axiom 33 (modus_ponens_2) }
105.53/105.74	  $$fresh59(is_a_theorem(Y), $$true, Z)
105.53/105.74	= { by axiom 33 (modus_ponens_2) }
105.53/105.74	  $$fresh60(?, ?, Y, Z)
105.53/105.74	
105.53/105.74	Lemma 132: or(not(X), Y) = implies(X, Y).
105.53/105.74	Proof:
105.53/105.74	  or(not(X), Y)
105.53/105.74	= { by axiom 116 (op_implies_or) }
105.53/105.74	  $$fresh21(op_implies_or, $$true, X, Y)
105.53/105.74	= { by axiom 125 (principia_op_implies_or) }
105.53/105.74	  $$fresh21($$true, $$true, X, Y)
105.53/105.74	= { by axiom 40 (op_implies_or) }
105.53/105.74	  implies(X, Y)
105.53/105.74	
105.53/105.74	Lemma 133: not(and(X, not(Y))) = implies(X, Y).
105.53/105.74	Proof:
105.53/105.74	  not(and(X, not(Y)))
105.53/105.74	= { by axiom 115 (op_implies_and) }
105.53/105.74	  $$fresh22(op_implies_and, $$true, X, Y)
105.53/105.74	= { by axiom 129 (hilbert_op_implies_and) }
105.53/105.74	  $$fresh22($$true, $$true, X, Y)
105.53/105.74	= { by axiom 39 (op_implies_and) }
105.53/105.74	  implies(X, Y)
105.53/105.74	
105.53/105.74	Lemma 134: implies(not(X), Y) = or(X, Y).
105.53/105.74	Proof:
105.53/105.74	  implies(not(X), Y)
105.53/105.74	= { by lemma 133 }
105.53/105.74	  not(and(not(X), not(Y)))
105.53/105.74	= { by axiom 113 (op_or) }
105.53/105.74	  $$fresh20(op_or, $$true, X, Y)
105.53/105.74	= { by axiom 130 (hilbert_op_or) }
105.53/105.74	  $$fresh20($$true, $$true, X, Y)
105.53/105.74	= { by axiom 41 (op_or) }
105.53/105.74	  or(X, Y)
105.53/105.74	
105.53/105.74	Lemma 135: not(implies(X, not(Y))) = and(X, Y).
105.53/105.74	Proof:
105.53/105.74	  not(implies(X, not(Y)))
105.53/105.74	= { by lemma 132 }
105.53/105.74	  not(or(not(X), not(Y)))
105.53/105.74	= { by axiom 114 (op_and) }
105.53/105.74	  $$fresh24(op_and, $$true, X, Y)
105.53/105.74	= { by axiom 126 (principia_op_and) }
105.53/105.74	  $$fresh24($$true, $$true, X, Y)
105.53/105.74	= { by axiom 37 (op_and) }
105.53/105.74	  and(X, Y)
105.53/105.74	
105.53/105.74	Lemma 136: not(or(X, not(Y))) = and(not(X), Y).
105.53/105.74	Proof:
105.53/105.74	  not(or(X, not(Y)))
105.53/105.74	= { by lemma 134 }
105.53/105.74	  not(implies(not(X), not(Y)))
105.53/105.74	= { by lemma 135 }
105.53/105.74	  and(not(X), Y)
105.53/105.74	
105.53/105.74	Lemma 137: $$fresh18(or_1, $$true, X, Y) = $$fresh49(cn2, $$true, X, Y).
105.53/105.74	Proof:
105.53/105.74	  $$fresh18(or_1, $$true, X, Y)
105.53/105.74	= { by axiom 79 (or_1_1) }
105.53/105.74	  is_a_theorem(implies(X, or(X, Y)))
105.53/105.74	= { by lemma 134 }
105.53/105.74	  is_a_theorem(implies(X, implies(not(X), Y)))
105.53/105.74	= { by axiom 77 (cn2_1) }
105.53/105.74	  $$fresh49(cn2, $$true, X, Y)
105.53/105.74	
105.53/105.74	Lemma 138: implies(implies(X, not(Y)), Z) = or(and(X, Y), Z).
105.53/105.74	Proof:
105.53/105.74	  implies(implies(X, not(Y)), Z)
105.53/105.74	= { by lemma 132 }
105.53/105.74	  implies(or(not(X), not(Y)), Z)
105.53/105.74	= { by lemma 132 }
105.53/105.74	  or(not(or(not(X), not(Y))), Z)
105.53/105.74	= { by axiom 114 (op_and) }
105.53/105.74	  or($$fresh24(op_and, $$true, X, Y), Z)
105.53/105.74	= { by axiom 126 (principia_op_and) }
105.53/105.74	  or($$fresh24($$true, $$true, X, Y), Z)
105.53/105.74	= { by axiom 37 (op_and) }
105.53/105.74	  or(and(X, Y), Z)
105.53/105.74	
105.53/105.74	Lemma 139: $$fresh60(?, ?, implies(or(X, X), X), Y) = $$true.
105.53/105.74	Proof:
105.53/105.74	  $$fresh60(?, ?, implies(or(X, X), X), Y)
105.53/105.74	= { by axiom 33 (modus_ponens_2) }
105.53/105.74	  $$fresh59(is_a_theorem(implies(or(X, X), X)), $$true, Y)
105.53/105.74	= { by axiom 87 (r1_1) }
105.53/105.74	  $$fresh59($$fresh12(r1, $$true, X), $$true, Y)
105.53/105.74	= { by axiom 121 (principia_r1) }
105.53/105.74	  $$fresh59($$fresh12($$true, $$true, X), $$true, Y)
105.53/105.74	= { by axiom 49 (r1_1) }
105.53/105.74	  $$fresh59($$true, $$true, Y)
105.53/105.74	= { by axiom 32 (modus_ponens_2) }
105.53/105.74	  $$true
105.53/105.74	
105.53/105.74	Lemma 140: $$fresh60(?, ?, implies(X, or(Y, X)), Z) = $$true.
105.53/105.74	Proof:
105.53/105.74	  $$fresh60(?, ?, implies(X, or(Y, X)), Z)
105.53/105.74	= { by axiom 33 (modus_ponens_2) }
105.53/105.74	  $$fresh59(is_a_theorem(implies(X, or(Y, X))), $$true, Z)
105.53/105.74	= { by axiom 91 (r2_1) }
105.53/105.74	  $$fresh59($$fresh10(r2, $$true, Y, X), $$true, Z)
105.53/105.74	= { by axiom 120 (principia_r2) }
105.53/105.74	  $$fresh59($$fresh10($$true, $$true, Y, X), $$true, Z)
105.53/105.74	= { by axiom 51 (r2_1) }
105.53/105.74	  $$fresh59($$true, $$true, Z)
105.53/105.74	= { by axiom 32 (modus_ponens_2) }
105.53/105.74	  $$true
105.53/105.74	
105.53/105.74	Lemma 141: $$fresh60(?, ?, or(X, not(Y)), implies(Y, X)) = is_a_theorem(implies(Y, X)).
105.53/105.74	Proof:
105.53/105.74	  $$fresh60(?, ?, or(X, not(Y)), implies(Y, X))
105.53/105.74	= { by lemma 131 }
105.53/105.74	  $$fresh60($$true, $$true, or(X, not(Y)), implies(Y, X))
105.53/105.74	= { by axiom 123 (principia_modus_ponens) }
105.53/105.74	  $$fresh60(modus_ponens, $$true, or(X, not(Y)), implies(Y, X))
105.53/105.74	= { by axiom 84 (modus_ponens_2) }
105.53/105.74	  $$fresh28(is_a_theorem(implies(or(X, not(Y)), implies(Y, X))), $$true, or(X, not(Y)), implies(Y, X))
105.53/105.74	= { by lemma 132 }
105.53/105.74	  $$fresh28(is_a_theorem(implies(or(X, not(Y)), or(not(Y), X))), $$true, or(X, not(Y)), implies(Y, X))
105.53/105.74	= { by axiom 97 (r3_1) }
105.53/105.74	  $$fresh28($$fresh8(r3, $$true, X, not(Y)), $$true, or(X, not(Y)), implies(Y, X))
105.53/105.74	= { by axiom 122 (principia_r3) }
105.53/105.74	  $$fresh28($$fresh8($$true, $$true, X, not(Y)), $$true, or(X, not(Y)), implies(Y, X))
105.53/105.74	= { by axiom 53 (r3_1) }
105.53/105.74	  $$fresh28($$true, $$true, or(X, not(Y)), implies(Y, X))
105.53/105.74	= { by axiom 31 (modus_ponens_2) }
105.53/105.74	  is_a_theorem(implies(Y, X))
105.53/105.74	
105.53/105.74	Lemma 142: $$fresh60(?, ?, implies(or(X, Y), or(Y, X)), Z) = $$true.
105.53/105.74	Proof:
105.53/105.74	  $$fresh60(?, ?, implies(or(X, Y), or(Y, X)), Z)
105.53/105.74	= { by axiom 33 (modus_ponens_2) }
105.53/105.74	  $$fresh59(is_a_theorem(implies(or(X, Y), or(Y, X))), $$true, Z)
105.53/105.74	= { by axiom 97 (r3_1) }
105.53/105.74	  $$fresh59($$fresh8(r3, $$true, X, Y), $$true, Z)
105.53/105.74	= { by axiom 122 (principia_r3) }
105.53/105.74	  $$fresh59($$fresh8($$true, $$true, X, Y), $$true, Z)
105.53/105.74	= { by axiom 53 (r3_1) }
105.53/105.74	  $$fresh59($$true, $$true, Z)
105.53/105.74	= { by axiom 32 (modus_ponens_2) }
105.53/105.74	  $$true
105.53/105.74	
105.53/105.74	Lemma 143: $$fresh(is_a_theorem(equiv(X, Y)), $$true, X, Y) = Y.
105.53/105.74	Proof:
105.53/105.74	  $$fresh(is_a_theorem(equiv(X, Y)), $$true, X, Y)
105.53/105.74	= { by axiom 65 (substitution_of_equivalents_2) }
105.53/105.74	  $$fresh2(substitution_of_equivalents, $$true, X, Y)
105.53/105.74	= { by axiom 127 (substitution_of_equivalents) }
105.53/105.74	  $$fresh2($$true, $$true, X, Y)
105.53/105.74	= { by axiom 60 (substitution_of_equivalents_2) }
105.53/105.74	  Y
105.53/105.74	
105.53/105.74	Lemma 144: is_a_theorem(implies(implies(Y, Z), implies(or(X, Y), or(X, Z)))) = $$true.
105.53/105.74	Proof:
105.53/105.74	  is_a_theorem(implies(implies(Y, Z), implies(or(X, Y), or(X, Z))))
105.53/105.74	= { by axiom 69 (r5_1) }
105.53/105.74	  $$fresh4(r5, $$true, X, Y, Z)
105.53/105.74	= { by axiom 124 (principia_r5) }
105.53/105.74	  $$fresh4($$true, $$true, X, Y, Z)
105.53/105.74	= { by axiom 57 (r5_1) }
105.53/105.74	  $$true
105.53/105.74	
105.53/105.74	Lemma 145: is_a_theorem(implies(or(X, or(Y, Z)), or(Y, or(X, Z)))) = $$true.
105.53/105.74	Proof:
105.53/105.74	  is_a_theorem(implies(or(X, or(Y, Z)), or(Y, or(X, Z))))
105.53/105.74	= { by axiom 75 (r4_1) }
105.53/105.74	  $$fresh6(r4, $$true, X, Y, Z)
105.53/105.74	= { by axiom 118 (principia_r4) }
105.53/105.74	  $$fresh6($$true, $$true, X, Y, Z)
105.53/105.74	= { by axiom 55 (r4_1) }
105.53/105.74	  $$true
105.53/105.74	
105.53/105.74	Lemma 146: $$fresh28(is_a_theorem(implies(X, Y)), $$true, X, Y) = $$fresh60(?, ?, X, Y).
105.53/105.74	Proof:
105.53/105.74	  $$fresh28(is_a_theorem(implies(X, Y)), $$true, X, Y)
105.53/105.74	= { by axiom 84 (modus_ponens_2) }
105.53/105.74	  $$fresh60(modus_ponens, $$true, X, Y)
105.53/105.74	= { by axiom 123 (principia_modus_ponens) }
105.53/105.74	  $$fresh60($$true, $$true, X, Y)
105.53/105.74	= { by lemma 131 }
105.53/105.74	  $$fresh60(?, ?, X, Y)
105.53/105.74	
105.53/105.74	Lemma 147: and(implies(X, Y), implies(Y, X)) = equiv(X, Y).
105.53/105.74	Proof:
105.53/105.74	  and(implies(X, Y), implies(Y, X))
105.53/105.74	= { by axiom 117 (op_equiv) }
105.53/105.74	  $$fresh23(op_equiv, $$true, X, Y)
105.53/105.74	= { by axiom 119 (principia_op_equiv) }
105.53/105.74	  $$fresh23($$true, $$true, X, Y)
105.53/105.74	= { by axiom 38 (op_equiv) }
105.53/105.74	  equiv(X, Y)
105.53/105.74	
105.53/105.75	Lemma 148: $$fresh60(?, ?, or(X, or(Y, Z)), or(Y, or(X, Z))) = is_a_theorem(or(Y, or(X, Z))).
105.53/105.75	Proof:
105.53/105.75	  $$fresh60(?, ?, or(X, or(Y, Z)), or(Y, or(X, Z)))
105.53/105.75	= { by lemma 146 }
105.53/105.75	  $$fresh28(is_a_theorem(implies(or(X, or(Y, Z)), or(Y, or(X, Z)))), $$true, or(X, or(Y, Z)), or(Y, or(X, Z)))
105.53/105.75	= { by lemma 145 }
105.53/105.75	  $$fresh28($$true, $$true, or(X, or(Y, Z)), or(Y, or(X, Z)))
105.53/105.75	= { by axiom 31 (modus_ponens_2) }
105.53/105.75	  is_a_theorem(or(Y, or(X, Z)))
105.53/105.75	
105.53/105.75	Lemma 149: $$fresh60(?, ?, implies(or(X, or(Y, Z)), or(Y, or(X, Z))), W) = $$true.
105.53/105.75	Proof:
105.53/105.75	  $$fresh60(?, ?, implies(or(X, or(Y, Z)), or(Y, or(X, Z))), W)
105.53/105.75	= { by axiom 33 (modus_ponens_2) }
105.53/105.75	  $$fresh59(is_a_theorem(implies(or(X, or(Y, Z)), or(Y, or(X, Z)))), $$true, W)
105.53/105.75	= { by lemma 145 }
105.53/105.75	  $$fresh59($$true, $$true, W)
105.53/105.75	= { by axiom 32 (modus_ponens_2) }
105.53/105.75	  $$true
105.53/105.75	
105.53/105.75	Lemma 150: $$fresh60(?, ?, implies(X, Y), implies(or(Z, X), or(Z, Y))) = is_a_theorem(implies(or(Z, X), or(Z, Y))).
105.53/105.75	Proof:
105.53/105.75	  $$fresh60(?, ?, implies(X, Y), implies(or(Z, X), or(Z, Y)))
105.53/105.75	= { by lemma 146 }
105.53/105.75	  $$fresh28(is_a_theorem(implies(implies(X, Y), implies(or(Z, X), or(Z, Y)))), $$true, implies(X, Y), implies(or(Z, X), or(Z, Y)))
105.53/105.75	= { by lemma 144 }
105.53/105.75	  $$fresh28($$true, $$true, implies(X, Y), implies(or(Z, X), or(Z, Y)))
105.53/105.75	= { by axiom 31 (modus_ponens_2) }
105.53/105.75	  is_a_theorem(implies(or(Z, X), or(Z, Y)))
105.53/105.75	
105.53/105.75	Lemma 151: $$fresh60(?, ?, implies(Y, or(X, Z)), or(X, implies(Y, Z))) = is_a_theorem(or(X, implies(Y, Z))).
105.53/105.75	Proof:
105.53/105.75	  $$fresh60(?, ?, implies(Y, or(X, Z)), or(X, implies(Y, Z)))
105.53/105.75	= { by lemma 132 }
105.53/105.75	  $$fresh60(?, ?, or(not(Y), or(X, Z)), or(X, implies(Y, Z)))
105.53/105.75	= { by lemma 132 }
105.53/105.75	  $$fresh60(?, ?, or(not(Y), or(X, Z)), or(X, or(not(Y), Z)))
105.53/105.75	= { by lemma 148 }
105.53/105.75	  is_a_theorem(or(X, or(not(Y), Z)))
105.53/105.75	= { by lemma 132 }
105.53/105.75	  is_a_theorem(or(X, implies(Y, Z)))
105.53/105.75	
105.53/105.75	Lemma 152: is_a_theorem(implies(X, implies(Y, and(Y, X)))) = $$true.
