TSTP Solution File: LCL457+1 by Twee---2.4.2
View Problem
- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : LCL457+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 08:19:04 EDT 2023
% Result : Theorem 66.64s 9.07s
% Output : Proof 69.32s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.13 % Problem : LCL457+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
% 0.14/0.35 % Computer : n016.cluster.edu
% 0.14/0.35 % Model : x86_64 x86_64
% 0.14/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.14/0.35 % Memory : 8042.1875MB
% 0.14/0.35 % OS : Linux 3.10.0-693.el7.x86_64
% 0.14/0.35 % CPULimit : 300
% 0.14/0.35 % WCLimit : 300
% 0.14/0.35 % DateTime : Fri Aug 25 00:45:40 EDT 2023
% 0.14/0.35 % CPUTime :
% 66.64/9.07 Command-line arguments: --lhs-weight 9 --flip-ordering --complete-subsets --normalise-queue-percent 10 --cp-renormalise-threshold 10
% 66.64/9.07
% 66.64/9.07 % SZS status Theorem
% 66.64/9.07
% 68.37/9.25 % SZS output start Proof
% 68.37/9.25 Take the following subset of the input axioms:
% 68.37/9.26 fof(and_1, axiom, and_1 <=> ![X, Y]: is_a_theorem(implies(and(X, Y), X))).
% 68.37/9.26 fof(and_2, axiom, and_2 <=> ![X2, Y2]: is_a_theorem(implies(and(X2, Y2), Y2))).
% 68.37/9.26 fof(and_3, axiom, and_3 <=> ![X2, Y2]: is_a_theorem(implies(X2, implies(Y2, and(X2, Y2))))).
% 68.37/9.26 fof(hilbert_and_1, axiom, and_1).
% 68.37/9.26 fof(hilbert_and_2, axiom, and_2).
% 68.37/9.26 fof(hilbert_and_3, axiom, and_3).
% 68.37/9.26 fof(hilbert_implies_1, axiom, implies_1).
% 68.37/9.26 fof(hilbert_implies_2, axiom, implies_2).
% 68.37/9.26 fof(hilbert_implies_3, axiom, implies_3).
% 68.37/9.26 fof(hilbert_modus_ponens, axiom, modus_ponens).
% 68.37/9.26 fof(hilbert_modus_tollens, axiom, modus_tollens).
% 68.37/9.26 fof(hilbert_op_equiv, axiom, op_equiv).
% 68.37/9.26 fof(hilbert_op_implies_and, axiom, op_implies_and).
% 68.37/9.26 fof(hilbert_op_or, axiom, op_or).
% 68.37/9.26 fof(hilbert_or_1, axiom, or_1).
% 68.37/9.26 fof(hilbert_or_2, axiom, or_2).
% 68.37/9.26 fof(hilbert_or_3, axiom, or_3).
% 68.37/9.26 fof(implies_1, axiom, implies_1 <=> ![X2, Y2]: is_a_theorem(implies(X2, implies(Y2, X2)))).
% 68.37/9.26 fof(implies_2, axiom, implies_2 <=> ![X2, Y2]: is_a_theorem(implies(implies(X2, implies(X2, Y2)), implies(X2, Y2)))).
% 68.37/9.26 fof(implies_3, axiom, implies_3 <=> ![Z, X2, Y2]: is_a_theorem(implies(implies(X2, Y2), implies(implies(Y2, Z), implies(X2, Z))))).
% 68.37/9.26 fof(modus_ponens, axiom, modus_ponens <=> ![X2, Y2]: ((is_a_theorem(X2) & is_a_theorem(implies(X2, Y2))) => is_a_theorem(Y2))).
% 68.37/9.26 fof(modus_tollens, axiom, modus_tollens <=> ![X2, Y2]: is_a_theorem(implies(implies(not(Y2), not(X2)), implies(X2, Y2)))).
% 68.37/9.26 fof(op_and, axiom, op_and => ![X2, Y2]: and(X2, Y2)=not(or(not(X2), not(Y2)))).
% 68.37/9.26 fof(op_equiv, axiom, op_equiv => ![X2, Y2]: equiv(X2, Y2)=and(implies(X2, Y2), implies(Y2, X2))).
% 68.37/9.26 fof(op_implies_and, axiom, op_implies_and => ![X2, Y2]: implies(X2, Y2)=not(and(X2, not(Y2)))).
% 68.37/9.26 fof(op_implies_or, axiom, op_implies_or => ![X2, Y2]: implies(X2, Y2)=or(not(X2), Y2)).
% 68.37/9.26 fof(op_or, axiom, op_or => ![X2, Y2]: or(X2, Y2)=not(and(not(X2), not(Y2)))).
% 68.37/9.26 fof(or_1, axiom, or_1 <=> ![X2, Y2]: is_a_theorem(implies(X2, or(X2, Y2)))).
% 68.37/9.26 fof(or_2, axiom, or_2 <=> ![X2, Y2]: is_a_theorem(implies(Y2, or(X2, Y2)))).
% 68.37/9.26 fof(or_3, axiom, or_3 <=> ![X2, Y2, Z2]: is_a_theorem(implies(implies(X2, Z2), implies(implies(Y2, Z2), implies(or(X2, Y2), Z2))))).
% 68.37/9.26 fof(principia_op_and, axiom, op_and).
% 68.37/9.26 fof(principia_op_implies_or, axiom, op_implies_or).
% 68.37/9.26 fof(principia_r4, conjecture, r4).
% 68.37/9.26 fof(r3, axiom, r3 <=> ![P, Q]: is_a_theorem(implies(or(P, Q), or(Q, P)))).
% 68.37/9.26 fof(r4, axiom, r4 <=> ![R, P2, Q2]: is_a_theorem(implies(or(P2, or(Q2, R)), or(Q2, or(P2, R))))).
% 68.37/9.26 fof(r5, axiom, r5 <=> ![P2, Q2, R2]: is_a_theorem(implies(implies(Q2, R2), implies(or(P2, Q2), or(P2, R2))))).
% 68.37/9.26 fof(substitution_of_equivalents, axiom, substitution_of_equivalents <=> ![X2, Y2]: (is_a_theorem(equiv(X2, Y2)) => X2=Y2)).
% 68.37/9.26 fof(substitution_of_equivalents, axiom, substitution_of_equivalents).
% 68.37/9.26
% 68.37/9.26 Now clausify the problem and encode Horn clauses using encoding 3 of
% 68.37/9.26 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 68.37/9.26 We repeatedly replace C & s=t => u=v by the two clauses:
% 68.37/9.26 fresh(y, y, x1...xn) = u
% 68.37/9.26 C => fresh(s, t, x1...xn) = v
% 68.37/9.26 where fresh is a fresh function symbol and x1..xn are the free
% 68.37/9.26 variables of u and v.
% 68.37/9.26 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 68.37/9.26 input problem has no model of domain size 1).
% 68.37/9.26
% 68.37/9.26 The encoding turns the above axioms into the following unit equations and goals:
% 68.37/9.26
% 68.37/9.26 Axiom 1 (hilbert_modus_ponens): modus_ponens = true.
% 68.37/9.26 Axiom 2 (substitution_of_equivalents): substitution_of_equivalents = true.
% 68.37/9.26 Axiom 3 (hilbert_modus_tollens): modus_tollens = true.
% 68.37/9.26 Axiom 4 (hilbert_implies_1): implies_1 = true.
% 68.37/9.26 Axiom 5 (hilbert_implies_2): implies_2 = true.
% 68.37/9.26 Axiom 6 (hilbert_implies_3): implies_3 = true.
% 68.37/9.26 Axiom 7 (hilbert_and_1): and_1 = true.
% 68.37/9.26 Axiom 8 (hilbert_and_2): and_2 = true.
% 68.37/9.26 Axiom 9 (hilbert_and_3): and_3 = true.
% 68.37/9.26 Axiom 10 (hilbert_or_1): or_1 = true.
% 68.37/9.26 Axiom 11 (hilbert_or_2): or_2 = true.
% 68.37/9.26 Axiom 12 (hilbert_or_3): or_3 = true.
% 68.37/9.26 Axiom 13 (hilbert_op_equiv): op_equiv = true.
% 68.37/9.26 Axiom 14 (hilbert_op_or): op_or = true.
% 68.37/9.26 Axiom 15 (principia_op_and): op_and = true.
% 68.37/9.26 Axiom 16 (hilbert_op_implies_and): op_implies_and = true.
% 68.37/9.26 Axiom 17 (principia_op_implies_or): op_implies_or = true.
% 68.37/9.26 Axiom 18 (r4): fresh7(X, X) = true.
% 68.37/9.26 Axiom 19 (modus_ponens_2): fresh60(X, X, Y) = true.
% 68.37/9.26 Axiom 20 (modus_ponens_2): fresh28(X, X, Y) = is_a_theorem(Y).
% 68.37/9.26 Axiom 21 (substitution_of_equivalents_2): fresh(X, X, Y, Z) = Z.
% 68.37/9.26 Axiom 22 (modus_ponens_2): fresh59(X, X, Y, Z) = fresh60(modus_ponens, true, Z).
% 68.37/9.26 Axiom 23 (and_1_1): fresh58(X, X, Y, Z) = true.
% 68.37/9.26 Axiom 24 (and_2_1): fresh55(X, X, Y, Z) = true.
% 68.37/9.26 Axiom 25 (and_3_1): fresh53(X, X, Y, Z) = true.
% 68.37/9.26 Axiom 26 (implies_1_1): fresh39(X, X, Y, Z) = true.
% 68.37/9.26 Axiom 27 (implies_2_1): fresh37(X, X, Y, Z) = true.
% 68.37/9.26 Axiom 28 (modus_tollens_1): fresh25(X, X, Y, Z) = true.
% 68.37/9.26 Axiom 29 (op_and): fresh24(X, X, Y, Z) = and(Y, Z).
% 68.37/9.26 Axiom 30 (op_equiv): fresh23(X, X, Y, Z) = equiv(Y, Z).
% 68.37/9.26 Axiom 31 (op_implies_and): fresh22(X, X, Y, Z) = implies(Y, Z).
% 68.37/9.26 Axiom 32 (op_implies_and): fresh22(op_implies_and, true, X, Y) = not(and(X, not(Y))).
% 68.37/9.26 Axiom 33 (op_implies_or): fresh21(X, X, Y, Z) = implies(Y, Z).
% 68.37/9.26 Axiom 34 (op_implies_or): fresh21(op_implies_or, true, X, Y) = or(not(X), Y).
% 68.37/9.26 Axiom 35 (op_or): fresh20(X, X, Y, Z) = or(Y, Z).
% 68.37/9.26 Axiom 36 (or_1_1): fresh18(X, X, Y, Z) = true.
% 68.37/9.26 Axiom 37 (or_2_1): fresh16(X, X, Y, Z) = true.
% 68.37/9.26 Axiom 38 (substitution_of_equivalents_2): fresh2(X, X, Y, Z) = Y.
% 68.37/9.26 Axiom 39 (implies_1_1): fresh39(implies_1, true, X, Y) = is_a_theorem(implies(X, implies(Y, X))).
% 68.37/9.26 Axiom 40 (or_1_1): fresh18(or_1, true, X, Y) = is_a_theorem(implies(X, or(X, Y))).
% 68.37/9.26 Axiom 41 (or_2_1): fresh16(or_2, true, X, Y) = is_a_theorem(implies(Y, or(X, Y))).
% 68.37/9.26 Axiom 42 (and_1_1): fresh58(and_1, true, X, Y) = is_a_theorem(implies(and(X, Y), X)).
% 68.37/9.26 Axiom 43 (and_2_1): fresh55(and_2, true, X, Y) = is_a_theorem(implies(and(X, Y), Y)).
% 68.37/9.26 Axiom 44 (op_and): fresh24(op_and, true, X, Y) = not(or(not(X), not(Y))).
% 68.37/9.26 Axiom 45 (op_or): fresh20(op_or, true, X, Y) = not(and(not(X), not(Y))).
% 68.37/9.26 Axiom 46 (implies_3_1): fresh35(X, X, Y, Z, W) = true.
% 68.37/9.26 Axiom 47 (or_3_1): fresh14(X, X, Y, Z, W) = true.
% 68.37/9.26 Axiom 48 (op_equiv): fresh23(op_equiv, true, X, Y) = and(implies(X, Y), implies(Y, X)).
% 68.37/9.26 Axiom 49 (and_3_1): fresh53(and_3, true, X, Y) = is_a_theorem(implies(X, implies(Y, and(X, Y)))).
% 68.37/9.26 Axiom 50 (r3_1): fresh8(r3, true, X, Y) = is_a_theorem(implies(or(X, Y), or(Y, X))).
% 68.37/9.26 Axiom 51 (substitution_of_equivalents_2): fresh2(substitution_of_equivalents, true, X, Y) = fresh(is_a_theorem(equiv(X, Y)), true, X, Y).
% 68.37/9.26 Axiom 52 (modus_ponens_2): fresh59(is_a_theorem(implies(X, Y)), true, X, Y) = fresh28(is_a_theorem(X), true, Y).
% 68.37/9.26 Axiom 53 (implies_2_1): fresh37(implies_2, true, X, Y) = is_a_theorem(implies(implies(X, implies(X, Y)), implies(X, Y))).
% 68.37/9.26 Axiom 54 (modus_tollens_1): fresh25(modus_tollens, true, X, Y) = is_a_theorem(implies(implies(not(Y), not(X)), implies(X, Y))).
% 68.37/9.26 Axiom 55 (implies_3_1): fresh35(implies_3, true, X, Y, Z) = is_a_theorem(implies(implies(X, Y), implies(implies(Y, Z), implies(X, Z)))).
% 68.37/9.26 Axiom 56 (r5_1): fresh4(r5, true, X, Y, Z) = is_a_theorem(implies(implies(Y, Z), implies(or(X, Y), or(X, Z)))).
% 68.37/9.26 Axiom 57 (or_3_1): fresh14(or_3, true, X, Y, Z) = is_a_theorem(implies(implies(X, Z), implies(implies(Y, Z), implies(or(X, Y), Z)))).
% 68.37/9.26 Axiom 58 (r4): fresh7(is_a_theorem(implies(or(p2, or(q2, r6)), or(q2, or(p2, r6)))), true) = r4.
