TSTP Solution File: LCL501+1 by Twee---2.4.2
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- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : LCL501+1 : TPTP v8.1.2. Released v3.3.0.
% Transfm : none
% Format : tptp:raw
% Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% Computer : n006.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Thu Aug 31 08:19:14 EDT 2023
% Result : Theorem 111.45s 14.67s
% Output : Proof 112.49s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.13 % Problem : LCL501+1 : TPTP v8.1.2. Released v3.3.0.
% 0.00/0.14 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.15/0.36 % Computer : n006.cluster.edu
% 0.15/0.36 % Model : x86_64 x86_64
% 0.15/0.36 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.15/0.36 % Memory : 8042.1875MB
% 0.15/0.36 % OS : Linux 3.10.0-693.el7.x86_64
% 0.15/0.36 % CPULimit : 300
% 0.15/0.36 % WCLimit : 300
% 0.15/0.36 % DateTime : Thu Aug 24 20:33:07 EDT 2023
% 0.15/0.36 % CPUTime :
% 111.45/14.67 Command-line arguments: --lhs-weight 1 --flip-ordering --normalise-queue-percent 10 --cp-renormalise-threshold 10
% 111.45/14.67
% 111.45/14.67 % SZS status Theorem
% 111.45/14.67
% 111.45/14.73 % SZS output start Proof
% 111.45/14.73 Take the following subset of the input axioms:
% 111.45/14.74 fof(kn1, axiom, kn1 <=> ![P]: is_a_theorem(implies(P, and(P, P)))).
% 111.45/14.74 fof(kn3, axiom, kn3 <=> ![Q, R, P2]: is_a_theorem(implies(implies(P2, Q), implies(not(and(Q, R)), not(and(R, P2)))))).
% 111.45/14.74 fof(modus_ponens, axiom, modus_ponens <=> ![X, Y]: ((is_a_theorem(X) & is_a_theorem(implies(X, Y))) => is_a_theorem(Y))).
% 111.45/14.74 fof(op_and, axiom, op_and => ![X2, Y2]: and(X2, Y2)=not(or(not(X2), not(Y2)))).
% 111.45/14.74 fof(op_equiv, axiom, op_equiv => ![X2, Y2]: equiv(X2, Y2)=and(implies(X2, Y2), implies(Y2, X2))).
% 111.45/14.74 fof(op_implies_and, axiom, op_implies_and => ![X2, Y2]: implies(X2, Y2)=not(and(X2, not(Y2)))).
% 111.45/14.74 fof(op_implies_or, axiom, op_implies_or => ![X2, Y2]: implies(X2, Y2)=or(not(X2), Y2)).
% 111.45/14.74 fof(op_or, axiom, op_or => ![X2, Y2]: or(X2, Y2)=not(and(not(X2), not(Y2)))).
% 111.45/14.74 fof(principia_modus_ponens, axiom, modus_ponens).
% 111.45/14.74 fof(principia_op_and, axiom, op_and).
% 111.45/14.74 fof(principia_op_equiv, axiom, op_equiv).
% 111.45/14.74 fof(principia_op_implies_or, axiom, op_implies_or).
% 111.45/14.74 fof(principia_r1, axiom, r1).
% 111.45/14.74 fof(principia_r2, axiom, r2).
% 111.45/14.74 fof(principia_r3, axiom, r3).
% 111.45/14.74 fof(principia_r4, axiom, r4).
% 111.45/14.74 fof(principia_r5, axiom, r5).
% 111.45/14.74 fof(r1, axiom, r1 <=> ![P2]: is_a_theorem(implies(or(P2, P2), P2))).
% 111.45/14.74 fof(r2, axiom, r2 <=> ![P2, Q2]: is_a_theorem(implies(Q2, or(P2, Q2)))).
% 111.45/14.74 fof(r3, axiom, r3 <=> ![P2, Q2]: is_a_theorem(implies(or(P2, Q2), or(Q2, P2)))).
% 111.45/14.74 fof(r4, axiom, r4 <=> ![P2, Q2, R2]: is_a_theorem(implies(or(P2, or(Q2, R2)), or(Q2, or(P2, R2))))).
% 111.45/14.74 fof(r5, axiom, r5 <=> ![P2, Q2, R2]: is_a_theorem(implies(implies(Q2, R2), implies(or(P2, Q2), or(P2, R2))))).
% 111.45/14.74 fof(rosser_kn3, conjecture, kn3).
% 111.45/14.74 fof(rosser_op_implies_and, axiom, op_implies_and).
% 111.45/14.74 fof(rosser_op_or, axiom, op_or).
% 111.45/14.74 fof(substitution_of_equivalents, axiom, substitution_of_equivalents <=> ![X2, Y2]: (is_a_theorem(equiv(X2, Y2)) => X2=Y2)).
% 111.45/14.74 fof(substitution_of_equivalents, axiom, substitution_of_equivalents).
% 111.45/14.74
% 111.45/14.74 Now clausify the problem and encode Horn clauses using encoding 3 of
% 111.45/14.74 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 111.45/14.74 We repeatedly replace C & s=t => u=v by the two clauses:
% 111.45/14.74 fresh(y, y, x1...xn) = u
% 111.45/14.74 C => fresh(s, t, x1...xn) = v
% 111.45/14.74 where fresh is a fresh function symbol and x1..xn are the free
% 111.45/14.74 variables of u and v.
% 111.45/14.74 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 111.45/14.74 input problem has no model of domain size 1).
% 111.45/14.74
% 111.45/14.74 The encoding turns the above axioms into the following unit equations and goals:
% 111.45/14.74
% 111.45/14.74 Axiom 1 (principia_modus_ponens): modus_ponens = true.
% 111.45/14.74 Axiom 2 (substitution_of_equivalents): substitution_of_equivalents = true.
% 111.45/14.74 Axiom 3 (principia_r1): r1 = true.
% 111.45/14.74 Axiom 4 (principia_r2): r2 = true.
% 111.45/14.74 Axiom 5 (principia_r3): r3 = true.
% 111.45/14.74 Axiom 6 (principia_r4): r4 = true.
% 111.45/14.74 Axiom 7 (principia_r5): r5 = true.
% 111.45/14.74 Axiom 8 (principia_op_equiv): op_equiv = true.
% 111.45/14.74 Axiom 9 (rosser_op_or): op_or = true.
% 111.45/14.74 Axiom 10 (principia_op_and): op_and = true.
% 111.45/14.74 Axiom 11 (rosser_op_implies_and): op_implies_and = true.
% 111.45/14.74 Axiom 12 (principia_op_implies_or): op_implies_or = true.
% 111.45/14.74 Axiom 13 (kn3): fresh30(X, X) = true.
% 111.45/14.74 Axiom 14 (modus_ponens_2): fresh60(X, X, Y) = true.
% 111.45/14.74 Axiom 15 (modus_ponens_2): fresh28(X, X, Y) = is_a_theorem(Y).
% 111.45/14.74 Axiom 16 (r1_1): fresh12(X, X, Y) = true.
% 111.45/14.74 Axiom 17 (substitution_of_equivalents_2): fresh(X, X, Y, Z) = Z.
% 111.45/14.74 Axiom 18 (modus_ponens_2): fresh59(X, X, Y, Z) = fresh60(modus_ponens, true, Z).
% 111.45/14.74 Axiom 19 (op_and): fresh24(X, X, Y, Z) = and(Y, Z).
% 111.45/14.74 Axiom 20 (op_equiv): fresh23(X, X, Y, Z) = equiv(Y, Z).
% 111.45/14.74 Axiom 21 (op_implies_and): fresh22(X, X, Y, Z) = implies(Y, Z).
% 111.45/14.74 Axiom 22 (op_implies_and): fresh22(op_implies_and, true, X, Y) = not(and(X, not(Y))).
% 111.45/14.74 Axiom 23 (op_implies_or): fresh21(X, X, Y, Z) = implies(Y, Z).
% 111.45/14.74 Axiom 24 (op_implies_or): fresh21(op_implies_or, true, X, Y) = or(not(X), Y).
% 111.45/14.74 Axiom 25 (op_or): fresh20(X, X, Y, Z) = or(Y, Z).
% 111.45/14.74 Axiom 26 (r2_1): fresh10(X, X, Y, Z) = true.
% 111.45/14.74 Axiom 27 (r3_1): fresh8(X, X, Y, Z) = true.
% 111.45/14.74 Axiom 28 (substitution_of_equivalents_2): fresh2(X, X, Y, Z) = Y.
% 111.45/14.74 Axiom 29 (kn1_1): fresh33(kn1, true, X) = is_a_theorem(implies(X, and(X, X))).
% 111.45/14.74 Axiom 30 (r2_1): fresh10(r2, true, X, Y) = is_a_theorem(implies(Y, or(X, Y))).
% 111.45/14.74 Axiom 31 (r1_1): fresh12(r1, true, X) = is_a_theorem(implies(or(X, X), X)).
% 111.45/14.74 Axiom 32 (op_and): fresh24(op_and, true, X, Y) = not(or(not(X), not(Y))).
% 111.45/14.74 Axiom 33 (op_or): fresh20(op_or, true, X, Y) = not(and(not(X), not(Y))).
% 111.45/14.74 Axiom 34 (r4_1): fresh6(X, X, Y, Z, W) = true.
% 111.45/14.74 Axiom 35 (r5_1): fresh4(X, X, Y, Z, W) = true.
% 111.45/14.74 Axiom 36 (op_equiv): fresh23(op_equiv, true, X, Y) = and(implies(X, Y), implies(Y, X)).
% 111.45/14.74 Axiom 37 (r3_1): fresh8(r3, true, X, Y) = is_a_theorem(implies(or(X, Y), or(Y, X))).
