TSTP Solution File: LCL463+1 by Twee---2.4.2
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- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : LCL463+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 : n016.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 08:19:07 EDT 2023
% Result : Theorem 94.26s 12.67s
% Output : Proof 96.04s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.13 % Problem : LCL463+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
% 0.13/0.35 % Computer : n016.cluster.edu
% 0.13/0.35 % Model : x86_64 x86_64
% 0.13/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.35 % Memory : 8042.1875MB
% 0.13/0.35 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.35 % CPULimit : 300
% 0.13/0.35 % WCLimit : 300
% 0.13/0.35 % DateTime : Fri Aug 25 07:01:55 EDT 2023
% 0.13/0.35 % CPUTime :
% 94.26/12.67 Command-line arguments: --no-flatten-goal
% 94.26/12.67
% 94.26/12.67 % SZS status Theorem
% 94.26/12.67
% 95.44/12.73 % SZS output start Proof
% 95.44/12.73 Take the following subset of the input axioms:
% 95.44/12.73 fof(cn1, axiom, cn1 <=> ![P, Q, R]: is_a_theorem(implies(implies(P, Q), implies(implies(Q, R), implies(P, R))))).
% 95.44/12.73 fof(cn2, axiom, cn2 <=> ![P2, Q2]: is_a_theorem(implies(P2, implies(not(P2), Q2)))).
% 95.44/12.73 fof(cn3, axiom, cn3 <=> ![P2]: is_a_theorem(implies(implies(not(P2), P2), P2))).
% 95.44/12.73 fof(hilbert_implies_1, conjecture, implies_1).
% 95.44/12.73 fof(hilbert_op_implies_and, axiom, op_implies_and).
% 95.44/12.73 fof(implies_1, axiom, implies_1 <=> ![X, Y]: is_a_theorem(implies(X, implies(Y, X)))).
% 95.44/12.73 fof(luka_cn1, axiom, cn1).
% 95.44/12.73 fof(luka_cn2, axiom, cn2).
% 95.44/12.73 fof(luka_cn3, axiom, cn3).
% 95.44/12.73 fof(luka_modus_ponens, axiom, modus_ponens).
% 95.44/12.73 fof(luka_op_implies, axiom, op_implies).
% 95.44/12.73 fof(luka_op_or, axiom, op_or).
% 95.44/12.73 fof(modus_ponens, axiom, modus_ponens <=> ![X2, Y2]: ((is_a_theorem(X2) & is_a_theorem(implies(X2, Y2))) => is_a_theorem(Y2))).
% 95.44/12.73 fof(op_implies_and, axiom, op_implies_and => ![X2, Y2]: implies(X2, Y2)=not(and(X2, not(Y2)))).
% 95.44/12.73 fof(op_or, axiom, op_or => ![X2, Y2]: or(X2, Y2)=not(and(not(X2), not(Y2)))).
% 95.44/12.73 fof(or_1, axiom, or_1 <=> ![X2, Y2]: is_a_theorem(implies(X2, or(X2, Y2)))).
% 95.44/12.73 fof(r1, axiom, r1 <=> ![P2]: is_a_theorem(implies(or(P2, P2), P2))).
% 95.44/12.73
% 95.44/12.73 Now clausify the problem and encode Horn clauses using encoding 3 of
% 95.44/12.73 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 95.44/12.73 We repeatedly replace C & s=t => u=v by the two clauses:
% 95.44/12.73 fresh(y, y, x1...xn) = u
% 95.44/12.73 C => fresh(s, t, x1...xn) = v
% 95.44/12.73 where fresh is a fresh function symbol and x1..xn are the free
% 95.44/12.73 variables of u and v.
% 95.44/12.73 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 95.44/12.73 input problem has no model of domain size 1).
% 95.44/12.73
% 95.44/12.73 The encoding turns the above axioms into the following unit equations and goals:
% 95.44/12.73
% 95.44/12.73 Axiom 1 (luka_op_implies): op_implies = true.
% 95.44/12.73 Axiom 2 (hilbert_op_implies_and): op_implies_and = true.
% 95.44/12.73 Axiom 3 (luka_modus_ponens): modus_ponens = true.
% 95.44/12.73 Axiom 4 (luka_cn1): cn1 = true.
% 95.44/12.73 Axiom 5 (luka_cn2): cn2 = true.
% 95.44/12.73 Axiom 6 (luka_cn3): cn3 = true.
% 95.44/12.73 Axiom 7 (luka_op_or): op_or = true.
% 95.44/12.73 Axiom 8 (implies_1): fresh40(X, X) = true.
% 95.44/12.73 Axiom 9 (modus_ponens_2): fresh60(X, X, Y) = true.
% 95.44/12.73 Axiom 10 (cn3_1): fresh47(X, X, Y) = true.
% 95.44/12.73 Axiom 11 (modus_ponens_2): fresh28(X, X, Y) = is_a_theorem(Y).
% 95.44/12.73 Axiom 12 (modus_ponens_2): fresh59(X, X, Y, Z) = fresh60(modus_ponens, true, Z).
% 95.44/12.73 Axiom 13 (cn2_1): fresh49(X, X, Y, Z) = true.
% 95.44/12.73 Axiom 14 (op_implies_and): fresh22(X, X, Y, Z) = implies(Y, Z).
% 95.44/12.73 Axiom 15 (op_or): fresh20(X, X, Y, Z) = or(Y, Z).
% 95.44/12.73 Axiom 16 (op_implies_and): fresh22(op_implies_and, true, X, Y) = not(and(X, not(Y))).
% 95.44/12.73 Axiom 17 (cn1_1): fresh51(X, X, Y, Z, W) = true.
% 95.44/12.73 Axiom 18 (op_or): fresh20(op_or, true, X, Y) = not(and(not(X), not(Y))).
% 95.44/12.73 Axiom 19 (or_1_1): fresh18(or_1, true, X, Y) = is_a_theorem(implies(X, or(X, Y))).
% 95.44/12.73 Axiom 20 (implies_1_1): fresh39(implies_1, true, X, Y) = is_a_theorem(implies(X, implies(Y, X))).
% 95.44/12.73 Axiom 21 (r1_1): fresh12(r1, true, X) = is_a_theorem(implies(or(X, X), X)).
% 95.44/12.73 Axiom 22 (cn2_1): fresh49(cn2, true, X, Y) = is_a_theorem(implies(X, implies(not(X), Y))).
% 95.44/12.73 Axiom 23 (cn3_1): fresh47(cn3, true, X) = is_a_theorem(implies(implies(not(X), X), X)).
% 95.44/12.73 Axiom 24 (modus_ponens_2): fresh59(is_a_theorem(implies(X, Y)), true, X, Y) = fresh28(is_a_theorem(X), true, Y).
% 95.44/12.73 Axiom 25 (implies_1): fresh40(is_a_theorem(implies(x12, implies(y12, x12))), true) = implies_1.
% 95.44/12.73 Axiom 26 (cn1_1): fresh51(cn1, true, X, Y, Z) = is_a_theorem(implies(implies(X, Y), implies(implies(Y, Z), implies(X, Z)))).
% 95.44/12.73
% 95.44/12.73 Lemma 27: is_a_theorem(implies(or(X, X), X)) = fresh12(r1, op_implies, X).
% 95.44/12.73 Proof:
% 95.44/12.73 is_a_theorem(implies(or(X, X), X))
% 95.44/12.73 = { by axiom 21 (r1_1) R->L }
% 95.44/12.73 fresh12(r1, true, X)
% 95.44/12.73 = { by axiom 1 (luka_op_implies) R->L }
% 95.44/12.73 fresh12(r1, op_implies, X)
% 95.44/12.73
% 95.44/12.73 Lemma 28: implies(not(X), Y) = or(X, Y).
% 95.44/12.73 Proof:
% 95.44/12.73 implies(not(X), Y)
% 95.44/12.73 = { by axiom 14 (op_implies_and) R->L }
% 95.44/12.73 fresh22(op_implies, op_implies, not(X), Y)
% 95.44/12.73 = { by axiom 1 (luka_op_implies) }
% 95.44/12.73 fresh22(op_implies, true, not(X), Y)
% 95.44/12.73 = { by axiom 1 (luka_op_implies) }
% 95.44/12.73 fresh22(true, true, not(X), Y)
% 95.44/12.73 = { by axiom 2 (hilbert_op_implies_and) R->L }
% 95.44/12.73 fresh22(op_implies_and, true, not(X), Y)
% 95.44/12.73 = { by axiom 16 (op_implies_and) }
% 95.44/12.73 not(and(not(X), not(Y)))
% 95.44/12.73 = { by axiom 18 (op_or) R->L }
% 95.44/12.73 fresh20(op_or, true, X, Y)
% 95.44/12.73 = { by axiom 7 (luka_op_or) }
% 95.44/12.73 fresh20(true, true, X, Y)
% 95.44/12.73 = { by axiom 1 (luka_op_implies) R->L }
% 95.44/12.73 fresh20(op_implies, true, X, Y)
% 95.44/12.73 = { by axiom 1 (luka_op_implies) R->L }
% 95.44/12.73 fresh20(op_implies, op_implies, X, Y)
% 95.44/12.73 = { by axiom 15 (op_or) }
% 95.44/12.73 or(X, Y)
% 95.44/12.73
% 95.44/12.73 Lemma 29: fresh12(r1, op_implies, X) = op_implies.
% 95.44/12.73 Proof:
% 95.44/12.73 fresh12(r1, op_implies, X)
% 95.44/12.73 = { by lemma 27 R->L }
% 95.44/12.73 is_a_theorem(implies(or(X, X), X))
% 95.44/12.73 = { by lemma 28 R->L }
% 95.44/12.73 is_a_theorem(implies(implies(not(X), X), X))
% 95.44/12.73 = { by axiom 23 (cn3_1) R->L }
% 95.44/12.73 fresh47(cn3, true, X)
% 95.44/12.73 = { by axiom 6 (luka_cn3) }
% 95.44/12.73 fresh47(true, true, X)
% 95.44/12.73 = { by axiom 1 (luka_op_implies) R->L }
% 95.44/12.73 fresh47(op_implies, true, X)
% 95.44/12.73 = { by axiom 1 (luka_op_implies) R->L }
% 95.44/12.73 fresh47(op_implies, op_implies, X)
% 95.44/12.73 = { by axiom 10 (cn3_1) }
% 95.44/12.73 true
% 95.44/12.73 = { by axiom 1 (luka_op_implies) R->L }
% 95.44/12.73 op_implies
% 95.44/12.73
% 95.44/12.73 Lemma 30: fresh59(is_a_theorem(implies(X, Y)), op_implies, X, Y) = fresh28(is_a_theorem(X), op_implies, Y).
% 95.44/12.73 Proof:
% 95.44/12.73 fresh59(is_a_theorem(implies(X, Y)), op_implies, X, Y)
% 95.44/12.73 = { by axiom 1 (luka_op_implies) }
% 95.44/12.73 fresh59(is_a_theorem(implies(X, Y)), true, X, Y)
% 95.44/12.73 = { by axiom 24 (modus_ponens_2) }
% 95.44/12.73 fresh28(is_a_theorem(X), true, Y)
% 95.44/12.73 = { by axiom 1 (luka_op_implies) R->L }
% 95.44/12.73 fresh28(is_a_theorem(X), op_implies, Y)
% 95.44/12.73
% 95.44/12.73 Lemma 31: is_a_theorem(implies(X, or(X, Y))) = fresh18(or_1, op_implies, X, Y).
% 95.44/12.73 Proof:
% 95.44/12.73 is_a_theorem(implies(X, or(X, Y)))
% 95.44/12.73 = { by axiom 19 (or_1_1) R->L }
% 95.44/12.74 fresh18(or_1, true, X, Y)
% 95.44/12.74 = { by axiom 1 (luka_op_implies) R->L }
% 95.44/12.74 fresh18(or_1, op_implies, X, Y)
% 95.44/12.74
% 95.44/12.74 Lemma 32: fresh18(or_1, op_implies, X, Y) = op_implies.