105.53/105.75	Proof:
105.53/105.75	  is_a_theorem(implies(X, implies(Y, and(Y, X))))
105.53/105.75	= { by lemma 132 }
105.53/105.75	  is_a_theorem(implies(X, or(not(Y), and(Y, X))))
105.53/105.75	= { by axiom 31 (modus_ponens_2) }
105.53/105.75	  $$fresh28($$true, $$true, implies(X, or(and(Y, X), not(Y))), implies(X, or(not(Y), and(Y, X))))
105.53/105.75	= { by lemma 142 }
105.53/105.75	  $$fresh28($$fresh60(?, ?, implies(or(and(Y, X), not(Y)), or(not(Y), and(Y, X))), implies(or(not(X), or(and(Y, X), not(Y))), or(not(X), or(not(Y), and(Y, X))))), $$true, implies(X, or(and(Y, X), not(Y))), implies(X, or(not(Y), and(Y, X))))
105.53/105.75	= { by lemma 150 }
105.53/105.75	  $$fresh28(is_a_theorem(implies(or(not(X), or(and(Y, X), not(Y))), or(not(X), or(not(Y), and(Y, X))))), $$true, implies(X, or(and(Y, X), not(Y))), implies(X, or(not(Y), and(Y, X))))
105.53/105.75	= { by lemma 132 }
105.53/105.75	  $$fresh28(is_a_theorem(implies(implies(X, or(and(Y, X), not(Y))), or(not(X), or(not(Y), and(Y, X))))), $$true, implies(X, or(and(Y, X), not(Y))), implies(X, or(not(Y), and(Y, X))))
105.53/105.75	= { by lemma 132 }
105.53/105.75	  $$fresh28(is_a_theorem(implies(implies(X, or(and(Y, X), not(Y))), implies(X, or(not(Y), and(Y, X))))), $$true, implies(X, or(and(Y, X), not(Y))), implies(X, or(not(Y), and(Y, X))))
105.53/105.75	= { by lemma 146 }
105.53/105.75	  $$fresh60(?, ?, implies(X, or(and(Y, X), not(Y))), implies(X, or(not(Y), and(Y, X))))
105.53/105.75	= { by axiom 33 (modus_ponens_2) }
105.53/105.75	  $$fresh59(is_a_theorem(implies(X, or(and(Y, X), not(Y)))), $$true, implies(X, or(not(Y), and(Y, X))))
105.53/105.75	= { by lemma 138 }
105.53/105.75	  $$fresh59(is_a_theorem(implies(X, implies(implies(Y, not(X)), not(Y)))), $$true, implies(X, or(not(Y), and(Y, X))))
105.53/105.75	= { by lemma 132 }
105.53/105.75	  $$fresh59(is_a_theorem(or(not(X), implies(implies(Y, not(X)), not(Y)))), $$true, implies(X, or(not(Y), and(Y, X))))
105.53/105.75	= { by lemma 151 }
105.53/105.75	  $$fresh59($$fresh60(?, ?, implies(implies(Y, not(X)), or(not(X), not(Y))), or(not(X), implies(implies(Y, not(X)), not(Y)))), $$true, implies(X, or(not(Y), and(Y, X))))
105.53/105.75	= { by lemma 132 }
105.53/105.75	  $$fresh59($$fresh60(?, ?, implies(or(not(Y), not(X)), or(not(X), not(Y))), or(not(X), implies(implies(Y, not(X)), not(Y)))), $$true, implies(X, or(not(Y), and(Y, X))))
105.53/105.75	= { by lemma 142 }
105.53/105.75	  $$fresh59($$true, $$true, implies(X, or(not(Y), and(Y, X))))
105.53/105.75	= { by axiom 32 (modus_ponens_2) }
105.53/105.75	  $$true
105.53/105.75	
105.53/105.75	Lemma 153: $$fresh53(and_3, $$true, X, Y) = $$true.
105.53/105.75	Proof:
105.53/105.75	  $$fresh53(and_3, $$true, X, Y)
105.53/105.75	= { by axiom 89 (and_3_1) }
105.53/105.75	  is_a_theorem(implies(X, implies(Y, and(X, Y))))
105.53/105.75	= { by lemma 132 }
105.53/105.75	  is_a_theorem(or(not(X), implies(Y, and(X, Y))))
105.53/105.75	= { by lemma 151 }
105.53/105.75	  $$fresh60(?, ?, implies(Y, or(not(X), and(X, Y))), or(not(X), implies(Y, and(X, Y))))
105.53/105.75	= { by lemma 132 }
105.53/105.75	  $$fresh60(?, ?, implies(Y, implies(X, and(X, Y))), or(not(X), implies(Y, and(X, Y))))
105.53/105.75	= { by lemma 132 }
105.53/105.75	  $$fresh60(?, ?, implies(Y, implies(X, and(X, Y))), implies(X, implies(Y, and(X, Y))))
105.53/105.75	= { by axiom 33 (modus_ponens_2) }
105.53/105.75	  $$fresh59(is_a_theorem(implies(Y, implies(X, and(X, Y)))), $$true, implies(X, implies(Y, and(X, Y))))
105.53/105.75	= { by lemma 152 }
105.53/105.75	  $$fresh59($$true, $$true, implies(X, implies(Y, and(X, Y))))
105.53/105.75	= { by axiom 32 (modus_ponens_2) }
105.53/105.75	  $$true
105.53/105.75	
105.53/105.75	Lemma 154: or(X, Y) = or(Y, X).
105.53/105.75	Proof:
105.53/105.75	  or(X, Y)
105.53/105.75	= { by lemma 143 }
105.53/105.75	  $$fresh(is_a_theorem(equiv(or(Y, X), or(X, Y))), $$true, or(Y, X), or(X, Y))
105.53/105.75	= { by axiom 31 (modus_ponens_2) }
105.53/105.75	  $$fresh($$fresh28($$true, $$true, implies(or(X, Y), or(Y, X)), equiv(or(Y, X), or(X, Y))), $$true, or(Y, X), or(X, Y))
105.53/105.75	= { by lemma 142 }
105.53/105.75	  $$fresh($$fresh28($$fresh60(?, ?, implies(or(Y, X), or(X, Y)), implies(implies(or(X, Y), or(Y, X)), and(implies(or(Y, X), or(X, Y)), implies(or(X, Y), or(Y, X))))), $$true, implies(or(X, Y), or(Y, X)), equiv(or(Y, X), or(X, Y))), $$true, or(Y, X), or(X, Y))
105.53/105.75	= { by lemma 146 }
105.53/105.75	  $$fresh($$fresh28($$fresh28(is_a_theorem(implies(implies(or(Y, X), or(X, Y)), implies(implies(or(X, Y), or(Y, X)), and(implies(or(Y, X), or(X, Y)), implies(or(X, Y), or(Y, X)))))), $$true, implies(or(Y, X), or(X, Y)), implies(implies(or(X, Y), or(Y, X)), and(implies(or(Y, X), or(X, Y)), implies(or(X, Y), or(Y, X))))), $$true, implies(or(X, Y), or(Y, X)), equiv(or(Y, X), or(X, Y))), $$true, or(Y, X), or(X, Y))
105.53/105.75	= { by axiom 89 (and_3_1) }
105.53/105.75	  $$fresh($$fresh28($$fresh28($$fresh53(and_3, $$true, implies(or(Y, X), or(X, Y)), implies(or(X, Y), or(Y, X))), $$true, implies(or(Y, X), or(X, Y)), implies(implies(or(X, Y), or(Y, X)), and(implies(or(Y, X), or(X, Y)), implies(or(X, Y), or(Y, X))))), $$true, implies(or(X, Y), or(Y, X)), equiv(or(Y, X), or(X, Y))), $$true, or(Y, X), or(X, Y))
105.53/105.75	= { by lemma 153 }
105.53/105.75	  $$fresh($$fresh28($$fresh28($$true, $$true, implies(or(Y, X), or(X, Y)), implies(implies(or(X, Y), or(Y, X)), and(implies(or(Y, X), or(X, Y)), implies(or(X, Y), or(Y, X))))), $$true, implies(or(X, Y), or(Y, X)), equiv(or(Y, X), or(X, Y))), $$true, or(Y, X), or(X, Y))
105.53/105.75	= { by axiom 31 (modus_ponens_2) }
105.53/105.75	  $$fresh($$fresh28(is_a_theorem(implies(implies(or(X, Y), or(Y, X)), and(implies(or(Y, X), or(X, Y)), implies(or(X, Y), or(Y, X))))), $$true, implies(or(X, Y), or(Y, X)), equiv(or(Y, X), or(X, Y))), $$true, or(Y, X), or(X, Y))
105.53/105.75	= { by lemma 147 }
105.53/105.75	  $$fresh($$fresh28(is_a_theorem(implies(implies(or(X, Y), or(Y, X)), equiv(or(Y, X), or(X, Y)))), $$true, implies(or(X, Y), or(Y, X)), equiv(or(Y, X), or(X, Y))), $$true, or(Y, X), or(X, Y))
105.53/105.75	= { by lemma 146 }
105.53/105.75	  $$fresh($$fresh60(?, ?, implies(or(X, Y), or(Y, X)), equiv(or(Y, X), or(X, Y))), $$true, or(Y, X), or(X, Y))
105.53/105.75	= { by lemma 142 }
105.53/105.75	  $$fresh($$true, $$true, or(Y, X), or(X, Y))
105.53/105.75	= { by axiom 59 (substitution_of_equivalents_2) }
105.53/105.77	  or(Y, X)
105.53/105.77	
105.53/105.77	Lemma 155: or(X, implies(Y, Z)) = implies(Y, or(X, Z)).
105.53/105.77	Proof:
105.53/105.77	  or(X, implies(Y, Z))
105.53/105.77	= { by lemma 132 }
105.53/105.77	  or(X, or(not(Y), Z))
105.53/105.77	= { by axiom 59 (substitution_of_equivalents_2) }
105.53/105.77	  $$fresh($$true, $$true, or(X, or(not(Y), Z)), or(not(Y), or(X, Z)))
105.53/105.77	= { by lemma 149 }
105.53/105.77	  $$fresh($$fresh60(?, ?, implies(or(X, or(not(Y), Z)), or(not(Y), or(X, Z))), equiv(or(X, or(not(Y), Z)), or(not(Y), or(X, Z)))), $$true, or(X, or(not(Y), Z)), or(not(Y), or(X, Z)))
105.53/105.77	= { by lemma 146 }
105.53/105.77	  $$fresh($$fresh28(is_a_theorem(implies(implies(or(X, or(not(Y), Z)), or(not(Y), or(X, Z))), equiv(or(X, or(not(Y), Z)), or(not(Y), or(X, Z))))), $$true, implies(or(X, or(not(Y), Z)), or(not(Y), or(X, Z))), equiv(or(X, or(not(Y), Z)), or(not(Y), or(X, Z)))), $$true, or(X, or(not(Y), Z)), or(not(Y), or(X, Z)))
105.53/105.77	= { by lemma 147 }
105.53/105.77	  $$fresh($$fresh28(is_a_theorem(implies(implies(or(X, or(not(Y), Z)), or(not(Y), or(X, Z))), and(implies(or(X, or(not(Y), Z)), or(not(Y), or(X, Z))), implies(or(not(Y), or(X, Z)), or(X, or(not(Y), Z)))))), $$true, implies(or(X, or(not(Y), Z)), or(not(Y), or(X, Z))), equiv(or(X, or(not(Y), Z)), or(not(Y), or(X, Z)))), $$true, or(X, or(not(Y), Z)), or(not(Y), or(X, Z)))
105.53/105.77	= { by lemma 135 }
105.53/105.77	  $$fresh($$fresh28(is_a_theorem(implies(implies(or(X, or(not(Y), Z)), or(not(Y), or(X, Z))), not(implies(implies(or(X, or(not(Y), Z)), or(not(Y), or(X, Z))), not(implies(or(not(Y), or(X, Z)), or(X, or(not(Y), Z)))))))), $$true, implies(or(X, or(not(Y), Z)), or(not(Y), or(X, Z))), equiv(or(X, or(not(Y), Z)), or(not(Y), or(X, Z)))), $$true, or(X, or(not(Y), Z)), or(not(Y), or(X, Z)))
105.53/105.77	= { by lemma 132 }
105.53/105.77	  $$fresh($$fresh28(is_a_theorem(implies(implies(or(X, or(not(Y), Z)), or(not(Y), or(X, Z))), not(or(not(implies(or(X, or(not(Y), Z)), or(not(Y), or(X, Z)))), not(implies(or(not(Y), or(X, Z)), or(X, or(not(Y), Z)))))))), $$true, implies(or(X, or(not(Y), Z)), or(not(Y), or(X, Z))), equiv(or(X, or(not(Y), Z)), or(not(Y), or(X, Z)))), $$true, or(X, or(not(Y), Z)), or(not(Y), or(X, Z)))
105.53/105.77	= { by lemma 154 }
105.53/105.77	  $$fresh($$fresh28(is_a_theorem(implies(implies(or(X, or(not(Y), Z)), or(not(Y), or(X, Z))), not(or(not(implies(or(not(Y), or(X, Z)), or(X, or(not(Y), Z)))), not(implies(or(X, or(not(Y), Z)), or(not(Y), or(X, Z)))))))), $$true, implies(or(X, or(not(Y), Z)), or(not(Y), or(X, Z))), equiv(or(X, or(not(Y), Z)), or(not(Y), or(X, Z)))), $$true, or(X, or(not(Y), Z)), or(not(Y), or(X, Z)))
105.53/105.77	= { by lemma 132 }
105.53/105.77	  $$fresh($$fresh28(is_a_theorem(implies(implies(or(X, or(not(Y), Z)), or(not(Y), or(X, Z))), not(implies(implies(or(not(Y), or(X, Z)), or(X, or(not(Y), Z))), not(implies(or(X, or(not(Y), Z)), or(not(Y), or(X, Z)))))))), $$true, implies(or(X, or(not(Y), Z)), or(not(Y), or(X, Z))), equiv(or(X, or(not(Y), Z)), or(not(Y), or(X, Z)))), $$true, or(X, or(not(Y), Z)), or(not(Y), or(X, Z)))
105.53/105.77	= { by lemma 135 }
105.53/105.77	  $$fresh($$fresh28(is_a_theorem(implies(implies(or(X, or(not(Y), Z)), or(not(Y), or(X, Z))), and(implies(or(not(Y), or(X, Z)), or(X, or(not(Y), Z))), implies(or(X, or(not(Y), Z)), or(not(Y), or(X, Z)))))), $$true, implies(or(X, or(not(Y), Z)), or(not(Y), or(X, Z))), equiv(or(X, or(not(Y), Z)), or(not(Y), or(X, Z)))), $$true, or(X, or(not(Y), Z)), or(not(Y), or(X, Z)))
105.53/105.77	= { by lemma 147 }
105.53/105.77	  $$fresh($$fresh28(is_a_theorem(implies(implies(or(X, or(not(Y), Z)), or(not(Y), or(X, Z))), equiv(or(not(Y), or(X, Z)), or(X, or(not(Y), Z))))), $$true, implies(or(X, or(not(Y), Z)), or(not(Y), or(X, Z))), equiv(or(X, or(not(Y), Z)), or(not(Y), or(X, Z)))), $$true, or(X, or(not(Y), Z)), or(not(Y), or(X, Z)))
105.53/105.77	= { by axiom 31 (modus_ponens_2) }
105.53/105.77	  $$fresh($$fresh28($$fresh28($$true, $$true, implies(or(not(Y), or(X, Z)), or(X, or(not(Y), Z))), implies(implies(or(X, or(not(Y), Z)), or(not(Y), or(X, Z))), equiv(or(not(Y), or(X, Z)), or(X, or(not(Y), Z))))), $$true, implies(or(X, or(not(Y), Z)), or(not(Y), or(X, Z))), equiv(or(X, or(not(Y), Z)), or(not(Y), or(X, Z)))), $$true, or(X, or(not(Y), Z)), or(not(Y), or(X, Z)))
105.53/105.77	= { by lemma 153 }
105.53/105.77	  $$fresh($$fresh28($$fresh28($$fresh53(and_3, $$true, implies(or(not(Y), or(X, Z)), or(X, or(not(Y), Z))), implies(or(X, or(not(Y), Z)), or(not(Y), or(X, Z)))), $$true, implies(or(not(Y), or(X, Z)), or(X, or(not(Y), Z))), implies(implies(or(X, or(not(Y), Z)), or(not(Y), or(X, Z))), equiv(or(not(Y), or(X, Z)), or(X, or(not(Y), Z))))), $$true, implies(or(X, or(not(Y), Z)), or(not(Y), or(X, Z))), equiv(or(X, or(not(Y), Z)), or(not(Y), or(X, Z)))), $$true, or(X, or(not(Y), Z)), or(not(Y), or(X, Z)))
105.53/105.77	= { by axiom 89 (and_3_1) }
105.53/105.77	  $$fresh($$fresh28($$fresh28(is_a_theorem(implies(implies(or(not(Y), or(X, Z)), or(X, or(not(Y), Z))), implies(implies(or(X, or(not(Y), Z)), or(not(Y), or(X, Z))), and(implies(or(not(Y), or(X, Z)), or(X, or(not(Y), Z))), implies(or(X, or(not(Y), Z)), or(not(Y), or(X, Z))))))), $$true, implies(or(not(Y), or(X, Z)), or(X, or(not(Y), Z))), implies(implies(or(X, or(not(Y), Z)), or(not(Y), or(X, Z))), equiv(or(not(Y), or(X, Z)), or(X, or(not(Y), Z))))), $$true, implies(or(X, or(not(Y), Z)), or(not(Y), or(X, Z))), equiv(or(X, or(not(Y), Z)), or(not(Y), or(X, Z)))), $$true, or(X, or(not(Y), Z)), or(not(Y), or(X, Z)))
105.53/105.77	= { by lemma 147 }
105.53/105.78	  $$fresh($$fresh28($$fresh28(is_a_theorem(implies(implies(or(not(Y), or(X, Z)), or(X, or(not(Y), Z))), implies(implies(or(X, or(not(Y), Z)), or(not(Y), or(X, Z))), equiv(or(not(Y), or(X, Z)), or(X, or(not(Y), Z)))))), $$true, implies(or(not(Y), or(X, Z)), or(X, or(not(Y), Z))), implies(implies(or(X, or(not(Y), Z)), or(not(Y), or(X, Z))), equiv(or(not(Y), or(X, Z)), or(X, or(not(Y), Z))))), $$true, implies(or(X, or(not(Y), Z)), or(not(Y), or(X, Z))), equiv(or(X, or(not(Y), Z)), or(not(Y), or(X, Z)))), $$true, or(X, or(not(Y), Z)), or(not(Y), or(X, Z)))
105.53/105.78	= { by lemma 146 }
105.53/105.78	  $$fresh($$fresh28($$fresh60(?, ?, implies(or(not(Y), or(X, Z)), or(X, or(not(Y), Z))), implies(implies(or(X, or(not(Y), Z)), or(not(Y), or(X, Z))), equiv(or(not(Y), or(X, Z)), or(X, or(not(Y), Z))))), $$true, implies(or(X, or(not(Y), Z)), or(not(Y), or(X, Z))), equiv(or(X, or(not(Y), Z)), or(not(Y), or(X, Z)))), $$true, or(X, or(not(Y), Z)), or(not(Y), or(X, Z)))
105.53/105.78	= { by lemma 149 }
105.53/105.78	  $$fresh($$fresh28($$true, $$true, implies(or(X, or(not(Y), Z)), or(not(Y), or(X, Z))), equiv(or(X, or(not(Y), Z)), or(not(Y), or(X, Z)))), $$true, or(X, or(not(Y), Z)), or(not(Y), or(X, Z)))
105.53/105.78	= { by axiom 31 (modus_ponens_2) }
105.53/105.78	  $$fresh(is_a_theorem(equiv(or(X, or(not(Y), Z)), or(not(Y), or(X, Z)))), $$true, or(X, or(not(Y), Z)), or(not(Y), or(X, Z)))
105.53/105.78	= { by lemma 143 }
105.53/105.78	  or(not(Y), or(X, Z))
105.53/105.78	= { by lemma 132 }
105.53/105.78	  implies(Y, or(X, Z))
105.53/105.78	
105.53/105.78	Lemma 156: implies(X, implies(Y, Z)) = implies(Y, implies(X, Z)).