% 111.45/14.74 Axiom 38 (substitution_of_equivalents_2): fresh2(substitution_of_equivalents, true, X, Y) = fresh(is_a_theorem(equiv(X, Y)), true, X, Y).
% 111.45/14.74 Axiom 39 (modus_ponens_2): fresh59(is_a_theorem(implies(X, Y)), true, X, Y) = fresh28(is_a_theorem(X), true, Y).
% 111.45/14.74 Axiom 40 (r5_1): fresh4(r5, true, X, Y, Z) = is_a_theorem(implies(implies(Y, Z), implies(or(X, Y), or(X, Z)))).
% 112.14/14.74 Axiom 41 (r4_1): fresh6(r4, true, X, Y, Z) = is_a_theorem(implies(or(X, or(Y, Z)), or(Y, or(X, Z)))).
% 112.14/14.74 Axiom 42 (kn3_1): fresh29(kn3, true, X, Y, Z) = is_a_theorem(implies(implies(X, Y), implies(not(and(Y, Z)), not(and(Z, X))))).
% 112.14/14.74 Axiom 43 (kn3): fresh30(is_a_theorem(implies(implies(p9, q7), implies(not(and(q7, r8)), not(and(r8, p9))))), true) = kn3.
% 112.14/14.74
% 112.14/14.74 Lemma 44: or(not(X), Y) = implies(X, Y).
% 112.14/14.74 Proof:
% 112.14/14.74 or(not(X), Y)
% 112.14/14.74 = { by axiom 24 (op_implies_or) R->L }
% 112.14/14.74 fresh21(op_implies_or, true, X, Y)
% 112.14/14.74 = { by axiom 12 (principia_op_implies_or) }
% 112.14/14.74 fresh21(true, true, X, Y)
% 112.14/14.74 = { by axiom 23 (op_implies_or) }
% 112.14/14.74 implies(X, Y)
% 112.14/14.74
% 112.14/14.74 Lemma 45: is_a_theorem(implies(or(X, X), X)) = true.
% 112.14/14.74 Proof:
% 112.14/14.74 is_a_theorem(implies(or(X, X), X))
% 112.14/14.74 = { by axiom 31 (r1_1) R->L }
% 112.14/14.74 fresh12(r1, true, X)
% 112.14/14.74 = { by axiom 3 (principia_r1) }
% 112.14/14.74 fresh12(true, true, X)
% 112.14/14.74 = { by axiom 16 (r1_1) }
% 112.14/14.74 true
% 112.14/14.74
% 112.14/14.74 Lemma 46: not(implies(X, not(Y))) = and(X, Y).
% 112.14/14.74 Proof:
% 112.14/14.74 not(implies(X, not(Y)))
% 112.14/14.74 = { by lemma 44 R->L }
% 112.14/14.74 not(or(not(X), not(Y)))
% 112.14/14.74 = { by axiom 32 (op_and) R->L }
% 112.14/14.74 fresh24(op_and, true, X, Y)
% 112.14/14.74 = { by axiom 10 (principia_op_and) }
% 112.14/14.74 fresh24(true, true, X, Y)
% 112.14/14.74 = { by axiom 19 (op_and) }
% 112.14/14.74 and(X, Y)
% 112.14/14.74
% 112.14/14.74 Lemma 47: implies(implies(X, not(Y)), Z) = or(and(X, Y), Z).
% 112.14/14.74 Proof:
% 112.14/14.74 implies(implies(X, not(Y)), Z)
% 112.14/14.74 = { by lemma 44 R->L }
% 112.14/14.74 or(not(implies(X, not(Y))), Z)
% 112.14/14.74 = { by lemma 46 }
% 112.14/14.74 or(and(X, Y), Z)
% 112.14/14.74
% 112.14/14.74 Lemma 48: is_a_theorem(implies(or(X, Y), or(Y, X))) = true.
% 112.14/14.74 Proof:
% 112.14/14.74 is_a_theorem(implies(or(X, Y), or(Y, X)))
% 112.14/14.74 = { by axiom 37 (r3_1) R->L }
% 112.14/14.74 fresh8(r3, true, X, Y)
% 112.14/14.74 = { by axiom 5 (principia_r3) }
% 112.14/14.74 fresh8(true, true, X, Y)
% 112.14/14.74 = { by axiom 27 (r3_1) }
% 112.14/14.74 true
% 112.14/14.74
% 112.14/14.74 Lemma 49: fresh59(X, X, Y, Z) = true.
% 112.14/14.74 Proof:
% 112.14/14.74 fresh59(X, X, Y, Z)
% 112.14/14.74 = { by axiom 18 (modus_ponens_2) }
% 112.14/14.74 fresh60(modus_ponens, true, Z)
% 112.14/14.74 = { by axiom 1 (principia_modus_ponens) }
% 112.14/14.74 fresh60(true, true, Z)
% 112.14/14.74 = { by axiom 14 (modus_ponens_2) }
% 112.14/14.74 true
% 112.14/14.74
% 112.14/14.74 Lemma 50: fresh28(is_a_theorem(or(X, Y)), true, or(Y, X)) = true.
% 112.14/14.74 Proof:
% 112.14/14.74 fresh28(is_a_theorem(or(X, Y)), true, or(Y, X))
% 112.14/14.74 = { by axiom 39 (modus_ponens_2) R->L }
% 112.14/14.74 fresh59(is_a_theorem(implies(or(X, Y), or(Y, X))), true, or(X, Y), or(Y, X))
% 112.14/14.74 = { by lemma 48 }
% 112.14/14.74 fresh59(true, true, or(X, Y), or(Y, X))
% 112.14/14.74 = { by lemma 49 }
% 112.14/14.74 true
% 112.14/14.74
% 112.14/14.74 Lemma 51: fresh33(kn1, true, X) = true.
% 112.14/14.74 Proof:
% 112.14/14.74 fresh33(kn1, true, X)
% 112.14/14.74 = { by axiom 29 (kn1_1) }
% 112.14/14.74 is_a_theorem(implies(X, and(X, X)))
% 112.14/14.74 = { by lemma 44 R->L }
% 112.14/14.74 is_a_theorem(or(not(X), and(X, X)))
% 112.14/14.74 = { by axiom 15 (modus_ponens_2) R->L }
% 112.14/14.74 fresh28(true, true, or(not(X), and(X, X)))
% 112.14/14.74 = { by lemma 45 R->L }
% 112.14/14.74 fresh28(is_a_theorem(implies(or(not(X), not(X)), not(X))), true, or(not(X), and(X, X)))
% 112.14/14.74 = { by lemma 44 }
% 112.14/14.74 fresh28(is_a_theorem(implies(implies(X, not(X)), not(X))), true, or(not(X), and(X, X)))
% 112.14/14.74 = { by lemma 47 }
% 112.14/14.74 fresh28(is_a_theorem(or(and(X, X), not(X))), true, or(not(X), and(X, X)))
% 112.14/14.74 = { by lemma 50 }
% 112.14/14.74 true
% 112.14/14.74
% 112.14/14.74 Lemma 52: and(implies(X, Y), implies(Y, X)) = equiv(X, Y).
% 112.14/14.74 Proof:
% 112.14/14.74 and(implies(X, Y), implies(Y, X))
% 112.14/14.74 = { by axiom 36 (op_equiv) R->L }
% 112.14/14.74 fresh23(op_equiv, true, X, Y)
% 112.14/14.74 = { by axiom 8 (principia_op_equiv) }
% 112.14/14.74 fresh23(true, true, X, Y)
% 112.14/14.74 = { by axiom 20 (op_equiv) }
% 112.14/14.74 equiv(X, Y)
% 112.14/14.74
% 112.14/14.74 Lemma 53: not(and(X, not(Y))) = implies(X, Y).
% 112.14/14.74 Proof:
% 112.14/14.74 not(and(X, not(Y)))
% 112.14/14.74 = { by axiom 22 (op_implies_and) R->L }
% 112.14/14.74 fresh22(op_implies_and, true, X, Y)
% 112.14/14.74 = { by axiom 11 (rosser_op_implies_and) }
% 112.14/14.74 fresh22(true, true, X, Y)
% 112.14/14.74 = { by axiom 21 (op_implies_and) }
% 112.14/14.75 implies(X, Y)
% 112.14/14.75
% 112.14/14.75 Lemma 54: implies(not(X), Y) = or(X, Y).
% 112.14/14.75 Proof:
% 112.14/14.75 implies(not(X), Y)
% 112.14/14.75 = { by lemma 53 R->L }
% 112.14/14.75 not(and(not(X), not(Y)))
% 112.14/14.75 = { by axiom 33 (op_or) R->L }
% 112.14/14.75 fresh20(op_or, true, X, Y)
% 112.14/14.75 = { by axiom 9 (rosser_op_or) }
% 112.14/14.75 fresh20(true, true, X, Y)
% 112.14/14.75 = { by axiom 25 (op_or) }
% 112.14/14.75 or(X, Y)
% 112.14/14.75
% 112.14/14.75 Lemma 55: implies(implies(X, Y), and(Y, not(X))) = not(equiv(X, Y)).
% 112.14/14.75 Proof:
% 112.14/14.75 implies(implies(X, Y), and(Y, not(X)))
% 112.14/14.75 = { by lemma 53 R->L }
% 112.14/14.75 not(and(implies(X, Y), not(and(Y, not(X)))))
% 112.14/14.75 = { by lemma 53 }
% 112.14/14.75 not(and(implies(X, Y), implies(Y, X)))
% 112.14/14.75 = { by lemma 52 }
% 112.14/14.75 not(equiv(X, Y))
% 112.14/14.75
% 112.14/14.75 Lemma 56: equiv(not(not(X)), Y) = not(not(equiv(X, Y))).