% 95.44/12.74 Proof:
% 95.44/12.74 fresh18(or_1, op_implies, X, Y)
% 95.44/12.74 = { by lemma 31 R->L }
% 95.44/12.74 is_a_theorem(implies(X, or(X, Y)))
% 95.44/12.74 = { by lemma 28 R->L }
% 95.44/12.74 is_a_theorem(implies(X, implies(not(X), Y)))
% 95.44/12.74 = { by axiom 22 (cn2_1) R->L }
% 95.44/12.74 fresh49(cn2, true, X, Y)
% 95.44/12.74 = { by axiom 5 (luka_cn2) }
% 95.44/12.74 fresh49(true, true, X, Y)
% 95.44/12.74 = { by axiom 1 (luka_op_implies) R->L }
% 95.44/12.74 fresh49(op_implies, true, X, Y)
% 95.44/12.74 = { by axiom 1 (luka_op_implies) R->L }
% 95.44/12.74 fresh49(op_implies, op_implies, X, Y)
% 95.44/12.74 = { by axiom 13 (cn2_1) }
% 95.44/12.74 true
% 95.44/12.74 = { by axiom 1 (luka_op_implies) R->L }
% 95.44/12.74 op_implies
% 95.44/12.74
% 95.44/12.74 Lemma 33: is_a_theorem(implies(implies(X, Y), implies(implies(Y, Z), implies(X, Z)))) = op_implies.
% 95.44/12.74 Proof:
% 95.44/12.74 is_a_theorem(implies(implies(X, Y), implies(implies(Y, Z), implies(X, Z))))
% 95.44/12.74 = { by axiom 26 (cn1_1) R->L }
% 95.44/12.74 fresh51(cn1, true, X, Y, Z)
% 95.44/12.74 = { by axiom 4 (luka_cn1) }
% 95.44/12.74 fresh51(true, true, X, Y, Z)
% 95.44/12.74 = { by axiom 1 (luka_op_implies) R->L }
% 95.44/12.74 fresh51(op_implies, true, X, Y, Z)
% 95.44/12.74 = { by axiom 1 (luka_op_implies) R->L }
% 95.44/12.74 fresh51(op_implies, op_implies, X, Y, Z)
% 95.44/12.74 = { by axiom 17 (cn1_1) }
% 95.44/12.74 true
% 95.44/12.74 = { by axiom 1 (luka_op_implies) R->L }
% 95.44/12.74 op_implies
% 95.44/12.74
% 95.44/12.74 Lemma 34: fresh59(X, X, Y, Z) = op_implies.
% 95.44/12.74 Proof:
% 95.44/12.74 fresh59(X, X, Y, Z)
% 95.44/12.74 = { by axiom 12 (modus_ponens_2) }
% 95.44/12.74 fresh60(modus_ponens, true, Z)
% 95.44/12.74 = { by axiom 3 (luka_modus_ponens) }
% 95.44/12.74 fresh60(true, true, Z)
% 95.44/12.74 = { by axiom 1 (luka_op_implies) R->L }
% 95.44/12.74 fresh60(op_implies, true, Z)
% 95.44/12.74 = { by axiom 1 (luka_op_implies) R->L }
% 95.44/12.74 fresh60(op_implies, op_implies, Z)
% 95.44/12.74 = { by axiom 9 (modus_ponens_2) }
% 95.44/12.74 true
% 95.44/12.74 = { by axiom 1 (luka_op_implies) R->L }
% 95.44/12.74 op_implies
% 95.44/12.74
% 95.44/12.74 Lemma 35: fresh28(is_a_theorem(implies(X, Y)), op_implies, implies(implies(Y, Z), implies(X, Z))) = op_implies.
% 95.44/12.74 Proof:
% 95.44/12.74 fresh28(is_a_theorem(implies(X, Y)), op_implies, implies(implies(Y, Z), implies(X, Z)))
% 95.44/12.74 = { by lemma 30 R->L }
% 95.44/12.74 fresh59(is_a_theorem(implies(implies(X, Y), implies(implies(Y, Z), implies(X, Z)))), op_implies, implies(X, Y), implies(implies(Y, Z), implies(X, Z)))
% 95.44/12.74 = { by lemma 33 }
% 95.44/12.74 fresh59(op_implies, op_implies, implies(X, Y), implies(implies(Y, Z), implies(X, Z)))
% 95.44/12.74 = { by lemma 34 }
% 95.44/12.74 op_implies
% 95.44/12.74
% 95.44/12.74 Lemma 36: is_a_theorem(implies(implies(or(X, Y), Z), implies(X, Z))) = op_implies.
% 95.44/12.74 Proof:
% 95.44/12.74 is_a_theorem(implies(implies(or(X, Y), Z), implies(X, Z)))
% 95.44/12.74 = { by axiom 11 (modus_ponens_2) R->L }
% 95.44/12.74 fresh28(op_implies, op_implies, implies(implies(or(X, Y), Z), implies(X, Z)))
% 95.44/12.74 = { by lemma 32 R->L }
% 95.44/12.74 fresh28(fresh18(or_1, op_implies, X, Y), op_implies, implies(implies(or(X, Y), Z), implies(X, Z)))
% 95.44/12.74 = { by lemma 31 R->L }
% 95.44/12.74 fresh28(is_a_theorem(implies(X, or(X, Y))), op_implies, implies(implies(or(X, Y), Z), implies(X, Z)))
% 95.44/12.74 = { by lemma 35 }
% 95.44/12.74 op_implies
% 95.44/12.74
% 95.44/12.74 Lemma 37: fresh28(is_a_theorem(implies(or(X, Y), Z)), op_implies, implies(X, Z)) = op_implies.
% 95.44/12.74 Proof:
% 95.44/12.74 fresh28(is_a_theorem(implies(or(X, Y), Z)), op_implies, implies(X, Z))
% 95.44/12.74 = { by lemma 30 R->L }
% 95.44/12.74 fresh59(is_a_theorem(implies(implies(or(X, Y), Z), implies(X, Z))), op_implies, implies(or(X, Y), Z), implies(X, Z))
% 95.44/12.74 = { by lemma 36 }
% 95.44/12.74 fresh59(op_implies, op_implies, implies(or(X, Y), Z), implies(X, Z))
% 95.44/12.74 = { by lemma 34 }
% 95.44/12.74 op_implies
% 95.44/12.74
% 95.44/12.74 Lemma 38: is_a_theorem(implies(X, X)) = op_implies.
% 95.44/12.74 Proof:
% 95.44/12.74 is_a_theorem(implies(X, X))
% 95.44/12.74 = { by axiom 11 (modus_ponens_2) R->L }
% 95.44/12.74 fresh28(op_implies, op_implies, implies(X, X))
% 95.44/12.74 = { by lemma 29 R->L }
% 95.44/12.74 fresh28(fresh12(r1, op_implies, X), op_implies, implies(X, X))
% 95.44/12.74 = { by lemma 27 R->L }
% 95.44/12.74 fresh28(is_a_theorem(implies(or(X, X), X)), op_implies, implies(X, X))
% 95.44/12.74 = { by lemma 37 }
% 95.44/12.74 op_implies
% 95.44/12.74
% 95.44/12.74 Lemma 39: is_a_theorem(implies(X, implies(Y, X))) = fresh39(implies_1, op_implies, X, Y).
% 95.44/12.74 Proof:
% 95.44/12.74 is_a_theorem(implies(X, implies(Y, X)))
% 95.44/12.74 = { by axiom 20 (implies_1_1) R->L }
% 95.44/12.74 fresh39(implies_1, true, X, Y)
% 95.44/12.74 = { by axiom 1 (luka_op_implies) R->L }
% 95.44/12.74 fresh39(implies_1, op_implies, X, Y)
% 95.44/12.74
% 95.44/12.74 Lemma 40: is_a_theorem(implies(or(X, Y), implies(implies(Y, Z), or(X, Z)))) = op_implies.
% 95.44/12.74 Proof:
% 95.44/12.74 is_a_theorem(implies(or(X, Y), implies(implies(Y, Z), or(X, Z))))
% 95.44/12.74 = { by lemma 28 R->L }
% 95.44/12.74 is_a_theorem(implies(or(X, Y), implies(implies(Y, Z), implies(not(X), Z))))
% 95.44/12.74 = { by lemma 28 R->L }
% 95.44/12.74 is_a_theorem(implies(implies(not(X), Y), implies(implies(Y, Z), implies(not(X), Z))))
% 95.44/12.74 = { by lemma 33 }
% 95.44/12.74 op_implies
% 95.44/12.74
% 95.44/12.74 Lemma 41: is_a_theorem(implies(X, implies(implies(Y, Z), or(X, Z)))) = op_implies.
% 95.44/12.74 Proof:
% 95.44/12.74 is_a_theorem(implies(X, implies(implies(Y, Z), or(X, Z))))
% 95.44/12.74 = { by axiom 11 (modus_ponens_2) R->L }
% 95.44/12.74 fresh28(op_implies, op_implies, implies(X, implies(implies(Y, Z), or(X, Z))))
% 95.44/12.74 = { by lemma 40 R->L }
% 95.44/12.74 fresh28(is_a_theorem(implies(or(X, Y), implies(implies(Y, Z), or(X, Z)))), op_implies, implies(X, implies(implies(Y, Z), or(X, Z))))
% 95.44/12.74 = { by lemma 37 }
% 95.44/12.74 op_implies
% 95.44/12.74
% 95.44/12.74 Lemma 42: is_a_theorem(implies(implies(X, Y), implies(or(X, X), Y))) = op_implies.
% 95.44/12.74 Proof:
% 95.44/12.74 is_a_theorem(implies(implies(X, Y), implies(or(X, X), Y)))
% 95.44/12.74 = { by axiom 11 (modus_ponens_2) R->L }
% 95.44/12.74 fresh28(op_implies, op_implies, implies(implies(X, Y), implies(or(X, X), Y)))
% 95.44/12.74 = { by lemma 29 R->L }
% 95.44/12.74 fresh28(fresh12(r1, op_implies, X), op_implies, implies(implies(X, Y), implies(or(X, X), Y)))
% 95.44/12.74 = { by lemma 27 R->L }
% 95.44/12.74 fresh28(is_a_theorem(implies(or(X, X), X)), op_implies, implies(implies(X, Y), implies(or(X, X), Y)))
% 95.44/12.74 = { by lemma 35 }
% 95.44/12.74 op_implies
% 95.44/12.74
% 95.44/12.74 Lemma 43: is_a_theorem(implies(implies(implies(or(X, X), Y), Z), implies(implies(X, Y), Z))) = op_implies.
% 95.44/12.74 Proof:
% 95.44/12.74 is_a_theorem(implies(implies(implies(or(X, X), Y), Z), implies(implies(X, Y), Z)))
% 95.44/12.74 = { by axiom 11 (modus_ponens_2) R->L }
% 95.44/12.74 fresh28(op_implies, op_implies, implies(implies(implies(or(X, X), Y), Z), implies(implies(X, Y), Z)))
% 95.44/12.74 = { by lemma 42 R->L }
% 95.44/12.74 fresh28(is_a_theorem(implies(implies(X, Y), implies(or(X, X), Y))), op_implies, implies(implies(implies(or(X, X), Y), Z), implies(implies(X, Y), Z)))
% 95.44/12.74 = { by lemma 35 }
% 95.44/12.74 op_implies
% 95.44/12.74
% 95.44/12.74 Lemma 44: fresh28(is_a_theorem(implies(implies(or(X, X), Y), Z)), op_implies, implies(implies(X, Y), Z)) = op_implies.
% 95.44/12.74 Proof:
% 95.44/12.74 fresh28(is_a_theorem(implies(implies(or(X, X), Y), Z)), op_implies, implies(implies(X, Y), Z))
% 95.44/12.74 = { by lemma 30 R->L }
% 95.44/12.74 fresh59(is_a_theorem(implies(implies(implies(or(X, X), Y), Z), implies(implies(X, Y), Z))), op_implies, implies(implies(or(X, X), Y), Z), implies(implies(X, Y), Z))
% 95.44/12.74 = { by lemma 43 }
% 95.44/12.74 fresh59(op_implies, op_implies, implies(implies(or(X, X), Y), Z), implies(implies(X, Y), Z))
% 95.44/12.74 = { by lemma 34 }
% 95.44/12.74 op_implies
% 95.44/12.74
% 95.44/12.74 Lemma 45: fresh28(is_a_theorem(implies(implies(implies(X, Y), implies(Z, Y)), W)), op_implies, implies(implies(Z, X), W)) = op_implies.