% 68.37/9.26
% 68.37/9.26 Lemma 59: fresh59(X, X, Y, Z) = true.
% 68.37/9.26 Proof:
% 68.37/9.26 fresh59(X, X, Y, Z)
% 68.37/9.26 = { by axiom 22 (modus_ponens_2) }
% 68.37/9.26 fresh60(modus_ponens, true, Z)
% 68.37/9.26 = { by axiom 1 (hilbert_modus_ponens) }
% 68.37/9.26 fresh60(true, true, Z)
% 68.37/9.26 = { by axiom 19 (modus_ponens_2) }
% 68.37/9.26 true
% 68.37/9.26
% 68.37/9.26 Lemma 60: is_a_theorem(implies(X, implies(Y, and(X, Y)))) = true.
% 68.37/9.26 Proof:
% 68.37/9.26 is_a_theorem(implies(X, implies(Y, and(X, Y))))
% 68.37/9.26 = { by axiom 49 (and_3_1) R->L }
% 68.37/9.26 fresh53(and_3, true, X, Y)
% 68.37/9.26 = { by axiom 9 (hilbert_and_3) }
% 68.37/9.26 fresh53(true, true, X, Y)
% 68.37/9.26 = { by axiom 25 (and_3_1) }
% 68.37/9.26 true
% 68.37/9.26
% 68.37/9.26 Lemma 61: fresh28(is_a_theorem(X), true, implies(Y, and(X, Y))) = true.
% 68.37/9.26 Proof:
% 68.37/9.26 fresh28(is_a_theorem(X), true, implies(Y, and(X, Y)))
% 68.37/9.26 = { by axiom 52 (modus_ponens_2) R->L }
% 68.37/9.26 fresh59(is_a_theorem(implies(X, implies(Y, and(X, Y)))), true, X, implies(Y, and(X, Y)))
% 68.37/9.26 = { by lemma 60 }
% 68.37/9.26 fresh59(true, true, X, implies(Y, and(X, Y)))
% 68.37/9.26 = { by lemma 59 }
% 68.37/9.26 true
% 68.37/9.26
% 68.37/9.26 Lemma 62: is_a_theorem(implies(implies(X, implies(X, Y)), implies(X, Y))) = true.
% 68.37/9.26 Proof:
% 68.37/9.26 is_a_theorem(implies(implies(X, implies(X, Y)), implies(X, Y)))
% 68.37/9.26 = { by axiom 53 (implies_2_1) R->L }
% 68.37/9.26 fresh37(implies_2, true, X, Y)
% 68.37/9.26 = { by axiom 5 (hilbert_implies_2) }
% 68.37/9.26 fresh37(true, true, X, Y)
% 68.37/9.26 = { by axiom 27 (implies_2_1) }
% 68.37/9.26 true
% 68.37/9.26
% 68.37/9.26 Lemma 63: fresh28(is_a_theorem(implies(X, implies(X, Y))), true, implies(X, Y)) = true.
% 68.37/9.26 Proof:
% 68.37/9.26 fresh28(is_a_theorem(implies(X, implies(X, Y))), true, implies(X, Y))
% 68.37/9.26 = { by axiom 52 (modus_ponens_2) R->L }
% 68.37/9.26 fresh59(is_a_theorem(implies(implies(X, implies(X, Y)), implies(X, Y))), true, implies(X, implies(X, Y)), implies(X, Y))
% 68.37/9.26 = { by lemma 62 }
% 68.37/9.26 fresh59(true, true, implies(X, implies(X, Y)), implies(X, Y))
% 68.37/9.26 = { by lemma 59 }
% 68.37/9.26 true
% 68.37/9.26
% 68.37/9.26 Lemma 64: and(implies(X, Y), implies(Y, X)) = equiv(X, Y).
% 68.37/9.26 Proof:
% 68.37/9.26 and(implies(X, Y), implies(Y, X))
% 68.37/9.26 = { by axiom 48 (op_equiv) R->L }
% 68.37/9.26 fresh23(op_equiv, true, X, Y)
% 68.37/9.26 = { by axiom 13 (hilbert_op_equiv) }
% 68.37/9.26 fresh23(true, true, X, Y)
% 68.37/9.26 = { by axiom 30 (op_equiv) }
% 68.37/9.26 equiv(X, Y)
% 68.37/9.26
% 68.37/9.26 Lemma 65: fresh(is_a_theorem(equiv(X, Y)), true, X, Y) = X.
% 68.37/9.26 Proof:
% 68.37/9.26 fresh(is_a_theorem(equiv(X, Y)), true, X, Y)
% 68.37/9.26 = { by axiom 51 (substitution_of_equivalents_2) R->L }
% 68.37/9.26 fresh2(substitution_of_equivalents, true, X, Y)
% 68.37/9.26 = { by axiom 2 (substitution_of_equivalents) }
% 68.37/9.26 fresh2(true, true, X, Y)
% 68.37/9.26 = { by axiom 38 (substitution_of_equivalents_2) }
% 68.37/9.26 X
% 68.37/9.26
% 68.37/9.26 Lemma 66: and(X, X) = X.
% 68.37/9.26 Proof:
% 68.37/9.26 and(X, X)
% 68.37/9.26 = { by axiom 21 (substitution_of_equivalents_2) R->L }
% 68.37/9.26 fresh(true, true, X, and(X, X))
% 68.37/9.26 = { by lemma 59 R->L }
% 68.37/9.26 fresh(fresh59(true, true, implies(and(X, X), X), equiv(X, and(X, X))), true, X, and(X, X))
% 68.37/9.26 = { by lemma 61 R->L }
% 68.37/9.26 fresh(fresh59(fresh28(is_a_theorem(implies(X, and(X, X))), true, implies(implies(and(X, X), X), and(implies(X, and(X, X)), implies(and(X, X), X)))), true, implies(and(X, X), X), equiv(X, and(X, X))), true, X, and(X, X))
% 68.37/9.26 = { by axiom 20 (modus_ponens_2) R->L }
% 68.37/9.26 fresh(fresh59(fresh28(fresh28(true, true, implies(X, and(X, X))), true, implies(implies(and(X, X), X), and(implies(X, and(X, X)), implies(and(X, X), X)))), true, implies(and(X, X), X), equiv(X, and(X, X))), true, X, and(X, X))
% 68.37/9.26 = { by lemma 60 R->L }
% 68.37/9.26 fresh(fresh59(fresh28(fresh28(is_a_theorem(implies(X, implies(X, and(X, X)))), true, implies(X, and(X, X))), true, implies(implies(and(X, X), X), and(implies(X, and(X, X)), implies(and(X, X), X)))), true, implies(and(X, X), X), equiv(X, and(X, X))), true, X, and(X, X))
% 68.37/9.26 = { by lemma 63 }
% 68.37/9.26 fresh(fresh59(fresh28(true, true, implies(implies(and(X, X), X), and(implies(X, and(X, X)), implies(and(X, X), X)))), true, implies(and(X, X), X), equiv(X, and(X, X))), true, X, and(X, X))
% 68.37/9.26 = { by axiom 20 (modus_ponens_2) }
% 68.37/9.26 fresh(fresh59(is_a_theorem(implies(implies(and(X, X), X), and(implies(X, and(X, X)), implies(and(X, X), X)))), true, implies(and(X, X), X), equiv(X, and(X, X))), true, X, and(X, X))
% 68.37/9.26 = { by lemma 64 }
% 68.37/9.26 fresh(fresh59(is_a_theorem(implies(implies(and(X, X), X), equiv(X, and(X, X)))), true, implies(and(X, X), X), equiv(X, and(X, X))), true, X, and(X, X))
% 68.37/9.26 = { by axiom 52 (modus_ponens_2) }
% 68.37/9.26 fresh(fresh28(is_a_theorem(implies(and(X, X), X)), true, equiv(X, and(X, X))), true, X, and(X, X))
% 68.37/9.26 = { by axiom 42 (and_1_1) R->L }
% 68.37/9.26 fresh(fresh28(fresh58(and_1, true, X, X), true, equiv(X, and(X, X))), true, X, and(X, X))
% 68.37/9.26 = { by axiom 7 (hilbert_and_1) }
% 68.37/9.26 fresh(fresh28(fresh58(true, true, X, X), true, equiv(X, and(X, X))), true, X, and(X, X))
% 68.37/9.26 = { by axiom 23 (and_1_1) }
% 68.37/9.26 fresh(fresh28(true, true, equiv(X, and(X, X))), true, X, and(X, X))
% 68.37/9.26 = { by axiom 20 (modus_ponens_2) }
% 68.37/9.26 fresh(is_a_theorem(equiv(X, and(X, X))), true, X, and(X, X))
% 68.37/9.26 = { by lemma 65 }
% 68.37/9.26 X
% 68.37/9.26
% 68.37/9.26 Lemma 67: is_a_theorem(implies(X, implies(Y, X))) = true.
% 68.37/9.26 Proof:
% 68.37/9.26 is_a_theorem(implies(X, implies(Y, X)))
% 68.37/9.26 = { by axiom 39 (implies_1_1) R->L }
% 68.37/9.26 fresh39(implies_1, true, X, Y)
% 68.37/9.26 = { by axiom 4 (hilbert_implies_1) }
% 68.37/9.26 fresh39(true, true, X, Y)
% 68.37/9.26 = { by axiom 26 (implies_1_1) }
% 68.37/9.26 true
% 68.37/9.26
% 68.37/9.26 Lemma 68: is_a_theorem(implies(X, X)) = true.
% 68.37/9.26 Proof:
% 68.37/9.26 is_a_theorem(implies(X, X))
% 68.37/9.26 = { by axiom 20 (modus_ponens_2) R->L }
% 68.37/9.26 fresh28(true, true, implies(X, X))
% 68.37/9.26 = { by lemma 67 R->L }
% 68.37/9.26 fresh28(is_a_theorem(implies(X, implies(X, X))), true, implies(X, X))
% 68.37/9.26 = { by lemma 63 }
% 68.37/9.26 true
% 68.37/9.26
% 68.37/9.26 Lemma 69: fresh28(is_a_theorem(implies(implies(X, Y), Y)), true, equiv(Y, implies(X, Y))) = true.
% 68.37/9.26 Proof:
% 68.37/9.27 fresh28(is_a_theorem(implies(implies(X, Y), Y)), true, equiv(Y, implies(X, Y)))
% 68.37/9.27 = { by axiom 52 (modus_ponens_2) R->L }
% 68.37/9.27 fresh59(is_a_theorem(implies(implies(implies(X, Y), Y), equiv(Y, implies(X, Y)))), true, implies(implies(X, Y), Y), equiv(Y, implies(X, Y)))
% 68.37/9.27 = { by lemma 64 R->L }
% 68.37/9.27 fresh59(is_a_theorem(implies(implies(implies(X, Y), Y), and(implies(Y, implies(X, Y)), implies(implies(X, Y), Y)))), true, implies(implies(X, Y), Y), equiv(Y, implies(X, Y)))
% 68.37/9.27 = { by axiom 20 (modus_ponens_2) R->L }
% 68.37/9.27 fresh59(fresh28(true, true, implies(implies(implies(X, Y), Y), and(implies(Y, implies(X, Y)), implies(implies(X, Y), Y)))), true, implies(implies(X, Y), Y), equiv(Y, implies(X, Y)))
% 68.37/9.27 = { by lemma 67 R->L }
% 68.37/9.27 fresh59(fresh28(is_a_theorem(implies(Y, implies(X, Y))), true, implies(implies(implies(X, Y), Y), and(implies(Y, implies(X, Y)), implies(implies(X, Y), Y)))), true, implies(implies(X, Y), Y), equiv(Y, implies(X, Y)))
% 68.37/9.27 = { by lemma 61 }
% 68.37/9.27 fresh59(true, true, implies(implies(X, Y), Y), equiv(Y, implies(X, Y)))
% 68.37/9.27 = { by lemma 59 }
% 68.37/9.27 true
% 68.37/9.27
% 68.37/9.27 Lemma 70: implies(X, implies(Y, Y)) = implies(Y, Y).
% 68.37/9.27 Proof:
% 68.37/9.27 implies(X, implies(Y, Y))
% 68.37/9.27 = { by axiom 21 (substitution_of_equivalents_2) R->L }
% 68.37/9.27 fresh(true, true, implies(Y, Y), implies(X, implies(Y, Y)))
% 68.37/9.27 = { by lemma 69 R->L }
% 68.37/9.27 fresh(fresh28(is_a_theorem(implies(implies(X, implies(Y, Y)), implies(Y, Y))), true, equiv(implies(Y, Y), implies(X, implies(Y, Y)))), true, implies(Y, Y), implies(X, implies(Y, Y)))
% 68.37/9.27 = { by axiom 20 (modus_ponens_2) R->L }
% 68.37/9.27 fresh(fresh28(fresh28(true, true, implies(implies(X, implies(Y, Y)), implies(Y, Y))), true, equiv(implies(Y, Y), implies(X, implies(Y, Y)))), true, implies(Y, Y), implies(X, implies(Y, Y)))
% 68.37/9.27 = { by lemma 68 R->L }
% 68.37/9.27 fresh(fresh28(fresh28(is_a_theorem(implies(Y, Y)), true, implies(implies(X, implies(Y, Y)), implies(Y, Y))), true, equiv(implies(Y, Y), implies(X, implies(Y, Y)))), true, implies(Y, Y), implies(X, implies(Y, Y)))
% 68.37/9.27 = { by axiom 52 (modus_ponens_2) R->L }
% 68.37/9.27 fresh(fresh28(fresh59(is_a_theorem(implies(implies(Y, Y), implies(implies(X, implies(Y, Y)), implies(Y, Y)))), true, implies(Y, Y), implies(implies(X, implies(Y, Y)), implies(Y, Y))), true, equiv(implies(Y, Y), implies(X, implies(Y, Y)))), true, implies(Y, Y), implies(X, implies(Y, Y)))
% 68.37/9.27 = { by lemma 67 }
% 68.37/9.27 fresh(fresh28(fresh59(true, true, implies(Y, Y), implies(implies(X, implies(Y, Y)), implies(Y, Y))), true, equiv(implies(Y, Y), implies(X, implies(Y, Y)))), true, implies(Y, Y), implies(X, implies(Y, Y)))
% 68.37/9.27 = { by lemma 59 }
% 68.37/9.27 fresh(fresh28(true, true, equiv(implies(Y, Y), implies(X, implies(Y, Y)))), true, implies(Y, Y), implies(X, implies(Y, Y)))
% 68.37/9.27 = { by axiom 20 (modus_ponens_2) }
% 68.37/9.27 fresh(is_a_theorem(equiv(implies(Y, Y), implies(X, implies(Y, Y)))), true, implies(Y, Y), implies(X, implies(Y, Y)))
% 68.37/9.27 = { by lemma 65 }
% 68.37/9.27 implies(Y, Y)
% 68.37/9.27
% 68.37/9.27 Lemma 71: and(implies(X, X), implies(implies(X, X), Y)) = equiv(Y, implies(X, X)).