105.53/105.78	Proof:
105.53/105.78	  implies(X, implies(Y, Z))
105.53/105.78	= { by lemma 132 }
105.53/105.78	  implies(X, or(not(Y), Z))
105.53/105.78	= { by lemma 155 }
105.53/105.78	  or(not(Y), implies(X, Z))
105.53/105.78	= { by lemma 132 }
118.63/118.86	  implies(Y, implies(X, Z))
118.63/118.86	
118.63/118.86	Goal 1 (hilbert_or_3): or_3 = $$true.
118.63/118.86	Proof:
118.63/118.86	  or_3
118.63/118.86	= { by axiom 102 (or_3) }
118.63/118.86	  $$fresh15(is_a_theorem(implies(implies(sK13_or_3_X, sK11_or_3_Z), implies(implies(sK12_or_3_Y, sK11_or_3_Z), implies(or(sK13_or_3_X, sK12_or_3_Y), sK11_or_3_Z)))), $$true)
118.63/118.86	= { by axiom 101 (or_3_1) }
118.63/118.86	  $$fresh15($$fresh14(or_3, $$true, sK13_or_3_X, sK12_or_3_Y, sK11_or_3_Z), $$true)
118.63/118.86	= { by axiom 59 (substitution_of_equivalents_2) }
118.63/118.86	  $$fresh15($$fresh14(or_3, $$true, $$fresh($$true, $$true, sK13_or_3_X, not(not(sK13_or_3_X))), sK12_or_3_Y, sK11_or_3_Z), $$true)
118.63/118.86	= { by axiom 32 (modus_ponens_2) }
118.63/118.86	  $$fresh15($$fresh14(or_3, $$true, $$fresh($$fresh59($$true, $$true, and(implies(sK13_or_3_X, sK13_or_3_X), implies(sK13_or_3_X, sK13_or_3_X))), $$true, sK13_or_3_X, not(not(sK13_or_3_X))), sK12_or_3_Y, sK11_or_3_Z), $$true)
118.63/118.86	= { by axiom 32 (modus_ponens_2) }
118.63/118.86	  $$fresh15($$fresh14(or_3, $$true, $$fresh($$fresh59($$fresh59($$true, $$true, not(not(or(not(sK13_or_3_X), not(not(sK13_or_3_X)))))), $$true, and(implies(sK13_or_3_X, sK13_or_3_X), implies(sK13_or_3_X, sK13_or_3_X))), $$true, sK13_or_3_X, not(not(sK13_or_3_X))), sK12_or_3_Y, sK11_or_3_Z), $$true)
118.63/118.86	= { by lemma 140 }
118.63/118.86	  $$fresh15($$fresh14(or_3, $$true, $$fresh($$fresh59($$fresh59($$fresh60(?, ?, implies(not(not(sK13_or_3_X)), or(or(not(sK13_or_3_X), not(not(sK13_or_3_X))), not(not(sK13_or_3_X)))), or(or(not(sK13_or_3_X), not(not(sK13_or_3_X))), or(not(sK13_or_3_X), not(not(sK13_or_3_X))))), $$true, not(not(or(not(sK13_or_3_X), not(not(sK13_or_3_X)))))), $$true, and(implies(sK13_or_3_X, sK13_or_3_X), implies(sK13_or_3_X, sK13_or_3_X))), $$true, sK13_or_3_X, not(not(sK13_or_3_X))), sK12_or_3_Y, sK11_or_3_Z), $$true)
118.63/118.86	= { by lemma 134 }
118.63/118.86	  $$fresh15($$fresh14(or_3, $$true, $$fresh($$fresh59($$fresh59($$fresh60(?, ?, or(not(sK13_or_3_X), or(or(not(sK13_or_3_X), not(not(sK13_or_3_X))), not(not(sK13_or_3_X)))), or(or(not(sK13_or_3_X), not(not(sK13_or_3_X))), or(not(sK13_or_3_X), not(not(sK13_or_3_X))))), $$true, not(not(or(not(sK13_or_3_X), not(not(sK13_or_3_X)))))), $$true, and(implies(sK13_or_3_X, sK13_or_3_X), implies(sK13_or_3_X, sK13_or_3_X))), $$true, sK13_or_3_X, not(not(sK13_or_3_X))), sK12_or_3_Y, sK11_or_3_Z), $$true)
118.63/118.86	= { by lemma 148 }
118.63/118.86	  $$fresh15($$fresh14(or_3, $$true, $$fresh($$fresh59($$fresh59(is_a_theorem(or(or(not(sK13_or_3_X), not(not(sK13_or_3_X))), or(not(sK13_or_3_X), not(not(sK13_or_3_X))))), $$true, not(not(or(not(sK13_or_3_X), not(not(sK13_or_3_X)))))), $$true, and(implies(sK13_or_3_X, sK13_or_3_X), implies(sK13_or_3_X, sK13_or_3_X))), $$true, sK13_or_3_X, not(not(sK13_or_3_X))), sK12_or_3_Y, sK11_or_3_Z), $$true)
118.63/118.86	= { by axiom 33 (modus_ponens_2) }
118.63/118.86	  $$fresh15($$fresh14(or_3, $$true, $$fresh($$fresh59($$fresh60(?, ?, or(or(not(sK13_or_3_X), not(not(sK13_or_3_X))), or(not(sK13_or_3_X), not(not(sK13_or_3_X)))), not(not(or(not(sK13_or_3_X), not(not(sK13_or_3_X)))))), $$true, and(implies(sK13_or_3_X, sK13_or_3_X), implies(sK13_or_3_X, sK13_or_3_X))), $$true, sK13_or_3_X, not(not(sK13_or_3_X))), sK12_or_3_Y, sK11_or_3_Z), $$true)
118.63/118.86	= { by lemma 131 }
118.63/118.86	  $$fresh15($$fresh14(or_3, $$true, $$fresh($$fresh59($$fresh60($$true, $$true, or(or(not(sK13_or_3_X), not(not(sK13_or_3_X))), or(not(sK13_or_3_X), not(not(sK13_or_3_X)))), not(not(or(not(sK13_or_3_X), not(not(sK13_or_3_X)))))), $$true, and(implies(sK13_or_3_X, sK13_or_3_X), implies(sK13_or_3_X, sK13_or_3_X))), $$true, sK13_or_3_X, not(not(sK13_or_3_X))), sK12_or_3_Y, sK11_or_3_Z), $$true)
118.63/118.86	= { by axiom 123 (principia_modus_ponens) }
118.63/118.86	  $$fresh15($$fresh14(or_3, $$true, $$fresh($$fresh59($$fresh60(modus_ponens, $$true, or(or(not(sK13_or_3_X), not(not(sK13_or_3_X))), or(not(sK13_or_3_X), not(not(sK13_or_3_X)))), not(not(or(not(sK13_or_3_X), not(not(sK13_or_3_X)))))), $$true, and(implies(sK13_or_3_X, sK13_or_3_X), implies(sK13_or_3_X, sK13_or_3_X))), $$true, sK13_or_3_X, not(not(sK13_or_3_X))), sK12_or_3_Y, sK11_or_3_Z), $$true)
118.63/118.86	= { by axiom 84 (modus_ponens_2) }
118.63/118.86	  $$fresh15($$fresh14(or_3, $$true, $$fresh($$fresh59($$fresh28(is_a_theorem(implies(or(or(not(sK13_or_3_X), not(not(sK13_or_3_X))), or(not(sK13_or_3_X), not(not(sK13_or_3_X)))), not(not(or(not(sK13_or_3_X), not(not(sK13_or_3_X))))))), $$true, or(or(not(sK13_or_3_X), not(not(sK13_or_3_X))), or(not(sK13_or_3_X), not(not(sK13_or_3_X)))), not(not(or(not(sK13_or_3_X), not(not(sK13_or_3_X)))))), $$true, and(implies(sK13_or_3_X, sK13_or_3_X), implies(sK13_or_3_X, sK13_or_3_X))), $$true, sK13_or_3_X, not(not(sK13_or_3_X))), sK12_or_3_Y, sK11_or_3_Z), $$true)
118.63/118.86	= { by lemma 134 }
118.63/118.86	  $$fresh15($$fresh14(or_3, $$true, $$fresh($$fresh59($$fresh28(is_a_theorem(implies(implies(not(or(not(sK13_or_3_X), not(not(sK13_or_3_X)))), or(not(sK13_or_3_X), not(not(sK13_or_3_X)))), not(not(or(not(sK13_or_3_X), not(not(sK13_or_3_X))))))), $$true, or(or(not(sK13_or_3_X), not(not(sK13_or_3_X))), or(not(sK13_or_3_X), not(not(sK13_or_3_X)))), not(not(or(not(sK13_or_3_X), not(not(sK13_or_3_X)))))), $$true, and(implies(sK13_or_3_X, sK13_or_3_X), implies(sK13_or_3_X, sK13_or_3_X))), $$true, sK13_or_3_X, not(not(sK13_or_3_X))), sK12_or_3_Y, sK11_or_3_Z), $$true)
118.63/118.86	= { by lemma 133 }
118.63/118.86	  $$fresh15($$fresh14(or_3, $$true, $$fresh($$fresh59($$fresh28(is_a_theorem(not(and(implies(not(or(not(sK13_or_3_X), not(not(sK13_or_3_X)))), or(not(sK13_or_3_X), not(not(sK13_or_3_X)))), not(not(not(or(not(sK13_or_3_X), not(not(sK13_or_3_X))))))))), $$true, or(or(not(sK13_or_3_X), not(not(sK13_or_3_X))), or(not(sK13_or_3_X), not(not(sK13_or_3_X)))), not(not(or(not(sK13_or_3_X), not(not(sK13_or_3_X)))))), $$true, and(implies(sK13_or_3_X, sK13_or_3_X), implies(sK13_or_3_X, sK13_or_3_X))), $$true, sK13_or_3_X, not(not(sK13_or_3_X))), sK12_or_3_Y, sK11_or_3_Z), $$true)
118.63/118.86	= { by lemma 133 }
118.63/118.86	  $$fresh15($$fresh14(or_3, $$true, $$fresh($$fresh59($$fresh28(is_a_theorem(not(and(not(and(not(or(not(sK13_or_3_X), not(not(sK13_or_3_X)))), not(or(not(sK13_or_3_X), not(not(sK13_or_3_X)))))), not(not(not(or(not(sK13_or_3_X), not(not(sK13_or_3_X))))))))), $$true, or(or(not(sK13_or_3_X), not(not(sK13_or_3_X))), or(not(sK13_or_3_X), not(not(sK13_or_3_X)))), not(not(or(not(sK13_or_3_X), not(not(sK13_or_3_X)))))), $$true, and(implies(sK13_or_3_X, sK13_or_3_X), implies(sK13_or_3_X, sK13_or_3_X))), $$true, sK13_or_3_X, not(not(sK13_or_3_X))), sK12_or_3_Y, sK11_or_3_Z), $$true)
118.63/118.86	= { by axiom 113 (op_or) }
118.63/118.86	  $$fresh15($$fresh14(or_3, $$true, $$fresh($$fresh59($$fresh28(is_a_theorem($$fresh20(op_or, $$true, and(not(or(not(sK13_or_3_X), not(not(sK13_or_3_X)))), not(or(not(sK13_or_3_X), not(not(sK13_or_3_X))))), not(not(or(not(sK13_or_3_X), not(not(sK13_or_3_X))))))), $$true, or(or(not(sK13_or_3_X), not(not(sK13_or_3_X))), or(not(sK13_or_3_X), not(not(sK13_or_3_X)))), not(not(or(not(sK13_or_3_X), not(not(sK13_or_3_X)))))), $$true, and(implies(sK13_or_3_X, sK13_or_3_X), implies(sK13_or_3_X, sK13_or_3_X))), $$true, sK13_or_3_X, not(not(sK13_or_3_X))), sK12_or_3_Y, sK11_or_3_Z), $$true)
118.63/118.86	= { by axiom 130 (hilbert_op_or) }
118.63/118.86	  $$fresh15($$fresh14(or_3, $$true, $$fresh($$fresh59($$fresh28(is_a_theorem($$fresh20($$true, $$true, and(not(or(not(sK13_or_3_X), not(not(sK13_or_3_X)))), not(or(not(sK13_or_3_X), not(not(sK13_or_3_X))))), not(not(or(not(sK13_or_3_X), not(not(sK13_or_3_X))))))), $$true, or(or(not(sK13_or_3_X), not(not(sK13_or_3_X))), or(not(sK13_or_3_X), not(not(sK13_or_3_X)))), not(not(or(not(sK13_or_3_X), not(not(sK13_or_3_X)))))), $$true, and(implies(sK13_or_3_X, sK13_or_3_X), implies(sK13_or_3_X, sK13_or_3_X))), $$true, sK13_or_3_X, not(not(sK13_or_3_X))), sK12_or_3_Y, sK11_or_3_Z), $$true)
118.63/118.86	= { by axiom 41 (op_or) }
118.63/118.86	  $$fresh15($$fresh14(or_3, $$true, $$fresh($$fresh59($$fresh28(is_a_theorem(or(and(not(or(not(sK13_or_3_X), not(not(sK13_or_3_X)))), not(or(not(sK13_or_3_X), not(not(sK13_or_3_X))))), not(not(or(not(sK13_or_3_X), not(not(sK13_or_3_X))))))), $$true, or(or(not(sK13_or_3_X), not(not(sK13_or_3_X))), or(not(sK13_or_3_X), not(not(sK13_or_3_X)))), not(not(or(not(sK13_or_3_X), not(not(sK13_or_3_X)))))), $$true, and(implies(sK13_or_3_X, sK13_or_3_X), implies(sK13_or_3_X, sK13_or_3_X))), $$true, sK13_or_3_X, not(not(sK13_or_3_X))), sK12_or_3_Y, sK11_or_3_Z), $$true)
118.63/118.86	= { by lemma 138 }
118.63/118.86	  $$fresh15($$fresh14(or_3, $$true, $$fresh($$fresh59($$fresh28(is_a_theorem(implies(implies(not(or(not(sK13_or_3_X), not(not(sK13_or_3_X)))), not(not(or(not(sK13_or_3_X), not(not(sK13_or_3_X)))))), not(not(or(not(sK13_or_3_X), not(not(sK13_or_3_X))))))), $$true, or(or(not(sK13_or_3_X), not(not(sK13_or_3_X))), or(not(sK13_or_3_X), not(not(sK13_or_3_X)))), not(not(or(not(sK13_or_3_X), not(not(sK13_or_3_X)))))), $$true, and(implies(sK13_or_3_X, sK13_or_3_X), implies(sK13_or_3_X, sK13_or_3_X))), $$true, sK13_or_3_X, not(not(sK13_or_3_X))), sK12_or_3_Y, sK11_or_3_Z), $$true)
118.63/118.86	= { by lemma 132 }
118.63/118.86	  $$fresh15($$fresh14(or_3, $$true, $$fresh($$fresh59($$fresh28(is_a_theorem(implies(or(not(not(or(not(sK13_or_3_X), not(not(sK13_or_3_X))))), not(not(or(not(sK13_or_3_X), not(not(sK13_or_3_X)))))), not(not(or(not(sK13_or_3_X), not(not(sK13_or_3_X))))))), $$true, or(or(not(sK13_or_3_X), not(not(sK13_or_3_X))), or(not(sK13_or_3_X), not(not(sK13_or_3_X)))), not(not(or(not(sK13_or_3_X), not(not(sK13_or_3_X)))))), $$true, and(implies(sK13_or_3_X, sK13_or_3_X), implies(sK13_or_3_X, sK13_or_3_X))), $$true, sK13_or_3_X, not(not(sK13_or_3_X))), sK12_or_3_Y, sK11_or_3_Z), $$true)
118.63/118.86	= { by axiom 87 (r1_1) }
118.63/118.86	  $$fresh15($$fresh14(or_3, $$true, $$fresh($$fresh59($$fresh28($$fresh12(r1, $$true, not(not(or(not(sK13_or_3_X), not(not(sK13_or_3_X)))))), $$true, or(or(not(sK13_or_3_X), not(not(sK13_or_3_X))), or(not(sK13_or_3_X), not(not(sK13_or_3_X)))), not(not(or(not(sK13_or_3_X), not(not(sK13_or_3_X)))))), $$true, and(implies(sK13_or_3_X, sK13_or_3_X), implies(sK13_or_3_X, sK13_or_3_X))), $$true, sK13_or_3_X, not(not(sK13_or_3_X))), sK12_or_3_Y, sK11_or_3_Z), $$true)
118.63/118.86	= { by axiom 121 (principia_r1) }
118.63/118.86	  $$fresh15($$fresh14(or_3, $$true, $$fresh($$fresh59($$fresh28($$fresh12($$true, $$true, not(not(or(not(sK13_or_3_X), not(not(sK13_or_3_X)))))), $$true, or(or(not(sK13_or_3_X), not(not(sK13_or_3_X))), or(not(sK13_or_3_X), not(not(sK13_or_3_X)))), not(not(or(not(sK13_or_3_X), not(not(sK13_or_3_X)))))), $$true, and(implies(sK13_or_3_X, sK13_or_3_X), implies(sK13_or_3_X, sK13_or_3_X))), $$true, sK13_or_3_X, not(not(sK13_or_3_X))), sK12_or_3_Y, sK11_or_3_Z), $$true)
118.63/118.86	= { by axiom 49 (r1_1) }
118.63/118.86	  $$fresh15($$fresh14(or_3, $$true, $$fresh($$fresh59($$fresh28($$true, $$true, or(or(not(sK13_or_3_X), not(not(sK13_or_3_X))), or(not(sK13_or_3_X), not(not(sK13_or_3_X)))), not(not(or(not(sK13_or_3_X), not(not(sK13_or_3_X)))))), $$true, and(implies(sK13_or_3_X, sK13_or_3_X), implies(sK13_or_3_X, sK13_or_3_X))), $$true, sK13_or_3_X, not(not(sK13_or_3_X))), sK12_or_3_Y, sK11_or_3_Z), $$true)
118.63/118.86	= { by axiom 31 (modus_ponens_2) }