% 95.44/12.74 Proof:
% 95.44/12.74 fresh28(is_a_theorem(implies(implies(implies(X, Y), implies(Z, Y)), W)), op_implies, implies(implies(Z, X), W))
% 95.44/12.74 = { by lemma 30 R->L }
% 95.44/12.74 fresh59(is_a_theorem(implies(implies(implies(implies(X, Y), implies(Z, Y)), W), implies(implies(Z, X), W))), op_implies, implies(implies(implies(X, Y), implies(Z, Y)), W), implies(implies(Z, X), W))
% 95.44/12.74 = { by axiom 11 (modus_ponens_2) R->L }
% 95.44/12.74 fresh59(fresh28(op_implies, op_implies, implies(implies(implies(implies(X, Y), implies(Z, Y)), W), implies(implies(Z, X), W))), op_implies, implies(implies(implies(X, Y), implies(Z, Y)), W), implies(implies(Z, X), W))
% 95.44/12.74 = { by lemma 33 R->L }
% 95.44/12.74 fresh59(fresh28(is_a_theorem(implies(implies(Z, X), implies(implies(X, Y), implies(Z, Y)))), op_implies, implies(implies(implies(implies(X, Y), implies(Z, Y)), W), implies(implies(Z, X), W))), op_implies, implies(implies(implies(X, Y), implies(Z, Y)), W), implies(implies(Z, X), W))
% 95.44/12.74 = { by lemma 35 }
% 95.44/12.74 fresh59(op_implies, op_implies, implies(implies(implies(X, Y), implies(Z, Y)), W), implies(implies(Z, X), W))
% 95.44/12.74 = { by lemma 34 }
% 95.44/12.74 op_implies
% 95.44/12.74
% 95.44/12.74 Goal 1 (hilbert_implies_1): implies_1 = true.
% 95.44/12.74 Proof:
% 95.44/12.74 implies_1
% 95.44/12.74 = { by axiom 25 (implies_1) R->L }
% 95.44/12.74 fresh40(is_a_theorem(implies(x12, implies(y12, x12))), true)
% 95.44/12.74 = { by lemma 39 }
% 95.44/12.74 fresh40(fresh39(implies_1, op_implies, x12, y12), true)
% 95.44/12.74 = { by axiom 1 (luka_op_implies) R->L }
% 95.44/12.74 fresh40(fresh39(implies_1, op_implies, x12, y12), op_implies)
% 95.44/12.74 = { by lemma 39 R->L }
% 95.44/12.74 fresh40(is_a_theorem(implies(x12, implies(y12, x12))), op_implies)
% 95.44/12.74 = { by axiom 11 (modus_ponens_2) R->L }
% 95.44/12.74 fresh40(fresh28(op_implies, op_implies, implies(x12, implies(y12, x12))), op_implies)
% 95.44/12.74 = { by lemma 34 R->L }
% 95.44/12.74 fresh40(fresh28(fresh59(op_implies, op_implies, implies(implies(implies(y12, x12), implies(y12, x12)), implies(x12, implies(y12, x12))), implies(or(not(implies(x12, implies(y12, x12))), X), implies(x12, implies(y12, x12)))), op_implies, implies(x12, implies(y12, x12))), op_implies)
% 95.44/12.74 = { by lemma 44 R->L }
% 95.44/12.75 fresh40(fresh28(fresh59(fresh28(is_a_theorem(implies(implies(or(implies(implies(y12, x12), implies(y12, x12)), implies(implies(y12, x12), implies(y12, x12))), implies(x12, implies(y12, x12))), implies(or(not(implies(x12, implies(y12, x12))), X), implies(x12, implies(y12, x12))))), op_implies, implies(implies(implies(implies(y12, x12), implies(y12, x12)), implies(x12, implies(y12, x12))), implies(or(not(implies(x12, implies(y12, x12))), X), implies(x12, implies(y12, x12))))), op_implies, implies(implies(implies(y12, x12), implies(y12, x12)), implies(x12, implies(y12, x12))), implies(or(not(implies(x12, implies(y12, x12))), X), implies(x12, implies(y12, x12)))), op_implies, implies(x12, implies(y12, x12))), op_implies)
% 95.44/12.75 = { by axiom 11 (modus_ponens_2) R->L }
% 95.44/12.75 fresh40(fresh28(fresh59(fresh28(fresh28(op_implies, op_implies, implies(implies(or(implies(implies(y12, x12), implies(y12, x12)), implies(implies(y12, x12), implies(y12, x12))), implies(x12, implies(y12, x12))), implies(or(not(implies(x12, implies(y12, x12))), X), implies(x12, implies(y12, x12))))), op_implies, implies(implies(implies(implies(y12, x12), implies(y12, x12)), implies(x12, implies(y12, x12))), implies(or(not(implies(x12, implies(y12, x12))), X), implies(x12, implies(y12, x12))))), op_implies, implies(implies(implies(y12, x12), implies(y12, x12)), implies(x12, implies(y12, x12))), implies(or(not(implies(x12, implies(y12, x12))), X), implies(x12, implies(y12, x12)))), op_implies, implies(x12, implies(y12, x12))), op_implies)
% 95.44/12.75 = { by lemma 37 R->L }
% 95.44/12.75 fresh40(fresh28(fresh59(fresh28(fresh28(fresh28(is_a_theorem(implies(or(or(not(implies(x12, implies(y12, x12))), X), implies(implies(y12, x12), implies(y12, x12))), or(implies(implies(y12, x12), implies(y12, x12)), implies(implies(y12, x12), implies(y12, x12))))), op_implies, implies(or(not(implies(x12, implies(y12, x12))), X), or(implies(implies(y12, x12), implies(y12, x12)), implies(implies(y12, x12), implies(y12, x12))))), op_implies, implies(implies(or(implies(implies(y12, x12), implies(y12, x12)), implies(implies(y12, x12), implies(y12, x12))), implies(x12, implies(y12, x12))), implies(or(not(implies(x12, implies(y12, x12))), X), implies(x12, implies(y12, x12))))), op_implies, implies(implies(implies(implies(y12, x12), implies(y12, x12)), implies(x12, implies(y12, x12))), implies(or(not(implies(x12, implies(y12, x12))), X), implies(x12, implies(y12, x12))))), op_implies, implies(implies(implies(y12, x12), implies(y12, x12)), implies(x12, implies(y12, x12))), implies(or(not(implies(x12, implies(y12, x12))), X), implies(x12, implies(y12, x12)))), op_implies, implies(x12, implies(y12, x12))), op_implies)
% 95.44/12.75 = { by lemma 28 R->L }
% 95.44/12.75 fresh40(fresh28(fresh59(fresh28(fresh28(fresh28(is_a_theorem(implies(implies(not(or(not(implies(x12, implies(y12, x12))), X)), implies(implies(y12, x12), implies(y12, x12))), or(implies(implies(y12, x12), implies(y12, x12)), implies(implies(y12, x12), implies(y12, x12))))), op_implies, implies(or(not(implies(x12, implies(y12, x12))), X), or(implies(implies(y12, x12), implies(y12, x12)), implies(implies(y12, x12), implies(y12, x12))))), op_implies, implies(implies(or(implies(implies(y12, x12), implies(y12, x12)), implies(implies(y12, x12), implies(y12, x12))), implies(x12, implies(y12, x12))), implies(or(not(implies(x12, implies(y12, x12))), X), implies(x12, implies(y12, x12))))), op_implies, implies(implies(implies(implies(y12, x12), implies(y12, x12)), implies(x12, implies(y12, x12))), implies(or(not(implies(x12, implies(y12, x12))), X), implies(x12, implies(y12, x12))))), op_implies, implies(implies(implies(y12, x12), implies(y12, x12)), implies(x12, implies(y12, x12))), implies(or(not(implies(x12, implies(y12, x12))), X), implies(x12, implies(y12, x12)))), op_implies, implies(x12, implies(y12, x12))), op_implies)
% 95.44/12.75 = { by axiom 11 (modus_ponens_2) R->L }
% 95.44/12.75 fresh40(fresh28(fresh59(fresh28(fresh28(fresh28(fresh28(op_implies, op_implies, implies(implies(not(or(not(implies(x12, implies(y12, x12))), X)), implies(implies(y12, x12), implies(y12, x12))), or(implies(implies(y12, x12), implies(y12, x12)), implies(implies(y12, x12), implies(y12, x12))))), op_implies, implies(or(not(implies(x12, implies(y12, x12))), X), or(implies(implies(y12, x12), implies(y12, x12)), implies(implies(y12, x12), implies(y12, x12))))), op_implies, implies(implies(or(implies(implies(y12, x12), implies(y12, x12)), implies(implies(y12, x12), implies(y12, x12))), implies(x12, implies(y12, x12))), implies(or(not(implies(x12, implies(y12, x12))), X), implies(x12, implies(y12, x12))))), op_implies, implies(implies(implies(implies(y12, x12), implies(y12, x12)), implies(x12, implies(y12, x12))), implies(or(not(implies(x12, implies(y12, x12))), X), implies(x12, implies(y12, x12))))), op_implies, implies(implies(implies(y12, x12), implies(y12, x12)), implies(x12, implies(y12, x12))), implies(or(not(implies(x12, implies(y12, x12))), X), implies(x12, implies(y12, x12)))), op_implies, implies(x12, implies(y12, x12))), op_implies)
% 95.44/12.75 = { by lemma 38 R->L }
% 95.44/12.75 fresh40(fresh28(fresh59(fresh28(fresh28(fresh28(fresh28(is_a_theorem(implies(implies(y12, x12), implies(y12, x12))), op_implies, implies(implies(not(or(not(implies(x12, implies(y12, x12))), X)), implies(implies(y12, x12), implies(y12, x12))), or(implies(implies(y12, x12), implies(y12, x12)), implies(implies(y12, x12), implies(y12, x12))))), op_implies, implies(or(not(implies(x12, implies(y12, x12))), X), or(implies(implies(y12, x12), implies(y12, x12)), implies(implies(y12, x12), implies(y12, x12))))), op_implies, implies(implies(or(implies(implies(y12, x12), implies(y12, x12)), implies(implies(y12, x12), implies(y12, x12))), implies(x12, implies(y12, x12))), implies(or(not(implies(x12, implies(y12, x12))), X), implies(x12, implies(y12, x12))))), op_implies, implies(implies(implies(implies(y12, x12), implies(y12, x12)), implies(x12, implies(y12, x12))), implies(or(not(implies(x12, implies(y12, x12))), X), implies(x12, implies(y12, x12))))), op_implies, implies(implies(implies(y12, x12), implies(y12, x12)), implies(x12, implies(y12, x12))), implies(or(not(implies(x12, implies(y12, x12))), X), implies(x12, implies(y12, x12)))), op_implies, implies(x12, implies(y12, x12))), op_implies)
% 95.44/12.75 = { by lemma 30 R->L }
% 95.44/12.75 fresh40(fresh28(fresh59(fresh28(fresh28(fresh28(fresh59(is_a_theorem(implies(implies(implies(y12, x12), implies(y12, x12)), implies(implies(not(or(not(implies(x12, implies(y12, x12))), X)), implies(implies(y12, x12), implies(y12, x12))), or(implies(implies(y12, x12), implies(y12, x12)), implies(implies(y12, x12), implies(y12, x12)))))), op_implies, implies(implies(y12, x12), implies(y12, x12)), implies(implies(not(or(not(implies(x12, implies(y12, x12))), X)), implies(implies(y12, x12), implies(y12, x12))), or(implies(implies(y12, x12), implies(y12, x12)), implies(implies(y12, x12), implies(y12, x12))))), op_implies, implies(or(not(implies(x12, implies(y12, x12))), X), or(implies(implies(y12, x12), implies(y12, x12)), implies(implies(y12, x12), implies(y12, x12))))), op_implies, implies(implies(or(implies(implies(y12, x12), implies(y12, x12)), implies(implies(y12, x12), implies(y12, x12))), implies(x12, implies(y12, x12))), implies(or(not(implies(x12, implies(y12, x12))), X), implies(x12, implies(y12, x12))))), op_implies, implies(implies(implies(implies(y12, x12), implies(y12, x12)), implies(x12, implies(y12, x12))), implies(or(not(implies(x12, implies(y12, x12))), X), implies(x12, implies(y12, x12))))), op_implies, implies(implies(implies(y12, x12), implies(y12, x12)), implies(x12, implies(y12, x12))), implies(or(not(implies(x12, implies(y12, x12))), X), implies(x12, implies(y12, x12)))), op_implies, implies(x12, implies(y12, x12))), op_implies)
