TSTP Solution File: LCL498+1 by Twee---2.4.2
View Problem
- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : LCL498+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 : n025.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 22.87s 3.32s
% Output : Proof 23.28s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : LCL498+1 : TPTP v8.1.2. Released v3.3.0.
% 0.00/0.13 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.13/0.34 % Computer : n025.cluster.edu
% 0.13/0.34 % Model : x86_64 x86_64
% 0.13/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.34 % Memory : 8042.1875MB
% 0.13/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.34 % CPULimit : 300
% 0.13/0.34 % WCLimit : 300
% 0.13/0.34 % DateTime : Fri Aug 25 06:11:37 EDT 2023
% 0.13/0.34 % CPUTime :
% 22.87/3.32 Command-line arguments: --lhs-weight 1 --flip-ordering --normalise-queue-percent 10 --cp-renormalise-threshold 10
% 22.87/3.32
% 22.87/3.32 % SZS status Theorem
% 22.87/3.32
% 23.28/3.36 % SZS output start Proof
% 23.28/3.36 Take the following subset of the input axioms:
% 23.28/3.36 fof(cn3, axiom, cn3 <=> ![P]: is_a_theorem(implies(implies(not(P), P), P))).
% 23.28/3.36 fof(luka_cn3, conjecture, cn3).
% 23.28/3.36 fof(luka_op_or, axiom, op_or).
% 23.28/3.36 fof(modus_ponens, axiom, modus_ponens <=> ![X, Y]: ((is_a_theorem(X) & is_a_theorem(implies(X, Y))) => is_a_theorem(Y))).
% 23.28/3.36 fof(op_and, axiom, op_and => ![X2, Y2]: and(X2, Y2)=not(or(not(X2), not(Y2)))).
% 23.28/3.36 fof(op_equiv, axiom, op_equiv => ![X2, Y2]: equiv(X2, Y2)=and(implies(X2, Y2), implies(Y2, X2))).
% 23.28/3.36 fof(op_implies_or, axiom, op_implies_or => ![X2, Y2]: implies(X2, Y2)=or(not(X2), Y2)).
% 23.28/3.36 fof(op_or, axiom, op_or => ![X2, Y2]: or(X2, Y2)=not(and(not(X2), not(Y2)))).
% 23.28/3.36 fof(or_2, axiom, or_2 <=> ![X2, Y2]: is_a_theorem(implies(Y2, or(X2, Y2)))).
% 23.28/3.36 fof(principia_modus_ponens, axiom, modus_ponens).
% 23.28/3.36 fof(principia_op_and, axiom, op_and).
% 23.28/3.36 fof(principia_op_equiv, axiom, op_equiv).
% 23.28/3.36 fof(principia_op_implies_or, axiom, op_implies_or).
% 23.28/3.36 fof(principia_r1, axiom, r1).
% 23.28/3.36 fof(principia_r2, axiom, r2).
% 23.28/3.36 fof(principia_r3, axiom, r3).
% 23.28/3.36 fof(principia_r4, axiom, r4).
% 23.28/3.36 fof(principia_r5, axiom, r5).
% 23.28/3.36 fof(r1, axiom, r1 <=> ![P2]: is_a_theorem(implies(or(P2, P2), P2))).
% 23.28/3.36 fof(r2, axiom, r2 <=> ![Q, P2]: is_a_theorem(implies(Q, or(P2, Q)))).
% 23.28/3.36 fof(r3, axiom, r3 <=> ![P2, Q2]: is_a_theorem(implies(or(P2, Q2), or(Q2, P2)))).
% 23.28/3.36 fof(r4, axiom, r4 <=> ![R, P2, Q2]: is_a_theorem(implies(or(P2, or(Q2, R)), or(Q2, or(P2, R))))).
% 23.28/3.36 fof(r5, axiom, r5 <=> ![P2, Q2, R2]: is_a_theorem(implies(implies(Q2, R2), implies(or(P2, Q2), or(P2, R2))))).
% 23.28/3.36 fof(substitution_of_equivalents, axiom, substitution_of_equivalents <=> ![X2, Y2]: (is_a_theorem(equiv(X2, Y2)) => X2=Y2)).
% 23.28/3.36 fof(substitution_of_equivalents, axiom, substitution_of_equivalents).
% 23.28/3.36
% 23.28/3.36 Now clausify the problem and encode Horn clauses using encoding 3 of
% 23.28/3.36 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 23.28/3.36 We repeatedly replace C & s=t => u=v by the two clauses:
% 23.28/3.36 fresh(y, y, x1...xn) = u
% 23.28/3.36 C => fresh(s, t, x1...xn) = v
% 23.28/3.36 where fresh is a fresh function symbol and x1..xn are the free
% 23.28/3.36 variables of u and v.
% 23.28/3.36 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 23.28/3.36 input problem has no model of domain size 1).
% 23.28/3.36
% 23.28/3.36 The encoding turns the above axioms into the following unit equations and goals:
% 23.28/3.36
% 23.28/3.36 Axiom 1 (principia_modus_ponens): modus_ponens = true.
% 23.28/3.36 Axiom 2 (substitution_of_equivalents): substitution_of_equivalents = true.
% 23.28/3.36 Axiom 3 (principia_r1): r1 = true.
% 23.28/3.36 Axiom 4 (principia_r2): r2 = true.
% 23.28/3.36 Axiom 5 (principia_r3): r3 = true.
% 23.28/3.36 Axiom 6 (principia_r4): r4 = true.
% 23.28/3.36 Axiom 7 (principia_r5): r5 = true.
% 23.28/3.36 Axiom 8 (principia_op_equiv): op_equiv = true.
% 23.28/3.36 Axiom 9 (luka_op_or): op_or = true.
% 23.28/3.36 Axiom 10 (principia_op_and): op_and = true.
% 23.28/3.36 Axiom 11 (principia_op_implies_or): op_implies_or = true.
% 23.28/3.36 Axiom 12 (cn3): fresh48(X, X) = true.
% 23.28/3.36 Axiom 13 (modus_ponens_2): fresh60(X, X, Y) = true.
% 23.28/3.36 Axiom 14 (modus_ponens_2): fresh28(X, X, Y) = is_a_theorem(Y).
% 23.28/3.36 Axiom 15 (r1_1): fresh12(X, X, Y) = true.
% 23.28/3.36 Axiom 16 (substitution_of_equivalents_2): fresh(X, X, Y, Z) = Z.