118.63/118.86	  $$fresh15($$fresh14(or_3, $$true, $$fresh($$fresh59(is_a_theorem(not(not(or(not(sK13_or_3_X), not(not(sK13_or_3_X)))))), $$true, and(implies(sK13_or_3_X, sK13_or_3_X), implies(sK13_or_3_X, sK13_or_3_X))), $$true, sK13_or_3_X, not(not(sK13_or_3_X))), sK12_or_3_Y, sK11_or_3_Z), $$true)
118.63/118.86	= { by lemma 136 }
118.63/118.86	  $$fresh15($$fresh14(or_3, $$true, $$fresh($$fresh59(is_a_theorem(not(and(not(not(sK13_or_3_X)), not(sK13_or_3_X)))), $$true, and(implies(sK13_or_3_X, sK13_or_3_X), implies(sK13_or_3_X, sK13_or_3_X))), $$true, sK13_or_3_X, not(not(sK13_or_3_X))), sK12_or_3_Y, sK11_or_3_Z), $$true)
118.63/118.86	= { by lemma 133 }
118.63/118.86	  $$fresh15($$fresh14(or_3, $$true, $$fresh($$fresh59(is_a_theorem(implies(not(not(sK13_or_3_X)), sK13_or_3_X)), $$true, and(implies(sK13_or_3_X, sK13_or_3_X), implies(sK13_or_3_X, sK13_or_3_X))), $$true, sK13_or_3_X, not(not(sK13_or_3_X))), sK12_or_3_Y, sK11_or_3_Z), $$true)
118.63/118.86	= { by lemma 134 }
118.63/118.86	  $$fresh15($$fresh14(or_3, $$true, $$fresh($$fresh59(is_a_theorem(or(not(sK13_or_3_X), sK13_or_3_X)), $$true, and(implies(sK13_or_3_X, sK13_or_3_X), implies(sK13_or_3_X, sK13_or_3_X))), $$true, sK13_or_3_X, not(not(sK13_or_3_X))), sK12_or_3_Y, sK11_or_3_Z), $$true)
118.63/118.86	= { by lemma 132 }
118.63/118.86	  $$fresh15($$fresh14(or_3, $$true, $$fresh($$fresh59(is_a_theorem(implies(sK13_or_3_X, sK13_or_3_X)), $$true, and(implies(sK13_or_3_X, sK13_or_3_X), implies(sK13_or_3_X, sK13_or_3_X))), $$true, sK13_or_3_X, not(not(sK13_or_3_X))), sK12_or_3_Y, sK11_or_3_Z), $$true)
118.63/118.86	= { by axiom 33 (modus_ponens_2) }
118.63/118.86	  $$fresh15($$fresh14(or_3, $$true, $$fresh($$fresh60(?, ?, implies(sK13_or_3_X, sK13_or_3_X), and(implies(sK13_or_3_X, sK13_or_3_X), implies(sK13_or_3_X, sK13_or_3_X))), $$true, sK13_or_3_X, not(not(sK13_or_3_X))), sK12_or_3_Y, sK11_or_3_Z), $$true)
118.63/118.86	= { by lemma 131 }
118.63/118.86	  $$fresh15($$fresh14(or_3, $$true, $$fresh($$fresh60($$true, $$true, implies(sK13_or_3_X, sK13_or_3_X), and(implies(sK13_or_3_X, sK13_or_3_X), implies(sK13_or_3_X, sK13_or_3_X))), $$true, sK13_or_3_X, not(not(sK13_or_3_X))), sK12_or_3_Y, sK11_or_3_Z), $$true)
118.63/118.86	= { by axiom 123 (principia_modus_ponens) }
118.63/118.86	  $$fresh15($$fresh14(or_3, $$true, $$fresh($$fresh60(modus_ponens, $$true, implies(sK13_or_3_X, sK13_or_3_X), and(implies(sK13_or_3_X, sK13_or_3_X), implies(sK13_or_3_X, sK13_or_3_X))), $$true, sK13_or_3_X, not(not(sK13_or_3_X))), sK12_or_3_Y, sK11_or_3_Z), $$true)
118.63/118.86	= { by axiom 84 (modus_ponens_2) }
118.63/118.86	  $$fresh15($$fresh14(or_3, $$true, $$fresh($$fresh28(is_a_theorem(implies(implies(sK13_or_3_X, sK13_or_3_X), and(implies(sK13_or_3_X, sK13_or_3_X), implies(sK13_or_3_X, sK13_or_3_X)))), $$true, implies(sK13_or_3_X, sK13_or_3_X), and(implies(sK13_or_3_X, sK13_or_3_X), implies(sK13_or_3_X, sK13_or_3_X))), $$true, sK13_or_3_X, not(not(sK13_or_3_X))), sK12_or_3_Y, sK11_or_3_Z), $$true)
118.63/118.86	= { by lemma 141 }
118.63/118.86	  $$fresh15($$fresh14(or_3, $$true, $$fresh($$fresh28($$fresh60(?, ?, or(and(implies(sK13_or_3_X, sK13_or_3_X), implies(sK13_or_3_X, sK13_or_3_X)), not(implies(sK13_or_3_X, sK13_or_3_X))), implies(implies(sK13_or_3_X, sK13_or_3_X), and(implies(sK13_or_3_X, sK13_or_3_X), implies(sK13_or_3_X, sK13_or_3_X)))), $$true, implies(sK13_or_3_X, sK13_or_3_X), and(implies(sK13_or_3_X, sK13_or_3_X), implies(sK13_or_3_X, sK13_or_3_X))), $$true, sK13_or_3_X, not(not(sK13_or_3_X))), sK12_or_3_Y, sK11_or_3_Z), $$true)
118.63/118.86	= { by lemma 138 }
118.63/118.86	  $$fresh15($$fresh14(or_3, $$true, $$fresh($$fresh28($$fresh60(?, ?, implies(implies(implies(sK13_or_3_X, sK13_or_3_X), not(implies(sK13_or_3_X, sK13_or_3_X))), not(implies(sK13_or_3_X, sK13_or_3_X))), implies(implies(sK13_or_3_X, sK13_or_3_X), and(implies(sK13_or_3_X, sK13_or_3_X), implies(sK13_or_3_X, sK13_or_3_X)))), $$true, implies(sK13_or_3_X, sK13_or_3_X), and(implies(sK13_or_3_X, sK13_or_3_X), implies(sK13_or_3_X, sK13_or_3_X))), $$true, sK13_or_3_X, not(not(sK13_or_3_X))), sK12_or_3_Y, sK11_or_3_Z), $$true)
118.63/118.86	= { by lemma 132 }
118.63/118.86	  $$fresh15($$fresh14(or_3, $$true, $$fresh($$fresh28($$fresh60(?, ?, implies(or(not(implies(sK13_or_3_X, sK13_or_3_X)), not(implies(sK13_or_3_X, sK13_or_3_X))), not(implies(sK13_or_3_X, sK13_or_3_X))), implies(implies(sK13_or_3_X, sK13_or_3_X), and(implies(sK13_or_3_X, sK13_or_3_X), implies(sK13_or_3_X, sK13_or_3_X)))), $$true, implies(sK13_or_3_X, sK13_or_3_X), and(implies(sK13_or_3_X, sK13_or_3_X), implies(sK13_or_3_X, sK13_or_3_X))), $$true, sK13_or_3_X, not(not(sK13_or_3_X))), sK12_or_3_Y, sK11_or_3_Z), $$true)
118.63/118.86	= { by lemma 139 }
118.63/118.86	  $$fresh15($$fresh14(or_3, $$true, $$fresh($$fresh28($$true, $$true, implies(sK13_or_3_X, sK13_or_3_X), and(implies(sK13_or_3_X, sK13_or_3_X), implies(sK13_or_3_X, sK13_or_3_X))), $$true, sK13_or_3_X, not(not(sK13_or_3_X))), sK12_or_3_Y, sK11_or_3_Z), $$true)
118.63/118.86	= { by axiom 31 (modus_ponens_2) }
118.63/118.86	  $$fresh15($$fresh14(or_3, $$true, $$fresh(is_a_theorem(and(implies(sK13_or_3_X, sK13_or_3_X), implies(sK13_or_3_X, sK13_or_3_X))), $$true, sK13_or_3_X, not(not(sK13_or_3_X))), sK12_or_3_Y, sK11_or_3_Z), $$true)
118.63/118.86	= { by lemma 133 }
118.63/118.86	  $$fresh15($$fresh14(or_3, $$true, $$fresh(is_a_theorem(and(not(and(sK13_or_3_X, not(sK13_or_3_X))), implies(sK13_or_3_X, sK13_or_3_X))), $$true, sK13_or_3_X, not(not(sK13_or_3_X))), sK12_or_3_Y, sK11_or_3_Z), $$true)
118.63/118.86	= { by lemma 132 }
118.63/118.86	  $$fresh15($$fresh14(or_3, $$true, $$fresh(is_a_theorem(and(not(and(sK13_or_3_X, not(sK13_or_3_X))), or(not(sK13_or_3_X), sK13_or_3_X))), $$true, sK13_or_3_X, not(not(sK13_or_3_X))), sK12_or_3_Y, sK11_or_3_Z), $$true)
118.63/118.86	= { by lemma 134 }
118.63/118.86	  $$fresh15($$fresh14(or_3, $$true, $$fresh(is_a_theorem(and(not(and(sK13_or_3_X, not(sK13_or_3_X))), implies(not(not(sK13_or_3_X)), sK13_or_3_X))), $$true, sK13_or_3_X, not(not(sK13_or_3_X))), sK12_or_3_Y, sK11_or_3_Z), $$true)
118.63/118.86	= { by lemma 136 }
118.63/118.86	  $$fresh15($$fresh14(or_3, $$true, $$fresh(is_a_theorem(not(or(and(sK13_or_3_X, not(sK13_or_3_X)), not(implies(not(not(sK13_or_3_X)), sK13_or_3_X))))), $$true, sK13_or_3_X, not(not(sK13_or_3_X))), sK12_or_3_Y, sK11_or_3_Z), $$true)
118.63/118.86	= { by lemma 138 }
118.63/118.86	  $$fresh15($$fresh14(or_3, $$true, $$fresh(is_a_theorem(not(implies(implies(sK13_or_3_X, not(not(sK13_or_3_X))), not(implies(not(not(sK13_or_3_X)), sK13_or_3_X))))), $$true, sK13_or_3_X, not(not(sK13_or_3_X))), sK12_or_3_Y, sK11_or_3_Z), $$true)
118.63/118.86	= { by lemma 135 }
118.63/118.86	  $$fresh15($$fresh14(or_3, $$true, $$fresh(is_a_theorem(and(implies(sK13_or_3_X, not(not(sK13_or_3_X))), implies(not(not(sK13_or_3_X)), sK13_or_3_X))), $$true, sK13_or_3_X, not(not(sK13_or_3_X))), sK12_or_3_Y, sK11_or_3_Z), $$true)
118.63/118.86	= { by axiom 117 (op_equiv) }
118.63/118.86	  $$fresh15($$fresh14(or_3, $$true, $$fresh(is_a_theorem($$fresh23(op_equiv, $$true, sK13_or_3_X, not(not(sK13_or_3_X)))), $$true, sK13_or_3_X, not(not(sK13_or_3_X))), sK12_or_3_Y, sK11_or_3_Z), $$true)
118.63/118.86	= { by axiom 119 (principia_op_equiv) }
118.63/118.86	  $$fresh15($$fresh14(or_3, $$true, $$fresh(is_a_theorem($$fresh23($$true, $$true, sK13_or_3_X, not(not(sK13_or_3_X)))), $$true, sK13_or_3_X, not(not(sK13_or_3_X))), sK12_or_3_Y, sK11_or_3_Z), $$true)
118.63/118.86	= { by axiom 38 (op_equiv) }
118.63/118.86	  $$fresh15($$fresh14(or_3, $$true, $$fresh(is_a_theorem(equiv(sK13_or_3_X, not(not(sK13_or_3_X)))), $$true, sK13_or_3_X, not(not(sK13_or_3_X))), sK12_or_3_Y, sK11_or_3_Z), $$true)
118.63/118.86	= { by axiom 65 (substitution_of_equivalents_2) }
118.63/118.86	  $$fresh15($$fresh14(or_3, $$true, $$fresh2(substitution_of_equivalents, $$true, sK13_or_3_X, not(not(sK13_or_3_X))), sK12_or_3_Y, sK11_or_3_Z), $$true)
118.63/118.86	= { by axiom 127 (substitution_of_equivalents) }
118.63/118.86	  $$fresh15($$fresh14(or_3, $$true, $$fresh2($$true, $$true, sK13_or_3_X, not(not(sK13_or_3_X))), sK12_or_3_Y, sK11_or_3_Z), $$true)
118.63/118.86	= { by axiom 60 (substitution_of_equivalents_2) }
118.63/118.86	  $$fresh15($$fresh14(or_3, $$true, not(not(sK13_or_3_X)), sK12_or_3_Y, sK11_or_3_Z), $$true)
118.63/118.86	= { by axiom 101 (or_3_1) }
118.63/118.86	  $$fresh15(is_a_theorem(implies(implies(not(not(sK13_or_3_X)), sK11_or_3_Z), implies(implies(sK12_or_3_Y, sK11_or_3_Z), implies(or(not(not(sK13_or_3_X)), sK12_or_3_Y), sK11_or_3_Z)))), $$true)
118.63/118.86	= { by lemma 134 }
118.63/118.86	  $$fresh15(is_a_theorem(implies(or(not(sK13_or_3_X), sK11_or_3_Z), implies(implies(sK12_or_3_Y, sK11_or_3_Z), implies(or(not(not(sK13_or_3_X)), sK12_or_3_Y), sK11_or_3_Z)))), $$true)
118.63/118.86	= { by lemma 132 }
118.63/118.86	  $$fresh15(is_a_theorem(implies(or(not(sK13_or_3_X), sK11_or_3_Z), implies(implies(sK12_or_3_Y, sK11_or_3_Z), implies(implies(not(sK13_or_3_X), sK12_or_3_Y), sK11_or_3_Z)))), $$true)
118.63/118.86	= { by lemma 156 }
118.63/118.86	  $$fresh15(is_a_theorem(implies(implies(sK12_or_3_Y, sK11_or_3_Z), implies(or(not(sK13_or_3_X), sK11_or_3_Z), implies(implies(not(sK13_or_3_X), sK12_or_3_Y), sK11_or_3_Z)))), $$true)
118.63/118.86	= { by lemma 156 }
118.63/118.86	  $$fresh15(is_a_theorem(implies(implies(sK12_or_3_Y, sK11_or_3_Z), implies(implies(not(sK13_or_3_X), sK12_or_3_Y), implies(or(not(sK13_or_3_X), sK11_or_3_Z), sK11_or_3_Z)))), $$true)
118.63/118.86	= { by lemma 156 }
118.63/118.86	  $$fresh15(is_a_theorem(implies(implies(not(sK13_or_3_X), sK12_or_3_Y), implies(implies(sK12_or_3_Y, sK11_or_3_Z), implies(or(not(sK13_or_3_X), sK11_or_3_Z), sK11_or_3_Z)))), $$true)
118.63/118.86	= { by axiom 59 (substitution_of_equivalents_2) }
118.63/118.86	  $$fresh15(is_a_theorem(implies(implies(not(sK13_or_3_X), sK12_or_3_Y), implies(implies(sK12_or_3_Y, sK11_or_3_Z), implies(or(not(sK13_or_3_X), sK11_or_3_Z), $$fresh($$true, $$true, sK11_or_3_Z, or(sK11_or_3_Z, sK11_or_3_Z)))))), $$true)
118.63/118.86	= { by axiom 32 (modus_ponens_2) }
118.63/118.86	  $$fresh15(is_a_theorem(implies(implies(not(sK13_or_3_X), sK12_or_3_Y), implies(implies(sK12_or_3_Y, sK11_or_3_Z), implies(or(not(sK13_or_3_X), sK11_or_3_Z), $$fresh($$fresh59($$true, $$true, equiv(sK11_or_3_Z, or(sK11_or_3_Z, sK11_or_3_Z))), $$true, sK11_or_3_Z, or(sK11_or_3_Z, sK11_or_3_Z)))))), $$true)
118.63/118.86	= { by axiom 32 (modus_ponens_2) }
118.63/118.86	  $$fresh15(is_a_theorem(implies(implies(not(sK13_or_3_X), sK12_or_3_Y), implies(implies(sK12_or_3_Y, sK11_or_3_Z), implies(or(not(sK13_or_3_X), sK11_or_3_Z), $$fresh($$fresh59($$fresh59($$true, $$true, implies(sK11_or_3_Z, implies(not(sK11_or_3_Z), sK11_or_3_Z))), $$true, equiv(sK11_or_3_Z, or(sK11_or_3_Z, sK11_or_3_Z))), $$true, sK11_or_3_Z, or(sK11_or_3_Z, sK11_or_3_Z)))))), $$true)
118.63/118.86	= { by axiom 32 (modus_ponens_2) }
118.63/118.86	  $$fresh15(is_a_theorem(implies(implies(not(sK13_or_3_X), sK12_or_3_Y), implies(implies(sK12_or_3_Y, sK11_or_3_Z), implies(or(not(sK13_or_3_X), sK11_or_3_Z), $$fresh($$fresh59($$fresh59($$fresh59($$true, $$true, implies(and(not(sK11_or_3_Z), not(sK11_or_3_Z)), not(sK11_or_3_Z))), $$true, implies(sK11_or_3_Z, implies(not(sK11_or_3_Z), sK11_or_3_Z))), $$true, equiv(sK11_or_3_Z, or(sK11_or_3_Z, sK11_or_3_Z))), $$true, sK11_or_3_Z, or(sK11_or_3_Z, sK11_or_3_Z)))))), $$true)
118.63/118.86	= { by lemma 142 }