% 68.37/9.27 Proof:
% 68.37/9.27 and(implies(X, X), implies(implies(X, X), Y))
% 68.37/9.27 = { by lemma 70 R->L }
% 68.37/9.27 and(implies(Y, implies(X, X)), implies(implies(X, X), Y))
% 68.37/9.27 = { by lemma 64 }
% 68.37/9.27 equiv(Y, implies(X, X))
% 68.37/9.27
% 68.37/9.27 Lemma 72: or(not(X), Y) = implies(X, Y).
% 68.37/9.27 Proof:
% 68.37/9.27 or(not(X), Y)
% 68.37/9.27 = { by axiom 34 (op_implies_or) R->L }
% 68.37/9.27 fresh21(op_implies_or, true, X, Y)
% 68.37/9.27 = { by axiom 17 (principia_op_implies_or) }
% 68.37/9.27 fresh21(true, true, X, Y)
% 68.37/9.27 = { by axiom 33 (op_implies_or) }
% 68.37/9.27 implies(X, Y)
% 68.37/9.27
% 68.37/9.27 Lemma 73: not(implies(X, not(Y))) = and(X, Y).
% 68.37/9.27 Proof:
% 68.37/9.27 not(implies(X, not(Y)))
% 68.37/9.27 = { by lemma 72 R->L }
% 68.37/9.27 not(or(not(X), not(Y)))
% 68.37/9.27 = { by axiom 44 (op_and) R->L }
% 68.37/9.27 fresh24(op_and, true, X, Y)
% 68.37/9.27 = { by axiom 15 (principia_op_and) }
% 68.37/9.27 fresh24(true, true, X, Y)
% 68.37/9.27 = { by axiom 29 (op_and) }
% 68.37/9.27 and(X, Y)
% 68.37/9.27
% 68.37/9.27 Lemma 74: not(and(X, not(Y))) = implies(X, Y).
% 68.37/9.27 Proof:
% 68.37/9.27 not(and(X, not(Y)))
% 68.37/9.27 = { by axiom 32 (op_implies_and) R->L }
% 68.37/9.27 fresh22(op_implies_and, true, X, Y)
% 68.37/9.27 = { by axiom 16 (hilbert_op_implies_and) }
% 68.37/9.27 fresh22(true, true, X, Y)
% 68.37/9.27 = { by axiom 31 (op_implies_and) }
% 68.37/9.27 implies(X, Y)
% 68.37/9.27
% 68.37/9.27 Lemma 75: implies(not(X), Y) = or(X, Y).
% 68.37/9.27 Proof:
% 68.37/9.27 implies(not(X), Y)
% 68.37/9.27 = { by lemma 74 R->L }
% 68.37/9.27 not(and(not(X), not(Y)))
% 68.37/9.27 = { by axiom 45 (op_or) R->L }
% 68.37/9.27 fresh20(op_or, true, X, Y)
% 68.37/9.27 = { by axiom 14 (hilbert_op_or) }
% 68.37/9.27 fresh20(true, true, X, Y)
% 68.37/9.27 = { by axiom 35 (op_or) }
% 68.37/9.27 or(X, Y)
% 68.37/9.27
% 68.37/9.27 Lemma 76: or(and(X, not(Y)), Z) = implies(implies(X, Y), Z).
% 68.37/9.27 Proof:
% 68.37/9.27 or(and(X, not(Y)), Z)
% 68.37/9.27 = { by lemma 75 R->L }
% 68.37/9.27 implies(not(and(X, not(Y))), Z)
% 68.37/9.27 = { by lemma 74 }
% 68.37/9.27 implies(implies(X, Y), Z)
% 68.37/9.27
% 68.37/9.27 Lemma 77: not(or(X, not(Y))) = and(not(X), Y).
% 68.37/9.27 Proof:
% 68.37/9.27 not(or(X, not(Y)))
% 68.37/9.27 = { by lemma 75 R->L }
% 68.37/9.27 not(implies(not(X), not(Y)))
% 68.37/9.27 = { by lemma 73 }
% 68.37/9.27 and(not(X), Y)
% 68.37/9.27
% 68.37/9.27 Lemma 78: or(and(not(X), Y), Z) = implies(or(X, not(Y)), Z).
% 68.37/9.27 Proof:
% 68.37/9.27 or(and(not(X), Y), Z)
% 68.37/9.27 = { by lemma 77 R->L }
% 68.37/9.27 or(not(or(X, not(Y))), Z)
% 68.37/9.27 = { by lemma 72 }
% 68.37/9.27 implies(or(X, not(Y)), Z)
% 68.37/9.27
% 68.37/9.27 Lemma 79: implies(or(X, not(not(Y))), Z) = implies(or(X, Y), Z).
% 68.37/9.27 Proof:
% 68.37/9.27 implies(or(X, not(not(Y))), Z)
% 68.37/9.27 = { by lemma 78 R->L }
% 68.37/9.27 or(and(not(X), not(Y)), Z)
% 68.37/9.27 = { by lemma 76 }
% 68.37/9.27 implies(implies(not(X), Y), Z)
% 68.37/9.27 = { by lemma 75 }
% 68.37/9.27 implies(or(X, Y), Z)
% 68.37/9.27
% 68.37/9.27 Lemma 80: is_a_theorem(implies(or(X, not(Y)), implies(Y, X))) = true.
% 68.37/9.27 Proof:
% 68.37/9.27 is_a_theorem(implies(or(X, not(Y)), implies(Y, X)))
% 68.37/9.27 = { by lemma 75 R->L }
% 68.37/9.27 is_a_theorem(implies(implies(not(X), not(Y)), implies(Y, X)))
% 68.37/9.27 = { by axiom 54 (modus_tollens_1) R->L }
% 68.37/9.27 fresh25(modus_tollens, true, Y, X)
% 68.37/9.27 = { by axiom 3 (hilbert_modus_tollens) }
% 68.37/9.27 fresh25(true, true, Y, X)
% 68.37/9.27 = { by axiom 28 (modus_tollens_1) }
% 68.37/9.27 true
% 68.37/9.27
% 68.37/9.27 Lemma 81: fresh8(r3, true, X, Y) = true.
% 68.37/9.27 Proof:
% 68.37/9.27 fresh8(r3, true, X, Y)
% 68.37/9.27 = { by axiom 50 (r3_1) }
% 68.37/9.27 is_a_theorem(implies(or(X, Y), or(Y, X)))
% 68.37/9.27 = { by lemma 75 R->L }
% 68.37/9.27 is_a_theorem(implies(or(X, Y), implies(not(Y), X)))
% 68.37/9.27 = { by lemma 79 R->L }
% 68.37/9.27 is_a_theorem(implies(or(X, not(not(Y))), implies(not(Y), X)))
% 68.37/9.27 = { by lemma 80 }
% 68.37/9.27 true
% 68.37/9.27
% 68.37/9.27 Lemma 82: is_a_theorem(implies(implies(implies(X, X), Y), Y)) = true.
% 68.37/9.27 Proof:
% 68.37/9.27 is_a_theorem(implies(implies(implies(X, X), Y), Y))
% 68.37/9.27 = { by lemma 76 R->L }
% 68.37/9.27 is_a_theorem(or(and(implies(X, X), not(Y)), Y))
% 68.37/9.27 = { by axiom 20 (modus_ponens_2) R->L }
% 68.37/9.27 fresh28(true, true, or(and(implies(X, X), not(Y)), Y))
% 68.37/9.27 = { by lemma 61 R->L }
% 68.37/9.27 fresh28(fresh28(is_a_theorem(implies(X, X)), true, implies(not(Y), and(implies(X, X), not(Y)))), true, or(and(implies(X, X), not(Y)), Y))
% 68.37/9.27 = { by lemma 68 }
% 68.37/9.27 fresh28(fresh28(true, true, implies(not(Y), and(implies(X, X), not(Y)))), true, or(and(implies(X, X), not(Y)), Y))
% 68.37/9.27 = { by axiom 20 (modus_ponens_2) }
% 68.37/9.27 fresh28(is_a_theorem(implies(not(Y), and(implies(X, X), not(Y)))), true, or(and(implies(X, X), not(Y)), Y))
% 68.37/9.27 = { by lemma 75 }
% 68.37/9.27 fresh28(is_a_theorem(or(Y, and(implies(X, X), not(Y)))), true, or(and(implies(X, X), not(Y)), Y))
% 68.37/9.27 = { by axiom 52 (modus_ponens_2) R->L }
% 68.37/9.27 fresh59(is_a_theorem(implies(or(Y, and(implies(X, X), not(Y))), or(and(implies(X, X), not(Y)), Y))), true, or(Y, and(implies(X, X), not(Y))), or(and(implies(X, X), not(Y)), Y))
% 68.37/9.27 = { by axiom 50 (r3_1) R->L }
% 68.37/9.27 fresh59(fresh8(r3, true, Y, and(implies(X, X), not(Y))), true, or(Y, and(implies(X, X), not(Y))), or(and(implies(X, X), not(Y)), Y))
% 68.37/9.27 = { by lemma 81 }
% 68.37/9.27 fresh59(true, true, or(Y, and(implies(X, X), not(Y))), or(and(implies(X, X), not(Y)), Y))
% 68.37/9.27 = { by lemma 59 }
% 68.37/9.27 true
% 68.37/9.27
% 68.37/9.27 Lemma 83: implies(implies(X, X), Y) = Y.
% 68.37/9.27 Proof:
% 68.37/9.27 implies(implies(X, X), Y)
% 68.37/9.27 = { by axiom 21 (substitution_of_equivalents_2) R->L }
% 68.37/9.27 fresh(true, true, Y, implies(implies(X, X), Y))
% 68.37/9.27 = { by lemma 69 R->L }
% 68.37/9.27 fresh(fresh28(is_a_theorem(implies(implies(implies(X, X), Y), Y)), true, equiv(Y, implies(implies(X, X), Y))), true, Y, implies(implies(X, X), Y))
% 68.37/9.27 = { by lemma 82 }
% 68.37/9.27 fresh(fresh28(true, true, equiv(Y, implies(implies(X, X), Y))), true, Y, implies(implies(X, X), Y))
% 68.37/9.27 = { by axiom 20 (modus_ponens_2) }
% 68.37/9.27 fresh(is_a_theorem(equiv(Y, implies(implies(X, X), Y))), true, Y, implies(implies(X, X), Y))
% 68.37/9.27 = { by lemma 65 }
% 68.37/9.27 Y
% 68.37/9.27
% 68.37/9.27 Lemma 84: implies(implies(X, not(Y)), Z) = or(and(X, Y), Z).
% 68.37/9.27 Proof:
% 68.37/9.27 implies(implies(X, not(Y)), Z)
% 68.37/9.27 = { by lemma 72 R->L }
% 68.37/9.27 or(not(implies(X, not(Y))), Z)
% 68.37/9.27 = { by lemma 73 }
% 68.37/9.27 or(and(X, Y), Z)
% 68.37/9.27
% 68.37/9.27 Lemma 85: and(not(and(X, Y)), Z) = and(implies(X, not(Y)), Z).
% 68.37/9.27 Proof:
% 68.37/9.27 and(not(and(X, Y)), Z)
% 68.37/9.27 = { by lemma 77 R->L }
% 68.37/9.27 not(or(and(X, Y), not(Z)))
% 68.37/9.27 = { by lemma 84 R->L }
% 68.37/9.27 not(implies(implies(X, not(Y)), not(Z)))
% 68.37/9.27 = { by lemma 73 }
% 68.37/9.27 and(implies(X, not(Y)), Z)
% 68.37/9.27
% 68.37/9.27 Lemma 86: and(not(not(X)), Y) = and(X, Y).
% 68.37/9.27 Proof:
% 68.37/9.27 and(not(not(X)), Y)
% 68.37/9.27 = { by lemma 77 R->L }
% 68.37/9.27 not(or(not(X), not(Y)))
% 68.37/9.27 = { by lemma 72 }
% 68.37/9.27 not(implies(X, not(Y)))
% 68.37/9.27 = { by lemma 73 }
% 68.37/9.27 and(X, Y)
% 68.37/9.27
% 68.37/9.27 Lemma 87: implies(not(X), X) = not(not(X)).
% 68.37/9.27 Proof:
% 68.37/9.27 implies(not(X), X)
% 68.37/9.27 = { by lemma 74 R->L }
% 68.37/9.27 not(and(not(X), not(X)))
% 68.37/9.27 = { by lemma 66 }
% 68.37/9.27 not(not(X))
% 68.37/9.27
% 68.37/9.27 Lemma 88: not(not(X)) = or(X, X).
% 68.37/9.27 Proof:
% 68.37/9.27 not(not(X))
% 68.37/9.27 = { by lemma 87 R->L }
% 68.37/9.27 implies(not(X), X)
% 68.37/9.27 = { by lemma 75 }
% 68.37/9.27 or(X, X)
% 68.37/9.27
% 68.37/9.27 Lemma 89: or(X, X) = X.