% 112.14/14.75 Proof:
% 112.14/14.75 equiv(not(not(X)), Y)
% 112.14/14.75 = { by lemma 52 R->L }
% 112.14/14.75 and(implies(not(not(X)), Y), implies(Y, not(not(X))))
% 112.14/14.75 = { by lemma 54 }
% 112.14/14.75 and(or(not(X), Y), implies(Y, not(not(X))))
% 112.14/14.75 = { by lemma 46 R->L }
% 112.14/14.75 not(implies(or(not(X), Y), not(implies(Y, not(not(X))))))
% 112.14/14.75 = { by lemma 46 }
% 112.14/14.75 not(implies(or(not(X), Y), and(Y, not(X))))
% 112.14/14.75 = { by lemma 44 }
% 112.14/14.75 not(implies(implies(X, Y), and(Y, not(X))))
% 112.14/14.75 = { by lemma 55 }
% 112.14/14.75 not(not(equiv(X, Y)))
% 112.14/14.75
% 112.14/14.75 Lemma 57: is_a_theorem(or(X, or(Y, not(X)))) = true.
% 112.14/14.75 Proof:
% 112.14/14.75 is_a_theorem(or(X, or(Y, not(X))))
% 112.14/14.75 = { by lemma 54 R->L }
% 112.14/14.75 is_a_theorem(implies(not(X), or(Y, not(X))))
% 112.14/14.75 = { by axiom 30 (r2_1) R->L }
% 112.14/14.75 fresh10(r2, true, Y, not(X))
% 112.14/14.75 = { by axiom 4 (principia_r2) }
% 112.14/14.75 fresh10(true, true, Y, not(X))
% 112.14/14.75 = { by axiom 26 (r2_1) }
% 112.14/14.75 true
% 112.14/14.75
% 112.14/14.75 Lemma 58: is_a_theorem(implies(or(X, or(Y, Z)), or(Y, or(X, Z)))) = true.
% 112.14/14.75 Proof:
% 112.14/14.75 is_a_theorem(implies(or(X, or(Y, Z)), or(Y, or(X, Z))))
% 112.14/14.75 = { by axiom 41 (r4_1) R->L }
% 112.14/14.75 fresh6(r4, true, X, Y, Z)
% 112.14/14.75 = { by axiom 6 (principia_r4) }
% 112.14/14.75 fresh6(true, true, X, Y, Z)
% 112.14/14.75 = { by axiom 34 (r4_1) }
% 112.14/14.75 true
% 112.14/14.75
% 112.14/14.75 Lemma 59: not(or(X, not(Y))) = and(not(X), Y).
% 112.14/14.75 Proof:
% 112.14/14.75 not(or(X, not(Y)))
% 112.14/14.75 = { by lemma 54 R->L }
% 112.14/14.75 not(implies(not(X), not(Y)))
% 112.14/14.75 = { by lemma 46 }
% 112.14/14.75 and(not(X), Y)
% 112.14/14.75
% 112.14/14.75 Lemma 60: or(and(X, not(Y)), Z) = implies(implies(X, Y), Z).
% 112.14/14.75 Proof:
% 112.14/14.75 or(and(X, not(Y)), Z)
% 112.14/14.75 = { by lemma 54 R->L }
% 112.14/14.75 implies(not(and(X, not(Y))), Z)
% 112.14/14.75 = { by lemma 53 }
% 112.14/14.75 implies(implies(X, Y), Z)
% 112.14/14.75
% 112.14/14.75 Lemma 61: fresh(is_a_theorem(equiv(X, Y)), true, X, Y) = X.
% 112.14/14.75 Proof:
% 112.14/14.75 fresh(is_a_theorem(equiv(X, Y)), true, X, Y)
% 112.14/14.75 = { by axiom 38 (substitution_of_equivalents_2) R->L }
% 112.14/14.75 fresh2(substitution_of_equivalents, true, X, Y)
% 112.14/14.75 = { by axiom 2 (substitution_of_equivalents) }
% 112.14/14.75 fresh2(true, true, X, Y)
% 112.14/14.75 = { by axiom 28 (substitution_of_equivalents_2) }
% 112.14/14.75 X
% 112.14/14.75
% 112.14/14.75 Lemma 62: and(X, not(Y)) = not(implies(X, Y)).
% 112.14/14.75 Proof:
% 112.14/14.75 and(X, not(Y))
% 112.14/14.75 = { by axiom 17 (substitution_of_equivalents_2) R->L }
% 112.14/14.75 fresh(true, true, not(implies(X, Y)), and(X, not(Y)))
% 112.14/14.75 = { by lemma 49 R->L }
% 112.14/14.75 fresh(fresh59(true, true, implies(not(not(and(X, not(Y)))), not(not(and(X, not(Y))))), equiv(not(not(and(X, not(Y)))), not(not(and(X, not(Y)))))), true, not(implies(X, Y)), and(X, not(Y)))
% 112.14/14.75 = { by lemma 51 R->L }
% 112.14/14.75 fresh(fresh59(fresh33(kn1, true, implies(not(not(and(X, not(Y)))), not(not(and(X, not(Y)))))), true, implies(not(not(and(X, not(Y)))), not(not(and(X, not(Y))))), equiv(not(not(and(X, not(Y)))), not(not(and(X, not(Y)))))), true, not(implies(X, Y)), and(X, not(Y)))
% 112.14/14.75 = { by axiom 29 (kn1_1) }
% 112.14/14.75 fresh(fresh59(is_a_theorem(implies(implies(not(not(and(X, not(Y)))), not(not(and(X, not(Y))))), and(implies(not(not(and(X, not(Y)))), not(not(and(X, not(Y))))), implies(not(not(and(X, not(Y)))), not(not(and(X, not(Y)))))))), true, implies(not(not(and(X, not(Y)))), not(not(and(X, not(Y))))), equiv(not(not(and(X, not(Y)))), not(not(and(X, not(Y)))))), true, not(implies(X, Y)), and(X, not(Y)))
% 112.14/14.75 = { by lemma 52 }
% 112.14/14.75 fresh(fresh59(is_a_theorem(implies(implies(not(not(and(X, not(Y)))), not(not(and(X, not(Y))))), equiv(not(not(and(X, not(Y)))), not(not(and(X, not(Y))))))), true, implies(not(not(and(X, not(Y)))), not(not(and(X, not(Y))))), equiv(not(not(and(X, not(Y)))), not(not(and(X, not(Y)))))), true, not(implies(X, Y)), and(X, not(Y)))
% 112.14/14.75 = { by axiom 39 (modus_ponens_2) }
% 112.14/14.75 fresh(fresh28(is_a_theorem(implies(not(not(and(X, not(Y)))), not(not(and(X, not(Y)))))), true, equiv(not(not(and(X, not(Y)))), not(not(and(X, not(Y)))))), true, not(implies(X, Y)), and(X, not(Y)))
% 112.14/14.75 = { by lemma 56 }
% 112.14/14.75 fresh(fresh28(is_a_theorem(implies(not(not(and(X, not(Y)))), not(not(and(X, not(Y)))))), true, not(not(equiv(and(X, not(Y)), not(not(and(X, not(Y)))))))), true, not(implies(X, Y)), and(X, not(Y)))
% 112.14/14.76 = { by lemma 44 R->L }
% 112.14/14.76 fresh(fresh28(is_a_theorem(or(not(not(not(and(X, not(Y))))), not(not(and(X, not(Y)))))), true, not(not(equiv(and(X, not(Y)), not(not(and(X, not(Y)))))))), true, not(implies(X, Y)), and(X, not(Y)))
% 112.14/14.76 = { by axiom 15 (modus_ponens_2) R->L }
% 112.14/14.76 fresh(fresh28(fresh28(true, true, or(not(not(not(and(X, not(Y))))), not(not(and(X, not(Y)))))), true, not(not(equiv(and(X, not(Y)), not(not(and(X, not(Y)))))))), true, not(implies(X, Y)), and(X, not(Y)))
% 112.14/14.76 = { by lemma 49 R->L }
% 112.14/14.76 fresh(fresh28(fresh28(fresh59(true, true, or(or(not(not(and(X, not(Y)))), not(not(not(and(X, not(Y)))))), or(not(not(and(X, not(Y)))), not(not(not(and(X, not(Y))))))), or(not(not(and(X, not(Y)))), not(not(not(and(X, not(Y))))))), true, or(not(not(not(and(X, not(Y))))), not(not(and(X, not(Y)))))), true, not(not(equiv(and(X, not(Y)), not(not(and(X, not(Y)))))))), true, not(implies(X, Y)), and(X, not(Y)))
% 112.14/14.76 = { by lemma 45 R->L }
% 112.14/14.76 fresh(fresh28(fresh28(fresh59(is_a_theorem(implies(or(or(not(not(and(X, not(Y)))), not(not(not(and(X, not(Y)))))), or(not(not(and(X, not(Y)))), not(not(not(and(X, not(Y))))))), or(not(not(and(X, not(Y)))), not(not(not(and(X, not(Y)))))))), true, or(or(not(not(and(X, not(Y)))), not(not(not(and(X, not(Y)))))), or(not(not(and(X, not(Y)))), not(not(not(and(X, not(Y))))))), or(not(not(and(X, not(Y)))), not(not(not(and(X, not(Y))))))), true, or(not(not(not(and(X, not(Y))))), not(not(and(X, not(Y)))))), true, not(not(equiv(and(X, not(Y)), not(not(and(X, not(Y)))))))), true, not(implies(X, Y)), and(X, not(Y)))
% 112.14/14.76 = { by axiom 39 (modus_ponens_2) }