% 23.28/3.36 Axiom 17 (modus_ponens_2): fresh59(X, X, Y, Z) = fresh60(modus_ponens, true, Z).
% 23.28/3.36 Axiom 18 (op_and): fresh24(X, X, Y, Z) = and(Y, Z).
% 23.28/3.36 Axiom 19 (op_equiv): fresh23(X, X, Y, Z) = equiv(Y, Z).
% 23.28/3.36 Axiom 20 (op_implies_or): fresh21(X, X, Y, Z) = implies(Y, Z).
% 23.28/3.36 Axiom 21 (op_implies_or): fresh21(op_implies_or, true, X, Y) = or(not(X), Y).
% 23.28/3.37 Axiom 22 (op_or): fresh20(X, X, Y, Z) = or(Y, Z).
% 23.28/3.37 Axiom 23 (r2_1): fresh10(X, X, Y, Z) = true.
% 23.28/3.37 Axiom 24 (r3_1): fresh8(X, X, Y, Z) = true.
% 23.28/3.37 Axiom 25 (substitution_of_equivalents_2): fresh2(X, X, Y, Z) = Y.
% 23.28/3.37 Axiom 26 (or_2_1): fresh16(or_2, true, X, Y) = is_a_theorem(implies(Y, or(X, Y))).
% 23.28/3.37 Axiom 27 (r2_1): fresh10(r2, true, X, Y) = is_a_theorem(implies(Y, or(X, Y))).
% 23.28/3.37 Axiom 28 (r1_1): fresh12(r1, true, X) = is_a_theorem(implies(or(X, X), X)).
% 23.28/3.37 Axiom 29 (op_and): fresh24(op_and, true, X, Y) = not(or(not(X), not(Y))).
% 23.28/3.37 Axiom 30 (op_or): fresh20(op_or, true, X, Y) = not(and(not(X), not(Y))).
% 23.28/3.37 Axiom 31 (r4_1): fresh6(X, X, Y, Z, W) = true.
% 23.28/3.37 Axiom 32 (r5_1): fresh4(X, X, Y, Z, W) = true.
% 23.28/3.37 Axiom 33 (op_equiv): fresh23(op_equiv, true, X, Y) = and(implies(X, Y), implies(Y, X)).
% 23.28/3.37 Axiom 34 (r3_1): fresh8(r3, true, X, Y) = is_a_theorem(implies(or(X, Y), or(Y, X))).
% 23.28/3.37 Axiom 35 (substitution_of_equivalents_2): fresh2(substitution_of_equivalents, true, X, Y) = fresh(is_a_theorem(equiv(X, Y)), true, X, Y).
% 23.28/3.37 Axiom 36 (modus_ponens_2): fresh59(is_a_theorem(implies(X, Y)), true, X, Y) = fresh28(is_a_theorem(X), true, Y).
% 23.28/3.37 Axiom 37 (cn3): fresh48(is_a_theorem(implies(implies(not(p6), p6), p6)), true) = cn3.
% 23.28/3.37 Axiom 38 (r5_1): fresh4(r5, true, X, Y, Z) = is_a_theorem(implies(implies(Y, Z), implies(or(X, Y), or(X, Z)))).
% 23.28/3.37 Axiom 39 (r4_1): fresh6(r4, true, X, Y, Z) = is_a_theorem(implies(or(X, or(Y, Z)), or(Y, or(X, Z)))).
% 23.28/3.37
% 23.28/3.37 Lemma 40: fresh59(X, X, Y, Z) = true.
% 23.28/3.37 Proof:
% 23.28/3.37 fresh59(X, X, Y, Z)
% 23.28/3.37 = { by axiom 17 (modus_ponens_2) }
% 23.28/3.37 fresh60(modus_ponens, true, Z)
% 23.28/3.37 = { by axiom 1 (principia_modus_ponens) }
% 23.28/3.37 fresh60(true, true, Z)
% 23.28/3.37 = { by axiom 13 (modus_ponens_2) }
% 23.28/3.37 true
% 23.28/3.37
% 23.28/3.37 Lemma 41: is_a_theorem(implies(implies(X, Y), implies(or(Z, X), or(Z, Y)))) = true.
% 23.28/3.37 Proof:
% 23.28/3.37 is_a_theorem(implies(implies(X, Y), implies(or(Z, X), or(Z, Y))))
% 23.28/3.37 = { by axiom 38 (r5_1) R->L }
% 23.28/3.37 fresh4(r5, true, Z, X, Y)
% 23.28/3.37 = { by axiom 7 (principia_r5) }
% 23.28/3.37 fresh4(true, true, Z, X, Y)
% 23.28/3.37 = { by axiom 32 (r5_1) }
% 23.28/3.37 true
% 23.28/3.37
% 23.28/3.37 Lemma 42: or(not(X), Y) = implies(X, Y).
% 23.28/3.37 Proof:
% 23.28/3.37 or(not(X), Y)
% 23.28/3.37 = { by axiom 21 (op_implies_or) R->L }
% 23.28/3.37 fresh21(op_implies_or, true, X, Y)
% 23.28/3.37 = { by axiom 11 (principia_op_implies_or) }
% 23.28/3.37 fresh21(true, true, X, Y)
% 23.28/3.37 = { by axiom 20 (op_implies_or) }
% 23.28/3.37 implies(X, Y)
% 23.28/3.37
% 23.28/3.37 Lemma 43: is_a_theorem(implies(implies(X, or(Y, Z)), or(Y, implies(X, Z)))) = true.
% 23.28/3.37 Proof:
% 23.28/3.37 is_a_theorem(implies(implies(X, or(Y, Z)), or(Y, implies(X, Z))))
% 23.28/3.37 = { by lemma 42 R->L }
% 23.28/3.37 is_a_theorem(implies(implies(X, or(Y, Z)), or(Y, or(not(X), Z))))
% 23.28/3.37 = { by lemma 42 R->L }
% 23.28/3.37 is_a_theorem(implies(or(not(X), or(Y, Z)), or(Y, or(not(X), Z))))
% 23.28/3.37 = { by axiom 39 (r4_1) R->L }
% 23.28/3.37 fresh6(r4, true, not(X), Y, Z)
% 23.28/3.37 = { by axiom 6 (principia_r4) }
% 23.28/3.37 fresh6(true, true, not(X), Y, Z)
% 23.28/3.37 = { by axiom 31 (r4_1) }
% 23.28/3.37 true
% 23.28/3.37
% 23.28/3.37 Lemma 44: fresh28(is_a_theorem(implies(X, implies(Y, Z))), true, implies(Y, implies(X, Z))) = true.