% 68.37/9.27 Proof:
% 68.37/9.27 or(X, X)
% 68.37/9.27 = { by lemma 65 R->L }
% 68.37/9.27 fresh(is_a_theorem(equiv(or(X, X), X)), true, or(X, X), X)
% 68.37/9.27 = { by lemma 66 R->L }
% 68.37/9.27 fresh(is_a_theorem(equiv(or(X, X), and(X, X))), true, or(X, X), X)
% 68.37/9.27 = { by lemma 73 R->L }
% 68.37/9.27 fresh(is_a_theorem(equiv(or(X, X), not(implies(X, not(X))))), true, or(X, X), X)
% 68.37/9.27 = { by lemma 74 R->L }
% 68.37/9.27 fresh(is_a_theorem(equiv(or(X, X), not(not(and(X, not(not(X))))))), true, or(X, X), X)
% 68.37/9.27 = { by lemma 64 R->L }
% 68.37/9.27 fresh(is_a_theorem(and(implies(or(X, X), not(not(and(X, not(not(X)))))), implies(not(not(and(X, not(not(X))))), or(X, X)))), true, or(X, X), X)
% 68.37/9.27 = { by lemma 85 R->L }
% 68.37/9.27 fresh(is_a_theorem(and(not(and(or(X, X), not(and(X, not(not(X)))))), implies(not(not(and(X, not(not(X))))), or(X, X)))), true, or(X, X), X)
% 68.37/9.27 = { by lemma 74 }
% 68.37/9.27 fresh(is_a_theorem(and(implies(or(X, X), and(X, not(not(X)))), implies(not(not(and(X, not(not(X))))), or(X, X)))), true, or(X, X), X)
% 68.37/9.27 = { by lemma 75 }
% 68.37/9.27 fresh(is_a_theorem(and(implies(or(X, X), and(X, not(not(X)))), or(not(and(X, not(not(X)))), or(X, X)))), true, or(X, X), X)
% 68.37/9.27 = { by lemma 72 }
% 68.37/9.27 fresh(is_a_theorem(and(implies(or(X, X), and(X, not(not(X)))), implies(and(X, not(not(X))), or(X, X)))), true, or(X, X), X)
% 68.37/9.27 = { by lemma 64 }
% 68.37/9.27 fresh(is_a_theorem(equiv(or(X, X), and(X, not(not(X))))), true, or(X, X), X)
% 68.37/9.27 = { by lemma 86 R->L }
% 68.37/9.27 fresh(is_a_theorem(equiv(or(X, X), and(not(not(X)), not(not(X))))), true, or(X, X), X)
% 68.37/9.27 = { by lemma 66 }
% 68.37/9.27 fresh(is_a_theorem(equiv(or(X, X), not(not(X)))), true, or(X, X), X)
% 68.37/9.27 = { by lemma 88 }
% 68.37/9.27 fresh(is_a_theorem(equiv(or(X, X), or(X, X))), true, or(X, X), X)
% 68.37/9.27 = { by lemma 64 R->L }
% 68.37/9.27 fresh(is_a_theorem(and(implies(or(X, X), or(X, X)), implies(or(X, X), or(X, X)))), true, or(X, X), X)
% 68.37/9.27 = { by lemma 66 }
% 68.37/9.27 fresh(is_a_theorem(implies(or(X, X), or(X, X))), true, or(X, X), X)
% 68.37/9.27 = { by lemma 79 R->L }
% 68.37/9.27 fresh(is_a_theorem(implies(or(X, not(not(X))), or(X, X))), true, or(X, X), X)
% 68.37/9.27 = { by lemma 78 R->L }
% 68.37/9.27 fresh(is_a_theorem(or(and(not(X), not(X)), or(X, X))), true, or(X, X), X)
% 68.37/9.27 = { by lemma 66 }
% 68.37/9.27 fresh(is_a_theorem(or(not(X), or(X, X))), true, or(X, X), X)
% 68.37/9.27 = { by lemma 72 }
% 68.37/9.27 fresh(is_a_theorem(implies(X, or(X, X))), true, or(X, X), X)
% 68.37/9.27 = { by axiom 40 (or_1_1) R->L }
% 68.37/9.27 fresh(fresh18(or_1, true, X, X), true, or(X, X), X)
% 68.37/9.27 = { by axiom 10 (hilbert_or_1) }
% 68.37/9.27 fresh(fresh18(true, true, X, X), true, or(X, X), X)
% 68.37/9.27 = { by axiom 36 (or_1_1) }
% 68.37/9.27 fresh(true, true, or(X, X), X)
% 68.37/9.27 = { by axiom 21 (substitution_of_equivalents_2) }
% 68.37/9.27 X
% 68.37/9.27
% 68.37/9.27 Lemma 90: not(or(X, X)) = implies(X, not(X)).
% 68.37/9.27 Proof:
% 68.37/9.27 not(or(X, X))
% 68.37/9.27 = { by lemma 88 R->L }
% 68.37/9.27 not(not(not(X)))
% 68.37/9.27 = { by lemma 88 }
% 68.37/9.27 or(not(X), not(X))
% 68.37/9.27 = { by lemma 72 }
% 68.37/9.27 implies(X, not(X))
% 68.37/9.27
% 68.37/9.27 Lemma 91: not(not(X)) = X.
% 68.37/9.27 Proof:
% 68.37/9.27 not(not(X))
% 68.37/9.27 = { by lemma 89 R->L }
% 68.37/9.27 not(or(not(X), not(X)))
% 68.37/9.27 = { by lemma 77 }
% 68.37/9.27 and(not(not(X)), X)
% 68.37/9.27 = { by lemma 73 R->L }
% 68.37/9.27 not(implies(not(not(X)), not(X)))
% 68.37/9.27 = { by lemma 87 }
% 68.37/9.27 not(not(not(not(X))))
% 68.37/9.27 = { by lemma 88 }
% 68.37/9.27 not(not(or(X, X)))
% 68.37/9.27 = { by lemma 90 }
% 68.37/9.27 not(implies(X, not(X)))
% 68.37/9.27 = { by lemma 73 }
% 68.37/9.27 and(X, X)
% 68.37/9.27 = { by lemma 66 }
% 68.37/9.27 X
% 68.37/9.27
% 68.37/9.27 Lemma 92: and(implies(X, X), Y) = Y.
% 68.37/9.27 Proof:
% 68.37/9.27 and(implies(X, X), Y)
% 68.37/9.27 = { by lemma 73 R->L }
% 68.37/9.27 not(implies(implies(X, X), not(Y)))
% 68.37/9.27 = { by lemma 83 }
% 68.37/9.27 not(not(Y))
% 68.37/9.27 = { by lemma 91 }
% 68.37/9.27 Y
% 68.37/9.27
% 68.37/9.28 Lemma 93: equiv(X, implies(Y, Y)) = X.
% 68.37/9.28 Proof:
% 68.37/9.28 equiv(X, implies(Y, Y))
% 68.37/9.28 = { by lemma 71 R->L }
% 68.37/9.28 and(implies(Y, Y), implies(implies(Y, Y), X))
% 68.37/9.28 = { by lemma 92 }
% 68.37/9.28 implies(implies(Y, Y), X)
% 68.37/9.28 = { by lemma 83 }
% 68.37/9.28 X
% 68.37/9.28
% 68.37/9.28 Lemma 94: fresh(is_a_theorem(X), true, X, implies(Y, Y)) = X.
% 68.37/9.28 Proof:
% 68.37/9.28 fresh(is_a_theorem(X), true, X, implies(Y, Y))
% 68.37/9.28 = { by lemma 93 R->L }
% 68.37/9.28 fresh(is_a_theorem(equiv(X, implies(Y, Y))), true, X, implies(Y, Y))
% 68.37/9.28 = { by lemma 65 }
% 68.37/9.28 X
% 68.37/9.28
% 68.37/9.28 Lemma 95: or(implies(X, not(Y)), Z) = implies(and(X, Y), Z).
% 68.37/9.28 Proof:
% 68.37/9.28 or(implies(X, not(Y)), Z)
% 68.37/9.28 = { by lemma 75 R->L }
% 68.37/9.28 implies(not(implies(X, not(Y))), Z)
% 68.37/9.28 = { by lemma 73 }
% 68.37/9.28 implies(and(X, Y), Z)
% 68.37/9.28
% 68.37/9.28 Lemma 96: implies(and(X, Y), and(Y, X)) = implies(Z, Z).
% 68.37/9.28 Proof:
% 68.37/9.28 implies(and(X, Y), and(Y, X))
% 68.37/9.28 = { by lemma 94 R->L }
% 68.37/9.28 fresh(is_a_theorem(implies(and(X, Y), and(Y, X))), true, implies(and(X, Y), and(Y, X)), implies(Z, Z))
% 68.37/9.28 = { by lemma 95 R->L }
% 68.37/9.28 fresh(is_a_theorem(or(implies(X, not(Y)), and(Y, X))), true, implies(and(X, Y), and(Y, X)), implies(Z, Z))
% 68.37/9.28 = { by lemma 86 R->L }
% 68.37/9.28 fresh(is_a_theorem(or(implies(X, not(Y)), and(not(not(Y)), X))), true, implies(and(X, Y), and(Y, X)), implies(Z, Z))
% 68.37/9.28 = { by lemma 77 R->L }
% 68.37/9.28 fresh(is_a_theorem(or(implies(X, not(Y)), not(or(not(Y), not(X))))), true, implies(and(X, Y), and(Y, X)), implies(Z, Z))
% 68.37/9.28 = { by axiom 20 (modus_ponens_2) R->L }
% 68.37/9.28 fresh(fresh28(true, true, or(implies(X, not(Y)), not(or(not(Y), not(X))))), true, implies(and(X, Y), and(Y, X)), implies(Z, Z))
% 68.37/9.28 = { by lemma 80 R->L }
% 68.37/9.28 fresh(fresh28(is_a_theorem(implies(or(not(Y), not(X)), implies(X, not(Y)))), true, or(implies(X, not(Y)), not(or(not(Y), not(X))))), true, implies(and(X, Y), and(Y, X)), implies(Z, Z))
% 68.37/9.28 = { by axiom 52 (modus_ponens_2) R->L }
% 68.37/9.28 fresh(fresh59(is_a_theorem(implies(implies(or(not(Y), not(X)), implies(X, not(Y))), or(implies(X, not(Y)), not(or(not(Y), not(X)))))), true, implies(or(not(Y), not(X)), implies(X, not(Y))), or(implies(X, not(Y)), not(or(not(Y), not(X))))), true, implies(and(X, Y), and(Y, X)), implies(Z, Z))
% 68.37/9.28 = { by lemma 72 R->L }
% 68.37/9.28 fresh(fresh59(is_a_theorem(implies(or(not(or(not(Y), not(X))), implies(X, not(Y))), or(implies(X, not(Y)), not(or(not(Y), not(X)))))), true, implies(or(not(Y), not(X)), implies(X, not(Y))), or(implies(X, not(Y)), not(or(not(Y), not(X))))), true, implies(and(X, Y), and(Y, X)), implies(Z, Z))
% 68.37/9.28 = { by axiom 50 (r3_1) R->L }
% 68.37/9.28 fresh(fresh59(fresh8(r3, true, not(or(not(Y), not(X))), implies(X, not(Y))), true, implies(or(not(Y), not(X)), implies(X, not(Y))), or(implies(X, not(Y)), not(or(not(Y), not(X))))), true, implies(and(X, Y), and(Y, X)), implies(Z, Z))
% 68.37/9.28 = { by lemma 81 }
% 68.37/9.28 fresh(fresh59(true, true, implies(or(not(Y), not(X)), implies(X, not(Y))), or(implies(X, not(Y)), not(or(not(Y), not(X))))), true, implies(and(X, Y), and(Y, X)), implies(Z, Z))
% 68.37/9.28 = { by lemma 59 }
% 68.37/9.28 fresh(true, true, implies(and(X, Y), and(Y, X)), implies(Z, Z))
% 68.37/9.28 = { by axiom 21 (substitution_of_equivalents_2) }
% 68.37/9.28 implies(Z, Z)
% 68.37/9.28
% 68.37/9.28 Lemma 97: and(Y, X) = and(X, Y).
% 68.37/9.28 Proof:
% 68.37/9.28 and(Y, X)
% 68.37/9.28 = { by axiom 21 (substitution_of_equivalents_2) R->L }
% 68.37/9.28 fresh(true, true, and(X, Y), and(Y, X))
% 68.37/9.28 = { by lemma 68 R->L }
% 68.37/9.28 fresh(is_a_theorem(implies(W, W)), true, and(X, Y), and(Y, X))
% 68.37/9.28 = { by lemma 96 R->L }
% 68.37/9.28 fresh(is_a_theorem(implies(and(Y, X), and(X, Y))), true, and(X, Y), and(Y, X))
% 68.37/9.28 = { by lemma 92 R->L }
% 68.37/9.28 fresh(is_a_theorem(and(implies(Z, Z), implies(and(Y, X), and(X, Y)))), true, and(X, Y), and(Y, X))
% 68.37/9.28 = { by lemma 96 R->L }
% 68.37/9.28 fresh(is_a_theorem(and(implies(and(X, Y), and(Y, X)), implies(and(Y, X), and(X, Y)))), true, and(X, Y), and(Y, X))
% 68.37/9.28 = { by lemma 64 }
% 68.37/9.28 fresh(is_a_theorem(equiv(and(X, Y), and(Y, X))), true, and(X, Y), and(Y, X))
% 68.37/9.28 = { by lemma 65 }
% 68.37/9.28 and(X, Y)
% 68.37/9.28
% 68.37/9.28 Lemma 98: implies(equiv(X, Y), equiv(Y, X)) = implies(Z, Z).