% 112.14/14.76 fresh(fresh28(fresh28(fresh28(is_a_theorem(or(or(not(not(and(X, not(Y)))), not(not(not(and(X, not(Y)))))), or(not(not(and(X, not(Y)))), not(not(not(and(X, not(Y)))))))), true, or(not(not(and(X, not(Y)))), not(not(not(and(X, not(Y))))))), true, or(not(not(not(and(X, not(Y))))), not(not(and(X, not(Y)))))), true, not(not(equiv(and(X, not(Y)), not(not(and(X, not(Y)))))))), true, not(implies(X, Y)), and(X, not(Y)))
% 112.14/14.76 = { by axiom 15 (modus_ponens_2) R->L }
% 112.14/14.76 fresh(fresh28(fresh28(fresh28(fresh28(true, true, or(or(not(not(and(X, not(Y)))), not(not(not(and(X, not(Y)))))), or(not(not(and(X, not(Y)))), not(not(not(and(X, not(Y)))))))), true, or(not(not(and(X, not(Y)))), not(not(not(and(X, not(Y))))))), true, or(not(not(not(and(X, not(Y))))), not(not(and(X, not(Y)))))), true, not(not(equiv(and(X, not(Y)), not(not(and(X, not(Y)))))))), true, not(implies(X, Y)), and(X, not(Y)))
% 112.14/14.76 = { by lemma 57 R->L }
% 112.14/14.76 fresh(fresh28(fresh28(fresh28(fresh28(is_a_theorem(or(not(not(and(X, not(Y)))), or(or(not(not(and(X, not(Y)))), not(not(not(and(X, not(Y)))))), not(not(not(and(X, not(Y)))))))), true, or(or(not(not(and(X, not(Y)))), not(not(not(and(X, not(Y)))))), or(not(not(and(X, not(Y)))), not(not(not(and(X, not(Y)))))))), true, or(not(not(and(X, not(Y)))), not(not(not(and(X, not(Y))))))), true, or(not(not(not(and(X, not(Y))))), not(not(and(X, not(Y)))))), true, not(not(equiv(and(X, not(Y)), not(not(and(X, not(Y)))))))), true, not(implies(X, Y)), and(X, not(Y)))
% 112.14/14.76 = { by axiom 39 (modus_ponens_2) R->L }
% 112.14/14.76 fresh(fresh28(fresh28(fresh28(fresh59(is_a_theorem(implies(or(not(not(and(X, not(Y)))), or(or(not(not(and(X, not(Y)))), not(not(not(and(X, not(Y)))))), not(not(not(and(X, not(Y))))))), or(or(not(not(and(X, not(Y)))), not(not(not(and(X, not(Y)))))), or(not(not(and(X, not(Y)))), not(not(not(and(X, not(Y))))))))), true, or(not(not(and(X, not(Y)))), or(or(not(not(and(X, not(Y)))), not(not(not(and(X, not(Y)))))), not(not(not(and(X, not(Y))))))), or(or(not(not(and(X, not(Y)))), not(not(not(and(X, not(Y)))))), or(not(not(and(X, not(Y)))), not(not(not(and(X, not(Y)))))))), true, or(not(not(and(X, not(Y)))), not(not(not(and(X, not(Y))))))), true, or(not(not(not(and(X, not(Y))))), not(not(and(X, not(Y)))))), true, not(not(equiv(and(X, not(Y)), not(not(and(X, not(Y)))))))), true, not(implies(X, Y)), and(X, not(Y)))
% 112.14/14.76 = { by lemma 58 }
% 112.14/14.76 fresh(fresh28(fresh28(fresh28(fresh59(true, true, or(not(not(and(X, not(Y)))), or(or(not(not(and(X, not(Y)))), not(not(not(and(X, not(Y)))))), not(not(not(and(X, not(Y))))))), or(or(not(not(and(X, not(Y)))), not(not(not(and(X, not(Y)))))), or(not(not(and(X, not(Y)))), not(not(not(and(X, not(Y)))))))), true, or(not(not(and(X, not(Y)))), not(not(not(and(X, not(Y))))))), true, or(not(not(not(and(X, not(Y))))), not(not(and(X, not(Y)))))), true, not(not(equiv(and(X, not(Y)), not(not(and(X, not(Y)))))))), true, not(implies(X, Y)), and(X, not(Y)))
% 112.14/14.76 = { by lemma 49 }
% 112.14/14.76 fresh(fresh28(fresh28(fresh28(true, true, or(not(not(and(X, not(Y)))), not(not(not(and(X, not(Y))))))), true, or(not(not(not(and(X, not(Y))))), not(not(and(X, not(Y)))))), true, not(not(equiv(and(X, not(Y)), not(not(and(X, not(Y)))))))), true, not(implies(X, Y)), and(X, not(Y)))
% 112.14/14.76 = { by axiom 15 (modus_ponens_2) }
% 112.14/14.76 fresh(fresh28(fresh28(is_a_theorem(or(not(not(and(X, not(Y)))), not(not(not(and(X, not(Y))))))), true, or(not(not(not(and(X, not(Y))))), not(not(and(X, not(Y)))))), true, not(not(equiv(and(X, not(Y)), not(not(and(X, not(Y)))))))), true, not(implies(X, Y)), and(X, not(Y)))
% 112.14/14.76 = { by lemma 50 }
% 112.14/14.76 fresh(fresh28(true, true, not(not(equiv(and(X, not(Y)), not(not(and(X, not(Y)))))))), true, not(implies(X, Y)), and(X, not(Y)))
% 112.14/14.76 = { by axiom 15 (modus_ponens_2) }
% 112.14/14.76 fresh(is_a_theorem(not(not(equiv(and(X, not(Y)), not(not(and(X, not(Y)))))))), true, not(implies(X, Y)), and(X, not(Y)))
% 112.14/14.76 = { by lemma 55 R->L }
% 112.14/14.76 fresh(is_a_theorem(not(implies(implies(and(X, not(Y)), not(not(and(X, not(Y))))), and(not(not(and(X, not(Y)))), not(and(X, not(Y))))))), true, not(implies(X, Y)), and(X, not(Y)))
% 112.14/14.76 = { by lemma 59 R->L }
% 112.14/14.76 fresh(is_a_theorem(not(implies(implies(and(X, not(Y)), not(not(and(X, not(Y))))), not(or(not(and(X, not(Y))), not(not(and(X, not(Y))))))))), true, not(implies(X, Y)), and(X, not(Y)))
% 112.14/14.76 = { by lemma 44 }
% 112.14/14.76 fresh(is_a_theorem(not(implies(implies(and(X, not(Y)), not(not(and(X, not(Y))))), not(implies(and(X, not(Y)), not(not(and(X, not(Y))))))))), true, not(implies(X, Y)), and(X, not(Y)))
% 112.14/14.77 = { by lemma 46 }
% 112.14/14.77 fresh(is_a_theorem(not(implies(implies(and(X, not(Y)), not(not(and(X, not(Y))))), and(and(X, not(Y)), not(and(X, not(Y))))))), true, not(implies(X, Y)), and(X, not(Y)))
% 112.14/14.77 = { by lemma 47 }
% 112.14/14.77 fresh(is_a_theorem(not(or(and(and(X, not(Y)), not(and(X, not(Y)))), and(and(X, not(Y)), not(and(X, not(Y))))))), true, not(implies(X, Y)), and(X, not(Y)))
% 112.14/14.77 = { by lemma 60 }
% 112.14/14.77 fresh(is_a_theorem(not(implies(implies(and(X, not(Y)), and(X, not(Y))), and(and(X, not(Y)), not(and(X, not(Y))))))), true, not(implies(X, Y)), and(X, not(Y)))
% 112.14/14.77 = { by lemma 55 }
% 112.14/14.77 fresh(is_a_theorem(not(not(equiv(and(X, not(Y)), and(X, not(Y)))))), true, not(implies(X, Y)), and(X, not(Y)))
% 112.14/14.77 = { by lemma 56 R->L }
% 112.14/14.77 fresh(is_a_theorem(equiv(not(not(and(X, not(Y)))), and(X, not(Y)))), true, not(implies(X, Y)), and(X, not(Y)))
% 112.14/14.77 = { by lemma 53 }
% 112.14/14.77 fresh(is_a_theorem(equiv(not(implies(X, Y)), and(X, not(Y)))), true, not(implies(X, Y)), and(X, not(Y)))
% 112.14/14.77 = { by lemma 61 }
% 112.14/14.77 not(implies(X, Y))
% 112.14/14.77
% 112.14/14.77 Lemma 63: or(implies(X, not(Y)), Z) = implies(and(X, Y), Z).
% 112.14/14.77 Proof:
% 112.14/14.77 or(implies(X, not(Y)), Z)
% 112.14/14.77 = { by lemma 54 R->L }
% 112.14/14.77 implies(not(implies(X, not(Y))), Z)
% 112.14/14.77 = { by lemma 46 }
% 112.14/14.77 implies(and(X, Y), Z)
% 112.14/14.77
% 112.14/14.77 Lemma 64: is_a_theorem(implies(implies(X, Y), or(Y, not(X)))) = true.
% 112.14/14.77 Proof:
% 112.14/14.77 is_a_theorem(implies(implies(X, Y), or(Y, not(X))))
% 112.14/14.77 = { by lemma 44 R->L }
% 112.14/14.77 is_a_theorem(implies(or(not(X), Y), or(Y, not(X))))
% 112.14/14.77 = { by lemma 48 }
% 112.14/14.77 true
% 112.14/14.77
% 112.14/14.77 Lemma 65: is_a_theorem(implies(implies(X, Y), implies(or(Z, X), or(Z, Y)))) = true.