% 23.28/3.37 Proof:
% 23.28/3.37 fresh28(is_a_theorem(implies(X, implies(Y, Z))), true, implies(Y, implies(X, Z)))
% 23.28/3.37 = { by axiom 36 (modus_ponens_2) R->L }
% 23.28/3.37 fresh59(is_a_theorem(implies(implies(X, implies(Y, Z)), implies(Y, implies(X, Z)))), true, implies(X, implies(Y, Z)), implies(Y, implies(X, Z)))
% 23.28/3.37 = { by lemma 42 R->L }
% 23.28/3.37 fresh59(is_a_theorem(implies(implies(X, implies(Y, Z)), or(not(Y), implies(X, Z)))), true, implies(X, implies(Y, Z)), implies(Y, implies(X, Z)))
% 23.28/3.37 = { by lemma 42 R->L }
% 23.28/3.37 fresh59(is_a_theorem(implies(implies(X, or(not(Y), Z)), or(not(Y), implies(X, Z)))), true, implies(X, implies(Y, Z)), implies(Y, implies(X, Z)))
% 23.28/3.37 = { by lemma 43 }
% 23.28/3.37 fresh59(true, true, implies(X, implies(Y, Z)), implies(Y, implies(X, Z)))
% 23.28/3.37 = { by lemma 40 }
% 23.28/3.37 true
% 23.28/3.37
% 23.28/3.37 Lemma 45: fresh28(is_a_theorem(implies(X, Y)), true, implies(implies(Y, Z), implies(X, Z))) = true.
% 23.28/3.37 Proof:
% 23.28/3.37 fresh28(is_a_theorem(implies(X, Y)), true, implies(implies(Y, Z), implies(X, Z)))
% 23.28/3.37 = { by axiom 36 (modus_ponens_2) R->L }
% 23.28/3.37 fresh59(is_a_theorem(implies(implies(X, Y), implies(implies(Y, Z), implies(X, Z)))), true, implies(X, Y), implies(implies(Y, Z), implies(X, Z)))
% 23.28/3.37 = { by axiom 14 (modus_ponens_2) R->L }
% 23.28/3.37 fresh59(fresh28(true, true, implies(implies(X, Y), implies(implies(Y, Z), implies(X, Z)))), true, implies(X, Y), implies(implies(Y, Z), implies(X, Z)))
% 23.28/3.37 = { by lemma 41 R->L }
% 23.28/3.37 fresh59(fresh28(is_a_theorem(implies(implies(Y, Z), implies(or(not(X), Y), or(not(X), Z)))), true, implies(implies(X, Y), implies(implies(Y, Z), implies(X, Z)))), true, implies(X, Y), implies(implies(Y, Z), implies(X, Z)))
% 23.28/3.37 = { by lemma 42 }
% 23.28/3.37 fresh59(fresh28(is_a_theorem(implies(implies(Y, Z), implies(implies(X, Y), or(not(X), Z)))), true, implies(implies(X, Y), implies(implies(Y, Z), implies(X, Z)))), true, implies(X, Y), implies(implies(Y, Z), implies(X, Z)))
% 23.28/3.37 = { by lemma 42 }
% 23.28/3.37 fresh59(fresh28(is_a_theorem(implies(implies(Y, Z), implies(implies(X, Y), implies(X, Z)))), true, implies(implies(X, Y), implies(implies(Y, Z), implies(X, Z)))), true, implies(X, Y), implies(implies(Y, Z), implies(X, Z)))
% 23.28/3.37 = { by lemma 44 }
% 23.28/3.37 fresh59(true, true, implies(X, Y), implies(implies(Y, Z), implies(X, Z)))
% 23.28/3.37 = { by lemma 40 }
% 23.28/3.37 true
% 23.28/3.37
% 23.28/3.37 Lemma 46: fresh16(or_2, true, X, Y) = true.
% 23.28/3.37 Proof:
% 23.28/3.37 fresh16(or_2, true, X, Y)
% 23.28/3.37 = { by axiom 26 (or_2_1) }
% 23.28/3.37 is_a_theorem(implies(Y, or(X, Y)))
% 23.28/3.37 = { by axiom 27 (r2_1) R->L }
% 23.28/3.37 fresh10(r2, true, X, Y)
% 23.28/3.37 = { by axiom 4 (principia_r2) }
% 23.28/3.37 fresh10(true, true, X, Y)
% 23.28/3.37 = { by axiom 23 (r2_1) }
% 23.28/3.37 true
% 23.28/3.37
% 23.28/3.37 Lemma 47: is_a_theorem(implies(or(X, Y), or(Y, X))) = true.
% 23.28/3.37 Proof:
% 23.28/3.37 is_a_theorem(implies(or(X, Y), or(Y, X)))
% 23.28/3.37 = { by axiom 34 (r3_1) R->L }
% 23.28/3.37 fresh8(r3, true, X, Y)
% 23.28/3.37 = { by axiom 5 (principia_r3) }
% 23.28/3.37 fresh8(true, true, X, Y)
% 23.28/3.37 = { by axiom 24 (r3_1) }
% 23.28/3.37 true
% 23.28/3.37
% 23.28/3.37 Lemma 48: is_a_theorem(implies(or(X, X), X)) = true.
% 23.28/3.37 Proof:
% 23.28/3.37 is_a_theorem(implies(or(X, X), X))
% 23.28/3.37 = { by axiom 28 (r1_1) R->L }
% 23.28/3.37 fresh12(r1, true, X)
% 23.28/3.37 = { by axiom 3 (principia_r1) }
% 23.28/3.37 fresh12(true, true, X)
% 23.28/3.37 = { by axiom 15 (r1_1) }
% 23.28/3.37 true
% 23.28/3.37
% 23.28/3.37 Lemma 49: fresh28(is_a_theorem(implies(X, Y)), true, implies(or(Z, X), or(Z, Y))) = true.
% 23.28/3.37 Proof:
% 23.28/3.37 fresh28(is_a_theorem(implies(X, Y)), true, implies(or(Z, X), or(Z, Y)))
% 23.28/3.37 = { by axiom 36 (modus_ponens_2) R->L }
% 23.28/3.37 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)))
% 23.28/3.37 = { by lemma 41 }
% 23.28/3.37 fresh59(true, true, implies(X, Y), implies(or(Z, X), or(Z, Y)))
% 23.28/3.37 = { by lemma 40 }
% 23.28/3.37 true
% 23.28/3.37
% 23.28/3.37 Lemma 50: is_a_theorem(implies(implies(X, Y), or(Y, not(X)))) = true.