% 68.37/9.28 Proof:
% 68.37/9.28 implies(equiv(X, Y), equiv(Y, X))
% 68.37/9.28 = { by lemma 94 R->L }
% 68.37/9.28 fresh(is_a_theorem(implies(equiv(X, Y), equiv(Y, X))), true, implies(equiv(X, Y), equiv(Y, X)), implies(Z, Z))
% 68.37/9.28 = { by axiom 20 (modus_ponens_2) R->L }
% 68.37/9.28 fresh(fresh28(true, true, implies(equiv(X, Y), equiv(Y, X))), true, implies(equiv(X, Y), equiv(Y, X)), implies(Z, Z))
% 68.37/9.28 = { by lemma 80 R->L }
% 68.37/9.28 fresh(fresh28(is_a_theorem(implies(or(and(Y, not(X)), not(implies(X, Y))), implies(implies(X, Y), and(Y, not(X))))), true, implies(equiv(X, Y), equiv(Y, X))), true, implies(equiv(X, Y), equiv(Y, X)), implies(Z, Z))
% 68.37/9.28 = { by lemma 74 R->L }
% 68.37/9.28 fresh(fresh28(is_a_theorem(implies(or(and(Y, not(X)), not(implies(X, Y))), not(and(implies(X, Y), not(and(Y, not(X))))))), true, implies(equiv(X, Y), equiv(Y, X))), true, implies(equiv(X, Y), equiv(Y, X)), implies(Z, Z))
% 68.37/9.28 = { by lemma 74 }
% 68.37/9.28 fresh(fresh28(is_a_theorem(implies(or(and(Y, not(X)), not(implies(X, Y))), not(and(implies(X, Y), implies(Y, X))))), true, implies(equiv(X, Y), equiv(Y, X))), true, implies(equiv(X, Y), equiv(Y, X)), implies(Z, Z))
% 68.37/9.28 = { by lemma 64 }
% 68.37/9.28 fresh(fresh28(is_a_theorem(implies(or(and(Y, not(X)), not(implies(X, Y))), not(equiv(X, Y)))), true, implies(equiv(X, Y), equiv(Y, X))), true, implies(equiv(X, Y), equiv(Y, X)), implies(Z, Z))
% 68.37/9.28 = { by lemma 76 }
% 68.37/9.28 fresh(fresh28(is_a_theorem(implies(implies(implies(Y, X), not(implies(X, Y))), not(equiv(X, Y)))), true, implies(equiv(X, Y), equiv(Y, X))), true, implies(equiv(X, Y), equiv(Y, X)), implies(Z, Z))
% 68.37/9.28 = { by lemma 84 }
% 68.37/9.28 fresh(fresh28(is_a_theorem(or(and(implies(Y, X), implies(X, Y)), not(equiv(X, Y)))), true, implies(equiv(X, Y), equiv(Y, X))), true, implies(equiv(X, Y), equiv(Y, X)), implies(Z, Z))
% 68.37/9.28 = { by lemma 64 }
% 68.37/9.28 fresh(fresh28(is_a_theorem(or(equiv(Y, X), not(equiv(X, Y)))), true, implies(equiv(X, Y), equiv(Y, X))), true, implies(equiv(X, Y), equiv(Y, X)), implies(Z, Z))
% 68.37/9.28 = { by axiom 52 (modus_ponens_2) R->L }
% 68.37/9.28 fresh(fresh59(is_a_theorem(implies(or(equiv(Y, X), not(equiv(X, Y))), implies(equiv(X, Y), equiv(Y, X)))), true, or(equiv(Y, X), not(equiv(X, Y))), implies(equiv(X, Y), equiv(Y, X))), true, implies(equiv(X, Y), equiv(Y, X)), implies(Z, Z))
% 68.37/9.28 = { by lemma 80 }
% 68.37/9.28 fresh(fresh59(true, true, or(equiv(Y, X), not(equiv(X, Y))), implies(equiv(X, Y), equiv(Y, X))), true, implies(equiv(X, Y), equiv(Y, X)), implies(Z, Z))
% 68.37/9.28 = { by lemma 59 }
% 68.37/9.28 fresh(true, true, implies(equiv(X, Y), equiv(Y, X)), implies(Z, Z))
% 68.37/9.28 = { by axiom 21 (substitution_of_equivalents_2) }
% 68.37/9.28 implies(Z, Z)
% 68.37/9.28
% 68.37/9.28 Lemma 99: equiv(Y, X) = equiv(X, Y).
% 68.37/9.28 Proof:
% 68.37/9.28 equiv(Y, X)
% 68.37/9.28 = { by axiom 21 (substitution_of_equivalents_2) R->L }
% 68.37/9.28 fresh(true, true, equiv(X, Y), equiv(Y, X))
% 68.37/9.28 = { by lemma 68 R->L }
% 68.37/9.28 fresh(is_a_theorem(implies(W, W)), true, equiv(X, Y), equiv(Y, X))
% 68.37/9.28 = { by lemma 98 R->L }
% 68.37/9.28 fresh(is_a_theorem(implies(equiv(Y, X), equiv(X, Y))), true, equiv(X, Y), equiv(Y, X))
% 68.37/9.28 = { by lemma 92 R->L }
% 68.37/9.28 fresh(is_a_theorem(and(implies(Z, Z), implies(equiv(Y, X), equiv(X, Y)))), true, equiv(X, Y), equiv(Y, X))
% 68.37/9.28 = { by lemma 98 R->L }
% 68.37/9.28 fresh(is_a_theorem(and(implies(equiv(X, Y), equiv(Y, X)), implies(equiv(Y, X), equiv(X, Y)))), true, equiv(X, Y), equiv(Y, X))
% 68.37/9.28 = { by lemma 64 }
% 68.37/9.28 fresh(is_a_theorem(equiv(equiv(X, Y), equiv(Y, X))), true, equiv(X, Y), equiv(Y, X))
% 69.10/9.28 = { by lemma 65 }
% 69.10/9.28 equiv(X, Y)
% 69.10/9.28
% 69.10/9.28 Lemma 100: not(and(X, Y)) = implies(X, not(Y)).
% 69.10/9.28 Proof:
% 69.10/9.28 not(and(X, Y))
% 69.10/9.28 = { by lemma 91 R->L }
% 69.10/9.28 not(and(X, not(not(Y))))
% 69.10/9.28 = { by lemma 74 }
% 69.10/9.28 implies(X, not(Y))
% 69.10/9.28
% 69.10/9.28 Lemma 101: and(X, not(Y)) = not(implies(X, Y)).
% 69.10/9.28 Proof:
% 69.10/9.28 and(X, not(Y))
% 69.10/9.28 = { by lemma 91 R->L }
% 69.10/9.28 not(not(and(X, not(Y))))
% 69.10/9.28 = { by lemma 74 }
% 69.10/9.28 not(implies(X, Y))
% 69.10/9.28
% 69.10/9.28 Lemma 102: and(not(X), Y) = not(implies(Y, X)).
% 69.10/9.28 Proof:
% 69.10/9.28 and(not(X), Y)
% 69.10/9.28 = { by lemma 97 }
% 69.10/9.28 and(Y, not(X))
% 69.10/9.28 = { by lemma 101 }
% 69.10/9.28 not(implies(Y, X))
% 69.10/9.28
% 69.10/9.28 Lemma 103: or(X, not(Y)) = implies(Y, X).
% 69.10/9.28 Proof:
% 69.10/9.28 or(X, not(Y))
% 69.10/9.28 = { by lemma 75 R->L }
% 69.10/9.28 implies(not(X), not(Y))
% 69.10/9.28 = { by lemma 100 R->L }
% 69.10/9.28 not(and(not(X), Y))
% 69.10/9.28 = { by lemma 102 }
% 69.10/9.28 not(not(implies(Y, X)))
% 69.10/9.28 = { by lemma 91 }
% 69.10/9.28 implies(Y, X)
% 69.10/9.28
% 69.10/9.28 Lemma 104: or(Y, X) = or(X, Y).
% 69.10/9.28 Proof:
% 69.10/9.28 or(Y, X)
% 69.10/9.28 = { by lemma 91 R->L }
% 69.10/9.28 or(Y, not(not(X)))
% 69.10/9.28 = { by lemma 103 }
% 69.10/9.28 implies(not(X), Y)
% 69.10/9.28 = { by lemma 75 }
% 69.10/9.28 or(X, Y)
% 69.10/9.28
% 69.10/9.28 Lemma 105: and(implies(X, not(Y)), or(Y, X)) = equiv(X, not(Y)).
% 69.10/9.28 Proof:
% 69.10/9.28 and(implies(X, not(Y)), or(Y, X))
% 69.10/9.28 = { by lemma 75 R->L }
% 69.10/9.28 and(implies(X, not(Y)), implies(not(Y), X))
% 69.10/9.28 = { by lemma 64 }
% 69.10/9.28 equiv(X, not(Y))
% 69.10/9.28
% 69.10/9.28 Lemma 106: equiv(X, not(Y)) = equiv(Y, not(X)).
% 69.10/9.28 Proof:
% 69.10/9.28 equiv(X, not(Y))
% 69.10/9.28 = { by lemma 99 }
% 69.10/9.28 equiv(not(Y), X)
% 69.10/9.28 = { by lemma 64 R->L }
% 69.10/9.28 and(implies(not(Y), X), implies(X, not(Y)))
% 69.10/9.28 = { by lemma 103 R->L }
% 69.10/9.28 and(or(X, not(not(Y))), implies(X, not(Y)))
% 69.10/9.28 = { by lemma 75 R->L }
% 69.10/9.28 and(implies(not(X), not(not(Y))), implies(X, not(Y)))
% 69.10/9.28 = { by lemma 103 R->L }
% 69.10/9.28 and(implies(not(X), not(not(Y))), or(not(Y), not(X)))
% 69.10/9.28 = { by lemma 105 }
% 69.10/9.28 equiv(not(X), not(not(Y)))
% 69.10/9.28 = { by lemma 99 R->L }
% 69.10/9.28 equiv(not(not(Y)), not(X))
% 69.10/9.28 = { by lemma 91 }
% 69.10/9.28 equiv(Y, not(X))
% 69.10/9.28
% 69.10/9.28 Lemma 107: equiv(not(X), Y) = equiv(X, not(Y)).
% 69.10/9.28 Proof:
% 69.10/9.28 equiv(not(X), Y)
% 69.10/9.28 = { by lemma 99 }
% 69.10/9.28 equiv(Y, not(X))
% 69.10/9.28 = { by lemma 106 }
% 69.10/9.28 equiv(X, not(Y))
% 69.10/9.28
% 69.10/9.28 Lemma 108: implies(Y, not(X)) = implies(X, not(Y)).
% 69.10/9.28 Proof:
% 69.10/9.28 implies(Y, not(X))
% 69.10/9.28 = { by lemma 100 R->L }
% 69.10/9.28 not(and(Y, X))
% 69.10/9.28 = { by lemma 97 }
% 69.10/9.28 not(and(X, Y))
% 69.10/9.28 = { by lemma 100 }
% 69.10/9.28 implies(X, not(Y))
% 69.10/9.28
% 69.10/9.28 Lemma 109: implies(and(X, Y), X) = implies(Z, Z).
% 69.10/9.28 Proof:
% 69.10/9.28 implies(and(X, Y), X)
% 69.10/9.28 = { by lemma 97 }
% 69.10/9.28 implies(and(Y, X), X)
% 69.10/9.28 = { by lemma 94 R->L }
% 69.10/9.28 fresh(is_a_theorem(implies(and(Y, X), X)), true, implies(and(Y, X), X), implies(Z, Z))
% 69.10/9.28 = { by axiom 43 (and_2_1) R->L }
% 69.10/9.28 fresh(fresh55(and_2, true, Y, X), true, implies(and(Y, X), X), implies(Z, Z))
% 69.10/9.28 = { by axiom 8 (hilbert_and_2) }
% 69.10/9.28 fresh(fresh55(true, true, Y, X), true, implies(and(Y, X), X), implies(Z, Z))
% 69.10/9.28 = { by axiom 24 (and_2_1) }
% 69.10/9.28 fresh(true, true, implies(and(Y, X), X), implies(Z, Z))
% 69.10/9.28 = { by axiom 21 (substitution_of_equivalents_2) }
% 69.10/9.28 implies(Z, Z)
% 69.10/9.28
% 69.10/9.28 Lemma 110: implies(implies(X, Y), implies(or(X, Y), Y)) = implies(Z, Z).
% 69.10/9.28 Proof:
% 69.10/9.28 implies(implies(X, Y), implies(or(X, Y), Y))
% 69.10/9.28 = { by lemma 104 }
% 69.10/9.28 implies(implies(X, Y), implies(or(Y, X), Y))
% 69.10/9.28 = { by lemma 94 R->L }
% 69.10/9.28 fresh(is_a_theorem(implies(implies(X, Y), implies(or(Y, X), Y))), true, implies(implies(X, Y), implies(or(Y, X), Y)), implies(Z, Z))
% 69.10/9.28 = { by axiom 20 (modus_ponens_2) R->L }
% 69.10/9.28 fresh(fresh28(true, true, implies(implies(X, Y), implies(or(Y, X), Y))), true, implies(implies(X, Y), implies(or(Y, X), Y)), implies(Z, Z))
% 69.10/9.28 = { by axiom 47 (or_3_1) R->L }
% 69.10/9.28 fresh(fresh28(fresh14(true, true, Y, X, Y), true, implies(implies(X, Y), implies(or(Y, X), Y))), true, implies(implies(X, Y), implies(or(Y, X), Y)), implies(Z, Z))
% 69.10/9.28 = { by axiom 12 (hilbert_or_3) R->L }
% 69.10/9.28 fresh(fresh28(fresh14(or_3, true, Y, X, Y), true, implies(implies(X, Y), implies(or(Y, X), Y))), true, implies(implies(X, Y), implies(or(Y, X), Y)), implies(Z, Z))
% 69.10/9.28 = { by axiom 57 (or_3_1) }
% 69.10/9.28 fresh(fresh28(is_a_theorem(implies(implies(Y, Y), implies(implies(X, Y), implies(or(Y, X), Y)))), true, implies(implies(X, Y), implies(or(Y, X), Y))), true, implies(implies(X, Y), implies(or(Y, X), Y)), implies(Z, Z))
% 69.10/9.28 = { by axiom 52 (modus_ponens_2) R->L }
% 69.10/9.29 fresh(fresh59(is_a_theorem(implies(implies(implies(Y, Y), implies(implies(X, Y), implies(or(Y, X), Y))), implies(implies(X, Y), implies(or(Y, X), Y)))), true, implies(implies(Y, Y), implies(implies(X, Y), implies(or(Y, X), Y))), implies(implies(X, Y), implies(or(Y, X), Y))), true, implies(implies(X, Y), implies(or(Y, X), Y)), implies(Z, Z))
% 69.10/9.29 = { by lemma 82 }
% 69.10/9.29 fresh(fresh59(true, true, implies(implies(Y, Y), implies(implies(X, Y), implies(or(Y, X), Y))), implies(implies(X, Y), implies(or(Y, X), Y))), true, implies(implies(X, Y), implies(or(Y, X), Y)), implies(Z, Z))
% 69.10/9.29 = { by lemma 59 }
% 69.10/9.29 fresh(true, true, implies(implies(X, Y), implies(or(Y, X), Y)), implies(Z, Z))
% 69.10/9.29 = { by axiom 21 (substitution_of_equivalents_2) }
% 69.10/9.29 implies(Z, Z)
% 69.10/9.29
% 69.10/9.29 Lemma 111: is_a_theorem(implies(implies(X, Y), implies(implies(Y, Z), implies(X, Z)))) = true.