% 112.14/14.77 Proof:
% 112.14/14.77 is_a_theorem(implies(implies(X, Y), implies(or(Z, X), or(Z, Y))))
% 112.14/14.77 = { by axiom 40 (r5_1) R->L }
% 112.14/14.77 fresh4(r5, true, Z, X, Y)
% 112.14/14.77 = { by axiom 7 (principia_r5) }
% 112.14/14.77 fresh4(true, true, Z, X, Y)
% 112.14/14.77 = { by axiom 35 (r5_1) }
% 112.14/14.77 true
% 112.14/14.77
% 112.14/14.77 Lemma 66: fresh28(is_a_theorem(implies(X, Y)), true, implies(or(Z, X), or(Z, Y))) = true.
% 112.14/14.77 Proof:
% 112.14/14.77 fresh28(is_a_theorem(implies(X, Y)), true, implies(or(Z, X), or(Z, Y)))
% 112.14/14.77 = { by axiom 39 (modus_ponens_2) R->L }
% 112.14/14.77 fresh59(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)))
% 112.14/14.77 = { by lemma 65 }
% 112.14/14.77 fresh59(true, true, implies(X, Y), implies(or(Z, X), or(Z, Y)))
% 112.14/14.77 = { by lemma 49 }
% 112.14/14.77 true
% 112.14/14.77
% 112.14/14.77 Lemma 67: fresh28(is_a_theorem(implies(X, or(Y, Z))), true, or(Y, implies(X, Z))) = true.
% 112.14/14.77 Proof:
% 112.14/14.77 fresh28(is_a_theorem(implies(X, or(Y, Z))), true, or(Y, implies(X, Z)))
% 112.14/14.77 = { by axiom 39 (modus_ponens_2) R->L }
% 112.14/14.77 fresh59(is_a_theorem(implies(implies(X, or(Y, Z)), or(Y, implies(X, Z)))), true, implies(X, or(Y, Z)), or(Y, implies(X, Z)))
% 112.14/14.77 = { by lemma 44 R->L }
% 112.14/14.77 fresh59(is_a_theorem(implies(implies(X, or(Y, Z)), or(Y, or(not(X), Z)))), true, implies(X, or(Y, Z)), or(Y, implies(X, Z)))
% 112.14/14.77 = { by lemma 44 R->L }
% 112.14/14.77 fresh59(is_a_theorem(implies(or(not(X), or(Y, Z)), or(Y, or(not(X), Z)))), true, implies(X, or(Y, Z)), or(Y, implies(X, Z)))
% 112.14/14.77 = { by lemma 58 }
% 112.14/14.77 fresh59(true, true, implies(X, or(Y, Z)), or(Y, implies(X, Z)))
% 112.14/14.77 = { by lemma 49 }
% 112.14/14.77 true
% 112.14/14.77
% 112.14/14.77 Lemma 68: fresh28(is_a_theorem(implies(X, Y)), true, implies(implies(Y, X), equiv(X, Y))) = true.
% 112.14/14.77 Proof:
% 112.14/14.77 fresh28(is_a_theorem(implies(X, Y)), true, implies(implies(Y, X), equiv(X, Y)))
% 112.14/14.77 = { by axiom 39 (modus_ponens_2) R->L }
% 112.14/14.77 fresh59(is_a_theorem(implies(implies(X, Y), implies(implies(Y, X), equiv(X, Y)))), true, implies(X, Y), implies(implies(Y, X), equiv(X, Y)))
% 112.14/14.77 = { by lemma 52 R->L }
% 112.14/14.77 fresh59(is_a_theorem(implies(implies(X, Y), implies(implies(Y, X), and(implies(X, Y), implies(Y, X))))), true, implies(X, Y), implies(implies(Y, X), equiv(X, Y)))
% 112.14/14.77 = { by lemma 46 R->L }
% 112.14/14.77 fresh59(is_a_theorem(implies(implies(X, Y), implies(implies(Y, X), not(implies(implies(X, Y), not(implies(Y, X))))))), true, implies(X, Y), implies(implies(Y, X), equiv(X, Y)))
% 112.14/14.77 = { by lemma 44 R->L }
% 112.14/14.77 fresh59(is_a_theorem(implies(implies(X, Y), or(not(implies(Y, X)), not(implies(implies(X, Y), not(implies(Y, X))))))), true, implies(X, Y), implies(implies(Y, X), equiv(X, Y)))
% 112.14/14.77 = { by axiom 15 (modus_ponens_2) R->L }
% 112.14/14.77 fresh59(fresh28(true, true, implies(implies(X, Y), or(not(implies(Y, X)), not(implies(implies(X, Y), not(implies(Y, X))))))), true, implies(X, Y), implies(implies(Y, X), equiv(X, Y)))
% 112.14/14.77 = { by lemma 49 R->L }
% 112.14/14.77 fresh59(fresh28(fresh59(true, true, or(not(implies(Y, X)), implies(implies(implies(X, Y), not(implies(Y, X))), not(implies(X, Y)))), or(not(implies(Y, X)), or(not(implies(X, Y)), not(implies(implies(X, Y), not(implies(Y, X))))))), true, implies(implies(X, Y), or(not(implies(Y, X)), not(implies(implies(X, Y), not(implies(Y, X))))))), true, implies(X, Y), implies(implies(Y, X), equiv(X, Y)))
% 112.14/14.77 = { by lemma 66 R->L }
% 112.14/14.77 fresh59(fresh28(fresh59(fresh28(is_a_theorem(implies(implies(implies(implies(X, Y), not(implies(Y, X))), not(implies(X, Y))), or(not(implies(X, Y)), not(implies(implies(X, Y), not(implies(Y, X))))))), true, implies(or(not(implies(Y, X)), implies(implies(implies(X, Y), not(implies(Y, X))), not(implies(X, Y)))), or(not(implies(Y, X)), or(not(implies(X, Y)), not(implies(implies(X, Y), not(implies(Y, X)))))))), true, or(not(implies(Y, X)), implies(implies(implies(X, Y), not(implies(Y, X))), not(implies(X, Y)))), or(not(implies(Y, X)), or(not(implies(X, Y)), not(implies(implies(X, Y), not(implies(Y, X))))))), true, implies(implies(X, Y), or(not(implies(Y, X)), not(implies(implies(X, Y), not(implies(Y, X))))))), true, implies(X, Y), implies(implies(Y, X), equiv(X, Y)))
% 112.14/14.77 = { by lemma 64 }
% 112.14/14.77 fresh59(fresh28(fresh59(fresh28(true, true, implies(or(not(implies(Y, X)), implies(implies(implies(X, Y), not(implies(Y, X))), not(implies(X, Y)))), or(not(implies(Y, X)), or(not(implies(X, Y)), not(implies(implies(X, Y), not(implies(Y, X)))))))), true, or(not(implies(Y, X)), implies(implies(implies(X, Y), not(implies(Y, X))), not(implies(X, Y)))), or(not(implies(Y, X)), or(not(implies(X, Y)), not(implies(implies(X, Y), not(implies(Y, X))))))), true, implies(implies(X, Y), or(not(implies(Y, X)), not(implies(implies(X, Y), not(implies(Y, X))))))), true, implies(X, Y), implies(implies(Y, X), equiv(X, Y)))
% 112.14/14.78 = { by axiom 15 (modus_ponens_2) }
% 112.14/14.78 fresh59(fresh28(fresh59(is_a_theorem(implies(or(not(implies(Y, X)), implies(implies(implies(X, Y), not(implies(Y, X))), not(implies(X, Y)))), or(not(implies(Y, X)), or(not(implies(X, Y)), not(implies(implies(X, Y), not(implies(Y, X)))))))), true, or(not(implies(Y, X)), implies(implies(implies(X, Y), not(implies(Y, X))), not(implies(X, Y)))), or(not(implies(Y, X)), or(not(implies(X, Y)), not(implies(implies(X, Y), not(implies(Y, X))))))), true, implies(implies(X, Y), or(not(implies(Y, X)), not(implies(implies(X, Y), not(implies(Y, X))))))), true, implies(X, Y), implies(implies(Y, X), equiv(X, Y)))
% 112.14/14.78 = { by axiom 39 (modus_ponens_2) }
% 112.14/14.78 fresh59(fresh28(fresh28(is_a_theorem(or(not(implies(Y, X)), implies(implies(implies(X, Y), not(implies(Y, X))), not(implies(X, Y))))), true, or(not(implies(Y, X)), or(not(implies(X, Y)), not(implies(implies(X, Y), not(implies(Y, X))))))), true, implies(implies(X, Y), or(not(implies(Y, X)), not(implies(implies(X, Y), not(implies(Y, X))))))), true, implies(X, Y), implies(implies(Y, X), equiv(X, Y)))
% 112.14/14.78 = { by axiom 15 (modus_ponens_2) R->L }
% 112.14/14.78 fresh59(fresh28(fresh28(fresh28(true, true, or(not(implies(Y, X)), implies(implies(implies(X, Y), not(implies(Y, X))), not(implies(X, Y))))), true, or(not(implies(Y, X)), or(not(implies(X, Y)), not(implies(implies(X, Y), not(implies(Y, X))))))), true, implies(implies(X, Y), or(not(implies(Y, X)), not(implies(implies(X, Y), not(implies(Y, X))))))), true, implies(X, Y), implies(implies(Y, X), equiv(X, Y)))
% 112.14/14.78 = { by lemma 64 R->L }