% 23.28/3.37 Proof:
% 23.28/3.37 is_a_theorem(implies(implies(X, Y), or(Y, not(X))))
% 23.28/3.37 = { by lemma 42 R->L }
% 23.28/3.37 is_a_theorem(implies(or(not(X), Y), or(Y, not(X))))
% 23.28/3.37 = { by lemma 47 }
% 23.28/3.37 true
% 23.28/3.37
% 23.28/3.37 Lemma 51: or(X, X) = X.
% 23.28/3.37 Proof:
% 23.28/3.37 or(X, X)
% 23.28/3.37 = { by axiom 25 (substitution_of_equivalents_2) R->L }
% 23.28/3.37 fresh2(true, true, or(X, X), X)
% 23.28/3.37 = { by axiom 2 (substitution_of_equivalents) R->L }
% 23.28/3.37 fresh2(substitution_of_equivalents, true, or(X, X), X)
% 23.28/3.37 = { by axiom 35 (substitution_of_equivalents_2) }
% 23.28/3.37 fresh(is_a_theorem(equiv(or(X, X), X)), true, or(X, X), X)
% 23.28/3.37 = { by axiom 14 (modus_ponens_2) R->L }
% 23.28/3.37 fresh(fresh28(true, true, equiv(or(X, X), X)), true, or(X, X), X)
% 23.28/3.37 = { by lemma 40 R->L }
% 23.28/3.37 fresh(fresh28(fresh59(true, true, implies(or(X, X), or(X, X)), implies(X, or(X, X))), true, equiv(or(X, X), X)), true, or(X, X), X)
% 23.28/3.37 = { by lemma 45 R->L }
% 23.28/3.37 fresh(fresh28(fresh59(fresh28(is_a_theorem(implies(X, or(X, X))), true, implies(implies(or(X, X), or(X, X)), implies(X, or(X, X)))), true, implies(or(X, X), or(X, X)), implies(X, or(X, X))), true, equiv(or(X, X), X)), true, or(X, X), X)
% 23.28/3.37 = { by axiom 26 (or_2_1) R->L }
% 23.28/3.37 fresh(fresh28(fresh59(fresh28(fresh16(or_2, true, X, X), true, implies(implies(or(X, X), or(X, X)), implies(X, or(X, X)))), true, implies(or(X, X), or(X, X)), implies(X, or(X, X))), true, equiv(or(X, X), X)), true, or(X, X), X)
% 23.28/3.37 = { by lemma 46 }
% 23.28/3.37 fresh(fresh28(fresh59(fresh28(true, true, implies(implies(or(X, X), or(X, X)), implies(X, or(X, X)))), true, implies(or(X, X), or(X, X)), implies(X, or(X, X))), true, equiv(or(X, X), X)), true, or(X, X), X)
% 23.28/3.37 = { by axiom 14 (modus_ponens_2) }
% 23.28/3.37 fresh(fresh28(fresh59(is_a_theorem(implies(implies(or(X, X), or(X, X)), implies(X, or(X, X)))), true, implies(or(X, X), or(X, X)), implies(X, or(X, X))), true, equiv(or(X, X), X)), true, or(X, X), X)
% 23.28/3.37 = { by axiom 36 (modus_ponens_2) }
% 23.28/3.37 fresh(fresh28(fresh28(is_a_theorem(implies(or(X, X), or(X, X))), true, implies(X, or(X, X))), true, equiv(or(X, X), X)), true, or(X, X), X)
% 23.28/3.37 = { by lemma 47 }
% 23.28/3.37 fresh(fresh28(fresh28(true, true, implies(X, or(X, X))), true, equiv(or(X, X), X)), true, or(X, X), X)
% 23.28/3.37 = { by axiom 14 (modus_ponens_2) }
% 23.28/3.37 fresh(fresh28(is_a_theorem(implies(X, or(X, X))), true, equiv(or(X, X), X)), true, or(X, X), X)
% 23.28/3.37 = { by axiom 36 (modus_ponens_2) R->L }
% 23.28/3.37 fresh(fresh59(is_a_theorem(implies(implies(X, or(X, X)), equiv(or(X, X), X))), true, implies(X, or(X, X)), equiv(or(X, X), X)), true, or(X, X), X)
% 23.28/3.37 = { by axiom 19 (op_equiv) R->L }
% 23.28/3.37 fresh(fresh59(is_a_theorem(implies(implies(X, or(X, X)), fresh23(true, true, or(X, X), X))), true, implies(X, or(X, X)), equiv(or(X, X), X)), true, or(X, X), X)
% 23.28/3.37 = { by axiom 8 (principia_op_equiv) R->L }
% 23.28/3.37 fresh(fresh59(is_a_theorem(implies(implies(X, or(X, X)), fresh23(op_equiv, true, or(X, X), X))), true, implies(X, or(X, X)), equiv(or(X, X), X)), true, or(X, X), X)
% 23.28/3.37 = { by axiom 33 (op_equiv) }
% 23.28/3.37 fresh(fresh59(is_a_theorem(implies(implies(X, or(X, X)), and(implies(or(X, X), X), implies(X, or(X, X))))), true, implies(X, or(X, X)), equiv(or(X, X), X)), true, or(X, X), X)
% 23.28/3.37 = { by axiom 14 (modus_ponens_2) R->L }
% 23.28/3.37 fresh(fresh59(fresh28(true, true, implies(implies(X, or(X, X)), and(implies(or(X, X), X), implies(X, or(X, X))))), true, implies(X, or(X, X)), equiv(or(X, X), X)), true, or(X, X), X)
% 23.28/3.37 = { by lemma 48 R->L }
% 23.28/3.37 fresh(fresh59(fresh28(is_a_theorem(implies(or(X, X), X)), true, implies(implies(X, or(X, X)), and(implies(or(X, X), X), implies(X, or(X, X))))), true, implies(X, or(X, X)), equiv(or(X, X), X)), true, or(X, X), X)
% 23.28/3.37 = { by axiom 36 (modus_ponens_2) R->L }
% 23.28/3.37 fresh(fresh59(fresh59(is_a_theorem(implies(implies(or(X, X), X), implies(implies(X, or(X, X)), and(implies(or(X, X), X), implies(X, or(X, X)))))), true, implies(or(X, X), X), implies(implies(X, or(X, X)), and(implies(or(X, X), X), implies(X, or(X, X))))), true, implies(X, or(X, X)), equiv(or(X, X), X)), true, or(X, X), X)