% 69.10/9.29 Proof:
% 69.10/9.29 is_a_theorem(implies(implies(X, Y), implies(implies(Y, Z), implies(X, Z))))
% 69.10/9.29 = { by axiom 55 (implies_3_1) R->L }
% 69.10/9.29 fresh35(implies_3, true, X, Y, Z)
% 69.10/9.29 = { by axiom 6 (hilbert_implies_3) }
% 69.10/9.29 fresh35(true, true, X, Y, Z)
% 69.10/9.29 = { by axiom 46 (implies_3_1) }
% 69.10/9.29 true
% 69.10/9.29
% 69.10/9.29 Lemma 112: implies(X, implies(implies(X, Y), Y)) = implies(Z, Z).
% 69.10/9.29 Proof:
% 69.10/9.29 implies(X, implies(implies(X, Y), Y))
% 69.10/9.29 = { by lemma 94 R->L }
% 69.10/9.29 fresh(is_a_theorem(implies(X, implies(implies(X, Y), Y))), true, implies(X, implies(implies(X, Y), Y)), implies(Z, Z))
% 69.10/9.29 = { by lemma 83 R->L }
% 69.10/9.29 fresh(is_a_theorem(implies(X, implies(implies(X, Y), implies(implies(W, W), Y)))), true, implies(X, implies(implies(X, Y), Y)), implies(Z, Z))
% 69.10/9.29 = { by lemma 83 R->L }
% 69.10/9.29 fresh(is_a_theorem(implies(implies(implies(W, W), X), implies(implies(X, Y), implies(implies(W, W), Y)))), true, implies(X, implies(implies(X, Y), Y)), implies(Z, Z))
% 69.10/9.29 = { by lemma 111 }
% 69.10/9.29 fresh(true, true, implies(X, implies(implies(X, Y), Y)), implies(Z, Z))
% 69.10/9.29 = { by axiom 21 (substitution_of_equivalents_2) }
% 69.10/9.29 implies(Z, Z)
% 69.10/9.29
% 69.10/9.29 Lemma 113: or(X, implies(implies(X, Y), Y)) = implies(implies(X, Y), Y).
% 69.10/9.29 Proof:
% 69.10/9.29 or(X, implies(implies(X, Y), Y))
% 69.10/9.29 = { by axiom 21 (substitution_of_equivalents_2) R->L }
% 69.10/9.29 fresh(true, true, implies(implies(X, Y), Y), or(X, implies(implies(X, Y), Y)))
% 69.10/9.29 = { by lemma 68 R->L }
% 69.10/9.29 fresh(is_a_theorem(implies(Z, Z)), true, implies(implies(X, Y), Y), or(X, implies(implies(X, Y), Y)))
% 69.10/9.29 = { by axiom 21 (substitution_of_equivalents_2) R->L }
% 69.10/9.29 fresh(is_a_theorem(fresh(true, true, implies(implies(implies(X, Y), Y), or(X, implies(implies(X, Y), Y))), implies(Z, Z))), true, implies(implies(X, Y), Y), or(X, implies(implies(X, Y), Y)))
% 69.10/9.29 = { by axiom 37 (or_2_1) R->L }
% 69.10/9.29 fresh(is_a_theorem(fresh(fresh16(true, true, X, implies(implies(X, Y), Y)), true, implies(implies(implies(X, Y), Y), or(X, implies(implies(X, Y), Y))), implies(Z, Z))), true, implies(implies(X, Y), Y), or(X, implies(implies(X, Y), Y)))
% 69.10/9.29 = { by axiom 11 (hilbert_or_2) R->L }
% 69.10/9.29 fresh(is_a_theorem(fresh(fresh16(or_2, true, X, implies(implies(X, Y), Y)), true, implies(implies(implies(X, Y), Y), or(X, implies(implies(X, Y), Y))), implies(Z, Z))), true, implies(implies(X, Y), Y), or(X, implies(implies(X, Y), Y)))
% 69.10/9.29 = { by axiom 41 (or_2_1) }
% 69.10/9.29 fresh(is_a_theorem(fresh(is_a_theorem(implies(implies(implies(X, Y), Y), or(X, implies(implies(X, Y), Y)))), true, implies(implies(implies(X, Y), Y), or(X, implies(implies(X, Y), Y))), implies(Z, Z))), true, implies(implies(X, Y), Y), or(X, implies(implies(X, Y), Y)))
% 69.10/9.29 = { by lemma 94 }
% 69.10/9.29 fresh(is_a_theorem(implies(implies(implies(X, Y), Y), or(X, implies(implies(X, Y), Y)))), true, implies(implies(X, Y), Y), or(X, implies(implies(X, Y), Y)))
% 69.10/9.29 = { by lemma 92 R->L }
% 69.10/9.29 fresh(is_a_theorem(and(implies(W, W), implies(implies(implies(X, Y), Y), or(X, implies(implies(X, Y), Y))))), true, implies(implies(X, Y), Y), or(X, implies(implies(X, Y), Y)))
% 69.10/9.29 = { by lemma 110 R->L }
% 69.14/9.29 fresh(is_a_theorem(and(implies(implies(X, implies(implies(X, Y), Y)), implies(or(X, implies(implies(X, Y), Y)), implies(implies(X, Y), Y))), implies(implies(implies(X, Y), Y), or(X, implies(implies(X, Y), Y))))), true, implies(implies(X, Y), Y), or(X, implies(implies(X, Y), Y)))
% 69.14/9.29 = { by lemma 112 }
% 69.14/9.29 fresh(is_a_theorem(and(implies(implies(V, V), implies(or(X, implies(implies(X, Y), Y)), implies(implies(X, Y), Y))), implies(implies(implies(X, Y), Y), or(X, implies(implies(X, Y), Y))))), true, implies(implies(X, Y), Y), or(X, implies(implies(X, Y), Y)))
% 69.14/9.29 = { by lemma 83 }
% 69.14/9.29 fresh(is_a_theorem(and(implies(or(X, implies(implies(X, Y), Y)), implies(implies(X, Y), Y)), implies(implies(implies(X, Y), Y), or(X, implies(implies(X, Y), Y))))), true, implies(implies(X, Y), Y), or(X, implies(implies(X, Y), Y)))
% 69.14/9.29 = { by lemma 64 }
% 69.14/9.29 fresh(is_a_theorem(equiv(or(X, implies(implies(X, Y), Y)), implies(implies(X, Y), Y))), true, implies(implies(X, Y), Y), or(X, implies(implies(X, Y), Y)))
% 69.14/9.29 = { by lemma 99 R->L }
% 69.14/9.29 fresh(is_a_theorem(equiv(implies(implies(X, Y), Y), or(X, implies(implies(X, Y), Y)))), true, implies(implies(X, Y), Y), or(X, implies(implies(X, Y), Y)))
% 69.14/9.29 = { by lemma 65 }
% 69.14/9.29 implies(implies(X, Y), Y)
% 69.14/9.29
% 69.14/9.29 Lemma 114: or(X, and(X, Y)) = X.
% 69.14/9.29 Proof:
% 69.14/9.29 or(X, and(X, Y))
% 69.14/9.29 = { by lemma 104 }
% 69.14/9.29 or(and(X, Y), X)
% 69.14/9.29 = { by lemma 83 R->L }
% 69.14/9.29 or(and(X, Y), implies(implies(Z, Z), X))
% 69.14/9.29 = { by lemma 109 R->L }
% 69.14/9.29 or(and(X, Y), implies(implies(and(X, Y), X), X))
% 69.14/9.29 = { by lemma 113 }
% 69.14/9.29 implies(implies(and(X, Y), X), X)
% 69.14/9.29 = { by lemma 109 }
% 69.14/9.29 implies(implies(W, W), X)
% 69.14/9.29 = { by lemma 83 }
% 69.14/9.29 X
% 69.14/9.29
% 69.14/9.29 Lemma 115: implies(implies(X, Y), X) = X.
% 69.14/9.29 Proof:
% 69.14/9.29 implies(implies(X, Y), X)
% 69.14/9.29 = { by lemma 103 R->L }
% 69.14/9.29 or(X, not(implies(X, Y)))
% 69.14/9.29 = { by lemma 101 R->L }
% 69.14/9.29 or(X, and(X, not(Y)))
% 69.14/9.29 = { by lemma 114 }
% 69.14/9.29 X
% 69.14/9.29
% 69.14/9.29 Lemma 116: implies(X, implies(X, Y)) = implies(X, Y).
% 69.14/9.29 Proof:
% 69.14/9.29 implies(X, implies(X, Y))
% 69.14/9.29 = { by axiom 21 (substitution_of_equivalents_2) R->L }
% 69.14/9.29 fresh(true, true, implies(X, Y), implies(X, implies(X, Y)))
% 69.14/9.29 = { by lemma 69 R->L }
% 69.14/9.29 fresh(fresh28(is_a_theorem(implies(implies(X, implies(X, Y)), implies(X, Y))), true, equiv(implies(X, Y), implies(X, implies(X, Y)))), true, implies(X, Y), implies(X, implies(X, Y)))
% 69.14/9.29 = { by lemma 62 }
% 69.14/9.29 fresh(fresh28(true, true, equiv(implies(X, Y), implies(X, implies(X, Y)))), true, implies(X, Y), implies(X, implies(X, Y)))
% 69.14/9.29 = { by axiom 20 (modus_ponens_2) }
% 69.14/9.29 fresh(is_a_theorem(equiv(implies(X, Y), implies(X, implies(X, Y)))), true, implies(X, Y), implies(X, implies(X, Y)))
% 69.14/9.29 = { by lemma 65 }
% 69.14/9.29 implies(X, Y)
% 69.14/9.29
% 69.14/9.29 Lemma 117: equiv(X, implies(X, Y)) = and(X, implies(X, Y)).
% 69.14/9.29 Proof:
% 69.14/9.29 equiv(X, implies(X, Y))
% 69.14/9.29 = { by lemma 99 }
% 69.14/9.29 equiv(implies(X, Y), X)
% 69.14/9.29 = { by lemma 64 R->L }
% 69.14/9.29 and(implies(implies(X, Y), X), implies(X, implies(X, Y)))
% 69.14/9.29 = { by lemma 115 }
% 69.14/9.29 and(X, implies(X, implies(X, Y)))
% 69.14/9.29 = { by lemma 116 }
% 69.14/9.29 and(X, implies(X, Y))
% 69.14/9.29
% 69.14/9.29 Lemma 118: or(X, implies(Y, not(Z))) = implies(and(Y, Z), X).
% 69.14/9.29 Proof:
% 69.14/9.29 or(X, implies(Y, not(Z)))
% 69.14/9.29 = { by lemma 108 }
% 69.14/9.29 or(X, implies(Z, not(Y)))
% 69.14/9.29 = { by lemma 100 R->L }
% 69.14/9.29 or(X, not(and(Z, Y)))
% 69.14/9.29 = { by lemma 103 }
% 69.14/9.29 implies(and(Z, Y), X)
% 69.14/9.29 = { by lemma 97 R->L }
% 69.14/9.29 implies(and(Y, Z), X)
% 69.14/9.29
% 69.14/9.29 Lemma 119: and(X, implies(Y, not(Z))) = not(implies(X, and(Y, Z))).
% 69.14/9.29 Proof:
% 69.14/9.29 and(X, implies(Y, not(Z)))
% 69.14/9.29 = { by lemma 73 R->L }
% 69.14/9.29 not(implies(X, not(implies(Y, not(Z)))))
% 69.14/9.29 = { by lemma 73 }
% 69.14/9.29 not(implies(X, and(Y, Z)))
% 69.14/9.29
% 69.14/9.29 Lemma 120: implies(implies(X, Y), not(implies(Y, X))) = not(equiv(X, Y)).
% 69.14/9.29 Proof:
% 69.14/9.29 implies(implies(X, Y), not(implies(Y, X)))
% 69.14/9.29 = { by lemma 66 R->L }
% 69.14/9.29 and(implies(implies(X, Y), not(implies(Y, X))), implies(implies(X, Y), not(implies(Y, X))))
% 69.14/9.29 = { by lemma 85 R->L }
% 69.14/9.29 and(not(and(implies(X, Y), implies(Y, X))), implies(implies(X, Y), not(implies(Y, X))))
% 69.14/9.29 = { by lemma 64 }
% 69.14/9.29 and(not(equiv(X, Y)), implies(implies(X, Y), not(implies(Y, X))))
% 69.14/9.29 = { by lemma 119 }
% 69.14/9.29 not(implies(not(equiv(X, Y)), and(implies(X, Y), implies(Y, X))))
% 69.14/9.29 = { by lemma 75 }
% 69.14/9.29 not(or(equiv(X, Y), and(implies(X, Y), implies(Y, X))))
% 69.14/9.29 = { by lemma 64 }
% 69.14/9.29 not(or(equiv(X, Y), equiv(X, Y)))
% 69.14/9.29 = { by lemma 90 }
% 69.14/9.29 implies(equiv(X, Y), not(equiv(X, Y)))
% 69.14/9.29 = { by lemma 72 R->L }
% 69.14/9.29 or(not(equiv(X, Y)), not(equiv(X, Y)))
% 69.14/9.29 = { by lemma 89 }
% 69.14/9.29 not(equiv(X, Y))
% 69.14/9.29
% 69.14/9.29 Lemma 121: implies(implies(X, Y), not(implies(implies(X, Y), X))) = not(equiv(X, implies(X, Y))).
% 69.14/9.29 Proof:
% 69.14/9.29 implies(implies(X, Y), not(implies(implies(X, Y), X)))
% 69.14/9.29 = { by lemma 116 R->L }
% 69.14/9.29 implies(implies(X, implies(X, Y)), not(implies(implies(X, Y), X)))
% 69.14/9.29 = { by lemma 120 }
% 69.14/9.29 not(equiv(X, implies(X, Y)))
% 69.14/9.29
% 69.14/9.29 Lemma 122: implies(X, implies(Y, X)) = implies(Z, Z).