% 112.14/14.78 fresh59(fresh28(fresh28(fresh28(is_a_theorem(implies(implies(implies(X, Y), not(implies(Y, X))), or(not(implies(Y, X)), not(implies(X, Y))))), true, or(not(implies(Y, X)), implies(implies(implies(X, Y), not(implies(Y, X))), not(implies(X, Y))))), true, or(not(implies(Y, X)), or(not(implies(X, Y)), not(implies(implies(X, Y), not(implies(Y, X))))))), true, implies(implies(X, Y), or(not(implies(Y, X)), not(implies(implies(X, Y), not(implies(Y, X))))))), true, implies(X, Y), implies(implies(Y, X), equiv(X, Y)))
% 112.14/14.78 = { by lemma 67 }
% 112.14/14.78 fresh59(fresh28(fresh28(true, true, or(not(implies(Y, X)), or(not(implies(X, Y)), not(implies(implies(X, Y), not(implies(Y, X))))))), true, implies(implies(X, Y), or(not(implies(Y, X)), not(implies(implies(X, Y), not(implies(Y, X))))))), true, implies(X, Y), implies(implies(Y, X), equiv(X, Y)))
% 112.14/14.78 = { by axiom 15 (modus_ponens_2) }
% 112.14/14.78 fresh59(fresh28(is_a_theorem(or(not(implies(Y, X)), or(not(implies(X, Y)), not(implies(implies(X, Y), not(implies(Y, X))))))), true, implies(implies(X, Y), or(not(implies(Y, X)), not(implies(implies(X, Y), not(implies(Y, X))))))), true, implies(X, Y), implies(implies(Y, X), equiv(X, Y)))
% 112.14/14.78 = { by lemma 44 }
% 112.14/14.78 fresh59(fresh28(is_a_theorem(or(not(implies(Y, X)), implies(implies(X, Y), not(implies(implies(X, Y), not(implies(Y, X))))))), true, implies(implies(X, Y), or(not(implies(Y, X)), not(implies(implies(X, Y), not(implies(Y, X))))))), true, implies(X, Y), implies(implies(Y, X), equiv(X, Y)))
% 112.14/14.78 = { by axiom 39 (modus_ponens_2) R->L }
% 112.14/14.78 fresh59(fresh59(is_a_theorem(implies(or(not(implies(Y, X)), implies(implies(X, Y), not(implies(implies(X, Y), not(implies(Y, X)))))), implies(implies(X, Y), or(not(implies(Y, X)), not(implies(implies(X, Y), not(implies(Y, X)))))))), true, or(not(implies(Y, X)), implies(implies(X, Y), not(implies(implies(X, Y), not(implies(Y, X)))))), implies(implies(X, Y), or(not(implies(Y, X)), not(implies(implies(X, Y), not(implies(Y, X))))))), true, implies(X, Y), implies(implies(Y, X), equiv(X, Y)))
% 112.14/14.78 = { by lemma 44 R->L }
% 112.14/14.78 fresh59(fresh59(is_a_theorem(implies(or(not(implies(Y, X)), implies(implies(X, Y), not(implies(implies(X, Y), not(implies(Y, X)))))), or(not(implies(X, Y)), or(not(implies(Y, X)), not(implies(implies(X, Y), not(implies(Y, X)))))))), true, or(not(implies(Y, X)), implies(implies(X, Y), not(implies(implies(X, Y), not(implies(Y, X)))))), implies(implies(X, Y), or(not(implies(Y, X)), not(implies(implies(X, Y), not(implies(Y, X))))))), true, implies(X, Y), implies(implies(Y, X), equiv(X, Y)))
% 112.14/14.78 = { by lemma 44 R->L }
% 112.14/14.78 fresh59(fresh59(is_a_theorem(implies(or(not(implies(Y, X)), or(not(implies(X, Y)), not(implies(implies(X, Y), not(implies(Y, X)))))), or(not(implies(X, Y)), or(not(implies(Y, X)), not(implies(implies(X, Y), not(implies(Y, X)))))))), true, or(not(implies(Y, X)), implies(implies(X, Y), not(implies(implies(X, Y), not(implies(Y, X)))))), implies(implies(X, Y), or(not(implies(Y, X)), not(implies(implies(X, Y), not(implies(Y, X))))))), true, implies(X, Y), implies(implies(Y, X), equiv(X, Y)))
% 112.14/14.78 = { by lemma 58 }
% 112.14/14.78 fresh59(fresh59(true, true, or(not(implies(Y, X)), implies(implies(X, Y), not(implies(implies(X, Y), not(implies(Y, X)))))), implies(implies(X, Y), or(not(implies(Y, X)), not(implies(implies(X, Y), not(implies(Y, X))))))), true, implies(X, Y), implies(implies(Y, X), equiv(X, Y)))
% 112.14/14.78 = { by lemma 49 }
% 112.14/14.78 fresh59(true, true, implies(X, Y), implies(implies(Y, X), equiv(X, Y)))
% 112.14/14.78 = { by lemma 49 }
% 112.49/14.78 true
% 112.49/14.78
% 112.49/14.78 Goal 1 (rosser_kn3): kn3 = true.
% 112.49/14.78 Proof:
% 112.49/14.78 kn3
% 112.49/14.78 = { by axiom 43 (kn3) R->L }
% 112.49/14.78 fresh30(is_a_theorem(implies(implies(p9, q7), implies(not(and(q7, r8)), not(and(r8, p9))))), true)
% 112.49/14.78 = { by axiom 42 (kn3_1) R->L }
% 112.49/14.78 fresh30(fresh29(kn3, true, p9, q7, r8), true)
% 112.49/14.78 = { by lemma 61 R->L }
% 112.49/14.78 fresh30(fresh29(kn3, true, p9, q7, fresh(is_a_theorem(equiv(r8, and(r8, r8))), true, r8, and(r8, r8))), true)
% 112.49/14.79 = { by axiom 15 (modus_ponens_2) R->L }
% 112.49/14.79 fresh30(fresh29(kn3, true, p9, q7, fresh(fresh28(true, true, equiv(r8, and(r8, r8))), true, r8, and(r8, r8))), true)
% 112.49/14.79 = { by lemma 49 R->L }
% 112.49/14.79 fresh30(fresh29(kn3, true, p9, q7, fresh(fresh28(fresh59(true, true, or(implies(r8, not(r8)), and(not(not(r8)), r8)), or(implies(r8, not(r8)), r8)), true, equiv(r8, and(r8, r8))), true, r8, and(r8, r8))), true)
% 112.49/14.79 = { by lemma 66 R->L }
% 112.49/14.79 fresh30(fresh29(kn3, true, p9, q7, fresh(fresh28(fresh59(fresh28(is_a_theorem(implies(and(not(not(r8)), r8), r8)), true, implies(or(implies(r8, not(r8)), and(not(not(r8)), r8)), or(implies(r8, not(r8)), r8))), true, or(implies(r8, not(r8)), and(not(not(r8)), r8)), or(implies(r8, not(r8)), r8)), true, equiv(r8, and(r8, r8))), true, r8, and(r8, r8))), true)
% 112.49/14.79 = { by lemma 63 R->L }
% 112.49/14.79 fresh30(fresh29(kn3, true, p9, q7, fresh(fresh28(fresh59(fresh28(is_a_theorem(or(implies(not(not(r8)), not(r8)), r8)), true, implies(or(implies(r8, not(r8)), and(not(not(r8)), r8)), or(implies(r8, not(r8)), r8))), true, or(implies(r8, not(r8)), and(not(not(r8)), r8)), or(implies(r8, not(r8)), r8)), true, equiv(r8, and(r8, r8))), true, r8, and(r8, r8))), true)
% 112.49/14.79 = { by axiom 15 (modus_ponens_2) R->L }
% 112.49/14.79 fresh30(fresh29(kn3, true, p9, q7, fresh(fresh28(fresh59(fresh28(fresh28(true, true, or(implies(not(not(r8)), not(r8)), r8)), true, implies(or(implies(r8, not(r8)), and(not(not(r8)), r8)), or(implies(r8, not(r8)), r8))), true, or(implies(r8, not(r8)), and(not(not(r8)), r8)), or(implies(r8, not(r8)), r8)), true, equiv(r8, and(r8, r8))), true, r8, and(r8, r8))), true)
% 112.49/14.79 = { by lemma 57 R->L }
% 112.49/14.79 fresh30(fresh29(kn3, true, p9, q7, fresh(fresh28(fresh59(fresh28(fresh28(is_a_theorem(or(r8, or(not(not(not(r8))), not(r8)))), true, or(implies(not(not(r8)), not(r8)), r8)), true, implies(or(implies(r8, not(r8)), and(not(not(r8)), r8)), or(implies(r8, not(r8)), r8))), true, or(implies(r8, not(r8)), and(not(not(r8)), r8)), or(implies(r8, not(r8)), r8)), true, equiv(r8, and(r8, r8))), true, r8, and(r8, r8))), true)
% 112.49/14.79 = { by lemma 44 }
% 112.49/14.79 fresh30(fresh29(kn3, true, p9, q7, fresh(fresh28(fresh59(fresh28(fresh28(is_a_theorem(or(r8, implies(not(not(r8)), not(r8)))), true, or(implies(not(not(r8)), not(r8)), r8)), true, implies(or(implies(r8, not(r8)), and(not(not(r8)), r8)), or(implies(r8, not(r8)), r8))), true, or(implies(r8, not(r8)), and(not(not(r8)), r8)), or(implies(r8, not(r8)), r8)), true, equiv(r8, and(r8, r8))), true, r8, and(r8, r8))), true)
% 112.49/14.79 = { by lemma 50 }