118.63/118.86	  $$fresh15(is_a_theorem(implies(implies(not(sK13_or_3_X), sK12_or_3_Y), implies(implies(sK12_or_3_Y, sK11_or_3_Z), implies(or(not(sK13_or_3_X), sK11_or_3_Z), $$fresh($$fresh59($$fresh59($$fresh59($$fresh60(?, ?, implies(or(not(not(sK11_or_3_Z)), not(not(sK11_or_3_Z))), or(not(not(sK11_or_3_Z)), not(not(sK11_or_3_Z)))), or(or(not(not(sK11_or_3_Z)), not(not(sK11_or_3_Z))), not(or(not(not(sK11_or_3_Z)), not(not(sK11_or_3_Z)))))), $$true, implies(and(not(sK11_or_3_Z), not(sK11_or_3_Z)), not(sK11_or_3_Z))), $$true, implies(sK11_or_3_Z, implies(not(sK11_or_3_Z), sK11_or_3_Z))), $$true, equiv(sK11_or_3_Z, or(sK11_or_3_Z, sK11_or_3_Z))), $$true, sK11_or_3_Z, or(sK11_or_3_Z, sK11_or_3_Z)))))), $$true)
118.63/118.86	= { by lemma 131 }
118.63/118.86	  $$fresh15(is_a_theorem(implies(implies(not(sK13_or_3_X), sK12_or_3_Y), implies(implies(sK12_or_3_Y, sK11_or_3_Z), implies(or(not(sK13_or_3_X), sK11_or_3_Z), $$fresh($$fresh59($$fresh59($$fresh59($$fresh60($$true, $$true, implies(or(not(not(sK11_or_3_Z)), not(not(sK11_or_3_Z))), or(not(not(sK11_or_3_Z)), not(not(sK11_or_3_Z)))), or(or(not(not(sK11_or_3_Z)), not(not(sK11_or_3_Z))), not(or(not(not(sK11_or_3_Z)), not(not(sK11_or_3_Z)))))), $$true, implies(and(not(sK11_or_3_Z), not(sK11_or_3_Z)), not(sK11_or_3_Z))), $$true, implies(sK11_or_3_Z, implies(not(sK11_or_3_Z), sK11_or_3_Z))), $$true, equiv(sK11_or_3_Z, or(sK11_or_3_Z, sK11_or_3_Z))), $$true, sK11_or_3_Z, or(sK11_or_3_Z, sK11_or_3_Z)))))), $$true)
118.63/118.86	= { by axiom 123 (principia_modus_ponens) }
118.63/118.87	  $$fresh15(is_a_theorem(implies(implies(not(sK13_or_3_X), sK12_or_3_Y), implies(implies(sK12_or_3_Y, sK11_or_3_Z), implies(or(not(sK13_or_3_X), sK11_or_3_Z), $$fresh($$fresh59($$fresh59($$fresh59($$fresh60(modus_ponens, $$true, implies(or(not(not(sK11_or_3_Z)), not(not(sK11_or_3_Z))), or(not(not(sK11_or_3_Z)), not(not(sK11_or_3_Z)))), or(or(not(not(sK11_or_3_Z)), not(not(sK11_or_3_Z))), not(or(not(not(sK11_or_3_Z)), not(not(sK11_or_3_Z)))))), $$true, implies(and(not(sK11_or_3_Z), not(sK11_or_3_Z)), not(sK11_or_3_Z))), $$true, implies(sK11_or_3_Z, implies(not(sK11_or_3_Z), sK11_or_3_Z))), $$true, equiv(sK11_or_3_Z, or(sK11_or_3_Z, sK11_or_3_Z))), $$true, sK11_or_3_Z, or(sK11_or_3_Z, sK11_or_3_Z)))))), $$true)
118.63/118.87	= { by axiom 84 (modus_ponens_2) }
118.63/118.87	  $$fresh15(is_a_theorem(implies(implies(not(sK13_or_3_X), sK12_or_3_Y), implies(implies(sK12_or_3_Y, sK11_or_3_Z), implies(or(not(sK13_or_3_X), sK11_or_3_Z), $$fresh($$fresh59($$fresh59($$fresh59($$fresh28(is_a_theorem(implies(implies(or(not(not(sK11_or_3_Z)), not(not(sK11_or_3_Z))), or(not(not(sK11_or_3_Z)), not(not(sK11_or_3_Z)))), or(or(not(not(sK11_or_3_Z)), not(not(sK11_or_3_Z))), not(or(not(not(sK11_or_3_Z)), not(not(sK11_or_3_Z))))))), $$true, implies(or(not(not(sK11_or_3_Z)), not(not(sK11_or_3_Z))), or(not(not(sK11_or_3_Z)), not(not(sK11_or_3_Z)))), or(or(not(not(sK11_or_3_Z)), not(not(sK11_or_3_Z))), not(or(not(not(sK11_or_3_Z)), not(not(sK11_or_3_Z)))))), $$true, implies(and(not(sK11_or_3_Z), not(sK11_or_3_Z)), not(sK11_or_3_Z))), $$true, implies(sK11_or_3_Z, implies(not(sK11_or_3_Z), sK11_or_3_Z))), $$true, equiv(sK11_or_3_Z, or(sK11_or_3_Z, sK11_or_3_Z))), $$true, sK11_or_3_Z, or(sK11_or_3_Z, sK11_or_3_Z)))))), $$true)
118.63/118.87	= { by lemma 132 }
118.63/118.87	  $$fresh15(is_a_theorem(implies(implies(not(sK13_or_3_X), sK12_or_3_Y), implies(implies(sK12_or_3_Y, sK11_or_3_Z), implies(or(not(sK13_or_3_X), sK11_or_3_Z), $$fresh($$fresh59($$fresh59($$fresh59($$fresh28(is_a_theorem(implies(or(not(or(not(not(sK11_or_3_Z)), not(not(sK11_or_3_Z)))), or(not(not(sK11_or_3_Z)), not(not(sK11_or_3_Z)))), or(or(not(not(sK11_or_3_Z)), not(not(sK11_or_3_Z))), not(or(not(not(sK11_or_3_Z)), not(not(sK11_or_3_Z))))))), $$true, implies(or(not(not(sK11_or_3_Z)), not(not(sK11_or_3_Z))), or(not(not(sK11_or_3_Z)), not(not(sK11_or_3_Z)))), or(or(not(not(sK11_or_3_Z)), not(not(sK11_or_3_Z))), not(or(not(not(sK11_or_3_Z)), not(not(sK11_or_3_Z)))))), $$true, implies(and(not(sK11_or_3_Z), not(sK11_or_3_Z)), not(sK11_or_3_Z))), $$true, implies(sK11_or_3_Z, implies(not(sK11_or_3_Z), sK11_or_3_Z))), $$true, equiv(sK11_or_3_Z, or(sK11_or_3_Z, sK11_or_3_Z))), $$true, sK11_or_3_Z, or(sK11_or_3_Z, sK11_or_3_Z)))))), $$true)
118.63/118.87	= { by axiom 97 (r3_1) }
118.63/118.87	  $$fresh15(is_a_theorem(implies(implies(not(sK13_or_3_X), sK12_or_3_Y), implies(implies(sK12_or_3_Y, sK11_or_3_Z), implies(or(not(sK13_or_3_X), sK11_or_3_Z), $$fresh($$fresh59($$fresh59($$fresh59($$fresh28($$fresh8(r3, $$true, not(or(not(not(sK11_or_3_Z)), not(not(sK11_or_3_Z)))), or(not(not(sK11_or_3_Z)), not(not(sK11_or_3_Z)))), $$true, implies(or(not(not(sK11_or_3_Z)), not(not(sK11_or_3_Z))), or(not(not(sK11_or_3_Z)), not(not(sK11_or_3_Z)))), or(or(not(not(sK11_or_3_Z)), not(not(sK11_or_3_Z))), not(or(not(not(sK11_or_3_Z)), not(not(sK11_or_3_Z)))))), $$true, implies(and(not(sK11_or_3_Z), not(sK11_or_3_Z)), not(sK11_or_3_Z))), $$true, implies(sK11_or_3_Z, implies(not(sK11_or_3_Z), sK11_or_3_Z))), $$true, equiv(sK11_or_3_Z, or(sK11_or_3_Z, sK11_or_3_Z))), $$true, sK11_or_3_Z, or(sK11_or_3_Z, sK11_or_3_Z)))))), $$true)
118.63/118.87	= { by axiom 122 (principia_r3) }
118.63/118.87	  $$fresh15(is_a_theorem(implies(implies(not(sK13_or_3_X), sK12_or_3_Y), implies(implies(sK12_or_3_Y, sK11_or_3_Z), implies(or(not(sK13_or_3_X), sK11_or_3_Z), $$fresh($$fresh59($$fresh59($$fresh59($$fresh28($$fresh8($$true, $$true, not(or(not(not(sK11_or_3_Z)), not(not(sK11_or_3_Z)))), or(not(not(sK11_or_3_Z)), not(not(sK11_or_3_Z)))), $$true, implies(or(not(not(sK11_or_3_Z)), not(not(sK11_or_3_Z))), or(not(not(sK11_or_3_Z)), not(not(sK11_or_3_Z)))), or(or(not(not(sK11_or_3_Z)), not(not(sK11_or_3_Z))), not(or(not(not(sK11_or_3_Z)), not(not(sK11_or_3_Z)))))), $$true, implies(and(not(sK11_or_3_Z), not(sK11_or_3_Z)), not(sK11_or_3_Z))), $$true, implies(sK11_or_3_Z, implies(not(sK11_or_3_Z), sK11_or_3_Z))), $$true, equiv(sK11_or_3_Z, or(sK11_or_3_Z, sK11_or_3_Z))), $$true, sK11_or_3_Z, or(sK11_or_3_Z, sK11_or_3_Z)))))), $$true)
118.63/118.87	= { by axiom 53 (r3_1) }
118.63/118.87	  $$fresh15(is_a_theorem(implies(implies(not(sK13_or_3_X), sK12_or_3_Y), implies(implies(sK12_or_3_Y, sK11_or_3_Z), implies(or(not(sK13_or_3_X), sK11_or_3_Z), $$fresh($$fresh59($$fresh59($$fresh59($$fresh28($$true, $$true, implies(or(not(not(sK11_or_3_Z)), not(not(sK11_or_3_Z))), or(not(not(sK11_or_3_Z)), not(not(sK11_or_3_Z)))), or(or(not(not(sK11_or_3_Z)), not(not(sK11_or_3_Z))), not(or(not(not(sK11_or_3_Z)), not(not(sK11_or_3_Z)))))), $$true, implies(and(not(sK11_or_3_Z), not(sK11_or_3_Z)), not(sK11_or_3_Z))), $$true, implies(sK11_or_3_Z, implies(not(sK11_or_3_Z), sK11_or_3_Z))), $$true, equiv(sK11_or_3_Z, or(sK11_or_3_Z, sK11_or_3_Z))), $$true, sK11_or_3_Z, or(sK11_or_3_Z, sK11_or_3_Z)))))), $$true)
118.63/118.87	= { by axiom 31 (modus_ponens_2) }
118.63/118.87	  $$fresh15(is_a_theorem(implies(implies(not(sK13_or_3_X), sK12_or_3_Y), implies(implies(sK12_or_3_Y, sK11_or_3_Z), implies(or(not(sK13_or_3_X), sK11_or_3_Z), $$fresh($$fresh59($$fresh59($$fresh59(is_a_theorem(or(or(not(not(sK11_or_3_Z)), not(not(sK11_or_3_Z))), not(or(not(not(sK11_or_3_Z)), not(not(sK11_or_3_Z)))))), $$true, implies(and(not(sK11_or_3_Z), not(sK11_or_3_Z)), not(sK11_or_3_Z))), $$true, implies(sK11_or_3_Z, implies(not(sK11_or_3_Z), sK11_or_3_Z))), $$true, equiv(sK11_or_3_Z, or(sK11_or_3_Z, sK11_or_3_Z))), $$true, sK11_or_3_Z, or(sK11_or_3_Z, sK11_or_3_Z)))))), $$true)
118.63/118.87	= { by axiom 114 (op_and) }
118.63/118.87	  $$fresh15(is_a_theorem(implies(implies(not(sK13_or_3_X), sK12_or_3_Y), implies(implies(sK12_or_3_Y, sK11_or_3_Z), implies(or(not(sK13_or_3_X), sK11_or_3_Z), $$fresh($$fresh59($$fresh59($$fresh59(is_a_theorem(or(or(not(not(sK11_or_3_Z)), not(not(sK11_or_3_Z))), $$fresh24(op_and, $$true, not(sK11_or_3_Z), not(sK11_or_3_Z)))), $$true, implies(and(not(sK11_or_3_Z), not(sK11_or_3_Z)), not(sK11_or_3_Z))), $$true, implies(sK11_or_3_Z, implies(not(sK11_or_3_Z), sK11_or_3_Z))), $$true, equiv(sK11_or_3_Z, or(sK11_or_3_Z, sK11_or_3_Z))), $$true, sK11_or_3_Z, or(sK11_or_3_Z, sK11_or_3_Z)))))), $$true)
118.63/118.87	= { by axiom 41 (op_or) }
118.63/118.87	  $$fresh15(is_a_theorem(implies(implies(not(sK13_or_3_X), sK12_or_3_Y), implies(implies(sK12_or_3_Y, sK11_or_3_Z), implies(or(not(sK13_or_3_X), sK11_or_3_Z), $$fresh($$fresh59($$fresh59($$fresh59(is_a_theorem($$fresh20($$true, $$true, or(not(not(sK11_or_3_Z)), not(not(sK11_or_3_Z))), $$fresh24(op_and, $$true, not(sK11_or_3_Z), not(sK11_or_3_Z)))), $$true, implies(and(not(sK11_or_3_Z), not(sK11_or_3_Z)), not(sK11_or_3_Z))), $$true, implies(sK11_or_3_Z, implies(not(sK11_or_3_Z), sK11_or_3_Z))), $$true, equiv(sK11_or_3_Z, or(sK11_or_3_Z, sK11_or_3_Z))), $$true, sK11_or_3_Z, or(sK11_or_3_Z, sK11_or_3_Z)))))), $$true)
118.63/118.87	= { by axiom 130 (hilbert_op_or) }
118.63/118.87	  $$fresh15(is_a_theorem(implies(implies(not(sK13_or_3_X), sK12_or_3_Y), implies(implies(sK12_or_3_Y, sK11_or_3_Z), implies(or(not(sK13_or_3_X), sK11_or_3_Z), $$fresh($$fresh59($$fresh59($$fresh59(is_a_theorem($$fresh20(op_or, $$true, or(not(not(sK11_or_3_Z)), not(not(sK11_or_3_Z))), $$fresh24(op_and, $$true, not(sK11_or_3_Z), not(sK11_or_3_Z)))), $$true, implies(and(not(sK11_or_3_Z), not(sK11_or_3_Z)), not(sK11_or_3_Z))), $$true, implies(sK11_or_3_Z, implies(not(sK11_or_3_Z), sK11_or_3_Z))), $$true, equiv(sK11_or_3_Z, or(sK11_or_3_Z, sK11_or_3_Z))), $$true, sK11_or_3_Z, or(sK11_or_3_Z, sK11_or_3_Z)))))), $$true)
118.63/118.87	= { by axiom 113 (op_or) }
118.63/118.87	  $$fresh15(is_a_theorem(implies(implies(not(sK13_or_3_X), sK12_or_3_Y), implies(implies(sK12_or_3_Y, sK11_or_3_Z), implies(or(not(sK13_or_3_X), sK11_or_3_Z), $$fresh($$fresh59($$fresh59($$fresh59(is_a_theorem(not(and(not(or(not(not(sK11_or_3_Z)), not(not(sK11_or_3_Z)))), not($$fresh24(op_and, $$true, not(sK11_or_3_Z), not(sK11_or_3_Z)))))), $$true, implies(and(not(sK11_or_3_Z), not(sK11_or_3_Z)), not(sK11_or_3_Z))), $$true, implies(sK11_or_3_Z, implies(not(sK11_or_3_Z), sK11_or_3_Z))), $$true, equiv(sK11_or_3_Z, or(sK11_or_3_Z, sK11_or_3_Z))), $$true, sK11_or_3_Z, or(sK11_or_3_Z, sK11_or_3_Z)))))), $$true)
118.63/118.87	= { by lemma 136 }
118.63/118.87	  $$fresh15(is_a_theorem(implies(implies(not(sK13_or_3_X), sK12_or_3_Y), implies(implies(sK12_or_3_Y, sK11_or_3_Z), implies(or(not(sK13_or_3_X), sK11_or_3_Z), $$fresh($$fresh59($$fresh59($$fresh59(is_a_theorem(not(and(and(not(not(not(sK11_or_3_Z))), not(sK11_or_3_Z)), not($$fresh24(op_and, $$true, not(sK11_or_3_Z), not(sK11_or_3_Z)))))), $$true, implies(and(not(sK11_or_3_Z), not(sK11_or_3_Z)), not(sK11_or_3_Z))), $$true, implies(sK11_or_3_Z, implies(not(sK11_or_3_Z), sK11_or_3_Z))), $$true, equiv(sK11_or_3_Z, or(sK11_or_3_Z, sK11_or_3_Z))), $$true, sK11_or_3_Z, or(sK11_or_3_Z, sK11_or_3_Z)))))), $$true)
118.63/118.87	= { by lemma 133 }
118.63/118.87	  $$fresh15(is_a_theorem(implies(implies(not(sK13_or_3_X), sK12_or_3_Y), implies(implies(sK12_or_3_Y, sK11_or_3_Z), implies(or(not(sK13_or_3_X), sK11_or_3_Z), $$fresh($$fresh59($$fresh59($$fresh59(is_a_theorem(implies(and(not(not(not(sK11_or_3_Z))), not(sK11_or_3_Z)), $$fresh24(op_and, $$true, not(sK11_or_3_Z), not(sK11_or_3_Z)))), $$true, implies(and(not(sK11_or_3_Z), not(sK11_or_3_Z)), not(sK11_or_3_Z))), $$true, implies(sK11_or_3_Z, implies(not(sK11_or_3_Z), sK11_or_3_Z))), $$true, equiv(sK11_or_3_Z, or(sK11_or_3_Z, sK11_or_3_Z))), $$true, sK11_or_3_Z, or(sK11_or_3_Z, sK11_or_3_Z)))))), $$true)
118.63/118.87	= { by lemma 136 }
118.63/118.87	  $$fresh15(is_a_theorem(implies(implies(not(sK13_or_3_X), sK12_or_3_Y), implies(implies(sK12_or_3_Y, sK11_or_3_Z), implies(or(not(sK13_or_3_X), sK11_or_3_Z), $$fresh($$fresh59($$fresh59($$fresh59(is_a_theorem(implies(not(or(not(not(sK11_or_3_Z)), not(not(sK11_or_3_Z)))), $$fresh24(op_and, $$true, not(sK11_or_3_Z), not(sK11_or_3_Z)))), $$true, implies(and(not(sK11_or_3_Z), not(sK11_or_3_Z)), not(sK11_or_3_Z))), $$true, implies(sK11_or_3_Z, implies(not(sK11_or_3_Z), sK11_or_3_Z))), $$true, equiv(sK11_or_3_Z, or(sK11_or_3_Z, sK11_or_3_Z))), $$true, sK11_or_3_Z, or(sK11_or_3_Z, sK11_or_3_Z)))))), $$true)