% 69.14/9.29 Proof:
% 69.14/9.29 implies(X, implies(Y, X))
% 69.14/9.29 = { by lemma 94 R->L }
% 69.14/9.29 fresh(is_a_theorem(implies(X, implies(Y, X))), true, implies(X, implies(Y, X)), implies(Z, Z))
% 69.14/9.29 = { by lemma 67 }
% 69.14/9.29 fresh(true, true, implies(X, implies(Y, X)), implies(Z, Z))
% 69.14/9.29 = { by axiom 21 (substitution_of_equivalents_2) }
% 69.14/9.29 implies(Z, Z)
% 69.14/9.29
% 69.14/9.29 Lemma 123: is_a_theorem(implies(implies(implies(X, Y), Z), implies(Y, Z))) = true.
% 69.14/9.29 Proof:
% 69.14/9.29 is_a_theorem(implies(implies(implies(X, Y), Z), implies(Y, Z)))
% 69.14/9.29 = { by lemma 83 R->L }
% 69.14/9.29 is_a_theorem(implies(implies(W, W), implies(implies(implies(X, Y), Z), implies(Y, Z))))
% 69.14/9.29 = { by lemma 122 R->L }
% 69.14/9.29 is_a_theorem(implies(implies(Y, implies(X, Y)), implies(implies(implies(X, Y), Z), implies(Y, Z))))
% 69.14/9.29 = { by lemma 111 }
% 69.14/9.29 true
% 69.14/9.29
% 69.14/9.29 Lemma 124: implies(implies(implies(X, Y), Z), implies(Y, Z)) = implies(W, W).
% 69.14/9.29 Proof:
% 69.14/9.29 implies(implies(implies(X, Y), Z), implies(Y, Z))
% 69.14/9.29 = { by lemma 94 R->L }
% 69.14/9.29 fresh(is_a_theorem(implies(implies(implies(X, Y), Z), implies(Y, Z))), true, implies(implies(implies(X, Y), Z), implies(Y, Z)), implies(W, W))
% 69.14/9.29 = { by lemma 123 }
% 69.14/9.29 fresh(true, true, implies(implies(implies(X, Y), Z), implies(Y, Z)), implies(W, W))
% 69.14/9.29 = { by axiom 21 (substitution_of_equivalents_2) }
% 69.14/9.29 implies(W, W)
% 69.14/9.29
% 69.14/9.29 Lemma 125: implies(and(X, Y), and(X, implies(X, Y))) = implies(Z, Z).
% 69.14/9.29 Proof:
% 69.14/9.29 implies(and(X, Y), and(X, implies(X, Y)))
% 69.14/9.29 = { by lemma 97 }
% 69.14/9.29 implies(and(Y, X), and(X, implies(X, Y)))
% 69.14/9.29 = { by lemma 117 R->L }
% 69.14/9.29 implies(and(Y, X), equiv(X, implies(X, Y)))
% 69.14/9.29 = { by lemma 115 R->L }
% 69.14/9.29 implies(and(Y, implies(implies(X, Y), X)), equiv(X, implies(X, Y)))
% 69.14/9.29 = { by lemma 118 R->L }
% 69.14/9.29 or(equiv(X, implies(X, Y)), implies(Y, not(implies(implies(X, Y), X))))
% 69.14/9.29 = { by lemma 75 R->L }
% 69.14/9.29 implies(not(equiv(X, implies(X, Y))), implies(Y, not(implies(implies(X, Y), X))))
% 69.14/9.29 = { by lemma 121 R->L }
% 69.14/9.29 implies(implies(implies(X, Y), not(implies(implies(X, Y), X))), implies(Y, not(implies(implies(X, Y), X))))
% 69.14/9.30 = { by lemma 124 }
% 69.14/9.30 implies(Z, Z)
% 69.14/9.30
% 69.14/9.30 Lemma 126: implies(and(X, implies(X, Y)), and(X, Y)) = implies(Z, Z).
% 69.14/9.30 Proof:
% 69.14/9.30 implies(and(X, implies(X, Y)), and(X, Y))
% 69.14/9.30 = { by lemma 97 }
% 69.14/9.30 implies(and(X, implies(X, Y)), and(Y, X))
% 69.14/9.30 = { by lemma 97 }
% 69.14/9.30 implies(and(implies(X, Y), X), and(Y, X))
% 69.14/9.30 = { by lemma 118 R->L }
% 69.14/9.30 or(and(Y, X), implies(implies(X, Y), not(X)))
% 69.14/9.30 = { by lemma 104 }
% 69.14/9.30 or(implies(implies(X, Y), not(X)), and(Y, X))
% 69.14/9.30 = { by lemma 103 R->L }
% 69.14/9.30 or(implies(or(Y, not(X)), not(X)), and(Y, X))
% 69.14/9.30 = { by lemma 104 }
% 69.14/9.30 or(and(Y, X), implies(or(Y, not(X)), not(X)))
% 69.14/9.30 = { by lemma 73 R->L }
% 69.14/9.30 or(not(implies(Y, not(X))), implies(or(Y, not(X)), not(X)))
% 69.14/9.30 = { by lemma 72 }
% 69.14/9.30 implies(implies(Y, not(X)), implies(or(Y, not(X)), not(X)))
% 69.14/9.30 = { by lemma 110 }
% 69.14/9.30 implies(Z, Z)
% 69.14/9.30
% 69.14/9.30 Lemma 127: and(X, implies(X, Y)) = and(X, Y).
% 69.14/9.30 Proof:
% 69.32/9.31 and(X, implies(X, Y))
% 69.32/9.31 = { by lemma 83 R->L }
% 69.32/9.31 implies(implies(Z, Z), and(X, implies(X, Y)))
% 69.32/9.31 = { by lemma 125 R->L }
% 69.32/9.31 implies(implies(and(X, Y), and(X, implies(X, Y))), and(X, implies(X, Y)))
% 69.32/9.31 = { by lemma 113 R->L }
% 69.32/9.31 or(and(X, Y), implies(implies(and(X, Y), and(X, implies(X, Y))), and(X, implies(X, Y))))
% 69.32/9.31 = { by lemma 125 }
% 69.32/9.32 or(and(X, Y), implies(implies(W, W), and(X, implies(X, Y))))
% 69.32/9.32 = { by lemma 83 }
% 69.32/9.32 or(and(X, Y), and(X, implies(X, Y)))
% 69.32/9.32 = { by lemma 104 }
% 69.32/9.32 or(and(X, implies(X, Y)), and(X, Y))
% 69.32/9.32 = { by lemma 83 R->L }
% 69.32/9.32 or(and(X, implies(X, Y)), implies(implies(V, V), and(X, Y)))
% 69.32/9.32 = { by lemma 126 R->L }
% 69.32/9.32 or(and(X, implies(X, Y)), implies(implies(and(X, implies(X, Y)), and(X, Y)), and(X, Y)))
% 69.32/9.32 = { by lemma 113 }
% 69.32/9.32 implies(implies(and(X, implies(X, Y)), and(X, Y)), and(X, Y))
% 69.32/9.32 = { by lemma 126 }
% 69.32/9.32 implies(implies(U, U), and(X, Y))
% 69.32/9.32 = { by lemma 83 }
% 69.32/9.32 and(X, Y)
% 69.32/9.32
% 69.32/9.32 Lemma 128: and(X, or(X, Y)) = X.
% 69.32/9.32 Proof:
% 69.32/9.32 and(X, or(X, Y))
% 69.32/9.32 = { by lemma 97 }
% 69.32/9.32 and(or(X, Y), X)
% 69.32/9.32 = { by lemma 75 R->L }
% 69.32/9.32 and(implies(not(X), Y), X)
% 69.32/9.32 = { by lemma 73 R->L }
% 69.32/9.32 not(implies(implies(not(X), Y), not(X)))
% 69.32/9.32 = { by lemma 115 }
% 69.32/9.32 not(not(X))
% 69.32/9.32 = { by lemma 91 }
% 69.32/9.32 X
% 69.32/9.32
% 69.32/9.32 Lemma 129: and(implies(X, implies(Y, X)), Z) = Z.
% 69.32/9.32 Proof:
% 69.32/9.32 and(implies(X, implies(Y, X)), Z)
% 69.32/9.32 = { by lemma 83 R->L }
% 69.32/9.32 and(implies(X, implies(Y, X)), implies(implies(W, W), Z))
% 69.32/9.32 = { by lemma 94 R->L }
% 69.32/9.32 and(fresh(is_a_theorem(implies(X, implies(Y, X))), true, implies(X, implies(Y, X)), implies(W, W)), implies(implies(W, W), Z))
% 69.32/9.32 = { by lemma 67 }
% 69.32/9.32 and(fresh(true, true, implies(X, implies(Y, X)), implies(W, W)), implies(implies(W, W), Z))
% 69.32/9.32 = { by axiom 21 (substitution_of_equivalents_2) }
% 69.32/9.32 and(implies(W, W), implies(implies(W, W), Z))
% 69.32/9.32 = { by lemma 71 }
% 69.32/9.32 equiv(Z, implies(W, W))
% 69.32/9.32 = { by lemma 93 }
% 69.32/9.32 Z
% 69.32/9.32
% 69.32/9.32 Lemma 130: implies(or(X, Y), X) = implies(Y, X).
% 69.32/9.32 Proof:
% 69.32/9.32 implies(or(X, Y), X)
% 69.32/9.32 = { by lemma 91 R->L }
% 69.32/9.32 not(not(implies(or(X, Y), X)))
% 69.32/9.32 = { by lemma 102 R->L }
% 69.32/9.32 not(and(not(X), or(X, Y)))
% 69.32/9.32 = { by lemma 75 R->L }
% 69.32/9.32 not(and(not(X), implies(not(X), Y)))
% 69.32/9.32 = { by lemma 127 }
% 69.32/9.32 not(and(not(X), Y))
% 69.32/9.32 = { by lemma 102 }
% 69.32/9.32 not(not(implies(Y, X)))
% 69.32/9.32 = { by lemma 91 }
% 69.32/9.32 implies(Y, X)
% 69.32/9.32
% 69.32/9.32 Lemma 131: implies(implies(X, Y), Y) = or(X, Y).
% 69.32/9.32 Proof:
% 69.32/9.32 implies(implies(X, Y), Y)
% 69.32/9.32 = { by lemma 103 R->L }
% 69.32/9.32 implies(or(Y, not(X)), Y)
% 69.32/9.32 = { by lemma 130 }
% 69.32/9.32 implies(not(X), Y)
% 69.32/9.32 = { by lemma 75 }
% 69.32/9.32 or(X, Y)
% 69.32/9.32
% 69.32/9.32 Lemma 132: equiv(X, implies(Y, X)) = or(X, Y).
% 69.32/9.32 Proof:
% 69.32/9.32 equiv(X, implies(Y, X))
% 69.32/9.32 = { by lemma 64 R->L }
% 69.32/9.32 and(implies(X, implies(Y, X)), implies(implies(Y, X), X))
% 69.32/9.32 = { by lemma 129 }
% 69.32/9.32 implies(implies(Y, X), X)
% 69.32/9.32 = { by lemma 131 }
% 69.32/9.32 or(Y, X)
% 69.32/9.32 = { by lemma 104 R->L }
% 69.32/9.32 or(X, Y)
% 69.32/9.32
% 69.32/9.32 Lemma 133: implies(X, or(Y, implies(Z, X))) = implies(W, W).
% 69.32/9.32 Proof:
% 69.32/9.32 implies(X, or(Y, implies(Z, X)))
% 69.32/9.32 = { by lemma 75 R->L }
% 69.32/9.32 implies(X, implies(not(Y), implies(Z, X)))
% 69.32/9.32 = { by lemma 94 R->L }
% 69.32/9.32 fresh(is_a_theorem(implies(X, implies(not(Y), implies(Z, X)))), true, implies(X, implies(not(Y), implies(Z, X))), implies(W, W))
% 69.32/9.32 = { by lemma 83 R->L }
% 69.32/9.32 fresh(is_a_theorem(implies(implies(V, V), implies(X, implies(not(Y), implies(Z, X))))), true, implies(X, implies(not(Y), implies(Z, X))), implies(W, W))
% 69.32/9.32 = { by lemma 122 R->L }
% 69.32/9.32 fresh(is_a_theorem(implies(implies(implies(Z, X), implies(not(Y), implies(Z, X))), implies(X, implies(not(Y), implies(Z, X))))), true, implies(X, implies(not(Y), implies(Z, X))), implies(W, W))
% 69.32/9.32 = { by lemma 123 }
% 69.32/9.32 fresh(true, true, implies(X, implies(not(Y), implies(Z, X))), implies(W, W))
% 69.32/9.32 = { by axiom 21 (substitution_of_equivalents_2) }
% 69.32/9.32 implies(W, W)
% 69.32/9.32
% 69.32/9.32 Lemma 134: implies(and(X, implies(X, Y)), Y) = implies(Z, Z).
% 69.32/9.32 Proof:
% 69.32/9.32 implies(and(X, implies(X, Y)), Y)
% 69.32/9.32 = { by lemma 118 R->L }
% 69.32/9.32 or(Y, implies(X, not(implies(X, Y))))
% 69.32/9.32 = { by lemma 102 R->L }
% 69.32/9.32 or(Y, implies(X, and(not(Y), X)))
% 69.32/9.32 = { by lemma 94 R->L }
% 69.32/9.32 fresh(is_a_theorem(or(Y, implies(X, and(not(Y), X)))), true, or(Y, implies(X, and(not(Y), X))), implies(Z, Z))
% 69.32/9.32 = { by lemma 75 R->L }
% 69.32/9.32 fresh(is_a_theorem(implies(not(Y), implies(X, and(not(Y), X)))), true, or(Y, implies(X, and(not(Y), X))), implies(Z, Z))
% 69.32/9.32 = { by lemma 60 }
% 69.32/9.32 fresh(true, true, or(Y, implies(X, and(not(Y), X))), implies(Z, Z))
% 69.32/9.32 = { by axiom 21 (substitution_of_equivalents_2) }
% 69.32/9.32 implies(Z, Z)
% 69.32/9.32
% 69.32/9.32 Lemma 135: and(implies(X, Y), implies(implies(X, Y), X)) = equiv(X, implies(X, Y)).