% 95.44/12.75 = { by lemma 41 }
% 95.44/12.75 fresh40(fresh28(fresh59(fresh28(fresh28(fresh28(fresh59(op_implies, op_implies, implies(implies(y12, x12), implies(y12, x12)), implies(implies(not(or(not(implies(x12, implies(y12, x12))), X)), implies(implies(y12, x12), implies(y12, x12))), or(implies(implies(y12, x12), implies(y12, x12)), implies(implies(y12, x12), implies(y12, x12))))), op_implies, implies(or(not(implies(x12, implies(y12, x12))), X), or(implies(implies(y12, x12), implies(y12, x12)), implies(implies(y12, x12), implies(y12, x12))))), op_implies, implies(implies(or(implies(implies(y12, x12), implies(y12, x12)), implies(implies(y12, x12), implies(y12, x12))), implies(x12, implies(y12, x12))), implies(or(not(implies(x12, implies(y12, x12))), X), implies(x12, implies(y12, x12))))), op_implies, implies(implies(implies(implies(y12, x12), implies(y12, x12)), implies(x12, implies(y12, x12))), implies(or(not(implies(x12, implies(y12, x12))), X), implies(x12, implies(y12, x12))))), op_implies, implies(implies(implies(y12, x12), implies(y12, x12)), implies(x12, implies(y12, x12))), implies(or(not(implies(x12, implies(y12, x12))), X), implies(x12, implies(y12, x12)))), op_implies, implies(x12, implies(y12, x12))), op_implies)
% 95.44/12.75 = { by lemma 34 }
% 95.44/12.75 fresh40(fresh28(fresh59(fresh28(fresh28(fresh28(op_implies, op_implies, implies(or(not(implies(x12, implies(y12, x12))), X), or(implies(implies(y12, x12), implies(y12, x12)), implies(implies(y12, x12), implies(y12, x12))))), op_implies, implies(implies(or(implies(implies(y12, x12), implies(y12, x12)), implies(implies(y12, x12), implies(y12, x12))), implies(x12, implies(y12, x12))), implies(or(not(implies(x12, implies(y12, x12))), X), implies(x12, implies(y12, x12))))), op_implies, implies(implies(implies(implies(y12, x12), implies(y12, x12)), implies(x12, implies(y12, x12))), implies(or(not(implies(x12, implies(y12, x12))), X), implies(x12, implies(y12, x12))))), op_implies, implies(implies(implies(y12, x12), implies(y12, x12)), implies(x12, implies(y12, x12))), implies(or(not(implies(x12, implies(y12, x12))), X), implies(x12, implies(y12, x12)))), op_implies, implies(x12, implies(y12, x12))), op_implies)
% 95.44/12.75 = { by axiom 11 (modus_ponens_2) }
% 95.44/12.75 fresh40(fresh28(fresh59(fresh28(fresh28(is_a_theorem(implies(or(not(implies(x12, implies(y12, x12))), X), or(implies(implies(y12, x12), implies(y12, x12)), implies(implies(y12, x12), implies(y12, x12))))), op_implies, implies(implies(or(implies(implies(y12, x12), implies(y12, x12)), implies(implies(y12, x12), implies(y12, x12))), implies(x12, implies(y12, x12))), implies(or(not(implies(x12, implies(y12, x12))), X), implies(x12, implies(y12, x12))))), op_implies, implies(implies(implies(implies(y12, x12), implies(y12, x12)), implies(x12, implies(y12, x12))), implies(or(not(implies(x12, implies(y12, x12))), X), implies(x12, implies(y12, x12))))), op_implies, implies(implies(implies(y12, x12), implies(y12, x12)), implies(x12, implies(y12, x12))), implies(or(not(implies(x12, implies(y12, x12))), X), implies(x12, implies(y12, x12)))), op_implies, implies(x12, implies(y12, x12))), op_implies)
% 95.44/12.75 = { by lemma 35 }
% 95.44/12.75 fresh40(fresh28(fresh59(fresh28(op_implies, op_implies, implies(implies(implies(implies(y12, x12), implies(y12, x12)), implies(x12, implies(y12, x12))), implies(or(not(implies(x12, implies(y12, x12))), X), implies(x12, implies(y12, x12))))), op_implies, implies(implies(implies(y12, x12), implies(y12, x12)), implies(x12, implies(y12, x12))), implies(or(not(implies(x12, implies(y12, x12))), X), implies(x12, implies(y12, x12)))), op_implies, implies(x12, implies(y12, x12))), op_implies)
% 95.44/12.75 = { by axiom 11 (modus_ponens_2) }
% 95.44/12.75 fresh40(fresh28(fresh59(is_a_theorem(implies(implies(implies(implies(y12, x12), implies(y12, x12)), implies(x12, implies(y12, x12))), implies(or(not(implies(x12, implies(y12, x12))), X), implies(x12, implies(y12, x12))))), op_implies, implies(implies(implies(y12, x12), implies(y12, x12)), implies(x12, implies(y12, x12))), implies(or(not(implies(x12, implies(y12, x12))), X), implies(x12, implies(y12, x12)))), op_implies, implies(x12, implies(y12, x12))), op_implies)
% 95.44/12.75 = { by lemma 30 }
% 95.44/12.75 fresh40(fresh28(fresh28(is_a_theorem(implies(implies(implies(y12, x12), implies(y12, x12)), implies(x12, implies(y12, x12)))), op_implies, implies(or(not(implies(x12, implies(y12, x12))), X), implies(x12, implies(y12, x12)))), op_implies, implies(x12, implies(y12, x12))), op_implies)
% 95.44/12.75 = { by axiom 11 (modus_ponens_2) R->L }
% 95.44/12.75 fresh40(fresh28(fresh28(fresh28(op_implies, op_implies, implies(implies(implies(y12, x12), implies(y12, x12)), implies(x12, implies(y12, x12)))), op_implies, implies(or(not(implies(x12, implies(y12, x12))), X), implies(x12, implies(y12, x12)))), op_implies, implies(x12, implies(y12, x12))), op_implies)
% 95.44/12.75 = { by lemma 34 R->L }
% 95.44/12.76 fresh40(fresh28(fresh28(fresh28(fresh59(op_implies, op_implies, implies(implies(y12, or(x12, x12)), implies(not(implies(y12, x12)), implies(y12, x12))), implies(x12, implies(not(implies(y12, x12)), implies(y12, x12)))), op_implies, implies(implies(implies(y12, x12), implies(y12, x12)), implies(x12, implies(y12, x12)))), op_implies, implies(or(not(implies(x12, implies(y12, x12))), X), implies(x12, implies(y12, x12)))), op_implies, implies(x12, implies(y12, x12))), op_implies)
% 95.44/12.76 = { by lemma 35 R->L }
% 95.44/12.76 fresh40(fresh28(fresh28(fresh28(fresh59(fresh28(is_a_theorem(implies(x12, implies(y12, or(x12, x12)))), op_implies, implies(implies(implies(y12, or(x12, x12)), implies(not(implies(y12, x12)), implies(y12, x12))), implies(x12, implies(not(implies(y12, x12)), implies(y12, x12))))), op_implies, implies(implies(y12, or(x12, x12)), implies(not(implies(y12, x12)), implies(y12, x12))), implies(x12, implies(not(implies(y12, x12)), implies(y12, x12)))), op_implies, implies(implies(implies(y12, x12), implies(y12, x12)), implies(x12, implies(y12, x12)))), op_implies, implies(or(not(implies(x12, implies(y12, x12))), X), implies(x12, implies(y12, x12)))), op_implies, implies(x12, implies(y12, x12))), op_implies)
% 95.44/12.76 = { by axiom 11 (modus_ponens_2) R->L }
% 95.44/12.76 fresh40(fresh28(fresh28(fresh28(fresh59(fresh28(fresh28(op_implies, op_implies, implies(x12, implies(y12, or(x12, x12)))), op_implies, implies(implies(implies(y12, or(x12, x12)), implies(not(implies(y12, x12)), implies(y12, x12))), implies(x12, implies(not(implies(y12, x12)), implies(y12, x12))))), op_implies, implies(implies(y12, or(x12, x12)), implies(not(implies(y12, x12)), implies(y12, x12))), implies(x12, implies(not(implies(y12, x12)), implies(y12, x12)))), op_implies, implies(implies(implies(y12, x12), implies(y12, x12)), implies(x12, implies(y12, x12)))), op_implies, implies(or(not(implies(x12, implies(y12, x12))), X), implies(x12, implies(y12, x12)))), op_implies, implies(x12, implies(y12, x12))), op_implies)
% 95.44/12.76 = { by lemma 35 R->L }
% 95.44/12.76 fresh40(fresh28(fresh28(fresh28(fresh59(fresh28(fresh28(fresh28(is_a_theorem(implies(y12, implies(or(not(y12), not(y12)), x12))), op_implies, implies(implies(implies(or(not(y12), not(y12)), x12), or(x12, x12)), implies(y12, or(x12, x12)))), op_implies, implies(x12, implies(y12, or(x12, x12)))), op_implies, implies(implies(implies(y12, or(x12, x12)), implies(not(implies(y12, x12)), implies(y12, x12))), implies(x12, implies(not(implies(y12, x12)), implies(y12, x12))))), op_implies, implies(implies(y12, or(x12, x12)), implies(not(implies(y12, x12)), implies(y12, x12))), implies(x12, implies(not(implies(y12, x12)), implies(y12, x12)))), op_implies, implies(implies(implies(y12, x12), implies(y12, x12)), implies(x12, implies(y12, x12)))), op_implies, implies(or(not(implies(x12, implies(y12, x12))), X), implies(x12, implies(y12, x12)))), op_implies, implies(x12, implies(y12, x12))), op_implies)
% 95.44/12.76 = { by axiom 11 (modus_ponens_2) R->L }
% 95.44/12.76 fresh40(fresh28(fresh28(fresh28(fresh59(fresh28(fresh28(fresh28(fresh28(op_implies, op_implies, implies(y12, implies(or(not(y12), not(y12)), x12))), op_implies, implies(implies(implies(or(not(y12), not(y12)), x12), or(x12, x12)), implies(y12, or(x12, x12)))), op_implies, implies(x12, implies(y12, or(x12, x12)))), op_implies, implies(implies(implies(y12, or(x12, x12)), implies(not(implies(y12, x12)), implies(y12, x12))), implies(x12, implies(not(implies(y12, x12)), implies(y12, x12))))), op_implies, implies(implies(y12, or(x12, x12)), implies(not(implies(y12, x12)), implies(y12, x12))), implies(x12, implies(not(implies(y12, x12)), implies(y12, x12)))), op_implies, implies(implies(implies(y12, x12), implies(y12, x12)), implies(x12, implies(y12, x12)))), op_implies, implies(or(not(implies(x12, implies(y12, x12))), X), implies(x12, implies(y12, x12)))), op_implies, implies(x12, implies(y12, x12))), op_implies)
% 95.44/12.76 = { by lemma 42 R->L }
% 95.44/12.76 fresh40(fresh28(fresh28(fresh28(fresh59(fresh28(fresh28(fresh28(fresh28(is_a_theorem(implies(implies(not(y12), x12), implies(or(not(y12), not(y12)), x12))), op_implies, implies(y12, implies(or(not(y12), not(y12)), x12))), op_implies, implies(implies(implies(or(not(y12), not(y12)), x12), or(x12, x12)), implies(y12, or(x12, x12)))), op_implies, implies(x12, implies(y12, or(x12, x12)))), op_implies, implies(implies(implies(y12, or(x12, x12)), implies(not(implies(y12, x12)), implies(y12, x12))), implies(x12, implies(not(implies(y12, x12)), implies(y12, x12))))), op_implies, implies(implies(y12, or(x12, x12)), implies(not(implies(y12, x12)), implies(y12, x12))), implies(x12, implies(not(implies(y12, x12)), implies(y12, x12)))), op_implies, implies(implies(implies(y12, x12), implies(y12, x12)), implies(x12, implies(y12, x12)))), op_implies, implies(or(not(implies(x12, implies(y12, x12))), X), implies(x12, implies(y12, x12)))), op_implies, implies(x12, implies(y12, x12))), op_implies)
% 95.44/12.76 = { by lemma 28 }