% 23.28/3.37 = { by axiom 14 (modus_ponens_2) R->L }
% 23.28/3.37 fresh(fresh59(fresh59(fresh28(true, true, implies(implies(or(X, X), X), implies(implies(X, or(X, X)), and(implies(or(X, X), X), implies(X, or(X, X)))))), true, implies(or(X, X), X), implies(implies(X, or(X, X)), and(implies(or(X, X), X), implies(X, or(X, X))))), true, implies(X, or(X, X)), equiv(or(X, X), X)), true, or(X, X), X)
% 23.28/3.37 = { by lemma 40 R->L }
% 23.28/3.38 fresh(fresh59(fresh59(fresh28(fresh59(true, true, or(not(implies(X, or(X, X))), implies(implies(implies(or(X, X), X), not(implies(X, or(X, X)))), not(implies(or(X, X), X)))), or(not(implies(X, or(X, X))), or(not(implies(or(X, X), X)), not(implies(implies(or(X, X), X), not(implies(X, or(X, X)))))))), true, implies(implies(or(X, X), X), implies(implies(X, or(X, X)), and(implies(or(X, X), X), implies(X, or(X, X)))))), true, implies(or(X, X), X), implies(implies(X, or(X, X)), and(implies(or(X, X), X), implies(X, or(X, X))))), true, implies(X, or(X, X)), equiv(or(X, X), X)), true, or(X, X), X)
% 23.28/3.38 = { by lemma 49 R->L }
% 23.28/3.38 fresh(fresh59(fresh59(fresh28(fresh59(fresh28(is_a_theorem(implies(implies(implies(implies(or(X, X), X), not(implies(X, or(X, X)))), not(implies(or(X, X), X))), or(not(implies(or(X, X), X)), not(implies(implies(or(X, X), X), not(implies(X, or(X, X)))))))), true, implies(or(not(implies(X, or(X, X))), implies(implies(implies(or(X, X), X), not(implies(X, or(X, X)))), not(implies(or(X, X), X)))), or(not(implies(X, or(X, X))), or(not(implies(or(X, X), X)), not(implies(implies(or(X, X), X), not(implies(X, or(X, X))))))))), true, or(not(implies(X, or(X, X))), implies(implies(implies(or(X, X), X), not(implies(X, or(X, X)))), not(implies(or(X, X), X)))), or(not(implies(X, or(X, X))), or(not(implies(or(X, X), X)), not(implies(implies(or(X, X), X), not(implies(X, or(X, X)))))))), true, implies(implies(or(X, X), X), implies(implies(X, or(X, X)), and(implies(or(X, X), X), implies(X, or(X, X)))))), true, implies(or(X, X), X), implies(implies(X, or(X, X)), and(implies(or(X, X), X), implies(X, or(X, X))))), true, implies(X, or(X, X)), equiv(or(X, X), X)), true, or(X, X), X)
% 23.28/3.38 = { by lemma 50 }
% 23.28/3.38 fresh(fresh59(fresh59(fresh28(fresh59(fresh28(true, true, implies(or(not(implies(X, or(X, X))), implies(implies(implies(or(X, X), X), not(implies(X, or(X, X)))), not(implies(or(X, X), X)))), or(not(implies(X, or(X, X))), or(not(implies(or(X, X), X)), not(implies(implies(or(X, X), X), not(implies(X, or(X, X))))))))), true, or(not(implies(X, or(X, X))), implies(implies(implies(or(X, X), X), not(implies(X, or(X, X)))), not(implies(or(X, X), X)))), or(not(implies(X, or(X, X))), or(not(implies(or(X, X), X)), not(implies(implies(or(X, X), X), not(implies(X, or(X, X)))))))), true, implies(implies(or(X, X), X), implies(implies(X, or(X, X)), and(implies(or(X, X), X), implies(X, or(X, X)))))), true, implies(or(X, X), X), implies(implies(X, or(X, X)), and(implies(or(X, X), X), implies(X, or(X, X))))), true, implies(X, or(X, X)), equiv(or(X, X), X)), true, or(X, X), X)
% 23.28/3.38 = { by axiom 14 (modus_ponens_2) }
% 23.28/3.38 fresh(fresh59(fresh59(fresh28(fresh59(is_a_theorem(implies(or(not(implies(X, or(X, X))), implies(implies(implies(or(X, X), X), not(implies(X, or(X, X)))), not(implies(or(X, X), X)))), or(not(implies(X, or(X, X))), or(not(implies(or(X, X), X)), not(implies(implies(or(X, X), X), not(implies(X, or(X, X))))))))), true, or(not(implies(X, or(X, X))), implies(implies(implies(or(X, X), X), not(implies(X, or(X, X)))), not(implies(or(X, X), X)))), or(not(implies(X, or(X, X))), or(not(implies(or(X, X), X)), not(implies(implies(or(X, X), X), not(implies(X, or(X, X)))))))), true, implies(implies(or(X, X), X), implies(implies(X, or(X, X)), and(implies(or(X, X), X), implies(X, or(X, X)))))), true, implies(or(X, X), X), implies(implies(X, or(X, X)), and(implies(or(X, X), X), implies(X, or(X, X))))), true, implies(X, or(X, X)), equiv(or(X, X), X)), true, or(X, X), X)
% 23.28/3.38 = { by axiom 36 (modus_ponens_2) }
% 23.28/3.38 fresh(fresh59(fresh59(fresh28(fresh28(is_a_theorem(or(not(implies(X, or(X, X))), implies(implies(implies(or(X, X), X), not(implies(X, or(X, X)))), not(implies(or(X, X), X))))), true, or(not(implies(X, or(X, X))), or(not(implies(or(X, X), X)), not(implies(implies(or(X, X), X), not(implies(X, or(X, X)))))))), true, implies(implies(or(X, X), X), implies(implies(X, or(X, X)), and(implies(or(X, X), X), implies(X, or(X, X)))))), true, implies(or(X, X), X), implies(implies(X, or(X, X)), and(implies(or(X, X), X), implies(X, or(X, X))))), true, implies(X, or(X, X)), equiv(or(X, X), X)), true, or(X, X), X)