% 112.49/14.79 fresh30(fresh29(kn3, true, p9, q7, fresh(fresh28(fresh59(fresh28(true, true, implies(or(implies(r8, not(r8)), and(not(not(r8)), r8)), or(implies(r8, not(r8)), r8))), true, or(implies(r8, not(r8)), and(not(not(r8)), r8)), or(implies(r8, not(r8)), r8)), true, equiv(r8, and(r8, r8))), true, r8, and(r8, r8))), true)
% 112.49/14.79 = { by axiom 15 (modus_ponens_2) }
% 112.49/14.79 fresh30(fresh29(kn3, true, p9, q7, fresh(fresh28(fresh59(is_a_theorem(implies(or(implies(r8, not(r8)), and(not(not(r8)), r8)), or(implies(r8, not(r8)), r8))), true, or(implies(r8, not(r8)), and(not(not(r8)), r8)), or(implies(r8, not(r8)), r8)), true, equiv(r8, and(r8, r8))), true, r8, and(r8, r8))), true)
% 112.49/14.79 = { by axiom 39 (modus_ponens_2) }
% 112.49/14.79 fresh30(fresh29(kn3, true, p9, q7, fresh(fresh28(fresh28(is_a_theorem(or(implies(r8, not(r8)), and(not(not(r8)), r8))), true, or(implies(r8, not(r8)), r8)), true, equiv(r8, and(r8, r8))), true, r8, and(r8, r8))), true)
% 112.49/14.79 = { by lemma 59 R->L }
% 112.49/14.79 fresh30(fresh29(kn3, true, p9, q7, fresh(fresh28(fresh28(is_a_theorem(or(implies(r8, not(r8)), not(or(not(r8), not(r8))))), true, or(implies(r8, not(r8)), r8)), true, equiv(r8, and(r8, r8))), true, r8, and(r8, r8))), true)
% 112.49/14.79 = { by axiom 15 (modus_ponens_2) R->L }
% 112.49/14.79 fresh30(fresh29(kn3, true, p9, q7, fresh(fresh28(fresh28(fresh28(true, true, or(implies(r8, not(r8)), not(or(not(r8), not(r8))))), true, or(implies(r8, not(r8)), r8)), true, equiv(r8, and(r8, r8))), true, r8, and(r8, r8))), true)
% 112.49/14.79 = { by lemma 48 R->L }
% 112.49/14.79 fresh30(fresh29(kn3, true, p9, q7, fresh(fresh28(fresh28(fresh28(is_a_theorem(implies(or(not(r8), not(r8)), or(not(r8), not(r8)))), true, or(implies(r8, not(r8)), not(or(not(r8), not(r8))))), true, or(implies(r8, not(r8)), r8)), true, equiv(r8, and(r8, r8))), true, r8, and(r8, r8))), true)
% 112.49/14.79 = { by lemma 44 }
% 112.49/14.79 fresh30(fresh29(kn3, true, p9, q7, fresh(fresh28(fresh28(fresh28(is_a_theorem(implies(or(not(r8), not(r8)), implies(r8, not(r8)))), true, or(implies(r8, not(r8)), not(or(not(r8), not(r8))))), true, or(implies(r8, not(r8)), r8)), true, equiv(r8, and(r8, r8))), true, r8, and(r8, r8))), true)
% 112.49/14.79 = { by axiom 39 (modus_ponens_2) R->L }
% 112.49/14.79 fresh30(fresh29(kn3, true, p9, q7, fresh(fresh28(fresh28(fresh59(is_a_theorem(implies(implies(or(not(r8), not(r8)), implies(r8, not(r8))), or(implies(r8, not(r8)), not(or(not(r8), not(r8)))))), true, implies(or(not(r8), not(r8)), implies(r8, not(r8))), or(implies(r8, not(r8)), not(or(not(r8), not(r8))))), true, or(implies(r8, not(r8)), r8)), true, equiv(r8, and(r8, r8))), true, r8, and(r8, r8))), true)
% 112.49/14.79 = { by lemma 64 }
% 112.49/14.79 fresh30(fresh29(kn3, true, p9, q7, fresh(fresh28(fresh28(fresh59(true, true, implies(or(not(r8), not(r8)), implies(r8, not(r8))), or(implies(r8, not(r8)), not(or(not(r8), not(r8))))), true, or(implies(r8, not(r8)), r8)), true, equiv(r8, and(r8, r8))), true, r8, and(r8, r8))), true)
% 112.49/14.79 = { by lemma 49 }
% 112.49/14.79 fresh30(fresh29(kn3, true, p9, q7, fresh(fresh28(fresh28(true, true, or(implies(r8, not(r8)), r8)), true, equiv(r8, and(r8, r8))), true, r8, and(r8, r8))), true)
% 112.49/14.79 = { by axiom 15 (modus_ponens_2) }
% 112.49/14.79 fresh30(fresh29(kn3, true, p9, q7, fresh(fresh28(is_a_theorem(or(implies(r8, not(r8)), r8)), true, equiv(r8, and(r8, r8))), true, r8, and(r8, r8))), true)
% 112.49/14.79 = { by lemma 63 }
% 112.49/14.79 fresh30(fresh29(kn3, true, p9, q7, fresh(fresh28(is_a_theorem(implies(and(r8, r8), r8)), true, equiv(r8, and(r8, r8))), true, r8, and(r8, r8))), true)
% 112.49/14.79 = { by axiom 39 (modus_ponens_2) R->L }
% 112.49/14.80 fresh30(fresh29(kn3, true, p9, q7, fresh(fresh59(is_a_theorem(implies(implies(and(r8, r8), r8), equiv(r8, and(r8, r8)))), true, implies(and(r8, r8), r8), equiv(r8, and(r8, r8))), true, r8, and(r8, r8))), true)
% 112.49/14.80 = { by axiom 15 (modus_ponens_2) R->L }
% 112.49/14.80 fresh30(fresh29(kn3, true, p9, q7, fresh(fresh59(fresh28(true, true, implies(implies(and(r8, r8), r8), equiv(r8, and(r8, r8)))), true, implies(and(r8, r8), r8), equiv(r8, and(r8, r8))), true, r8, and(r8, r8))), true)
% 112.49/14.80 = { by lemma 51 R->L }
% 112.49/14.80 fresh30(fresh29(kn3, true, p9, q7, fresh(fresh59(fresh28(fresh33(kn1, true, r8), true, implies(implies(and(r8, r8), r8), equiv(r8, and(r8, r8)))), true, implies(and(r8, r8), r8), equiv(r8, and(r8, r8))), true, r8, and(r8, r8))), true)
% 112.49/14.80 = { by axiom 29 (kn1_1) }
% 112.49/14.80 fresh30(fresh29(kn3, true, p9, q7, fresh(fresh59(fresh28(is_a_theorem(implies(r8, and(r8, r8))), true, implies(implies(and(r8, r8), r8), equiv(r8, and(r8, r8)))), true, implies(and(r8, r8), r8), equiv(r8, and(r8, r8))), true, r8, and(r8, r8))), true)
% 112.49/14.80 = { by lemma 68 }
% 112.49/14.80 fresh30(fresh29(kn3, true, p9, q7, fresh(fresh59(true, true, implies(and(r8, r8), r8), equiv(r8, and(r8, r8))), true, r8, and(r8, r8))), true)
% 112.49/14.80 = { by lemma 49 }
% 112.49/14.80 fresh30(fresh29(kn3, true, p9, q7, fresh(true, true, r8, and(r8, r8))), true)
% 112.49/14.80 = { by axiom 17 (substitution_of_equivalents_2) }
% 112.49/14.80 fresh30(fresh29(kn3, true, p9, q7, and(r8, r8)), true)
% 112.49/14.80 = { by lemma 46 R->L }
% 112.49/14.80 fresh30(fresh29(kn3, true, p9, q7, not(implies(r8, not(r8)))), true)
% 112.49/14.80 = { by axiom 42 (kn3_1) }
% 112.49/14.80 fresh30(is_a_theorem(implies(implies(p9, q7), implies(not(and(q7, not(implies(r8, not(r8))))), not(and(not(implies(r8, not(r8))), p9))))), true)
% 112.49/14.80 = { by lemma 54 }
% 112.49/14.80 fresh30(is_a_theorem(implies(implies(p9, q7), or(and(q7, not(implies(r8, not(r8)))), not(and(not(implies(r8, not(r8))), p9))))), true)
% 112.49/14.80 = { by lemma 46 R->L }
% 112.49/14.80 fresh30(is_a_theorem(implies(implies(p9, q7), or(and(q7, not(implies(r8, not(r8)))), not(not(implies(not(implies(r8, not(r8))), not(p9))))))), true)
% 112.49/14.80 = { by lemma 62 R->L }
% 112.49/14.80 fresh30(is_a_theorem(implies(implies(p9, q7), or(and(q7, not(implies(r8, not(r8)))), not(and(not(implies(r8, not(r8))), not(not(p9))))))), true)
% 112.49/14.80 = { by axiom 22 (op_implies_and) R->L }
% 112.49/14.80 fresh30(is_a_theorem(implies(implies(p9, q7), or(and(q7, not(implies(r8, not(r8)))), fresh22(op_implies_and, true, not(implies(r8, not(r8))), not(p9))))), true)
% 112.49/14.80 = { by axiom 11 (rosser_op_implies_and) }
% 112.49/14.80 fresh30(is_a_theorem(implies(implies(p9, q7), or(and(q7, not(implies(r8, not(r8)))), fresh22(true, true, not(implies(r8, not(r8))), not(p9))))), true)
% 112.49/14.80 = { by axiom 21 (op_implies_and) }
% 112.49/14.80 fresh30(is_a_theorem(implies(implies(p9, q7), or(and(q7, not(implies(r8, not(r8)))), implies(not(implies(r8, not(r8))), not(p9))))), true)
% 112.49/14.80 = { by lemma 54 }