% 95.44/12.76 fresh40(fresh28(fresh28(fresh28(fresh59(fresh28(fresh28(fresh28(fresh28(is_a_theorem(implies(or(y12, x12), implies(or(not(y12), not(y12)), x12))), op_implies, implies(y12, implies(or(not(y12), not(y12)), x12))), op_implies, implies(implies(implies(or(not(y12), not(y12)), x12), or(x12, x12)), implies(y12, or(x12, x12)))), op_implies, implies(x12, implies(y12, or(x12, x12)))), op_implies, implies(implies(implies(y12, or(x12, x12)), implies(not(implies(y12, x12)), implies(y12, x12))), implies(x12, implies(not(implies(y12, x12)), implies(y12, x12))))), op_implies, implies(implies(y12, or(x12, x12)), implies(not(implies(y12, x12)), implies(y12, x12))), implies(x12, implies(not(implies(y12, x12)), implies(y12, x12)))), op_implies, implies(implies(implies(y12, x12), implies(y12, x12)), implies(x12, implies(y12, x12)))), op_implies, implies(or(not(implies(x12, implies(y12, x12))), X), implies(x12, implies(y12, x12)))), op_implies, implies(x12, implies(y12, x12))), op_implies)
% 95.44/12.76 = { by lemma 37 }
% 95.44/12.76 fresh40(fresh28(fresh28(fresh28(fresh59(fresh28(fresh28(fresh28(op_implies, op_implies, implies(implies(implies(or(not(y12), not(y12)), x12), or(x12, x12)), implies(y12, or(x12, x12)))), op_implies, implies(x12, implies(y12, or(x12, x12)))), op_implies, implies(implies(implies(y12, or(x12, x12)), implies(not(implies(y12, x12)), implies(y12, x12))), implies(x12, implies(not(implies(y12, x12)), implies(y12, x12))))), op_implies, implies(implies(y12, or(x12, x12)), implies(not(implies(y12, x12)), implies(y12, x12))), implies(x12, implies(not(implies(y12, x12)), implies(y12, x12)))), op_implies, implies(implies(implies(y12, x12), implies(y12, x12)), implies(x12, implies(y12, x12)))), op_implies, implies(or(not(implies(x12, implies(y12, x12))), X), implies(x12, implies(y12, x12)))), op_implies, implies(x12, implies(y12, x12))), op_implies)
% 95.44/12.76 = { by axiom 11 (modus_ponens_2) }
% 95.44/12.76 fresh40(fresh28(fresh28(fresh28(fresh59(fresh28(fresh28(is_a_theorem(implies(implies(implies(or(not(y12), not(y12)), x12), or(x12, x12)), implies(y12, or(x12, x12)))), op_implies, implies(x12, implies(y12, or(x12, x12)))), op_implies, implies(implies(implies(y12, or(x12, x12)), implies(not(implies(y12, x12)), implies(y12, x12))), implies(x12, implies(not(implies(y12, x12)), implies(y12, x12))))), op_implies, implies(implies(y12, or(x12, x12)), implies(not(implies(y12, x12)), implies(y12, x12))), implies(x12, implies(not(implies(y12, x12)), implies(y12, x12)))), op_implies, implies(implies(implies(y12, x12), implies(y12, x12)), implies(x12, implies(y12, x12)))), op_implies, implies(or(not(implies(x12, implies(y12, x12))), X), implies(x12, implies(y12, x12)))), op_implies, implies(x12, implies(y12, x12))), op_implies)
% 95.44/12.76 = { by lemma 30 R->L }
% 95.44/12.76 fresh40(fresh28(fresh28(fresh28(fresh59(fresh28(fresh59(is_a_theorem(implies(implies(implies(implies(or(not(y12), not(y12)), x12), or(x12, x12)), implies(y12, or(x12, x12))), implies(x12, implies(y12, or(x12, x12))))), op_implies, implies(implies(implies(or(not(y12), not(y12)), x12), or(x12, x12)), implies(y12, or(x12, x12))), implies(x12, implies(y12, or(x12, x12)))), op_implies, implies(implies(implies(y12, or(x12, x12)), implies(not(implies(y12, x12)), implies(y12, x12))), implies(x12, implies(not(implies(y12, x12)), implies(y12, x12))))), op_implies, implies(implies(y12, or(x12, x12)), implies(not(implies(y12, x12)), implies(y12, x12))), implies(x12, implies(not(implies(y12, x12)), implies(y12, x12)))), op_implies, implies(implies(implies(y12, x12), implies(y12, x12)), implies(x12, implies(y12, x12)))), op_implies, implies(or(not(implies(x12, implies(y12, x12))), X), implies(x12, implies(y12, x12)))), op_implies, implies(x12, implies(y12, x12))), op_implies)
% 95.44/12.76 = { by axiom 11 (modus_ponens_2) R->L }
% 95.44/12.76 fresh40(fresh28(fresh28(fresh28(fresh59(fresh28(fresh59(fresh28(op_implies, op_implies, implies(implies(implies(implies(or(not(y12), not(y12)), x12), or(x12, x12)), implies(y12, or(x12, x12))), implies(x12, implies(y12, or(x12, x12))))), op_implies, implies(implies(implies(or(not(y12), not(y12)), x12), or(x12, x12)), implies(y12, or(x12, x12))), implies(x12, implies(y12, or(x12, x12)))), op_implies, implies(implies(implies(y12, or(x12, x12)), implies(not(implies(y12, x12)), implies(y12, x12))), implies(x12, implies(not(implies(y12, x12)), implies(y12, x12))))), op_implies, implies(implies(y12, or(x12, x12)), implies(not(implies(y12, x12)), implies(y12, x12))), implies(x12, implies(not(implies(y12, x12)), implies(y12, x12)))), op_implies, implies(implies(implies(y12, x12), implies(y12, x12)), implies(x12, implies(y12, x12)))), op_implies, implies(or(not(implies(x12, implies(y12, x12))), X), implies(x12, implies(y12, x12)))), op_implies, implies(x12, implies(y12, x12))), op_implies)
% 95.44/12.76 = { by lemma 41 R->L }
% 95.44/12.76 fresh40(fresh28(fresh28(fresh28(fresh59(fresh28(fresh59(fresh28(is_a_theorem(implies(x12, implies(implies(or(not(y12), not(y12)), x12), or(x12, x12)))), op_implies, implies(implies(implies(implies(or(not(y12), not(y12)), x12), or(x12, x12)), implies(y12, or(x12, x12))), implies(x12, implies(y12, or(x12, x12))))), op_implies, implies(implies(implies(or(not(y12), not(y12)), x12), or(x12, x12)), implies(y12, or(x12, x12))), implies(x12, implies(y12, or(x12, x12)))), op_implies, implies(implies(implies(y12, or(x12, x12)), implies(not(implies(y12, x12)), implies(y12, x12))), implies(x12, implies(not(implies(y12, x12)), implies(y12, x12))))), op_implies, implies(implies(y12, or(x12, x12)), implies(not(implies(y12, x12)), implies(y12, x12))), implies(x12, implies(not(implies(y12, x12)), implies(y12, x12)))), op_implies, implies(implies(implies(y12, x12), implies(y12, x12)), implies(x12, implies(y12, x12)))), op_implies, implies(or(not(implies(x12, implies(y12, x12))), X), implies(x12, implies(y12, x12)))), op_implies, implies(x12, implies(y12, x12))), op_implies)
% 95.44/12.76 = { by lemma 35 }
% 95.44/12.76 fresh40(fresh28(fresh28(fresh28(fresh59(fresh28(fresh59(op_implies, op_implies, implies(implies(implies(or(not(y12), not(y12)), x12), or(x12, x12)), implies(y12, or(x12, x12))), implies(x12, implies(y12, or(x12, x12)))), op_implies, implies(implies(implies(y12, or(x12, x12)), implies(not(implies(y12, x12)), implies(y12, x12))), implies(x12, implies(not(implies(y12, x12)), implies(y12, x12))))), op_implies, implies(implies(y12, or(x12, x12)), implies(not(implies(y12, x12)), implies(y12, x12))), implies(x12, implies(not(implies(y12, x12)), implies(y12, x12)))), op_implies, implies(implies(implies(y12, x12), implies(y12, x12)), implies(x12, implies(y12, x12)))), op_implies, implies(or(not(implies(x12, implies(y12, x12))), X), implies(x12, implies(y12, x12)))), op_implies, implies(x12, implies(y12, x12))), op_implies)
% 95.44/12.76 = { by lemma 34 }
% 95.44/12.77 fresh40(fresh28(fresh28(fresh28(fresh59(fresh28(op_implies, op_implies, implies(implies(implies(y12, or(x12, x12)), implies(not(implies(y12, x12)), implies(y12, x12))), implies(x12, implies(not(implies(y12, x12)), implies(y12, x12))))), op_implies, implies(implies(y12, or(x12, x12)), implies(not(implies(y12, x12)), implies(y12, x12))), implies(x12, implies(not(implies(y12, x12)), implies(y12, x12)))), op_implies, implies(implies(implies(y12, x12), implies(y12, x12)), implies(x12, implies(y12, x12)))), op_implies, implies(or(not(implies(x12, implies(y12, x12))), X), implies(x12, implies(y12, x12)))), op_implies, implies(x12, implies(y12, x12))), op_implies)
% 95.44/12.77 = { by axiom 11 (modus_ponens_2) }
% 95.44/12.77 fresh40(fresh28(fresh28(fresh28(fresh59(is_a_theorem(implies(implies(implies(y12, or(x12, x12)), implies(not(implies(y12, x12)), implies(y12, x12))), implies(x12, implies(not(implies(y12, x12)), implies(y12, x12))))), op_implies, implies(implies(y12, or(x12, x12)), implies(not(implies(y12, x12)), implies(y12, x12))), implies(x12, implies(not(implies(y12, x12)), implies(y12, x12)))), op_implies, implies(implies(implies(y12, x12), implies(y12, x12)), implies(x12, implies(y12, x12)))), op_implies, implies(or(not(implies(x12, implies(y12, x12))), X), implies(x12, implies(y12, x12)))), op_implies, implies(x12, implies(y12, x12))), op_implies)
% 95.44/12.77 = { by lemma 30 }
% 95.44/12.77 fresh40(fresh28(fresh28(fresh28(fresh28(is_a_theorem(implies(implies(y12, or(x12, x12)), implies(not(implies(y12, x12)), implies(y12, x12)))), op_implies, implies(x12, implies(not(implies(y12, x12)), implies(y12, x12)))), op_implies, implies(implies(implies(y12, x12), implies(y12, x12)), implies(x12, implies(y12, x12)))), op_implies, implies(or(not(implies(x12, implies(y12, x12))), X), implies(x12, implies(y12, x12)))), op_implies, implies(x12, implies(y12, x12))), op_implies)
% 96.04/12.77 = { by axiom 11 (modus_ponens_2) R->L }
% 96.04/12.77 fresh40(fresh28(fresh28(fresh28(fresh28(fresh28(op_implies, op_implies, implies(implies(y12, or(x12, x12)), implies(not(implies(y12, x12)), implies(y12, x12)))), op_implies, implies(x12, implies(not(implies(y12, x12)), implies(y12, x12)))), op_implies, implies(implies(implies(y12, x12), implies(y12, x12)), implies(x12, implies(y12, x12)))), op_implies, implies(or(not(implies(x12, implies(y12, x12))), X), implies(x12, implies(y12, x12)))), op_implies, implies(x12, implies(y12, x12))), op_implies)
% 96.04/12.78 = { by lemma 44 R->L }
% 96.04/12.78 fresh40(fresh28(fresh28(fresh28(fresh28(fresh28(fresh28(is_a_theorem(implies(implies(or(implies(or(x12, x12), x12), implies(or(x12, x12), x12)), implies(y12, x12)), implies(not(implies(y12, x12)), implies(y12, x12)))), op_implies, implies(implies(implies(or(x12, x12), x12), implies(y12, x12)), implies(not(implies(y12, x12)), implies(y12, x12)))), op_implies, implies(implies(y12, or(x12, x12)), implies(not(implies(y12, x12)), implies(y12, x12)))), op_implies, implies(x12, implies(not(implies(y12, x12)), implies(y12, x12)))), op_implies, implies(implies(implies(y12, x12), implies(y12, x12)), implies(x12, implies(y12, x12)))), op_implies, implies(or(not(implies(x12, implies(y12, x12))), X), implies(x12, implies(y12, x12)))), op_implies, implies(x12, implies(y12, x12))), op_implies)
% 96.04/12.78 = { by axiom 11 (modus_ponens_2) R->L }