118.63/118.87	= { by axiom 114 (op_and) }
118.63/118.87	  $$fresh15(is_a_theorem(implies(implies(not(sK13_or_3_X), sK12_or_3_Y), implies(implies(sK12_or_3_Y, sK11_or_3_Z), implies(or(not(sK13_or_3_X), sK11_or_3_Z), $$fresh($$fresh59($$fresh59($$fresh59(is_a_theorem(implies($$fresh24(op_and, $$true, not(sK11_or_3_Z), not(sK11_or_3_Z)), $$fresh24(op_and, $$true, not(sK11_or_3_Z), not(sK11_or_3_Z)))), $$true, implies(and(not(sK11_or_3_Z), not(sK11_or_3_Z)), not(sK11_or_3_Z))), $$true, implies(sK11_or_3_Z, implies(not(sK11_or_3_Z), sK11_or_3_Z))), $$true, equiv(sK11_or_3_Z, or(sK11_or_3_Z, sK11_or_3_Z))), $$true, sK11_or_3_Z, or(sK11_or_3_Z, sK11_or_3_Z)))))), $$true)
118.63/118.87	= { by axiom 126 (principia_op_and) }
118.63/118.87	  $$fresh15(is_a_theorem(implies(implies(not(sK13_or_3_X), sK12_or_3_Y), implies(implies(sK12_or_3_Y, sK11_or_3_Z), implies(or(not(sK13_or_3_X), sK11_or_3_Z), $$fresh($$fresh59($$fresh59($$fresh59(is_a_theorem(implies($$fresh24($$true, $$true, not(sK11_or_3_Z), not(sK11_or_3_Z)), $$fresh24(op_and, $$true, not(sK11_or_3_Z), not(sK11_or_3_Z)))), $$true, implies(and(not(sK11_or_3_Z), not(sK11_or_3_Z)), not(sK11_or_3_Z))), $$true, implies(sK11_or_3_Z, implies(not(sK11_or_3_Z), sK11_or_3_Z))), $$true, equiv(sK11_or_3_Z, or(sK11_or_3_Z, sK11_or_3_Z))), $$true, sK11_or_3_Z, or(sK11_or_3_Z, sK11_or_3_Z)))))), $$true)
118.63/118.87	= { by axiom 37 (op_and) }
118.63/118.87	  $$fresh15(is_a_theorem(implies(implies(not(sK13_or_3_X), sK12_or_3_Y), implies(implies(sK12_or_3_Y, sK11_or_3_Z), implies(or(not(sK13_or_3_X), sK11_or_3_Z), $$fresh($$fresh59($$fresh59($$fresh59(is_a_theorem(implies(and(not(sK11_or_3_Z), not(sK11_or_3_Z)), $$fresh24(op_and, $$true, not(sK11_or_3_Z), not(sK11_or_3_Z)))), $$true, implies(and(not(sK11_or_3_Z), not(sK11_or_3_Z)), not(sK11_or_3_Z))), $$true, implies(sK11_or_3_Z, implies(not(sK11_or_3_Z), sK11_or_3_Z))), $$true, equiv(sK11_or_3_Z, or(sK11_or_3_Z, sK11_or_3_Z))), $$true, sK11_or_3_Z, or(sK11_or_3_Z, sK11_or_3_Z)))))), $$true)
118.63/118.87	= { by axiom 126 (principia_op_and) }
118.63/118.87	  $$fresh15(is_a_theorem(implies(implies(not(sK13_or_3_X), sK12_or_3_Y), implies(implies(sK12_or_3_Y, sK11_or_3_Z), implies(or(not(sK13_or_3_X), sK11_or_3_Z), $$fresh($$fresh59($$fresh59($$fresh59(is_a_theorem(implies(and(not(sK11_or_3_Z), not(sK11_or_3_Z)), $$fresh24($$true, $$true, not(sK11_or_3_Z), not(sK11_or_3_Z)))), $$true, implies(and(not(sK11_or_3_Z), not(sK11_or_3_Z)), not(sK11_or_3_Z))), $$true, implies(sK11_or_3_Z, implies(not(sK11_or_3_Z), sK11_or_3_Z))), $$true, equiv(sK11_or_3_Z, or(sK11_or_3_Z, sK11_or_3_Z))), $$true, sK11_or_3_Z, or(sK11_or_3_Z, sK11_or_3_Z)))))), $$true)
118.63/118.87	= { by axiom 37 (op_and) }
118.63/118.87	  $$fresh15(is_a_theorem(implies(implies(not(sK13_or_3_X), sK12_or_3_Y), implies(implies(sK12_or_3_Y, sK11_or_3_Z), implies(or(not(sK13_or_3_X), sK11_or_3_Z), $$fresh($$fresh59($$fresh59($$fresh59(is_a_theorem(implies(and(not(sK11_or_3_Z), not(sK11_or_3_Z)), and(not(sK11_or_3_Z), not(sK11_or_3_Z)))), $$true, implies(and(not(sK11_or_3_Z), not(sK11_or_3_Z)), not(sK11_or_3_Z))), $$true, implies(sK11_or_3_Z, implies(not(sK11_or_3_Z), sK11_or_3_Z))), $$true, equiv(sK11_or_3_Z, or(sK11_or_3_Z, sK11_or_3_Z))), $$true, sK11_or_3_Z, or(sK11_or_3_Z, sK11_or_3_Z)))))), $$true)
118.63/118.87	= { by axiom 33 (modus_ponens_2) }
118.63/118.87	  $$fresh15(is_a_theorem(implies(implies(not(sK13_or_3_X), sK12_or_3_Y), implies(implies(sK12_or_3_Y, sK11_or_3_Z), implies(or(not(sK13_or_3_X), sK11_or_3_Z), $$fresh($$fresh59($$fresh59($$fresh60(?, ?, implies(and(not(sK11_or_3_Z), not(sK11_or_3_Z)), and(not(sK11_or_3_Z), not(sK11_or_3_Z))), implies(and(not(sK11_or_3_Z), not(sK11_or_3_Z)), not(sK11_or_3_Z))), $$true, implies(sK11_or_3_Z, implies(not(sK11_or_3_Z), sK11_or_3_Z))), $$true, equiv(sK11_or_3_Z, or(sK11_or_3_Z, sK11_or_3_Z))), $$true, sK11_or_3_Z, or(sK11_or_3_Z, sK11_or_3_Z)))))), $$true)
118.63/118.87	= { by lemma 146 }
118.63/118.87	  $$fresh15(is_a_theorem(implies(implies(not(sK13_or_3_X), sK12_or_3_Y), implies(implies(sK12_or_3_Y, sK11_or_3_Z), implies(or(not(sK13_or_3_X), sK11_or_3_Z), $$fresh($$fresh59($$fresh59($$fresh28(is_a_theorem(implies(implies(and(not(sK11_or_3_Z), not(sK11_or_3_Z)), and(not(sK11_or_3_Z), not(sK11_or_3_Z))), implies(and(not(sK11_or_3_Z), not(sK11_or_3_Z)), not(sK11_or_3_Z)))), $$true, implies(and(not(sK11_or_3_Z), not(sK11_or_3_Z)), and(not(sK11_or_3_Z), not(sK11_or_3_Z))), implies(and(not(sK11_or_3_Z), not(sK11_or_3_Z)), not(sK11_or_3_Z))), $$true, implies(sK11_or_3_Z, implies(not(sK11_or_3_Z), sK11_or_3_Z))), $$true, equiv(sK11_or_3_Z, or(sK11_or_3_Z, sK11_or_3_Z))), $$true, sK11_or_3_Z, or(sK11_or_3_Z, sK11_or_3_Z)))))), $$true)
118.63/118.87	= { by lemma 132 }
118.63/118.87	  $$fresh15(is_a_theorem(implies(implies(not(sK13_or_3_X), sK12_or_3_Y), implies(implies(sK12_or_3_Y, sK11_or_3_Z), implies(or(not(sK13_or_3_X), sK11_or_3_Z), $$fresh($$fresh59($$fresh59($$fresh28(is_a_theorem(implies(or(not(and(not(sK11_or_3_Z), not(sK11_or_3_Z))), and(not(sK11_or_3_Z), not(sK11_or_3_Z))), implies(and(not(sK11_or_3_Z), not(sK11_or_3_Z)), not(sK11_or_3_Z)))), $$true, implies(and(not(sK11_or_3_Z), not(sK11_or_3_Z)), and(not(sK11_or_3_Z), not(sK11_or_3_Z))), implies(and(not(sK11_or_3_Z), not(sK11_or_3_Z)), not(sK11_or_3_Z))), $$true, implies(sK11_or_3_Z, implies(not(sK11_or_3_Z), sK11_or_3_Z))), $$true, equiv(sK11_or_3_Z, or(sK11_or_3_Z, sK11_or_3_Z))), $$true, sK11_or_3_Z, or(sK11_or_3_Z, sK11_or_3_Z)))))), $$true)
118.63/118.87	= { by lemma 132 }
118.63/118.87	  $$fresh15(is_a_theorem(implies(implies(not(sK13_or_3_X), sK12_or_3_Y), implies(implies(sK12_or_3_Y, sK11_or_3_Z), implies(or(not(sK13_or_3_X), sK11_or_3_Z), $$fresh($$fresh59($$fresh59($$fresh28(is_a_theorem(implies(or(not(and(not(sK11_or_3_Z), not(sK11_or_3_Z))), and(not(sK11_or_3_Z), not(sK11_or_3_Z))), or(not(and(not(sK11_or_3_Z), not(sK11_or_3_Z))), not(sK11_or_3_Z)))), $$true, implies(and(not(sK11_or_3_Z), not(sK11_or_3_Z)), and(not(sK11_or_3_Z), not(sK11_or_3_Z))), implies(and(not(sK11_or_3_Z), not(sK11_or_3_Z)), not(sK11_or_3_Z))), $$true, implies(sK11_or_3_Z, implies(not(sK11_or_3_Z), sK11_or_3_Z))), $$true, equiv(sK11_or_3_Z, or(sK11_or_3_Z, sK11_or_3_Z))), $$true, sK11_or_3_Z, or(sK11_or_3_Z, sK11_or_3_Z)))))), $$true)
118.63/118.87	= { by lemma 150 }
118.63/118.87	  $$fresh15(is_a_theorem(implies(implies(not(sK13_or_3_X), sK12_or_3_Y), implies(implies(sK12_or_3_Y, sK11_or_3_Z), implies(or(not(sK13_or_3_X), sK11_or_3_Z), $$fresh($$fresh59($$fresh59($$fresh28($$fresh60(?, ?, implies(and(not(sK11_or_3_Z), not(sK11_or_3_Z)), not(sK11_or_3_Z)), implies(or(not(and(not(sK11_or_3_Z), not(sK11_or_3_Z))), and(not(sK11_or_3_Z), not(sK11_or_3_Z))), or(not(and(not(sK11_or_3_Z), not(sK11_or_3_Z))), not(sK11_or_3_Z)))), $$true, implies(and(not(sK11_or_3_Z), not(sK11_or_3_Z)), and(not(sK11_or_3_Z), not(sK11_or_3_Z))), implies(and(not(sK11_or_3_Z), not(sK11_or_3_Z)), not(sK11_or_3_Z))), $$true, implies(sK11_or_3_Z, implies(not(sK11_or_3_Z), sK11_or_3_Z))), $$true, equiv(sK11_or_3_Z, or(sK11_or_3_Z, sK11_or_3_Z))), $$true, sK11_or_3_Z, or(sK11_or_3_Z, sK11_or_3_Z)))))), $$true)
118.63/118.87	= { by axiom 33 (modus_ponens_2) }
118.63/118.87	  $$fresh15(is_a_theorem(implies(implies(not(sK13_or_3_X), sK12_or_3_Y), implies(implies(sK12_or_3_Y, sK11_or_3_Z), implies(or(not(sK13_or_3_X), sK11_or_3_Z), $$fresh($$fresh59($$fresh59($$fresh28($$fresh59(is_a_theorem(implies(and(not(sK11_or_3_Z), not(sK11_or_3_Z)), not(sK11_or_3_Z))), $$true, implies(or(not(and(not(sK11_or_3_Z), not(sK11_or_3_Z))), and(not(sK11_or_3_Z), not(sK11_or_3_Z))), or(not(and(not(sK11_or_3_Z), not(sK11_or_3_Z))), not(sK11_or_3_Z)))), $$true, implies(and(not(sK11_or_3_Z), not(sK11_or_3_Z)), and(not(sK11_or_3_Z), not(sK11_or_3_Z))), implies(and(not(sK11_or_3_Z), not(sK11_or_3_Z)), not(sK11_or_3_Z))), $$true, implies(sK11_or_3_Z, implies(not(sK11_or_3_Z), sK11_or_3_Z))), $$true, equiv(sK11_or_3_Z, or(sK11_or_3_Z, sK11_or_3_Z))), $$true, sK11_or_3_Z, or(sK11_or_3_Z, sK11_or_3_Z)))))), $$true)
118.63/118.87	= { by axiom 37 (op_and) }
118.63/118.87	  $$fresh15(is_a_theorem(implies(implies(not(sK13_or_3_X), sK12_or_3_Y), implies(implies(sK12_or_3_Y, sK11_or_3_Z), implies(or(not(sK13_or_3_X), sK11_or_3_Z), $$fresh($$fresh59($$fresh59($$fresh28($$fresh59(is_a_theorem(implies($$fresh24($$true, $$true, not(sK11_or_3_Z), not(sK11_or_3_Z)), not(sK11_or_3_Z))), $$true, implies(or(not(and(not(sK11_or_3_Z), not(sK11_or_3_Z))), and(not(sK11_or_3_Z), not(sK11_or_3_Z))), or(not(and(not(sK11_or_3_Z), not(sK11_or_3_Z))), not(sK11_or_3_Z)))), $$true, implies(and(not(sK11_or_3_Z), not(sK11_or_3_Z)), and(not(sK11_or_3_Z), not(sK11_or_3_Z))), implies(and(not(sK11_or_3_Z), not(sK11_or_3_Z)), not(sK11_or_3_Z))), $$true, implies(sK11_or_3_Z, implies(not(sK11_or_3_Z), sK11_or_3_Z))), $$true, equiv(sK11_or_3_Z, or(sK11_or_3_Z, sK11_or_3_Z))), $$true, sK11_or_3_Z, or(sK11_or_3_Z, sK11_or_3_Z)))))), $$true)
118.63/118.87	= { by axiom 126 (principia_op_and) }
118.63/118.87	  $$fresh15(is_a_theorem(implies(implies(not(sK13_or_3_X), sK12_or_3_Y), implies(implies(sK12_or_3_Y, sK11_or_3_Z), implies(or(not(sK13_or_3_X), sK11_or_3_Z), $$fresh($$fresh59($$fresh59($$fresh28($$fresh59(is_a_theorem(implies($$fresh24(op_and, $$true, not(sK11_or_3_Z), not(sK11_or_3_Z)), not(sK11_or_3_Z))), $$true, implies(or(not(and(not(sK11_or_3_Z), not(sK11_or_3_Z))), and(not(sK11_or_3_Z), not(sK11_or_3_Z))), or(not(and(not(sK11_or_3_Z), not(sK11_or_3_Z))), not(sK11_or_3_Z)))), $$true, implies(and(not(sK11_or_3_Z), not(sK11_or_3_Z)), and(not(sK11_or_3_Z), not(sK11_or_3_Z))), implies(and(not(sK11_or_3_Z), not(sK11_or_3_Z)), not(sK11_or_3_Z))), $$true, implies(sK11_or_3_Z, implies(not(sK11_or_3_Z), sK11_or_3_Z))), $$true, equiv(sK11_or_3_Z, or(sK11_or_3_Z, sK11_or_3_Z))), $$true, sK11_or_3_Z, or(sK11_or_3_Z, sK11_or_3_Z)))))), $$true)
118.63/118.87	= { by axiom 114 (op_and) }
118.63/118.87	  $$fresh15(is_a_theorem(implies(implies(not(sK13_or_3_X), sK12_or_3_Y), implies(implies(sK12_or_3_Y, sK11_or_3_Z), implies(or(not(sK13_or_3_X), sK11_or_3_Z), $$fresh($$fresh59($$fresh59($$fresh28($$fresh59(is_a_theorem(implies(not(or(not(not(sK11_or_3_Z)), not(not(sK11_or_3_Z)))), not(sK11_or_3_Z))), $$true, implies(or(not(and(not(sK11_or_3_Z), not(sK11_or_3_Z))), and(not(sK11_or_3_Z), not(sK11_or_3_Z))), or(not(and(not(sK11_or_3_Z), not(sK11_or_3_Z))), not(sK11_or_3_Z)))), $$true, implies(and(not(sK11_or_3_Z), not(sK11_or_3_Z)), and(not(sK11_or_3_Z), not(sK11_or_3_Z))), implies(and(not(sK11_or_3_Z), not(sK11_or_3_Z)), not(sK11_or_3_Z))), $$true, implies(sK11_or_3_Z, implies(not(sK11_or_3_Z), sK11_or_3_Z))), $$true, equiv(sK11_or_3_Z, or(sK11_or_3_Z, sK11_or_3_Z))), $$true, sK11_or_3_Z, or(sK11_or_3_Z, sK11_or_3_Z)))))), $$true)
118.63/118.87	= { by lemma 134 }
118.63/118.88	  $$fresh15(is_a_theorem(implies(implies(not(sK13_or_3_X), sK12_or_3_Y), implies(implies(sK12_or_3_Y, sK11_or_3_Z), implies(or(not(sK13_or_3_X), sK11_or_3_Z), $$fresh($$fresh59($$fresh59($$fresh28($$fresh59(is_a_theorem(or(or(not(not(sK11_or_3_Z)), not(not(sK11_or_3_Z))), not(sK11_or_3_Z))), $$true, implies(or(not(and(not(sK11_or_3_Z), not(sK11_or_3_Z))), and(not(sK11_or_3_Z), not(sK11_or_3_Z))), or(not(and(not(sK11_or_3_Z), not(sK11_or_3_Z))), not(sK11_or_3_Z)))), $$true, implies(and(not(sK11_or_3_Z), not(sK11_or_3_Z)), and(not(sK11_or_3_Z), not(sK11_or_3_Z))), implies(and(not(sK11_or_3_Z), not(sK11_or_3_Z)), not(sK11_or_3_Z))), $$true, implies(sK11_or_3_Z, implies(not(sK11_or_3_Z), sK11_or_3_Z))), $$true, equiv(sK11_or_3_Z, or(sK11_or_3_Z, sK11_or_3_Z))), $$true, sK11_or_3_Z, or(sK11_or_3_Z, sK11_or_3_Z)))))), $$true)