% 23.28/3.38 = { by axiom 14 (modus_ponens_2) R->L }
% 23.28/3.38 fresh(fresh59(fresh59(fresh28(fresh28(fresh28(true, true, or(not(implies(X, or(X, X))), implies(implies(implies(or(X, X), X), not(implies(X, or(X, X)))), not(implies(or(X, X), X))))), true, or(not(implies(X, or(X, X))), or(not(implies(or(X, X), X)), not(implies(implies(or(X, X), X), not(implies(X, or(X, X)))))))), true, implies(implies(or(X, X), X), implies(implies(X, or(X, X)), and(implies(or(X, X), X), implies(X, or(X, X)))))), true, implies(or(X, X), X), implies(implies(X, or(X, X)), and(implies(or(X, X), X), implies(X, or(X, X))))), true, implies(X, or(X, X)), equiv(or(X, X), X)), true, or(X, X), X)
% 23.28/3.38 = { by lemma 50 R->L }
% 23.28/3.38 fresh(fresh59(fresh59(fresh28(fresh28(fresh28(is_a_theorem(implies(implies(implies(or(X, X), X), not(implies(X, or(X, X)))), or(not(implies(X, or(X, X))), not(implies(or(X, X), X))))), true, or(not(implies(X, or(X, X))), implies(implies(implies(or(X, X), X), not(implies(X, or(X, X)))), not(implies(or(X, X), X))))), true, or(not(implies(X, or(X, X))), or(not(implies(or(X, X), X)), not(implies(implies(or(X, X), X), not(implies(X, or(X, X)))))))), true, implies(implies(or(X, X), X), implies(implies(X, or(X, X)), and(implies(or(X, X), X), implies(X, or(X, X)))))), true, implies(or(X, X), X), implies(implies(X, or(X, X)), and(implies(or(X, X), X), implies(X, or(X, X))))), true, implies(X, or(X, X)), equiv(or(X, X), X)), true, or(X, X), X)
% 23.28/3.38 = { by axiom 36 (modus_ponens_2) R->L }
% 23.28/3.38 fresh(fresh59(fresh59(fresh28(fresh28(fresh59(is_a_theorem(implies(implies(implies(implies(or(X, X), X), not(implies(X, or(X, X)))), or(not(implies(X, or(X, X))), not(implies(or(X, X), X)))), or(not(implies(X, or(X, X))), implies(implies(implies(or(X, X), X), not(implies(X, or(X, X)))), not(implies(or(X, X), X)))))), true, implies(implies(implies(or(X, X), X), not(implies(X, or(X, X)))), or(not(implies(X, or(X, X))), not(implies(or(X, X), X)))), or(not(implies(X, or(X, X))), implies(implies(implies(or(X, X), X), not(implies(X, or(X, X)))), not(implies(or(X, X), X))))), true, or(not(implies(X, or(X, X))), or(not(implies(or(X, X), X)), not(implies(implies(or(X, X), X), not(implies(X, or(X, X)))))))), true, implies(implies(or(X, X), X), implies(implies(X, or(X, X)), and(implies(or(X, X), X), implies(X, or(X, X)))))), true, implies(or(X, X), X), implies(implies(X, or(X, X)), and(implies(or(X, X), X), implies(X, or(X, X))))), true, implies(X, or(X, X)), equiv(or(X, X), X)), true, or(X, X), X)
% 23.28/3.38 = { by lemma 43 }
% 23.28/3.38 fresh(fresh59(fresh59(fresh28(fresh28(fresh59(true, true, implies(implies(implies(or(X, X), X), not(implies(X, or(X, X)))), or(not(implies(X, or(X, X))), not(implies(or(X, X), X)))), or(not(implies(X, or(X, X))), implies(implies(implies(or(X, X), X), not(implies(X, or(X, X)))), not(implies(or(X, X), X))))), true, or(not(implies(X, or(X, X))), or(not(implies(or(X, X), X)), not(implies(implies(or(X, X), X), not(implies(X, or(X, X)))))))), true, implies(implies(or(X, X), X), implies(implies(X, or(X, X)), and(implies(or(X, X), X), implies(X, or(X, X)))))), true, implies(or(X, X), X), implies(implies(X, or(X, X)), and(implies(or(X, X), X), implies(X, or(X, X))))), true, implies(X, or(X, X)), equiv(or(X, X), X)), true, or(X, X), X)
% 23.28/3.38 = { by lemma 40 }
% 23.28/3.38 fresh(fresh59(fresh59(fresh28(fresh28(true, true, or(not(implies(X, or(X, X))), or(not(implies(or(X, X), X)), not(implies(implies(or(X, X), X), not(implies(X, or(X, X)))))))), true, implies(implies(or(X, X), X), implies(implies(X, or(X, X)), and(implies(or(X, X), X), implies(X, or(X, X)))))), true, implies(or(X, X), X), implies(implies(X, or(X, X)), and(implies(or(X, X), X), implies(X, or(X, X))))), true, implies(X, or(X, X)), equiv(or(X, X), X)), true, or(X, X), X)
% 23.28/3.38 = { by axiom 14 (modus_ponens_2) }