% 96.04/12.78 fresh40(fresh28(fresh28(fresh28(fresh28(fresh28(fresh28(fresh28(op_implies, op_implies, implies(implies(or(implies(or(x12, x12), x12), implies(or(x12, x12), x12)), implies(y12, x12)), implies(not(implies(y12, x12)), implies(y12, x12)))), op_implies, implies(implies(implies(or(x12, x12), x12), implies(y12, x12)), implies(not(implies(y12, x12)), implies(y12, x12)))), op_implies, implies(implies(y12, or(x12, x12)), implies(not(implies(y12, x12)), implies(y12, x12)))), op_implies, implies(x12, implies(not(implies(y12, x12)), implies(y12, x12)))), op_implies, implies(implies(implies(y12, x12), implies(y12, x12)), implies(x12, implies(y12, x12)))), op_implies, implies(or(not(implies(x12, implies(y12, x12))), X), implies(x12, implies(y12, x12)))), op_implies, implies(x12, implies(y12, x12))), op_implies)
% 96.04/12.78 = { by lemma 37 R->L }
% 96.04/12.78 fresh40(fresh28(fresh28(fresh28(fresh28(fresh28(fresh28(fresh28(fresh28(is_a_theorem(implies(or(not(implies(y12, x12)), implies(or(x12, x12), x12)), or(implies(or(x12, x12), x12), implies(or(x12, x12), x12)))), op_implies, implies(not(implies(y12, x12)), or(implies(or(x12, x12), x12), implies(or(x12, x12), x12)))), op_implies, implies(implies(or(implies(or(x12, x12), x12), implies(or(x12, x12), x12)), implies(y12, x12)), implies(not(implies(y12, x12)), implies(y12, x12)))), op_implies, implies(implies(implies(or(x12, x12), x12), implies(y12, x12)), implies(not(implies(y12, x12)), implies(y12, x12)))), op_implies, implies(implies(y12, or(x12, x12)), implies(not(implies(y12, x12)), implies(y12, x12)))), op_implies, implies(x12, implies(not(implies(y12, x12)), implies(y12, x12)))), op_implies, implies(implies(implies(y12, x12), implies(y12, x12)), implies(x12, implies(y12, x12)))), op_implies, implies(or(not(implies(x12, implies(y12, x12))), X), implies(x12, implies(y12, x12)))), op_implies, implies(x12, implies(y12, x12))), op_implies)
% 96.04/12.78 = { by lemma 28 R->L }
% 96.04/12.78 fresh40(fresh28(fresh28(fresh28(fresh28(fresh28(fresh28(fresh28(fresh28(is_a_theorem(implies(implies(not(not(implies(y12, x12))), implies(or(x12, x12), x12)), or(implies(or(x12, x12), x12), implies(or(x12, x12), x12)))), op_implies, implies(not(implies(y12, x12)), or(implies(or(x12, x12), x12), implies(or(x12, x12), x12)))), op_implies, implies(implies(or(implies(or(x12, x12), x12), implies(or(x12, x12), x12)), implies(y12, x12)), implies(not(implies(y12, x12)), implies(y12, x12)))), op_implies, implies(implies(implies(or(x12, x12), x12), implies(y12, x12)), implies(not(implies(y12, x12)), implies(y12, x12)))), op_implies, implies(implies(y12, or(x12, x12)), implies(not(implies(y12, x12)), implies(y12, x12)))), op_implies, implies(x12, implies(not(implies(y12, x12)), implies(y12, x12)))), op_implies, implies(implies(implies(y12, x12), implies(y12, x12)), implies(x12, implies(y12, x12)))), op_implies, implies(or(not(implies(x12, implies(y12, x12))), X), implies(x12, implies(y12, x12)))), op_implies, implies(x12, implies(y12, x12))), op_implies)
% 96.04/12.78 = { by axiom 11 (modus_ponens_2) R->L }
% 96.04/12.78 fresh40(fresh28(fresh28(fresh28(fresh28(fresh28(fresh28(fresh28(fresh28(fresh28(op_implies, op_implies, implies(implies(not(not(implies(y12, x12))), implies(or(x12, x12), x12)), or(implies(or(x12, x12), x12), implies(or(x12, x12), x12)))), op_implies, implies(not(implies(y12, x12)), or(implies(or(x12, x12), x12), implies(or(x12, x12), x12)))), op_implies, implies(implies(or(implies(or(x12, x12), x12), implies(or(x12, x12), x12)), implies(y12, x12)), implies(not(implies(y12, x12)), implies(y12, x12)))), op_implies, implies(implies(implies(or(x12, x12), x12), implies(y12, x12)), implies(not(implies(y12, x12)), implies(y12, x12)))), op_implies, implies(implies(y12, or(x12, x12)), implies(not(implies(y12, x12)), implies(y12, x12)))), op_implies, implies(x12, implies(not(implies(y12, x12)), implies(y12, x12)))), op_implies, implies(implies(implies(y12, x12), implies(y12, x12)), implies(x12, implies(y12, x12)))), op_implies, implies(or(not(implies(x12, implies(y12, x12))), X), implies(x12, implies(y12, x12)))), op_implies, implies(x12, implies(y12, x12))), op_implies)
% 96.04/12.78 = { by lemma 34 R->L }
% 96.04/12.78 fresh40(fresh28(fresh28(fresh28(fresh28(fresh28(fresh28(fresh28(fresh28(fresh28(fresh59(op_implies, op_implies, implies(or(x12, x12), x12), or(implies(or(x12, x12), x12), not(not(implies(y12, x12))))), op_implies, implies(implies(not(not(implies(y12, x12))), implies(or(x12, x12), x12)), or(implies(or(x12, x12), x12), implies(or(x12, x12), x12)))), op_implies, implies(not(implies(y12, x12)), or(implies(or(x12, x12), x12), implies(or(x12, x12), x12)))), op_implies, implies(implies(or(implies(or(x12, x12), x12), implies(or(x12, x12), x12)), implies(y12, x12)), implies(not(implies(y12, x12)), implies(y12, x12)))), op_implies, implies(implies(implies(or(x12, x12), x12), implies(y12, x12)), implies(not(implies(y12, x12)), implies(y12, x12)))), op_implies, implies(implies(y12, or(x12, x12)), implies(not(implies(y12, x12)), implies(y12, x12)))), op_implies, implies(x12, implies(not(implies(y12, x12)), implies(y12, x12)))), op_implies, implies(implies(implies(y12, x12), implies(y12, x12)), implies(x12, implies(y12, x12)))), op_implies, implies(or(not(implies(x12, implies(y12, x12))), X), implies(x12, implies(y12, x12)))), op_implies, implies(x12, implies(y12, x12))), op_implies)
% 96.04/12.78 = { by lemma 32 R->L }
% 96.04/12.78 fresh40(fresh28(fresh28(fresh28(fresh28(fresh28(fresh28(fresh28(fresh28(fresh28(fresh59(fresh18(or_1, op_implies, implies(or(x12, x12), x12), not(not(implies(y12, x12)))), op_implies, implies(or(x12, x12), x12), or(implies(or(x12, x12), x12), not(not(implies(y12, x12))))), op_implies, implies(implies(not(not(implies(y12, x12))), implies(or(x12, x12), x12)), or(implies(or(x12, x12), x12), implies(or(x12, x12), x12)))), op_implies, implies(not(implies(y12, x12)), or(implies(or(x12, x12), x12), implies(or(x12, x12), x12)))), op_implies, implies(implies(or(implies(or(x12, x12), x12), implies(or(x12, x12), x12)), implies(y12, x12)), implies(not(implies(y12, x12)), implies(y12, x12)))), op_implies, implies(implies(implies(or(x12, x12), x12), implies(y12, x12)), implies(not(implies(y12, x12)), implies(y12, x12)))), op_implies, implies(implies(y12, or(x12, x12)), implies(not(implies(y12, x12)), implies(y12, x12)))), op_implies, implies(x12, implies(not(implies(y12, x12)), implies(y12, x12)))), op_implies, implies(implies(implies(y12, x12), implies(y12, x12)), implies(x12, implies(y12, x12)))), op_implies, implies(or(not(implies(x12, implies(y12, x12))), X), implies(x12, implies(y12, x12)))), op_implies, implies(x12, implies(y12, x12))), op_implies)
% 96.04/12.78 = { by lemma 31 R->L }
% 96.04/12.78 fresh40(fresh28(fresh28(fresh28(fresh28(fresh28(fresh28(fresh28(fresh28(fresh28(fresh59(is_a_theorem(implies(implies(or(x12, x12), x12), or(implies(or(x12, x12), x12), not(not(implies(y12, x12)))))), op_implies, implies(or(x12, x12), x12), or(implies(or(x12, x12), x12), not(not(implies(y12, x12))))), op_implies, implies(implies(not(not(implies(y12, x12))), implies(or(x12, x12), x12)), or(implies(or(x12, x12), x12), implies(or(x12, x12), x12)))), op_implies, implies(not(implies(y12, x12)), or(implies(or(x12, x12), x12), implies(or(x12, x12), x12)))), op_implies, implies(implies(or(implies(or(x12, x12), x12), implies(or(x12, x12), x12)), implies(y12, x12)), implies(not(implies(y12, x12)), implies(y12, x12)))), op_implies, implies(implies(implies(or(x12, x12), x12), implies(y12, x12)), implies(not(implies(y12, x12)), implies(y12, x12)))), op_implies, implies(implies(y12, or(x12, x12)), implies(not(implies(y12, x12)), implies(y12, x12)))), op_implies, implies(x12, implies(not(implies(y12, x12)), implies(y12, x12)))), op_implies, implies(implies(implies(y12, x12), implies(y12, x12)), implies(x12, implies(y12, x12)))), op_implies, implies(or(not(implies(x12, implies(y12, x12))), X), implies(x12, implies(y12, x12)))), op_implies, implies(x12, implies(y12, x12))), op_implies)
% 96.04/12.78 = { by lemma 30 }
% 96.04/12.78 fresh40(fresh28(fresh28(fresh28(fresh28(fresh28(fresh28(fresh28(fresh28(fresh28(fresh28(is_a_theorem(implies(or(x12, x12), x12)), op_implies, or(implies(or(x12, x12), x12), not(not(implies(y12, x12))))), op_implies, implies(implies(not(not(implies(y12, x12))), implies(or(x12, x12), x12)), or(implies(or(x12, x12), x12), implies(or(x12, x12), x12)))), op_implies, implies(not(implies(y12, x12)), or(implies(or(x12, x12), x12), implies(or(x12, x12), x12)))), op_implies, implies(implies(or(implies(or(x12, x12), x12), implies(or(x12, x12), x12)), implies(y12, x12)), implies(not(implies(y12, x12)), implies(y12, x12)))), op_implies, implies(implies(implies(or(x12, x12), x12), implies(y12, x12)), implies(not(implies(y12, x12)), implies(y12, x12)))), op_implies, implies(implies(y12, or(x12, x12)), implies(not(implies(y12, x12)), implies(y12, x12)))), op_implies, implies(x12, implies(not(implies(y12, x12)), implies(y12, x12)))), op_implies, implies(implies(implies(y12, x12), implies(y12, x12)), implies(x12, implies(y12, x12)))), op_implies, implies(or(not(implies(x12, implies(y12, x12))), X), implies(x12, implies(y12, x12)))), op_implies, implies(x12, implies(y12, x12))), op_implies)
% 96.04/12.78 = { by lemma 27 }
% 96.04/12.78 fresh40(fresh28(fresh28(fresh28(fresh28(fresh28(fresh28(fresh28(fresh28(fresh28(fresh28(fresh12(r1, op_implies, x12), op_implies, or(implies(or(x12, x12), x12), not(not(implies(y12, x12))))), op_implies, implies(implies(not(not(implies(y12, x12))), implies(or(x12, x12), x12)), or(implies(or(x12, x12), x12), implies(or(x12, x12), x12)))), op_implies, implies(not(implies(y12, x12)), or(implies(or(x12, x12), x12), implies(or(x12, x12), x12)))), op_implies, implies(implies(or(implies(or(x12, x12), x12), implies(or(x12, x12), x12)), implies(y12, x12)), implies(not(implies(y12, x12)), implies(y12, x12)))), op_implies, implies(implies(implies(or(x12, x12), x12), implies(y12, x12)), implies(not(implies(y12, x12)), implies(y12, x12)))), op_implies, implies(implies(y12, or(x12, x12)), implies(not(implies(y12, x12)), implies(y12, x12)))), op_implies, implies(x12, implies(not(implies(y12, x12)), implies(y12, x12)))), op_implies, implies(implies(implies(y12, x12), implies(y12, x12)), implies(x12, implies(y12, x12)))), op_implies, implies(or(not(implies(x12, implies(y12, x12))), X), implies(x12, implies(y12, x12)))), op_implies, implies(x12, implies(y12, x12))), op_implies)