% 112.49/14.80 fresh30(is_a_theorem(implies(implies(p9, q7), or(and(q7, not(implies(r8, not(r8)))), or(implies(r8, not(r8)), not(p9))))), true)
% 112.49/14.80 = { by lemma 62 }
% 112.49/14.80 fresh30(is_a_theorem(implies(implies(p9, q7), or(not(implies(q7, implies(r8, not(r8)))), or(implies(r8, not(r8)), not(p9))))), true)
% 112.49/14.80 = { by lemma 44 }
% 112.49/14.80 fresh30(is_a_theorem(implies(implies(p9, q7), implies(implies(q7, implies(r8, not(r8))), or(implies(r8, not(r8)), not(p9))))), true)
% 112.49/14.80 = { by axiom 17 (substitution_of_equivalents_2) R->L }
% 112.49/14.80 fresh30(is_a_theorem(implies(implies(p9, q7), implies(implies(q7, implies(r8, not(r8))), fresh(true, true, or(not(p9), implies(r8, not(r8))), or(implies(r8, not(r8)), not(p9)))))), true)
% 112.49/14.80 = { by lemma 49 R->L }
% 112.49/14.80 fresh30(is_a_theorem(implies(implies(p9, q7), implies(implies(q7, implies(r8, not(r8))), fresh(fresh59(true, true, implies(or(implies(r8, not(r8)), not(p9)), or(not(p9), implies(r8, not(r8)))), equiv(or(not(p9), implies(r8, not(r8))), or(implies(r8, not(r8)), not(p9)))), true, or(not(p9), implies(r8, not(r8))), or(implies(r8, not(r8)), not(p9)))))), true)
% 112.49/14.80 = { by lemma 68 R->L }
% 112.49/14.80 fresh30(is_a_theorem(implies(implies(p9, q7), implies(implies(q7, implies(r8, not(r8))), fresh(fresh59(fresh28(is_a_theorem(implies(or(not(p9), implies(r8, not(r8))), or(implies(r8, not(r8)), not(p9)))), true, implies(implies(or(implies(r8, not(r8)), not(p9)), or(not(p9), implies(r8, not(r8)))), equiv(or(not(p9), implies(r8, not(r8))), or(implies(r8, not(r8)), not(p9))))), true, implies(or(implies(r8, not(r8)), not(p9)), or(not(p9), implies(r8, not(r8)))), equiv(or(not(p9), implies(r8, not(r8))), or(implies(r8, not(r8)), not(p9)))), true, or(not(p9), implies(r8, not(r8))), or(implies(r8, not(r8)), not(p9)))))), true)
% 112.49/14.80 = { by lemma 48 }
% 112.49/14.80 fresh30(is_a_theorem(implies(implies(p9, q7), implies(implies(q7, implies(r8, not(r8))), fresh(fresh59(fresh28(true, true, implies(implies(or(implies(r8, not(r8)), not(p9)), or(not(p9), implies(r8, not(r8)))), equiv(or(not(p9), implies(r8, not(r8))), or(implies(r8, not(r8)), not(p9))))), true, implies(or(implies(r8, not(r8)), not(p9)), or(not(p9), implies(r8, not(r8)))), equiv(or(not(p9), implies(r8, not(r8))), or(implies(r8, not(r8)), not(p9)))), true, or(not(p9), implies(r8, not(r8))), or(implies(r8, not(r8)), not(p9)))))), true)
% 112.49/14.80 = { by axiom 15 (modus_ponens_2) }
% 112.49/14.81 fresh30(is_a_theorem(implies(implies(p9, q7), implies(implies(q7, implies(r8, not(r8))), fresh(fresh59(is_a_theorem(implies(implies(or(implies(r8, not(r8)), not(p9)), or(not(p9), implies(r8, not(r8)))), equiv(or(not(p9), implies(r8, not(r8))), or(implies(r8, not(r8)), not(p9))))), true, implies(or(implies(r8, not(r8)), not(p9)), or(not(p9), implies(r8, not(r8)))), equiv(or(not(p9), implies(r8, not(r8))), or(implies(r8, not(r8)), not(p9)))), true, or(not(p9), implies(r8, not(r8))), or(implies(r8, not(r8)), not(p9)))))), true)
% 112.49/14.81 = { by axiom 39 (modus_ponens_2) }
% 112.49/14.81 fresh30(is_a_theorem(implies(implies(p9, q7), implies(implies(q7, implies(r8, not(r8))), fresh(fresh28(is_a_theorem(implies(or(implies(r8, not(r8)), not(p9)), or(not(p9), implies(r8, not(r8))))), true, equiv(or(not(p9), implies(r8, not(r8))), or(implies(r8, not(r8)), not(p9)))), true, or(not(p9), implies(r8, not(r8))), or(implies(r8, not(r8)), not(p9)))))), true)
% 112.49/14.81 = { by lemma 48 }
% 112.49/14.81 fresh30(is_a_theorem(implies(implies(p9, q7), implies(implies(q7, implies(r8, not(r8))), fresh(fresh28(true, true, equiv(or(not(p9), implies(r8, not(r8))), or(implies(r8, not(r8)), not(p9)))), true, or(not(p9), implies(r8, not(r8))), or(implies(r8, not(r8)), not(p9)))))), true)
% 112.49/14.81 = { by axiom 15 (modus_ponens_2) }
% 112.49/14.81 fresh30(is_a_theorem(implies(implies(p9, q7), implies(implies(q7, implies(r8, not(r8))), fresh(is_a_theorem(equiv(or(not(p9), implies(r8, not(r8))), or(implies(r8, not(r8)), not(p9)))), true, or(not(p9), implies(r8, not(r8))), or(implies(r8, not(r8)), not(p9)))))), true)
% 112.49/14.81 = { by lemma 61 }
% 112.49/14.81 fresh30(is_a_theorem(implies(implies(p9, q7), implies(implies(q7, implies(r8, not(r8))), or(not(p9), implies(r8, not(r8)))))), true)
% 112.49/14.81 = { by lemma 44 }
% 112.49/14.81 fresh30(is_a_theorem(implies(implies(p9, q7), implies(implies(q7, implies(r8, not(r8))), implies(p9, implies(r8, not(r8)))))), true)
% 112.49/14.81 = { by lemma 60 R->L }
% 112.49/14.81 fresh30(is_a_theorem(or(and(p9, not(q7)), implies(implies(q7, implies(r8, not(r8))), implies(p9, implies(r8, not(r8)))))), true)
% 112.49/14.81 = { by lemma 44 R->L }
% 112.49/14.81 fresh30(is_a_theorem(or(and(p9, not(q7)), implies(or(not(q7), implies(r8, not(r8))), implies(p9, implies(r8, not(r8)))))), true)
% 112.49/14.81 = { by axiom 15 (modus_ponens_2) R->L }
% 112.49/14.81 fresh30(fresh28(true, true, or(and(p9, not(q7)), implies(or(not(q7), implies(r8, not(r8))), implies(p9, implies(r8, not(r8)))))), true)
% 112.49/14.81 = { by lemma 65 R->L }
% 112.49/14.81 fresh30(fresh28(is_a_theorem(implies(implies(not(not(q7)), implies(r8, not(r8))), implies(or(not(p9), not(not(q7))), or(not(p9), implies(r8, not(r8)))))), true, or(and(p9, not(q7)), implies(or(not(q7), implies(r8, not(r8))), implies(p9, implies(r8, not(r8)))))), true)
% 112.49/14.81 = { by lemma 44 }
% 112.49/14.81 fresh30(fresh28(is_a_theorem(implies(implies(not(not(q7)), implies(r8, not(r8))), implies(implies(p9, not(not(q7))), or(not(p9), implies(r8, not(r8)))))), true, or(and(p9, not(q7)), implies(or(not(q7), implies(r8, not(r8))), implies(p9, implies(r8, not(r8)))))), true)
% 112.49/14.81 = { by lemma 44 }
% 112.49/14.81 fresh30(fresh28(is_a_theorem(implies(implies(not(not(q7)), implies(r8, not(r8))), implies(implies(p9, not(not(q7))), implies(p9, implies(r8, not(r8)))))), true, or(and(p9, not(q7)), implies(or(not(q7), implies(r8, not(r8))), implies(p9, implies(r8, not(r8)))))), true)
% 112.49/14.81 = { by lemma 54 }
% 112.49/14.81 fresh30(fresh28(is_a_theorem(implies(or(not(q7), implies(r8, not(r8))), implies(implies(p9, not(not(q7))), implies(p9, implies(r8, not(r8)))))), true, or(and(p9, not(q7)), implies(or(not(q7), implies(r8, not(r8))), implies(p9, implies(r8, not(r8)))))), true)
% 112.49/14.81 = { by lemma 47 }
% 112.49/14.81 fresh30(fresh28(is_a_theorem(implies(or(not(q7), implies(r8, not(r8))), or(and(p9, not(q7)), implies(p9, implies(r8, not(r8)))))), true, or(and(p9, not(q7)), implies(or(not(q7), implies(r8, not(r8))), implies(p9, implies(r8, not(r8)))))), true)
% 112.49/14.81 = { by lemma 67 }
% 112.49/14.81 fresh30(true, true)
% 112.49/14.81 = { by axiom 13 (kn3) }
% 112.49/14.81 true
% 112.49/14.81 % SZS output end Proof
% 112.49/14.81
% 112.49/14.81 RESULT: Theorem (the conjecture is true).
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