% 23.28/3.38 fresh(fresh59(fresh59(fresh28(is_a_theorem(or(not(implies(X, or(X, X))), or(not(implies(or(X, X), X)), not(implies(implies(or(X, X), X), not(implies(X, or(X, X)))))))), true, implies(implies(or(X, X), X), implies(implies(X, or(X, X)), and(implies(or(X, X), X), implies(X, or(X, X)))))), true, implies(or(X, X), X), implies(implies(X, or(X, X)), and(implies(or(X, X), X), implies(X, or(X, X))))), true, implies(X, or(X, X)), equiv(or(X, X), X)), true, or(X, X), X)
% 23.28/3.38 = { by lemma 42 }
% 23.28/3.38 fresh(fresh59(fresh59(fresh28(is_a_theorem(or(not(implies(X, or(X, X))), implies(implies(or(X, X), X), not(implies(implies(or(X, X), X), not(implies(X, or(X, X)))))))), true, implies(implies(or(X, X), X), implies(implies(X, or(X, X)), and(implies(or(X, X), X), implies(X, or(X, X)))))), true, implies(or(X, X), X), implies(implies(X, or(X, X)), and(implies(or(X, X), X), implies(X, or(X, X))))), true, implies(X, or(X, X)), equiv(or(X, X), X)), true, or(X, X), X)
% 23.28/3.38 = { by lemma 42 }
% 23.28/3.38 fresh(fresh59(fresh59(fresh28(is_a_theorem(implies(implies(X, or(X, X)), implies(implies(or(X, X), X), not(implies(implies(or(X, X), X), not(implies(X, or(X, X)))))))), true, implies(implies(or(X, X), X), implies(implies(X, or(X, X)), and(implies(or(X, X), X), implies(X, or(X, X)))))), true, implies(or(X, X), X), implies(implies(X, or(X, X)), and(implies(or(X, X), X), implies(X, or(X, X))))), true, implies(X, or(X, X)), equiv(or(X, X), X)), true, or(X, X), X)
% 23.28/3.38 = { by lemma 42 R->L }
% 23.28/3.38 fresh(fresh59(fresh59(fresh28(is_a_theorem(implies(implies(X, or(X, X)), implies(implies(or(X, X), X), not(or(not(implies(or(X, X), X)), not(implies(X, or(X, X)))))))), true, implies(implies(or(X, X), X), implies(implies(X, or(X, X)), and(implies(or(X, X), X), implies(X, or(X, X)))))), true, implies(or(X, X), X), implies(implies(X, or(X, X)), and(implies(or(X, X), X), implies(X, or(X, X))))), true, implies(X, or(X, X)), equiv(or(X, X), X)), true, or(X, X), X)
% 23.28/3.38 = { by axiom 29 (op_and) R->L }
% 23.28/3.38 fresh(fresh59(fresh59(fresh28(is_a_theorem(implies(implies(X, or(X, X)), implies(implies(or(X, X), X), fresh24(op_and, true, implies(or(X, X), X), implies(X, or(X, X)))))), true, implies(implies(or(X, X), X), implies(implies(X, or(X, X)), and(implies(or(X, X), X), implies(X, or(X, X)))))), true, implies(or(X, X), X), implies(implies(X, or(X, X)), and(implies(or(X, X), X), implies(X, or(X, X))))), true, implies(X, or(X, X)), equiv(or(X, X), X)), true, or(X, X), X)
% 23.28/3.38 = { by axiom 10 (principia_op_and) }
% 23.28/3.38 fresh(fresh59(fresh59(fresh28(is_a_theorem(implies(implies(X, or(X, X)), implies(implies(or(X, X), X), fresh24(true, true, implies(or(X, X), X), implies(X, or(X, X)))))), true, implies(implies(or(X, X), X), implies(implies(X, or(X, X)), and(implies(or(X, X), X), implies(X, or(X, X)))))), true, implies(or(X, X), X), implies(implies(X, or(X, X)), and(implies(or(X, X), X), implies(X, or(X, X))))), true, implies(X, or(X, X)), equiv(or(X, X), X)), true, or(X, X), X)
% 23.28/3.38 = { by axiom 18 (op_and) }
% 23.28/3.38 fresh(fresh59(fresh59(fresh28(is_a_theorem(implies(implies(X, or(X, X)), implies(implies(or(X, X), X), and(implies(or(X, X), X), implies(X, or(X, X)))))), true, implies(implies(or(X, X), X), implies(implies(X, or(X, X)), and(implies(or(X, X), X), implies(X, or(X, X)))))), true, implies(or(X, X), X), implies(implies(X, or(X, X)), and(implies(or(X, X), X), implies(X, or(X, X))))), true, implies(X, or(X, X)), equiv(or(X, X), X)), true, or(X, X), X)
% 23.28/3.38 = { by lemma 44 }
% 23.28/3.38 fresh(fresh59(fresh59(true, true, implies(or(X, X), X), implies(implies(X, or(X, X)), and(implies(or(X, X), X), implies(X, or(X, X))))), true, implies(X, or(X, X)), equiv(or(X, X), X)), true, or(X, X), X)
% 23.28/3.38 = { by lemma 40 }
% 23.28/3.38 fresh(fresh59(true, true, implies(X, or(X, X)), equiv(or(X, X), X)), true, or(X, X), X)
% 23.28/3.38 = { by lemma 40 }
% 23.28/3.38 fresh(true, true, or(X, X), X)
% 23.28/3.38 = { by axiom 16 (substitution_of_equivalents_2) }
% 23.28/3.38 X
% 23.28/3.38
% 23.28/3.38 Lemma 52: implies(and(not(X), not(Y)), Z) = or(or(X, Y), Z).
% 23.28/3.38 Proof:
% 23.28/3.38 implies(and(not(X), not(Y)), Z)
% 23.28/3.38 = { by lemma 42 R->L }
% 23.28/3.38 or(not(and(not(X), not(Y))), Z)
% 23.28/3.38 = { by axiom 30 (op_or) R->L }
% 23.28/3.38 or(fresh20(op_or, true, X, Y), Z)
% 23.28/3.38 = { by axiom 9 (luka_op_or) }
% 23.28/3.38 or(fresh20(true, true, X, Y), Z)
% 23.28/3.38 = { by axiom 22 (op_or) }
% 23.28/3.38 or(or(X, Y), Z)
% 23.28/3.38
% 23.28/3.38 Goal 1 (luka_cn3): cn3 = true.