% 96.04/12.78 = { by lemma 29 }
% 96.04/12.78 fresh40(fresh28(fresh28(fresh28(fresh28(fresh28(fresh28(fresh28(fresh28(fresh28(fresh28(op_implies, op_implies, or(implies(or(x12, x12), x12), not(not(implies(y12, x12))))), op_implies, implies(implies(not(not(implies(y12, x12))), implies(or(x12, x12), x12)), or(implies(or(x12, x12), x12), implies(or(x12, x12), x12)))), op_implies, implies(not(implies(y12, x12)), or(implies(or(x12, x12), x12), implies(or(x12, x12), x12)))), op_implies, implies(implies(or(implies(or(x12, x12), x12), implies(or(x12, x12), x12)), implies(y12, x12)), implies(not(implies(y12, x12)), implies(y12, x12)))), op_implies, implies(implies(implies(or(x12, x12), x12), implies(y12, x12)), implies(not(implies(y12, x12)), implies(y12, x12)))), op_implies, implies(implies(y12, or(x12, x12)), implies(not(implies(y12, x12)), implies(y12, x12)))), op_implies, implies(x12, implies(not(implies(y12, x12)), implies(y12, x12)))), op_implies, implies(implies(implies(y12, x12), implies(y12, x12)), implies(x12, implies(y12, x12)))), op_implies, implies(or(not(implies(x12, implies(y12, x12))), X), implies(x12, implies(y12, x12)))), op_implies, implies(x12, implies(y12, x12))), op_implies)
% 96.04/12.79 = { by axiom 11 (modus_ponens_2) }
% 96.04/12.79 fresh40(fresh28(fresh28(fresh28(fresh28(fresh28(fresh28(fresh28(fresh28(fresh28(is_a_theorem(or(implies(or(x12, x12), x12), not(not(implies(y12, x12))))), op_implies, implies(implies(not(not(implies(y12, x12))), implies(or(x12, x12), x12)), or(implies(or(x12, x12), x12), implies(or(x12, x12), x12)))), op_implies, implies(not(implies(y12, x12)), or(implies(or(x12, x12), x12), implies(or(x12, x12), x12)))), op_implies, implies(implies(or(implies(or(x12, x12), x12), implies(or(x12, x12), x12)), implies(y12, x12)), implies(not(implies(y12, x12)), implies(y12, x12)))), op_implies, implies(implies(implies(or(x12, x12), x12), implies(y12, x12)), implies(not(implies(y12, x12)), implies(y12, x12)))), op_implies, implies(implies(y12, or(x12, x12)), implies(not(implies(y12, x12)), implies(y12, x12)))), op_implies, implies(x12, implies(not(implies(y12, x12)), implies(y12, x12)))), op_implies, implies(implies(implies(y12, x12), implies(y12, x12)), implies(x12, implies(y12, x12)))), op_implies, implies(or(not(implies(x12, implies(y12, x12))), X), implies(x12, implies(y12, x12)))), op_implies, implies(x12, implies(y12, x12))), op_implies)
% 96.04/12.79 = { by lemma 30 R->L }
% 96.04/12.79 fresh40(fresh28(fresh28(fresh28(fresh28(fresh28(fresh28(fresh28(fresh28(fresh59(is_a_theorem(implies(or(implies(or(x12, x12), x12), not(not(implies(y12, x12)))), implies(implies(not(not(implies(y12, x12))), implies(or(x12, x12), x12)), or(implies(or(x12, x12), x12), implies(or(x12, x12), x12))))), op_implies, or(implies(or(x12, x12), x12), not(not(implies(y12, x12)))), implies(implies(not(not(implies(y12, x12))), implies(or(x12, x12), x12)), or(implies(or(x12, x12), x12), implies(or(x12, x12), x12)))), op_implies, implies(not(implies(y12, x12)), or(implies(or(x12, x12), x12), implies(or(x12, x12), x12)))), op_implies, implies(implies(or(implies(or(x12, x12), x12), implies(or(x12, x12), x12)), implies(y12, x12)), implies(not(implies(y12, x12)), implies(y12, x12)))), op_implies, implies(implies(implies(or(x12, x12), x12), implies(y12, x12)), implies(not(implies(y12, x12)), implies(y12, x12)))), op_implies, implies(implies(y12, or(x12, x12)), implies(not(implies(y12, x12)), implies(y12, x12)))), op_implies, implies(x12, implies(not(implies(y12, x12)), implies(y12, x12)))), op_implies, implies(implies(implies(y12, x12), implies(y12, x12)), implies(x12, implies(y12, x12)))), op_implies, implies(or(not(implies(x12, implies(y12, x12))), X), implies(x12, implies(y12, x12)))), op_implies, implies(x12, implies(y12, x12))), op_implies)
% 96.04/12.79 = { by lemma 40 }
% 96.04/12.79 fresh40(fresh28(fresh28(fresh28(fresh28(fresh28(fresh28(fresh28(fresh28(fresh59(op_implies, op_implies, or(implies(or(x12, x12), x12), not(not(implies(y12, x12)))), implies(implies(not(not(implies(y12, x12))), implies(or(x12, x12), x12)), or(implies(or(x12, x12), x12), implies(or(x12, x12), x12)))), op_implies, implies(not(implies(y12, x12)), or(implies(or(x12, x12), x12), implies(or(x12, x12), x12)))), op_implies, implies(implies(or(implies(or(x12, x12), x12), implies(or(x12, x12), x12)), implies(y12, x12)), implies(not(implies(y12, x12)), implies(y12, x12)))), op_implies, implies(implies(implies(or(x12, x12), x12), implies(y12, x12)), implies(not(implies(y12, x12)), implies(y12, x12)))), op_implies, implies(implies(y12, or(x12, x12)), implies(not(implies(y12, x12)), implies(y12, x12)))), op_implies, implies(x12, implies(not(implies(y12, x12)), implies(y12, x12)))), op_implies, implies(implies(implies(y12, x12), implies(y12, x12)), implies(x12, implies(y12, x12)))), op_implies, implies(or(not(implies(x12, implies(y12, x12))), X), implies(x12, implies(y12, x12)))), op_implies, implies(x12, implies(y12, x12))), op_implies)
% 96.04/12.79 = { by lemma 34 }
% 96.04/12.79 fresh40(fresh28(fresh28(fresh28(fresh28(fresh28(fresh28(fresh28(fresh28(op_implies, op_implies, implies(not(implies(y12, x12)), or(implies(or(x12, x12), x12), implies(or(x12, x12), x12)))), op_implies, implies(implies(or(implies(or(x12, x12), x12), implies(or(x12, x12), x12)), implies(y12, x12)), implies(not(implies(y12, x12)), implies(y12, x12)))), op_implies, implies(implies(implies(or(x12, x12), x12), implies(y12, x12)), implies(not(implies(y12, x12)), implies(y12, x12)))), op_implies, implies(implies(y12, or(x12, x12)), implies(not(implies(y12, x12)), implies(y12, x12)))), op_implies, implies(x12, implies(not(implies(y12, x12)), implies(y12, x12)))), op_implies, implies(implies(implies(y12, x12), implies(y12, x12)), implies(x12, implies(y12, x12)))), op_implies, implies(or(not(implies(x12, implies(y12, x12))), X), implies(x12, implies(y12, x12)))), op_implies, implies(x12, implies(y12, x12))), op_implies)
% 96.04/12.79 = { by axiom 11 (modus_ponens_2) }
% 96.04/12.79 fresh40(fresh28(fresh28(fresh28(fresh28(fresh28(fresh28(fresh28(is_a_theorem(implies(not(implies(y12, x12)), or(implies(or(x12, x12), x12), implies(or(x12, x12), x12)))), op_implies, implies(implies(or(implies(or(x12, x12), x12), implies(or(x12, x12), x12)), implies(y12, x12)), implies(not(implies(y12, x12)), implies(y12, x12)))), op_implies, implies(implies(implies(or(x12, x12), x12), implies(y12, x12)), implies(not(implies(y12, x12)), implies(y12, x12)))), op_implies, implies(implies(y12, or(x12, x12)), implies(not(implies(y12, x12)), implies(y12, x12)))), op_implies, implies(x12, implies(not(implies(y12, x12)), implies(y12, x12)))), op_implies, implies(implies(implies(y12, x12), implies(y12, x12)), implies(x12, implies(y12, x12)))), op_implies, implies(or(not(implies(x12, implies(y12, x12))), X), implies(x12, implies(y12, x12)))), op_implies, implies(x12, implies(y12, x12))), op_implies)
% 96.04/12.79 = { by lemma 35 }
% 96.04/12.79 fresh40(fresh28(fresh28(fresh28(fresh28(fresh28(fresh28(op_implies, op_implies, implies(implies(implies(or(x12, x12), x12), implies(y12, x12)), implies(not(implies(y12, x12)), implies(y12, x12)))), op_implies, implies(implies(y12, or(x12, x12)), implies(not(implies(y12, x12)), implies(y12, x12)))), op_implies, implies(x12, implies(not(implies(y12, x12)), implies(y12, x12)))), op_implies, implies(implies(implies(y12, x12), implies(y12, x12)), implies(x12, implies(y12, x12)))), op_implies, implies(or(not(implies(x12, implies(y12, x12))), X), implies(x12, implies(y12, x12)))), op_implies, implies(x12, implies(y12, x12))), op_implies)
% 96.04/12.79 = { by axiom 11 (modus_ponens_2) }
% 96.04/12.79 fresh40(fresh28(fresh28(fresh28(fresh28(fresh28(is_a_theorem(implies(implies(implies(or(x12, x12), x12), implies(y12, x12)), implies(not(implies(y12, x12)), implies(y12, x12)))), op_implies, implies(implies(y12, or(x12, x12)), implies(not(implies(y12, x12)), implies(y12, x12)))), op_implies, implies(x12, implies(not(implies(y12, x12)), implies(y12, x12)))), op_implies, implies(implies(implies(y12, x12), implies(y12, x12)), implies(x12, implies(y12, x12)))), op_implies, implies(or(not(implies(x12, implies(y12, x12))), X), implies(x12, implies(y12, x12)))), op_implies, implies(x12, implies(y12, x12))), op_implies)
% 96.04/12.79 = { by lemma 45 }
% 96.04/12.79 fresh40(fresh28(fresh28(fresh28(fresh28(op_implies, op_implies, implies(x12, implies(not(implies(y12, x12)), implies(y12, x12)))), op_implies, implies(implies(implies(y12, x12), implies(y12, x12)), implies(x12, implies(y12, x12)))), op_implies, implies(or(not(implies(x12, implies(y12, x12))), X), implies(x12, implies(y12, x12)))), op_implies, implies(x12, implies(y12, x12))), op_implies)
% 96.04/12.79 = { by axiom 11 (modus_ponens_2) }
% 96.04/12.79 fresh40(fresh28(fresh28(fresh28(is_a_theorem(implies(x12, implies(not(implies(y12, x12)), implies(y12, x12)))), op_implies, implies(implies(implies(y12, x12), implies(y12, x12)), implies(x12, implies(y12, x12)))), op_implies, implies(or(not(implies(x12, implies(y12, x12))), X), implies(x12, implies(y12, x12)))), op_implies, implies(x12, implies(y12, x12))), op_implies)
% 96.04/12.79 = { by lemma 28 }
% 96.04/12.79 fresh40(fresh28(fresh28(fresh28(is_a_theorem(implies(x12, or(implies(y12, x12), implies(y12, x12)))), op_implies, implies(implies(implies(y12, x12), implies(y12, x12)), implies(x12, implies(y12, x12)))), op_implies, implies(or(not(implies(x12, implies(y12, x12))), X), implies(x12, implies(y12, x12)))), op_implies, implies(x12, implies(y12, x12))), op_implies)
% 96.04/12.79 = { by lemma 30 R->L }
% 96.04/12.79 fresh40(fresh28(fresh28(fresh59(is_a_theorem(implies(implies(x12, or(implies(y12, x12), implies(y12, x12))), implies(implies(implies(y12, x12), implies(y12, x12)), implies(x12, implies(y12, x12))))), op_implies, implies(x12, or(implies(y12, x12), implies(y12, x12))), implies(implies(implies(y12, x12), implies(y12, x12)), implies(x12, implies(y12, x12)))), op_implies, implies(or(not(implies(x12, implies(y12, x12))), X), implies(x12, implies(y12, x12)))), op_implies, implies(x12, implies(y12, x12))), op_implies)