118.63/118.88	= { by lemma 132 }
118.63/118.88	  $$fresh15(is_a_theorem(implies(implies(not(sK13_or_3_X), sK12_or_3_Y), implies(implies(sK12_or_3_Y, sK11_or_3_Z), implies(or(not(sK13_or_3_X), sK11_or_3_Z), $$fresh($$fresh59($$fresh59($$fresh28($$fresh59(is_a_theorem(or(implies(not(sK11_or_3_Z), not(not(sK11_or_3_Z))), not(sK11_or_3_Z))), $$true, implies(or(not(and(not(sK11_or_3_Z), not(sK11_or_3_Z))), and(not(sK11_or_3_Z), not(sK11_or_3_Z))), or(not(and(not(sK11_or_3_Z), not(sK11_or_3_Z))), not(sK11_or_3_Z)))), $$true, implies(and(not(sK11_or_3_Z), not(sK11_or_3_Z)), and(not(sK11_or_3_Z), not(sK11_or_3_Z))), implies(and(not(sK11_or_3_Z), not(sK11_or_3_Z)), not(sK11_or_3_Z))), $$true, implies(sK11_or_3_Z, implies(not(sK11_or_3_Z), sK11_or_3_Z))), $$true, equiv(sK11_or_3_Z, or(sK11_or_3_Z, sK11_or_3_Z))), $$true, sK11_or_3_Z, or(sK11_or_3_Z, sK11_or_3_Z)))))), $$true)
118.63/118.88	= { by axiom 31 (modus_ponens_2) }
118.63/118.88	  $$fresh15(is_a_theorem(implies(implies(not(sK13_or_3_X), sK12_or_3_Y), implies(implies(sK12_or_3_Y, sK11_or_3_Z), implies(or(not(sK13_or_3_X), sK11_or_3_Z), $$fresh($$fresh59($$fresh59($$fresh28($$fresh59($$fresh28($$true, $$true, or(not(sK11_or_3_Z), implies(not(sK11_or_3_Z), not(not(sK11_or_3_Z)))), or(implies(not(sK11_or_3_Z), not(not(sK11_or_3_Z))), not(sK11_or_3_Z))), $$true, implies(or(not(and(not(sK11_or_3_Z), not(sK11_or_3_Z))), and(not(sK11_or_3_Z), not(sK11_or_3_Z))), or(not(and(not(sK11_or_3_Z), not(sK11_or_3_Z))), not(sK11_or_3_Z)))), $$true, implies(and(not(sK11_or_3_Z), not(sK11_or_3_Z)), and(not(sK11_or_3_Z), not(sK11_or_3_Z))), implies(and(not(sK11_or_3_Z), not(sK11_or_3_Z)), not(sK11_or_3_Z))), $$true, implies(sK11_or_3_Z, implies(not(sK11_or_3_Z), sK11_or_3_Z))), $$true, equiv(sK11_or_3_Z, or(sK11_or_3_Z, sK11_or_3_Z))), $$true, sK11_or_3_Z, or(sK11_or_3_Z, sK11_or_3_Z)))))), $$true)
118.63/118.88	= { by axiom 53 (r3_1) }
118.63/118.88	  $$fresh15(is_a_theorem(implies(implies(not(sK13_or_3_X), sK12_or_3_Y), implies(implies(sK12_or_3_Y, sK11_or_3_Z), implies(or(not(sK13_or_3_X), sK11_or_3_Z), $$fresh($$fresh59($$fresh59($$fresh28($$fresh59($$fresh28($$fresh8($$true, $$true, not(sK11_or_3_Z), implies(not(sK11_or_3_Z), not(not(sK11_or_3_Z)))), $$true, or(not(sK11_or_3_Z), implies(not(sK11_or_3_Z), not(not(sK11_or_3_Z)))), or(implies(not(sK11_or_3_Z), not(not(sK11_or_3_Z))), not(sK11_or_3_Z))), $$true, implies(or(not(and(not(sK11_or_3_Z), not(sK11_or_3_Z))), and(not(sK11_or_3_Z), not(sK11_or_3_Z))), or(not(and(not(sK11_or_3_Z), not(sK11_or_3_Z))), not(sK11_or_3_Z)))), $$true, implies(and(not(sK11_or_3_Z), not(sK11_or_3_Z)), and(not(sK11_or_3_Z), not(sK11_or_3_Z))), implies(and(not(sK11_or_3_Z), not(sK11_or_3_Z)), not(sK11_or_3_Z))), $$true, implies(sK11_or_3_Z, implies(not(sK11_or_3_Z), sK11_or_3_Z))), $$true, equiv(sK11_or_3_Z, or(sK11_or_3_Z, sK11_or_3_Z))), $$true, sK11_or_3_Z, or(sK11_or_3_Z, sK11_or_3_Z)))))), $$true)
118.63/118.88	= { by axiom 122 (principia_r3) }
118.63/118.88	  $$fresh15(is_a_theorem(implies(implies(not(sK13_or_3_X), sK12_or_3_Y), implies(implies(sK12_or_3_Y, sK11_or_3_Z), implies(or(not(sK13_or_3_X), sK11_or_3_Z), $$fresh($$fresh59($$fresh59($$fresh28($$fresh59($$fresh28($$fresh8(r3, $$true, not(sK11_or_3_Z), implies(not(sK11_or_3_Z), not(not(sK11_or_3_Z)))), $$true, or(not(sK11_or_3_Z), implies(not(sK11_or_3_Z), not(not(sK11_or_3_Z)))), or(implies(not(sK11_or_3_Z), not(not(sK11_or_3_Z))), not(sK11_or_3_Z))), $$true, implies(or(not(and(not(sK11_or_3_Z), not(sK11_or_3_Z))), and(not(sK11_or_3_Z), not(sK11_or_3_Z))), or(not(and(not(sK11_or_3_Z), not(sK11_or_3_Z))), not(sK11_or_3_Z)))), $$true, implies(and(not(sK11_or_3_Z), not(sK11_or_3_Z)), and(not(sK11_or_3_Z), not(sK11_or_3_Z))), implies(and(not(sK11_or_3_Z), not(sK11_or_3_Z)), not(sK11_or_3_Z))), $$true, implies(sK11_or_3_Z, implies(not(sK11_or_3_Z), sK11_or_3_Z))), $$true, equiv(sK11_or_3_Z, or(sK11_or_3_Z, sK11_or_3_Z))), $$true, sK11_or_3_Z, or(sK11_or_3_Z, sK11_or_3_Z)))))), $$true)
118.63/118.88	= { by axiom 97 (r3_1) }
118.63/118.88	  $$fresh15(is_a_theorem(implies(implies(not(sK13_or_3_X), sK12_or_3_Y), implies(implies(sK12_or_3_Y, sK11_or_3_Z), implies(or(not(sK13_or_3_X), sK11_or_3_Z), $$fresh($$fresh59($$fresh59($$fresh28($$fresh59($$fresh28(is_a_theorem(implies(or(not(sK11_or_3_Z), implies(not(sK11_or_3_Z), not(not(sK11_or_3_Z)))), or(implies(not(sK11_or_3_Z), not(not(sK11_or_3_Z))), not(sK11_or_3_Z)))), $$true, or(not(sK11_or_3_Z), implies(not(sK11_or_3_Z), not(not(sK11_or_3_Z)))), or(implies(not(sK11_or_3_Z), not(not(sK11_or_3_Z))), not(sK11_or_3_Z))), $$true, implies(or(not(and(not(sK11_or_3_Z), not(sK11_or_3_Z))), and(not(sK11_or_3_Z), not(sK11_or_3_Z))), or(not(and(not(sK11_or_3_Z), not(sK11_or_3_Z))), not(sK11_or_3_Z)))), $$true, implies(and(not(sK11_or_3_Z), not(sK11_or_3_Z)), and(not(sK11_or_3_Z), not(sK11_or_3_Z))), implies(and(not(sK11_or_3_Z), not(sK11_or_3_Z)), not(sK11_or_3_Z))), $$true, implies(sK11_or_3_Z, implies(not(sK11_or_3_Z), sK11_or_3_Z))), $$true, equiv(sK11_or_3_Z, or(sK11_or_3_Z, sK11_or_3_Z))), $$true, sK11_or_3_Z, or(sK11_or_3_Z, sK11_or_3_Z)))))), $$true)
118.63/118.88	= { by lemma 146 }
118.63/118.88	  $$fresh15(is_a_theorem(implies(implies(not(sK13_or_3_X), sK12_or_3_Y), implies(implies(sK12_or_3_Y, sK11_or_3_Z), implies(or(not(sK13_or_3_X), sK11_or_3_Z), $$fresh($$fresh59($$fresh59($$fresh28($$fresh59($$fresh60(?, ?, or(not(sK11_or_3_Z), implies(not(sK11_or_3_Z), not(not(sK11_or_3_Z)))), or(implies(not(sK11_or_3_Z), not(not(sK11_or_3_Z))), not(sK11_or_3_Z))), $$true, implies(or(not(and(not(sK11_or_3_Z), not(sK11_or_3_Z))), and(not(sK11_or_3_Z), not(sK11_or_3_Z))), or(not(and(not(sK11_or_3_Z), not(sK11_or_3_Z))), not(sK11_or_3_Z)))), $$true, implies(and(not(sK11_or_3_Z), not(sK11_or_3_Z)), and(not(sK11_or_3_Z), not(sK11_or_3_Z))), implies(and(not(sK11_or_3_Z), not(sK11_or_3_Z)), not(sK11_or_3_Z))), $$true, implies(sK11_or_3_Z, implies(not(sK11_or_3_Z), sK11_or_3_Z))), $$true, equiv(sK11_or_3_Z, or(sK11_or_3_Z, sK11_or_3_Z))), $$true, sK11_or_3_Z, or(sK11_or_3_Z, sK11_or_3_Z)))))), $$true)
118.63/118.88	= { by lemma 134 }
118.63/118.88	  $$fresh15(is_a_theorem(implies(implies(not(sK13_or_3_X), sK12_or_3_Y), implies(implies(sK12_or_3_Y, sK11_or_3_Z), implies(or(not(sK13_or_3_X), sK11_or_3_Z), $$fresh($$fresh59($$fresh59($$fresh28($$fresh59($$fresh60(?, ?, implies(not(not(sK11_or_3_Z)), implies(not(sK11_or_3_Z), not(not(sK11_or_3_Z)))), or(implies(not(sK11_or_3_Z), not(not(sK11_or_3_Z))), not(sK11_or_3_Z))), $$true, implies(or(not(and(not(sK11_or_3_Z), not(sK11_or_3_Z))), and(not(sK11_or_3_Z), not(sK11_or_3_Z))), or(not(and(not(sK11_or_3_Z), not(sK11_or_3_Z))), not(sK11_or_3_Z)))), $$true, implies(and(not(sK11_or_3_Z), not(sK11_or_3_Z)), and(not(sK11_or_3_Z), not(sK11_or_3_Z))), implies(and(not(sK11_or_3_Z), not(sK11_or_3_Z)), not(sK11_or_3_Z))), $$true, implies(sK11_or_3_Z, implies(not(sK11_or_3_Z), sK11_or_3_Z))), $$true, equiv(sK11_or_3_Z, or(sK11_or_3_Z, sK11_or_3_Z))), $$true, sK11_or_3_Z, or(sK11_or_3_Z, sK11_or_3_Z)))))), $$true)
118.63/118.88	= { by lemma 132 }
118.63/118.88	  $$fresh15(is_a_theorem(implies(implies(not(sK13_or_3_X), sK12_or_3_Y), implies(implies(sK12_or_3_Y, sK11_or_3_Z), implies(or(not(sK13_or_3_X), sK11_or_3_Z), $$fresh($$fresh59($$fresh59($$fresh28($$fresh59($$fresh60(?, ?, implies(not(not(sK11_or_3_Z)), or(not(not(sK11_or_3_Z)), not(not(sK11_or_3_Z)))), or(implies(not(sK11_or_3_Z), not(not(sK11_or_3_Z))), not(sK11_or_3_Z))), $$true, implies(or(not(and(not(sK11_or_3_Z), not(sK11_or_3_Z))), and(not(sK11_or_3_Z), not(sK11_or_3_Z))), or(not(and(not(sK11_or_3_Z), not(sK11_or_3_Z))), not(sK11_or_3_Z)))), $$true, implies(and(not(sK11_or_3_Z), not(sK11_or_3_Z)), and(not(sK11_or_3_Z), not(sK11_or_3_Z))), implies(and(not(sK11_or_3_Z), not(sK11_or_3_Z)), not(sK11_or_3_Z))), $$true, implies(sK11_or_3_Z, implies(not(sK11_or_3_Z), sK11_or_3_Z))), $$true, equiv(sK11_or_3_Z, or(sK11_or_3_Z, sK11_or_3_Z))), $$true, sK11_or_3_Z, or(sK11_or_3_Z, sK11_or_3_Z)))))), $$true)
118.63/118.88	= { by lemma 140 }
118.63/118.88	  $$fresh15(is_a_theorem(implies(implies(not(sK13_or_3_X), sK12_or_3_Y), implies(implies(sK12_or_3_Y, sK11_or_3_Z), implies(or(not(sK13_or_3_X), sK11_or_3_Z), $$fresh($$fresh59($$fresh59($$fresh28($$fresh59($$true, $$true, implies(or(not(and(not(sK11_or_3_Z), not(sK11_or_3_Z))), and(not(sK11_or_3_Z), not(sK11_or_3_Z))), or(not(and(not(sK11_or_3_Z), not(sK11_or_3_Z))), not(sK11_or_3_Z)))), $$true, implies(and(not(sK11_or_3_Z), not(sK11_or_3_Z)), and(not(sK11_or_3_Z), not(sK11_or_3_Z))), implies(and(not(sK11_or_3_Z), not(sK11_or_3_Z)), not(sK11_or_3_Z))), $$true, implies(sK11_or_3_Z, implies(not(sK11_or_3_Z), sK11_or_3_Z))), $$true, equiv(sK11_or_3_Z, or(sK11_or_3_Z, sK11_or_3_Z))), $$true, sK11_or_3_Z, or(sK11_or_3_Z, sK11_or_3_Z)))))), $$true)
118.63/118.88	= { by axiom 32 (modus_ponens_2) }
118.63/118.88	  $$fresh15(is_a_theorem(implies(implies(not(sK13_or_3_X), sK12_or_3_Y), implies(implies(sK12_or_3_Y, sK11_or_3_Z), implies(or(not(sK13_or_3_X), sK11_or_3_Z), $$fresh($$fresh59($$fresh59($$fresh28($$true, $$true, implies(and(not(sK11_or_3_Z), not(sK11_or_3_Z)), and(not(sK11_or_3_Z), not(sK11_or_3_Z))), implies(and(not(sK11_or_3_Z), not(sK11_or_3_Z)), not(sK11_or_3_Z))), $$true, implies(sK11_or_3_Z, implies(not(sK11_or_3_Z), sK11_or_3_Z))), $$true, equiv(sK11_or_3_Z, or(sK11_or_3_Z, sK11_or_3_Z))), $$true, sK11_or_3_Z, or(sK11_or_3_Z, sK11_or_3_Z)))))), $$true)
118.63/118.88	= { by axiom 31 (modus_ponens_2) }
118.63/118.88	  $$fresh15(is_a_theorem(implies(implies(not(sK13_or_3_X), sK12_or_3_Y), implies(implies(sK12_or_3_Y, sK11_or_3_Z), implies(or(not(sK13_or_3_X), sK11_or_3_Z), $$fresh($$fresh59($$fresh59(is_a_theorem(implies(and(not(sK11_or_3_Z), not(sK11_or_3_Z)), not(sK11_or_3_Z))), $$true, implies(sK11_or_3_Z, implies(not(sK11_or_3_Z), sK11_or_3_Z))), $$true, equiv(sK11_or_3_Z, or(sK11_or_3_Z, sK11_or_3_Z))), $$true, sK11_or_3_Z, or(sK11_or_3_Z, sK11_or_3_Z)))))), $$true)
118.63/118.88	= { by lemma 132 }
118.63/118.88	  $$fresh15(is_a_theorem(implies(implies(not(sK13_or_3_X), sK12_or_3_Y), implies(implies(sK12_or_3_Y, sK11_or_3_Z), implies(or(not(sK13_or_3_X), sK11_or_3_Z), $$fresh($$fresh59($$fresh59(is_a_theorem(or(not(and(not(sK11_or_3_Z), not(sK11_or_3_Z))), not(sK11_or_3_Z))), $$true, implies(sK11_or_3_Z, implies(not(sK11_or_3_Z), sK11_or_3_Z))), $$true, equiv(sK11_or_3_Z, or(sK11_or_3_Z, sK11_or_3_Z))), $$true, sK11_or_3_Z, or(sK11_or_3_Z, sK11_or_3_Z)))))), $$true)
118.63/118.88	= { by lemma 133 }
118.63/118.88	  $$fresh15(is_a_theorem(implies(implies(not(sK13_or_3_X), sK12_or_3_Y), implies(implies(sK12_or_3_Y, sK11_or_3_Z), implies(or(not(sK13_or_3_X), sK11_or_3_Z), $$fresh($$fresh59($$fresh59(is_a_theorem(or(implies(not(sK11_or_3_Z), sK11_or_3_Z), not(sK11_or_3_Z))), $$true, implies(sK11_or_3_Z, implies(not(sK11_or_3_Z), sK11_or_3_Z))), $$true, equiv(sK11_or_3_Z, or(sK11_or_3_Z, sK11_or_3_Z))), $$true, sK11_or_3_Z, or(sK11_or_3_Z, sK11_or_3_Z)))))), $$true)
118.63/118.88	= { by axiom 33 (modus_ponens_2) }
118.63/118.88	  $$fresh15(is_a_theorem(implies(implies(not(sK13_or_3_X), sK12_or_3_Y), implies(implies(sK12_or_3_Y, sK11_or_3_Z), implies(or(not(sK13_or_3_X), sK11_or_3_Z), $$fresh($$fresh59($$fresh60(?, ?, or(implies(not(sK11_or_3_Z), sK11_or_3_Z), not(sK11_or_3_Z)), implies(sK11_or_3_Z, implies(not(sK11_or_3_Z), sK11_or_3_Z))), $$true, equiv(sK11_or_3_Z, or(sK11_or_3_Z, sK11_or_3_Z))), $$true, sK11_or_3_Z, or(sK11_or_3_Z, sK11_or_3_Z)))))), $$true)
118.63/118.88	= { by lemma 141 }