% 23.28/3.38 Proof:
% 23.28/3.38 cn3
% 23.28/3.38 = { by axiom 37 (cn3) R->L }
% 23.28/3.38 fresh48(is_a_theorem(implies(implies(not(p6), p6), p6)), true)
% 23.28/3.38 = { by lemma 51 R->L }
% 23.28/3.38 fresh48(is_a_theorem(implies(implies(not(p6), p6), or(p6, p6))), true)
% 23.28/3.38 = { by lemma 51 R->L }
% 23.28/3.38 fresh48(is_a_theorem(implies(implies(not(p6), or(p6, p6)), or(p6, p6))), true)
% 23.28/3.38 = { by axiom 14 (modus_ponens_2) R->L }
% 23.28/3.38 fresh48(fresh28(true, true, implies(implies(not(p6), or(p6, p6)), or(p6, p6))), true)
% 23.28/3.38 = { by lemma 45 R->L }
% 23.28/3.38 fresh48(fresh28(fresh28(is_a_theorem(implies(and(not(p6), not(p6)), not(p6))), true, implies(implies(not(p6), or(p6, p6)), implies(and(not(p6), not(p6)), or(p6, p6)))), true, implies(implies(not(p6), or(p6, p6)), or(p6, p6))), true)
% 23.28/3.39 = { by lemma 52 }
% 23.28/3.39 fresh48(fresh28(fresh28(is_a_theorem(or(or(p6, p6), not(p6))), true, implies(implies(not(p6), or(p6, p6)), implies(and(not(p6), not(p6)), or(p6, p6)))), true, implies(implies(not(p6), or(p6, p6)), or(p6, p6))), true)
% 23.28/3.39 = { by lemma 52 }
% 23.28/3.39 fresh48(fresh28(fresh28(is_a_theorem(or(or(p6, p6), not(p6))), true, implies(implies(not(p6), or(p6, p6)), or(or(p6, p6), or(p6, p6)))), true, implies(implies(not(p6), or(p6, p6)), or(p6, p6))), true)
% 23.28/3.39 = { by axiom 14 (modus_ponens_2) R->L }
% 23.28/3.39 fresh48(fresh28(fresh28(fresh28(true, true, or(or(p6, p6), not(p6))), true, implies(implies(not(p6), or(p6, p6)), or(or(p6, p6), or(p6, p6)))), true, implies(implies(not(p6), or(p6, p6)), or(p6, p6))), true)
% 23.28/3.39 = { by lemma 46 R->L }
% 23.28/3.39 fresh48(fresh28(fresh28(fresh28(fresh16(or_2, true, p6, p6), true, or(or(p6, p6), not(p6))), true, implies(implies(not(p6), or(p6, p6)), or(or(p6, p6), or(p6, p6)))), true, implies(implies(not(p6), or(p6, p6)), or(p6, p6))), true)
% 23.28/3.39 = { by axiom 26 (or_2_1) }
% 23.28/3.39 fresh48(fresh28(fresh28(fresh28(is_a_theorem(implies(p6, or(p6, p6))), true, or(or(p6, p6), not(p6))), true, implies(implies(not(p6), or(p6, p6)), or(or(p6, p6), or(p6, p6)))), true, implies(implies(not(p6), or(p6, p6)), or(p6, p6))), true)
% 23.28/3.39 = { by axiom 36 (modus_ponens_2) R->L }
% 23.28/3.39 fresh48(fresh28(fresh28(fresh59(is_a_theorem(implies(implies(p6, or(p6, p6)), or(or(p6, p6), not(p6)))), true, implies(p6, or(p6, p6)), or(or(p6, p6), not(p6))), true, implies(implies(not(p6), or(p6, p6)), or(or(p6, p6), or(p6, p6)))), true, implies(implies(not(p6), or(p6, p6)), or(p6, p6))), true)
% 23.28/3.39 = { by lemma 50 }
% 23.28/3.39 fresh48(fresh28(fresh28(fresh59(true, true, implies(p6, or(p6, p6)), or(or(p6, p6), not(p6))), true, implies(implies(not(p6), or(p6, p6)), or(or(p6, p6), or(p6, p6)))), true, implies(implies(not(p6), or(p6, p6)), or(p6, p6))), true)
% 23.28/3.39 = { by lemma 40 }
% 23.28/3.39 fresh48(fresh28(fresh28(true, true, implies(implies(not(p6), or(p6, p6)), or(or(p6, p6), or(p6, p6)))), true, implies(implies(not(p6), or(p6, p6)), or(p6, p6))), true)
% 23.28/3.39 = { by axiom 14 (modus_ponens_2) }
% 23.28/3.39 fresh48(fresh28(is_a_theorem(implies(implies(not(p6), or(p6, p6)), or(or(p6, p6), or(p6, p6)))), true, implies(implies(not(p6), or(p6, p6)), or(p6, p6))), true)
% 23.28/3.39 = { by axiom 36 (modus_ponens_2) R->L }
% 23.28/3.39 fresh48(fresh59(is_a_theorem(implies(implies(implies(not(p6), or(p6, p6)), or(or(p6, p6), or(p6, p6))), implies(implies(not(p6), or(p6, p6)), or(p6, p6)))), true, implies(implies(not(p6), or(p6, p6)), or(or(p6, p6), or(p6, p6))), implies(implies(not(p6), or(p6, p6)), or(p6, p6))), true)
% 23.28/3.39 = { by lemma 42 R->L }
% 23.28/3.39 fresh48(fresh59(is_a_theorem(implies(implies(implies(not(p6), or(p6, p6)), or(or(p6, p6), or(p6, p6))), or(not(implies(not(p6), or(p6, p6))), or(p6, p6)))), true, implies(implies(not(p6), or(p6, p6)), or(or(p6, p6), or(p6, p6))), implies(implies(not(p6), or(p6, p6)), or(p6, p6))), true)
% 23.28/3.39 = { by lemma 42 R->L }
% 23.28/3.39 fresh48(fresh59(is_a_theorem(implies(or(not(implies(not(p6), or(p6, p6))), or(or(p6, p6), or(p6, p6))), or(not(implies(not(p6), or(p6, p6))), or(p6, p6)))), true, implies(implies(not(p6), or(p6, p6)), or(or(p6, p6), or(p6, p6))), implies(implies(not(p6), or(p6, p6)), or(p6, p6))), true)
% 23.28/3.39 = { by axiom 14 (modus_ponens_2) R->L }
% 23.28/3.39 fresh48(fresh59(fresh28(true, true, implies(or(not(implies(not(p6), or(p6, p6))), or(or(p6, p6), or(p6, p6))), or(not(implies(not(p6), or(p6, p6))), or(p6, p6)))), true, implies(implies(not(p6), or(p6, p6)), or(or(p6, p6), or(p6, p6))), implies(implies(not(p6), or(p6, p6)), or(p6, p6))), true)
% 23.28/3.39 = { by lemma 48 R->L }
% 23.28/3.39 fresh48(fresh59(fresh28(is_a_theorem(implies(or(or(p6, p6), or(p6, p6)), or(p6, p6))), true, implies(or(not(implies(not(p6), or(p6, p6))), or(or(p6, p6), or(p6, p6))), or(not(implies(not(p6), or(p6, p6))), or(p6, p6)))), true, implies(implies(not(p6), or(p6, p6)), or(or(p6, p6), or(p6, p6))), implies(implies(not(p6), or(p6, p6)), or(p6, p6))), true)
% 23.28/3.39 = { by lemma 49 }
% 23.28/3.39 fresh48(fresh59(true, true, implies(implies(not(p6), or(p6, p6)), or(or(p6, p6), or(p6, p6))), implies(implies(not(p6), or(p6, p6)), or(p6, p6))), true)
% 23.28/3.39 = { by lemma 40 }
% 23.28/3.39 fresh48(true, true)
% 23.28/3.39 = { by axiom 12 (cn3) }
% 23.28/3.39 true
% 23.28/3.39 % SZS output end Proof
% 23.28/3.39
% 23.28/3.39 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------