% 96.04/12.79 = { by axiom 11 (modus_ponens_2) R->L }
% 96.04/12.79 fresh40(fresh28(fresh28(fresh59(fresh28(op_implies, op_implies, implies(implies(x12, or(implies(y12, x12), implies(y12, x12))), implies(implies(implies(y12, x12), implies(y12, x12)), implies(x12, implies(y12, x12))))), op_implies, implies(x12, or(implies(y12, x12), implies(y12, x12))), implies(implies(implies(y12, x12), implies(y12, x12)), implies(x12, implies(y12, x12)))), op_implies, implies(or(not(implies(x12, implies(y12, x12))), X), implies(x12, implies(y12, x12)))), op_implies, implies(x12, implies(y12, x12))), op_implies)
% 96.04/12.79 = { by lemma 43 R->L }
% 96.04/12.79 fresh40(fresh28(fresh28(fresh59(fresh28(is_a_theorem(implies(implies(implies(or(implies(y12, x12), implies(y12, x12)), implies(y12, x12)), implies(x12, implies(y12, x12))), implies(implies(implies(y12, x12), implies(y12, x12)), implies(x12, implies(y12, x12))))), op_implies, implies(implies(x12, or(implies(y12, x12), implies(y12, x12))), implies(implies(implies(y12, x12), implies(y12, x12)), implies(x12, implies(y12, x12))))), op_implies, implies(x12, or(implies(y12, x12), implies(y12, x12))), implies(implies(implies(y12, x12), implies(y12, x12)), implies(x12, implies(y12, x12)))), op_implies, implies(or(not(implies(x12, implies(y12, x12))), X), implies(x12, implies(y12, x12)))), op_implies, implies(x12, implies(y12, x12))), op_implies)
% 96.04/12.79 = { by lemma 45 }
% 96.04/12.79 fresh40(fresh28(fresh28(fresh59(op_implies, op_implies, implies(x12, or(implies(y12, x12), implies(y12, x12))), implies(implies(implies(y12, x12), implies(y12, x12)), implies(x12, implies(y12, x12)))), op_implies, implies(or(not(implies(x12, implies(y12, x12))), X), implies(x12, implies(y12, x12)))), op_implies, implies(x12, implies(y12, x12))), op_implies)
% 96.04/12.79 = { by lemma 34 }
% 96.04/12.79 fresh40(fresh28(fresh28(op_implies, op_implies, implies(or(not(implies(x12, implies(y12, x12))), X), implies(x12, implies(y12, x12)))), op_implies, implies(x12, implies(y12, x12))), op_implies)
% 96.04/12.79 = { by axiom 11 (modus_ponens_2) }
% 96.04/12.79 fresh40(fresh28(is_a_theorem(implies(or(not(implies(x12, implies(y12, x12))), X), implies(x12, implies(y12, x12)))), op_implies, implies(x12, implies(y12, x12))), op_implies)
% 96.04/12.79 = { by lemma 30 R->L }
% 96.04/12.79 fresh40(fresh59(is_a_theorem(implies(implies(or(not(implies(x12, implies(y12, x12))), X), implies(x12, implies(y12, x12))), implies(x12, implies(y12, x12)))), op_implies, implies(or(not(implies(x12, implies(y12, x12))), X), implies(x12, implies(y12, x12))), implies(x12, implies(y12, x12))), op_implies)
% 96.04/12.79 = { by axiom 11 (modus_ponens_2) R->L }
% 96.04/12.79 fresh40(fresh59(fresh28(op_implies, op_implies, implies(implies(or(not(implies(x12, implies(y12, x12))), X), implies(x12, implies(y12, x12))), implies(x12, implies(y12, x12)))), op_implies, implies(or(not(implies(x12, implies(y12, x12))), X), implies(x12, implies(y12, x12))), implies(x12, implies(y12, x12))), op_implies)
% 96.04/12.79 = { by lemma 38 R->L }
% 96.04/12.79 fresh40(fresh59(fresh28(is_a_theorem(implies(implies(x12, implies(y12, x12)), implies(x12, implies(y12, x12)))), op_implies, implies(implies(or(not(implies(x12, implies(y12, x12))), X), implies(x12, implies(y12, x12))), implies(x12, implies(y12, x12)))), op_implies, implies(or(not(implies(x12, implies(y12, x12))), X), implies(x12, implies(y12, x12))), implies(x12, implies(y12, x12))), op_implies)
% 96.04/12.79 = { by lemma 30 R->L }
% 96.04/12.79 fresh40(fresh59(fresh59(is_a_theorem(implies(implies(implies(x12, implies(y12, x12)), implies(x12, implies(y12, x12))), implies(implies(or(not(implies(x12, implies(y12, x12))), X), implies(x12, implies(y12, x12))), implies(x12, implies(y12, x12))))), op_implies, implies(implies(x12, implies(y12, x12)), implies(x12, implies(y12, x12))), implies(implies(or(not(implies(x12, implies(y12, x12))), X), implies(x12, implies(y12, x12))), implies(x12, implies(y12, x12)))), op_implies, implies(or(not(implies(x12, implies(y12, x12))), X), implies(x12, implies(y12, x12))), implies(x12, implies(y12, x12))), op_implies)
% 96.04/12.79 = { by axiom 11 (modus_ponens_2) R->L }
% 96.04/12.79 fresh40(fresh59(fresh59(fresh28(op_implies, op_implies, implies(implies(implies(x12, implies(y12, x12)), implies(x12, implies(y12, x12))), implies(implies(or(not(implies(x12, implies(y12, x12))), X), implies(x12, implies(y12, x12))), implies(x12, implies(y12, x12))))), op_implies, implies(implies(x12, implies(y12, x12)), implies(x12, implies(y12, x12))), implies(implies(or(not(implies(x12, implies(y12, x12))), X), implies(x12, implies(y12, x12))), implies(x12, implies(y12, x12)))), op_implies, implies(or(not(implies(x12, implies(y12, x12))), X), implies(x12, implies(y12, x12))), implies(x12, implies(y12, x12))), op_implies)
% 96.04/12.79 = { by lemma 35 R->L }
% 96.04/12.79 fresh40(fresh59(fresh59(fresh28(fresh28(is_a_theorem(implies(implies(or(not(implies(x12, implies(y12, x12))), X), implies(x12, implies(y12, x12))), or(implies(x12, implies(y12, x12)), implies(x12, implies(y12, x12))))), op_implies, implies(implies(or(implies(x12, implies(y12, x12)), implies(x12, implies(y12, x12))), implies(x12, implies(y12, x12))), implies(implies(or(not(implies(x12, implies(y12, x12))), X), implies(x12, implies(y12, x12))), implies(x12, implies(y12, x12))))), op_implies, implies(implies(implies(x12, implies(y12, x12)), implies(x12, implies(y12, x12))), implies(implies(or(not(implies(x12, implies(y12, x12))), X), implies(x12, implies(y12, x12))), implies(x12, implies(y12, x12))))), op_implies, implies(implies(x12, implies(y12, x12)), implies(x12, implies(y12, x12))), implies(implies(or(not(implies(x12, implies(y12, x12))), X), implies(x12, implies(y12, x12))), implies(x12, implies(y12, x12)))), op_implies, implies(or(not(implies(x12, implies(y12, x12))), X), implies(x12, implies(y12, x12))), implies(x12, implies(y12, x12))), op_implies)
% 96.04/12.79 = { by lemma 28 R->L }
% 96.04/12.79 fresh40(fresh59(fresh59(fresh28(fresh28(is_a_theorem(implies(implies(or(not(implies(x12, implies(y12, x12))), X), implies(x12, implies(y12, x12))), implies(not(implies(x12, implies(y12, x12))), implies(x12, implies(y12, x12))))), op_implies, implies(implies(or(implies(x12, implies(y12, x12)), implies(x12, implies(y12, x12))), implies(x12, implies(y12, x12))), implies(implies(or(not(implies(x12, implies(y12, x12))), X), implies(x12, implies(y12, x12))), implies(x12, implies(y12, x12))))), op_implies, implies(implies(implies(x12, implies(y12, x12)), implies(x12, implies(y12, x12))), implies(implies(or(not(implies(x12, implies(y12, x12))), X), implies(x12, implies(y12, x12))), implies(x12, implies(y12, x12))))), op_implies, implies(implies(x12, implies(y12, x12)), implies(x12, implies(y12, x12))), implies(implies(or(not(implies(x12, implies(y12, x12))), X), implies(x12, implies(y12, x12))), implies(x12, implies(y12, x12)))), op_implies, implies(or(not(implies(x12, implies(y12, x12))), X), implies(x12, implies(y12, x12))), implies(x12, implies(y12, x12))), op_implies)
% 96.04/12.79 = { by lemma 36 }
% 96.04/12.79 fresh40(fresh59(fresh59(fresh28(fresh28(op_implies, op_implies, implies(implies(or(implies(x12, implies(y12, x12)), implies(x12, implies(y12, x12))), implies(x12, implies(y12, x12))), implies(implies(or(not(implies(x12, implies(y12, x12))), X), implies(x12, implies(y12, x12))), implies(x12, implies(y12, x12))))), op_implies, implies(implies(implies(x12, implies(y12, x12)), implies(x12, implies(y12, x12))), implies(implies(or(not(implies(x12, implies(y12, x12))), X), implies(x12, implies(y12, x12))), implies(x12, implies(y12, x12))))), op_implies, implies(implies(x12, implies(y12, x12)), implies(x12, implies(y12, x12))), implies(implies(or(not(implies(x12, implies(y12, x12))), X), implies(x12, implies(y12, x12))), implies(x12, implies(y12, x12)))), op_implies, implies(or(not(implies(x12, implies(y12, x12))), X), implies(x12, implies(y12, x12))), implies(x12, implies(y12, x12))), op_implies)
% 96.04/12.79 = { by axiom 11 (modus_ponens_2) }
% 96.04/12.79 fresh40(fresh59(fresh59(fresh28(is_a_theorem(implies(implies(or(implies(x12, implies(y12, x12)), implies(x12, implies(y12, x12))), implies(x12, implies(y12, x12))), implies(implies(or(not(implies(x12, implies(y12, x12))), X), implies(x12, implies(y12, x12))), implies(x12, implies(y12, x12))))), op_implies, implies(implies(implies(x12, implies(y12, x12)), implies(x12, implies(y12, x12))), implies(implies(or(not(implies(x12, implies(y12, x12))), X), implies(x12, implies(y12, x12))), implies(x12, implies(y12, x12))))), op_implies, implies(implies(x12, implies(y12, x12)), implies(x12, implies(y12, x12))), implies(implies(or(not(implies(x12, implies(y12, x12))), X), implies(x12, implies(y12, x12))), implies(x12, implies(y12, x12)))), op_implies, implies(or(not(implies(x12, implies(y12, x12))), X), implies(x12, implies(y12, x12))), implies(x12, implies(y12, x12))), op_implies)
% 96.04/12.79 = { by lemma 44 }
% 96.04/12.79 fresh40(fresh59(fresh59(op_implies, op_implies, implies(implies(x12, implies(y12, x12)), implies(x12, implies(y12, x12))), implies(implies(or(not(implies(x12, implies(y12, x12))), X), implies(x12, implies(y12, x12))), implies(x12, implies(y12, x12)))), op_implies, implies(or(not(implies(x12, implies(y12, x12))), X), implies(x12, implies(y12, x12))), implies(x12, implies(y12, x12))), op_implies)
% 96.04/12.79 = { by lemma 34 }
% 96.04/12.79 fresh40(fresh59(op_implies, op_implies, implies(or(not(implies(x12, implies(y12, x12))), X), implies(x12, implies(y12, x12))), implies(x12, implies(y12, x12))), op_implies)
% 96.04/12.79 = { by lemma 34 }
% 96.04/12.79 fresh40(op_implies, op_implies)
% 96.04/12.79 = { by axiom 8 (implies_1) }
% 96.04/12.79 true
% 96.04/12.79 % SZS output end Proof
% 96.04/12.79
% 96.04/12.79 RESULT: Theorem (the conjecture is true).
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