118.63/118.88	  $$fresh15(is_a_theorem(implies(implies(not(sK13_or_3_X), sK12_or_3_Y), implies(implies(sK12_or_3_Y, sK11_or_3_Z), implies(or(not(sK13_or_3_X), sK11_or_3_Z), $$fresh($$fresh59(is_a_theorem(implies(sK11_or_3_Z, implies(not(sK11_or_3_Z), sK11_or_3_Z))), $$true, equiv(sK11_or_3_Z, or(sK11_or_3_Z, sK11_or_3_Z))), $$true, sK11_or_3_Z, or(sK11_or_3_Z, sK11_or_3_Z)))))), $$true)
118.63/118.88	= { by lemma 134 }
118.63/118.88	  $$fresh15(is_a_theorem(implies(implies(not(sK13_or_3_X), sK12_or_3_Y), implies(implies(sK12_or_3_Y, sK11_or_3_Z), implies(or(not(sK13_or_3_X), sK11_or_3_Z), $$fresh($$fresh59(is_a_theorem(implies(sK11_or_3_Z, or(sK11_or_3_Z, sK11_or_3_Z))), $$true, equiv(sK11_or_3_Z, or(sK11_or_3_Z, sK11_or_3_Z))), $$true, sK11_or_3_Z, or(sK11_or_3_Z, sK11_or_3_Z)))))), $$true)
118.63/118.88	= { by axiom 33 (modus_ponens_2) }
118.63/118.88	  $$fresh15(is_a_theorem(implies(implies(not(sK13_or_3_X), sK12_or_3_Y), implies(implies(sK12_or_3_Y, sK11_or_3_Z), implies(or(not(sK13_or_3_X), sK11_or_3_Z), $$fresh($$fresh60(?, ?, implies(sK11_or_3_Z, or(sK11_or_3_Z, sK11_or_3_Z)), equiv(sK11_or_3_Z, or(sK11_or_3_Z, sK11_or_3_Z))), $$true, sK11_or_3_Z, or(sK11_or_3_Z, sK11_or_3_Z)))))), $$true)
118.63/118.88	= { by lemma 146 }
118.63/118.88	  $$fresh15(is_a_theorem(implies(implies(not(sK13_or_3_X), sK12_or_3_Y), implies(implies(sK12_or_3_Y, sK11_or_3_Z), implies(or(not(sK13_or_3_X), sK11_or_3_Z), $$fresh($$fresh28(is_a_theorem(implies(implies(sK11_or_3_Z, or(sK11_or_3_Z, sK11_or_3_Z)), equiv(sK11_or_3_Z, or(sK11_or_3_Z, sK11_or_3_Z)))), $$true, implies(sK11_or_3_Z, or(sK11_or_3_Z, sK11_or_3_Z)), equiv(sK11_or_3_Z, or(sK11_or_3_Z, sK11_or_3_Z))), $$true, sK11_or_3_Z, or(sK11_or_3_Z, sK11_or_3_Z)))))), $$true)
118.63/118.88	= { by lemma 147 }
118.63/118.88	  $$fresh15(is_a_theorem(implies(implies(not(sK13_or_3_X), sK12_or_3_Y), implies(implies(sK12_or_3_Y, sK11_or_3_Z), implies(or(not(sK13_or_3_X), sK11_or_3_Z), $$fresh($$fresh28(is_a_theorem(implies(implies(sK11_or_3_Z, or(sK11_or_3_Z, sK11_or_3_Z)), and(implies(sK11_or_3_Z, or(sK11_or_3_Z, sK11_or_3_Z)), implies(or(sK11_or_3_Z, sK11_or_3_Z), sK11_or_3_Z)))), $$true, implies(sK11_or_3_Z, or(sK11_or_3_Z, sK11_or_3_Z)), equiv(sK11_or_3_Z, or(sK11_or_3_Z, sK11_or_3_Z))), $$true, sK11_or_3_Z, or(sK11_or_3_Z, sK11_or_3_Z)))))), $$true)
118.63/118.88	= { by axiom 31 (modus_ponens_2) }
118.63/118.88	  $$fresh15(is_a_theorem(implies(implies(not(sK13_or_3_X), sK12_or_3_Y), implies(implies(sK12_or_3_Y, sK11_or_3_Z), implies(or(not(sK13_or_3_X), sK11_or_3_Z), $$fresh($$fresh28($$fresh28($$true, $$true, implies(or(sK11_or_3_Z, sK11_or_3_Z), sK11_or_3_Z), implies(implies(sK11_or_3_Z, or(sK11_or_3_Z, sK11_or_3_Z)), and(implies(sK11_or_3_Z, or(sK11_or_3_Z, sK11_or_3_Z)), implies(or(sK11_or_3_Z, sK11_or_3_Z), sK11_or_3_Z)))), $$true, implies(sK11_or_3_Z, or(sK11_or_3_Z, sK11_or_3_Z)), equiv(sK11_or_3_Z, or(sK11_or_3_Z, sK11_or_3_Z))), $$true, sK11_or_3_Z, or(sK11_or_3_Z, sK11_or_3_Z)))))), $$true)
118.63/118.88	= { by lemma 152 }
118.63/118.88	  $$fresh15(is_a_theorem(implies(implies(not(sK13_or_3_X), sK12_or_3_Y), implies(implies(sK12_or_3_Y, sK11_or_3_Z), implies(or(not(sK13_or_3_X), sK11_or_3_Z), $$fresh($$fresh28($$fresh28(is_a_theorem(implies(implies(or(sK11_or_3_Z, sK11_or_3_Z), sK11_or_3_Z), implies(implies(sK11_or_3_Z, or(sK11_or_3_Z, sK11_or_3_Z)), and(implies(sK11_or_3_Z, or(sK11_or_3_Z, sK11_or_3_Z)), implies(or(sK11_or_3_Z, sK11_or_3_Z), sK11_or_3_Z))))), $$true, implies(or(sK11_or_3_Z, sK11_or_3_Z), sK11_or_3_Z), implies(implies(sK11_or_3_Z, or(sK11_or_3_Z, sK11_or_3_Z)), and(implies(sK11_or_3_Z, or(sK11_or_3_Z, sK11_or_3_Z)), implies(or(sK11_or_3_Z, sK11_or_3_Z), sK11_or_3_Z)))), $$true, implies(sK11_or_3_Z, or(sK11_or_3_Z, sK11_or_3_Z)), equiv(sK11_or_3_Z, or(sK11_or_3_Z, sK11_or_3_Z))), $$true, sK11_or_3_Z, or(sK11_or_3_Z, sK11_or_3_Z)))))), $$true)
118.63/118.88	= { by lemma 146 }
118.63/118.88	  $$fresh15(is_a_theorem(implies(implies(not(sK13_or_3_X), sK12_or_3_Y), implies(implies(sK12_or_3_Y, sK11_or_3_Z), implies(or(not(sK13_or_3_X), sK11_or_3_Z), $$fresh($$fresh28($$fresh60(?, ?, implies(or(sK11_or_3_Z, sK11_or_3_Z), sK11_or_3_Z), implies(implies(sK11_or_3_Z, or(sK11_or_3_Z, sK11_or_3_Z)), and(implies(sK11_or_3_Z, or(sK11_or_3_Z, sK11_or_3_Z)), implies(or(sK11_or_3_Z, sK11_or_3_Z), sK11_or_3_Z)))), $$true, implies(sK11_or_3_Z, or(sK11_or_3_Z, sK11_or_3_Z)), equiv(sK11_or_3_Z, or(sK11_or_3_Z, sK11_or_3_Z))), $$true, sK11_or_3_Z, or(sK11_or_3_Z, sK11_or_3_Z)))))), $$true)
118.63/118.88	= { by lemma 139 }
118.63/118.88	  $$fresh15(is_a_theorem(implies(implies(not(sK13_or_3_X), sK12_or_3_Y), implies(implies(sK12_or_3_Y, sK11_or_3_Z), implies(or(not(sK13_or_3_X), sK11_or_3_Z), $$fresh($$fresh28($$true, $$true, implies(sK11_or_3_Z, or(sK11_or_3_Z, sK11_or_3_Z)), equiv(sK11_or_3_Z, or(sK11_or_3_Z, sK11_or_3_Z))), $$true, sK11_or_3_Z, or(sK11_or_3_Z, sK11_or_3_Z)))))), $$true)
118.63/118.88	= { by axiom 31 (modus_ponens_2) }
118.63/118.88	  $$fresh15(is_a_theorem(implies(implies(not(sK13_or_3_X), sK12_or_3_Y), implies(implies(sK12_or_3_Y, sK11_or_3_Z), implies(or(not(sK13_or_3_X), sK11_or_3_Z), $$fresh(is_a_theorem(equiv(sK11_or_3_Z, or(sK11_or_3_Z, sK11_or_3_Z))), $$true, sK11_or_3_Z, or(sK11_or_3_Z, sK11_or_3_Z)))))), $$true)
118.63/118.88	= { by lemma 143 }
118.63/118.88	  $$fresh15(is_a_theorem(implies(implies(not(sK13_or_3_X), sK12_or_3_Y), implies(implies(sK12_or_3_Y, sK11_or_3_Z), implies(or(not(sK13_or_3_X), sK11_or_3_Z), or(sK11_or_3_Z, sK11_or_3_Z))))), $$true)
118.63/118.88	= { by lemma 156 }
118.63/118.88	  $$fresh15(is_a_theorem(implies(implies(not(sK13_or_3_X), sK12_or_3_Y), implies(or(not(sK13_or_3_X), sK11_or_3_Z), implies(implies(sK12_or_3_Y, sK11_or_3_Z), or(sK11_or_3_Z, sK11_or_3_Z))))), $$true)
118.63/118.88	= { by lemma 156 }
118.63/118.88	  $$fresh15(is_a_theorem(implies(or(not(sK13_or_3_X), sK11_or_3_Z), implies(implies(not(sK13_or_3_X), sK12_or_3_Y), implies(implies(sK12_or_3_Y, sK11_or_3_Z), or(sK11_or_3_Z, sK11_or_3_Z))))), $$true)
118.63/118.88	= { by lemma 154 }
118.63/118.88	  $$fresh15(is_a_theorem(implies(or(sK11_or_3_Z, not(sK13_or_3_X)), implies(implies(not(sK13_or_3_X), sK12_or_3_Y), implies(implies(sK12_or_3_Y, sK11_or_3_Z), or(sK11_or_3_Z, sK11_or_3_Z))))), $$true)
118.63/118.88	= { by lemma 155 }
118.63/118.88	  $$fresh15(is_a_theorem(implies(or(sK11_or_3_Z, not(sK13_or_3_X)), implies(implies(not(sK13_or_3_X), sK12_or_3_Y), or(sK11_or_3_Z, implies(implies(sK12_or_3_Y, sK11_or_3_Z), sK11_or_3_Z))))), $$true)
118.63/118.88	= { by lemma 155 }
118.63/118.88	  $$fresh15(is_a_theorem(implies(or(sK11_or_3_Z, not(sK13_or_3_X)), or(sK11_or_3_Z, implies(implies(not(sK13_or_3_X), sK12_or_3_Y), implies(implies(sK12_or_3_Y, sK11_or_3_Z), sK11_or_3_Z))))), $$true)
118.63/118.88	= { by axiom 31 (modus_ponens_2) }
118.63/118.88	  $$fresh15($$fresh28($$true, $$true, implies(not(sK13_or_3_X), implies(implies(not(sK13_or_3_X), sK12_or_3_Y), implies(implies(sK12_or_3_Y, sK11_or_3_Z), sK11_or_3_Z))), implies(or(sK11_or_3_Z, not(sK13_or_3_X)), or(sK11_or_3_Z, implies(implies(not(sK13_or_3_X), sK12_or_3_Y), implies(implies(sK12_or_3_Y, sK11_or_3_Z), sK11_or_3_Z))))), $$true)
118.63/118.88	= { by lemma 144 }
118.63/118.88	  $$fresh15($$fresh28(is_a_theorem(implies(implies(not(sK13_or_3_X), implies(implies(not(sK13_or_3_X), sK12_or_3_Y), implies(implies(sK12_or_3_Y, sK11_or_3_Z), sK11_or_3_Z))), implies(or(sK11_or_3_Z, not(sK13_or_3_X)), or(sK11_or_3_Z, implies(implies(not(sK13_or_3_X), sK12_or_3_Y), implies(implies(sK12_or_3_Y, sK11_or_3_Z), sK11_or_3_Z)))))), $$true, implies(not(sK13_or_3_X), implies(implies(not(sK13_or_3_X), sK12_or_3_Y), implies(implies(sK12_or_3_Y, sK11_or_3_Z), sK11_or_3_Z))), implies(or(sK11_or_3_Z, not(sK13_or_3_X)), or(sK11_or_3_Z, implies(implies(not(sK13_or_3_X), sK12_or_3_Y), implies(implies(sK12_or_3_Y, sK11_or_3_Z), sK11_or_3_Z))))), $$true)
118.63/118.88	= { by lemma 146 }
118.63/118.88	  $$fresh15($$fresh60(?, ?, implies(not(sK13_or_3_X), implies(implies(not(sK13_or_3_X), sK12_or_3_Y), implies(implies(sK12_or_3_Y, sK11_or_3_Z), sK11_or_3_Z))), implies(or(sK11_or_3_Z, not(sK13_or_3_X)), or(sK11_or_3_Z, implies(implies(not(sK13_or_3_X), sK12_or_3_Y), implies(implies(sK12_or_3_Y, sK11_or_3_Z), sK11_or_3_Z))))), $$true)
118.63/118.88	= { by lemma 156 }
118.63/118.88	  $$fresh15($$fresh60(?, ?, implies(not(sK13_or_3_X), implies(implies(sK12_or_3_Y, sK11_or_3_Z), implies(implies(not(sK13_or_3_X), sK12_or_3_Y), sK11_or_3_Z))), implies(or(sK11_or_3_Z, not(sK13_or_3_X)), or(sK11_or_3_Z, implies(implies(not(sK13_or_3_X), sK12_or_3_Y), implies(implies(sK12_or_3_Y, sK11_or_3_Z), sK11_or_3_Z))))), $$true)
118.63/118.88	= { by lemma 156 }
118.63/118.88	  $$fresh15($$fresh60(?, ?, implies(implies(sK12_or_3_Y, sK11_or_3_Z), implies(not(sK13_or_3_X), implies(implies(not(sK13_or_3_X), sK12_or_3_Y), sK11_or_3_Z))), implies(or(sK11_or_3_Z, not(sK13_or_3_X)), or(sK11_or_3_Z, implies(implies(not(sK13_or_3_X), sK12_or_3_Y), implies(implies(sK12_or_3_Y, sK11_or_3_Z), sK11_or_3_Z))))), $$true)
118.63/118.88	= { by lemma 156 }
118.63/118.88	  $$fresh15($$fresh60(?, ?, implies(implies(sK12_or_3_Y, sK11_or_3_Z), implies(implies(not(sK13_or_3_X), sK12_or_3_Y), implies(not(sK13_or_3_X), sK11_or_3_Z))), implies(or(sK11_or_3_Z, not(sK13_or_3_X)), or(sK11_or_3_Z, implies(implies(not(sK13_or_3_X), sK12_or_3_Y), implies(implies(sK12_or_3_Y, sK11_or_3_Z), sK11_or_3_Z))))), $$true)
118.63/118.88	= { by lemma 132 }
118.63/118.88	  $$fresh15($$fresh60(?, ?, implies(implies(sK12_or_3_Y, sK11_or_3_Z), implies(or(not(not(sK13_or_3_X)), sK12_or_3_Y), implies(not(sK13_or_3_X), sK11_or_3_Z))), implies(or(sK11_or_3_Z, not(sK13_or_3_X)), or(sK11_or_3_Z, implies(implies(not(sK13_or_3_X), sK12_or_3_Y), implies(implies(sK12_or_3_Y, sK11_or_3_Z), sK11_or_3_Z))))), $$true)
118.63/118.88	= { by lemma 132 }
118.63/118.88	  $$fresh15($$fresh60(?, ?, implies(implies(sK12_or_3_Y, sK11_or_3_Z), implies(or(not(not(sK13_or_3_X)), sK12_or_3_Y), or(not(not(sK13_or_3_X)), sK11_or_3_Z))), implies(or(sK11_or_3_Z, not(sK13_or_3_X)), or(sK11_or_3_Z, implies(implies(not(sK13_or_3_X), sK12_or_3_Y), implies(implies(sK12_or_3_Y, sK11_or_3_Z), sK11_or_3_Z))))), $$true)
118.63/118.88	= { by lemma 154 }
118.63/118.88	  $$fresh15($$fresh60(?, ?, implies(implies(sK12_or_3_Y, sK11_or_3_Z), implies(or(not(not(sK13_or_3_X)), sK12_or_3_Y), or(not(not(sK13_or_3_X)), sK11_or_3_Z))), implies(or(not(sK13_or_3_X), sK11_or_3_Z), or(sK11_or_3_Z, implies(implies(not(sK13_or_3_X), sK12_or_3_Y), implies(implies(sK12_or_3_Y, sK11_or_3_Z), sK11_or_3_Z))))), $$true)
118.63/118.88	= { by lemma 154 }
118.63/118.88	  $$fresh15($$fresh60(?, ?, implies(implies(sK12_or_3_Y, sK11_or_3_Z), implies(or(not(not(sK13_or_3_X)), sK12_or_3_Y), or(not(not(sK13_or_3_X)), sK11_or_3_Z))), implies(or(not(sK13_or_3_X), sK11_or_3_Z), or(implies(implies(not(sK13_or_3_X), sK12_or_3_Y), implies(implies(sK12_or_3_Y, sK11_or_3_Z), sK11_or_3_Z)), sK11_or_3_Z))), $$true)
118.63/118.88	= { by axiom 33 (modus_ponens_2) }
118.63/118.88	  $$fresh15($$fresh59(is_a_theorem(implies(implies(sK12_or_3_Y, sK11_or_3_Z), implies(or(not(not(sK13_or_3_X)), sK12_or_3_Y), or(not(not(sK13_or_3_X)), sK11_or_3_Z)))), $$true, implies(or(not(sK13_or_3_X), sK11_or_3_Z), or(implies(implies(not(sK13_or_3_X), sK12_or_3_Y), implies(implies(sK12_or_3_Y, sK11_or_3_Z), sK11_or_3_Z)), sK11_or_3_Z))), $$true)
118.63/118.88	= { by lemma 144 }
118.63/118.88	  $$fresh15($$fresh59($$true, $$true, implies(or(not(sK13_or_3_X), sK11_or_3_Z), or(implies(implies(not(sK13_or_3_X), sK12_or_3_Y), implies(implies(sK12_or_3_Y, sK11_or_3_Z), sK11_or_3_Z)), sK11_or_3_Z))), $$true)
118.63/118.88	= { by axiom 32 (modus_ponens_2) }
118.63/118.88	  $$fresh15($$true, $$true)
118.63/118.88	= { by axiom 46 (or_3) }
118.63/118.88	  $$true
118.63/118.88	% SZS output end Proof
118.63/118.88	
118.63/118.88	RESULT: Theorem (the conjecture is true).
118.63/118.89	EOF