% 69.32/9.32 Proof:
% 69.32/9.32 and(implies(X, Y), implies(implies(X, Y), X))
% 69.32/9.32 = { by lemma 116 R->L }
% 69.32/9.32 and(implies(X, implies(X, Y)), implies(implies(X, Y), X))
% 69.32/9.32 = { by lemma 64 }
% 69.32/9.33 equiv(X, implies(X, Y))
% 69.32/9.33
% 69.32/9.33 Lemma 136: implies(X, implies(or(X, Y), or(Y, Z))) = implies(or(X, Y), or(Y, Z)).
% 69.32/9.33 Proof:
% 69.32/9.33 implies(X, implies(or(X, Y), or(Y, Z)))
% 69.32/9.33 = { by lemma 130 R->L }
% 69.32/9.33 implies(or(implies(or(X, Y), or(Y, Z)), X), implies(or(X, Y), or(Y, Z)))
% 69.32/9.33 = { by lemma 131 R->L }
% 69.32/9.33 implies(implies(implies(implies(or(X, Y), or(Y, Z)), X), X), implies(or(X, Y), or(Y, Z)))
% 69.32/9.33 = { by lemma 92 R->L }
% 69.32/9.33 and(implies(W, W), implies(implies(implies(implies(or(X, Y), or(Y, Z)), X), X), implies(or(X, Y), or(Y, Z))))
% 69.32/9.33 = { by lemma 112 R->L }
% 69.32/9.33 and(implies(implies(or(X, Y), or(Y, Z)), implies(implies(implies(or(X, Y), or(Y, Z)), X), X)), implies(implies(implies(implies(or(X, Y), or(Y, Z)), X), X), implies(or(X, Y), or(Y, Z))))
% 69.32/9.33 = { by lemma 64 }
% 69.32/9.33 equiv(implies(or(X, Y), or(Y, Z)), implies(implies(implies(or(X, Y), or(Y, Z)), X), X))
% 69.32/9.33 = { by lemma 130 R->L }
% 69.32/9.33 equiv(implies(or(X, Y), or(Y, Z)), implies(implies(or(X, implies(or(X, Y), or(Y, Z))), X), X))
% 69.32/9.33 = { by lemma 91 R->L }
% 69.32/9.33 equiv(implies(or(X, Y), or(Y, Z)), implies(not(not(implies(or(X, implies(or(X, Y), or(Y, Z))), X))), X))
% 69.32/9.33 = { by lemma 75 R->L }
% 69.32/9.33 equiv(implies(or(X, Y), or(Y, Z)), implies(not(not(implies(implies(not(X), implies(or(X, Y), or(Y, Z))), X))), X))
% 69.32/9.33 = { by lemma 102 R->L }
% 69.32/9.33 equiv(implies(or(X, Y), or(Y, Z)), implies(not(and(not(X), implies(not(X), implies(or(X, Y), or(Y, Z))))), X))
% 69.32/9.33 = { by lemma 117 R->L }
% 69.32/9.33 equiv(implies(or(X, Y), or(Y, Z)), implies(not(equiv(not(X), implies(not(X), implies(or(X, Y), or(Y, Z))))), X))
% 69.32/9.33 = { by lemma 75 }
% 69.32/9.33 equiv(implies(or(X, Y), or(Y, Z)), implies(not(equiv(not(X), or(X, implies(or(X, Y), or(Y, Z))))), X))
% 69.32/9.33 = { by lemma 107 }
% 69.32/9.33 equiv(implies(or(X, Y), or(Y, Z)), implies(not(equiv(X, not(or(X, implies(or(X, Y), or(Y, Z)))))), X))
% 69.32/9.33 = { by lemma 106 R->L }
% 69.32/9.33 equiv(implies(or(X, Y), or(Y, Z)), implies(not(equiv(or(X, implies(or(X, Y), or(Y, Z))), not(X))), X))
% 69.32/9.33 = { by lemma 105 R->L }
% 69.32/9.33 equiv(implies(or(X, Y), or(Y, Z)), implies(not(and(implies(or(X, implies(or(X, Y), or(Y, Z))), not(X)), or(X, or(X, implies(or(X, Y), or(Y, Z)))))), X))
% 69.32/9.33 = { by lemma 75 R->L }
% 69.32/9.33 equiv(implies(or(X, Y), or(Y, Z)), implies(not(and(implies(or(X, implies(or(X, Y), or(Y, Z))), not(X)), or(X, implies(not(X), implies(or(X, Y), or(Y, Z)))))), X))
% 69.32/9.33 = { by lemma 75 R->L }
% 69.32/9.33 equiv(implies(or(X, Y), or(Y, Z)), implies(not(and(implies(or(X, implies(or(X, Y), or(Y, Z))), not(X)), implies(not(X), implies(not(X), implies(or(X, Y), or(Y, Z)))))), X))
% 69.32/9.33 = { by lemma 116 }
% 69.32/9.33 equiv(implies(or(X, Y), or(Y, Z)), implies(not(and(implies(or(X, implies(or(X, Y), or(Y, Z))), not(X)), implies(not(X), implies(or(X, Y), or(Y, Z))))), X))
% 69.32/9.33 = { by lemma 75 }
% 69.32/9.33 equiv(implies(or(X, Y), or(Y, Z)), implies(not(and(implies(or(X, implies(or(X, Y), or(Y, Z))), not(X)), or(X, implies(or(X, Y), or(Y, Z))))), X))
% 69.32/9.33 = { by lemma 97 }
% 69.32/9.33 equiv(implies(or(X, Y), or(Y, Z)), implies(not(and(or(X, implies(or(X, Y), or(Y, Z))), implies(or(X, implies(or(X, Y), or(Y, Z))), not(X)))), X))
% 69.32/9.33 = { by lemma 119 }
% 69.32/9.33 equiv(implies(or(X, Y), or(Y, Z)), implies(not(not(implies(or(X, implies(or(X, Y), or(Y, Z))), and(or(X, implies(or(X, Y), or(Y, Z))), X)))), X))
% 69.32/9.33 = { by lemma 97 R->L }
% 69.32/9.33 equiv(implies(or(X, Y), or(Y, Z)), implies(not(not(implies(or(X, implies(or(X, Y), or(Y, Z))), and(X, or(X, implies(or(X, Y), or(Y, Z))))))), X))
% 69.32/9.33 = { by lemma 91 }
% 69.32/9.33 equiv(implies(or(X, Y), or(Y, Z)), implies(implies(or(X, implies(or(X, Y), or(Y, Z))), and(X, or(X, implies(or(X, Y), or(Y, Z))))), X))
% 69.32/9.33 = { by lemma 104 }
% 69.32/9.33 equiv(implies(or(X, Y), or(Y, Z)), implies(implies(or(X, implies(or(Y, X), or(Y, Z))), and(X, or(X, implies(or(X, Y), or(Y, Z))))), X))
% 69.32/9.34 = { by lemma 94 R->L }
% 69.32/9.34 equiv(implies(or(X, Y), or(Y, Z)), implies(implies(fresh(is_a_theorem(or(X, implies(or(Y, X), or(Y, Z)))), true, or(X, implies(or(Y, X), or(Y, Z))), implies(V, V)), and(X, or(X, implies(or(X, Y), or(Y, Z))))), X))
% 69.32/9.34 = { by axiom 20 (modus_ponens_2) R->L }
% 69.32/9.34 equiv(implies(or(X, Y), or(Y, Z)), implies(implies(fresh(fresh28(true, true, or(X, implies(or(Y, X), or(Y, Z)))), true, or(X, implies(or(Y, X), or(Y, Z))), implies(V, V)), and(X, or(X, implies(or(X, Y), or(Y, Z))))), X))
% 69.32/9.34 = { by lemma 111 R->L }
% 69.32/9.34 equiv(implies(or(X, Y), or(Y, Z)), implies(implies(fresh(fresh28(is_a_theorem(implies(implies(and(not(not(not(Z))), not(Z)), not(X)), implies(implies(not(X), not(not(Y))), implies(and(not(not(not(Z))), not(Z)), not(not(Y)))))), true, or(X, implies(or(Y, X), or(Y, Z)))), true, or(X, implies(or(Y, X), or(Y, Z))), implies(V, V)), and(X, or(X, implies(or(X, Y), or(Y, Z))))), X))
% 69.32/9.34 = { by lemma 75 }
% 69.32/9.34 equiv(implies(or(X, Y), or(Y, Z)), implies(implies(fresh(fresh28(is_a_theorem(implies(implies(and(not(not(not(Z))), not(Z)), not(X)), implies(or(X, not(not(Y))), implies(and(not(not(not(Z))), not(Z)), not(not(Y)))))), true, or(X, implies(or(Y, X), or(Y, Z)))), true, or(X, implies(or(Y, X), or(Y, Z))), implies(V, V)), and(X, or(X, implies(or(X, Y), or(Y, Z))))), X))
% 69.32/9.34 = { by lemma 84 }
% 69.32/9.34 equiv(implies(or(X, Y), or(Y, Z)), implies(implies(fresh(fresh28(is_a_theorem(or(and(and(not(not(not(Z))), not(Z)), X), implies(or(X, not(not(Y))), implies(and(not(not(not(Z))), not(Z)), not(not(Y)))))), true, or(X, implies(or(Y, X), or(Y, Z)))), true, or(X, implies(or(Y, X), or(Y, Z))), implies(V, V)), and(X, or(X, implies(or(X, Y), or(Y, Z))))), X))
% 69.32/9.34 = { by lemma 108 }
% 69.32/9.34 equiv(implies(or(X, Y), or(Y, Z)), implies(implies(fresh(fresh28(is_a_theorem(or(and(and(not(not(not(Z))), not(Z)), X), implies(or(X, not(not(Y))), implies(not(Y), not(and(not(not(not(Z))), not(Z))))))), true, or(X, implies(or(Y, X), or(Y, Z)))), true, or(X, implies(or(Y, X), or(Y, Z))), implies(V, V)), and(X, or(X, implies(or(X, Y), or(Y, Z))))), X))
% 69.32/9.34 = { by lemma 104 }
% 69.32/9.34 equiv(implies(or(X, Y), or(Y, Z)), implies(implies(fresh(fresh28(is_a_theorem(or(implies(or(X, not(not(Y))), implies(not(Y), not(and(not(not(not(Z))), not(Z))))), and(and(not(not(not(Z))), not(Z)), X))), true, or(X, implies(or(Y, X), or(Y, Z)))), true, or(X, implies(or(Y, X), or(Y, Z))), implies(V, V)), and(X, or(X, implies(or(X, Y), or(Y, Z))))), X))
% 69.32/9.34 = { by lemma 74 R->L }
% 69.32/9.34 equiv(implies(or(X, Y), or(Y, Z)), implies(implies(fresh(fresh28(is_a_theorem(or(not(and(or(X, not(not(Y))), not(implies(not(Y), not(and(not(not(not(Z))), not(Z))))))), and(and(not(not(not(Z))), not(Z)), X))), true, or(X, implies(or(Y, X), or(Y, Z)))), true, or(X, implies(or(Y, X), or(Y, Z))), implies(V, V)), and(X, or(X, implies(or(X, Y), or(Y, Z))))), X))
% 69.32/9.34 = { by lemma 72 }
% 69.32/9.34 equiv(implies(or(X, Y), or(Y, Z)), implies(implies(fresh(fresh28(is_a_theorem(implies(and(or(X, not(not(Y))), not(implies(not(Y), not(and(not(not(not(Z))), not(Z)))))), and(and(not(not(not(Z))), not(Z)), X))), true, or(X, implies(or(Y, X), or(Y, Z)))), true, or(X, implies(or(Y, X), or(Y, Z))), implies(V, V)), and(X, or(X, implies(or(X, Y), or(Y, Z))))), X))
% 69.32/9.34 = { by lemma 73 }
% 69.32/9.34 equiv(implies(or(X, Y), or(Y, Z)), implies(implies(fresh(fresh28(is_a_theorem(implies(and(or(X, not(not(Y))), and(not(Y), and(not(not(not(Z))), not(Z)))), and(and(not(not(not(Z))), not(Z)), X))), true, or(X, implies(or(Y, X), or(Y, Z)))), true, or(X, implies(or(Y, X), or(Y, Z))), implies(V, V)), and(X, or(X, implies(or(X, Y), or(Y, Z))))), X))
% 69.32/9.34 = { by lemma 97 R->L }
% 69.32/9.34 equiv(implies(or(X, Y), or(Y, Z)), implies(implies(fresh(fresh28(is_a_theorem(implies(and(or(X, not(not(Y))), and(and(not(not(not(Z))), not(Z)), not(Y))), and(and(not(not(not(Z))), not(Z)), X))), true, or(X, implies(or(Y, X), or(Y, Z)))), true, or(X, implies(or(Y, X), or(Y, Z))), implies(V, V)), and(X, or(X, implies(or(X, Y), or(Y, Z))))), X))
% 69.32/9.34 = { by lemma 103 }
% 69.32/9.34 equiv(implies(or(X, Y), or(Y, Z)), implies(implies(fresh(fresh28(is_a_theorem(implies(and(implies(not(Y), X), and(and(not(not(not(Z))), not(Z)), not(Y))), and(and(not(not(not(Z))), not(Z)), X))), true, or(X, implies(or(Y, X), or(Y, Z)))), true, or(X, implies(or(Y, X), or(Y, Z))), implies(V, V)), and(X, or(X, implies(or(X, Y), or(Y, Z))))), X))
% 69.32/9.34 = { by lemma 97 R->L }
% 69.32/9.34 equiv(implies(or(X, Y), or(Y, Z)), implies(implies(fresh(fresh28(is_a_theorem(implies(and(implies(not(Y), X), and(not(Y), and(not(not(not(Z))), not(Z)))), and(and(not(not(not(Z))), not(Z)), X))), true, or(X, implies(or(Y, X), or(Y, Z)))), true, or(X, implies(or(Y, X), or(Y, Z))), implies(V, V)), and(X, or(X, implies(or(X, Y), or(Y, Z))))), X))
% 69.32/9.35 = { by lemma 97 R->L }
% 69.32/9.35 equiv(implies(or(X, Y), or(Y, Z)), implies(implies(fresh(fresh28(is_a_theorem(implies(and(implies(not(Y), X), and(not(Y), and(not(not(not(Z))), not(Z)))), and(X, and(not(not(not(Z))), not(Z))))), true, or(X, implies(or(Y, X), or(Y, Z)))), true, or(X, implies(or(Y, X), or(Y, Z))), implies(V, V)), and(X, or(X, implies(or(X, Y), or(Y, Z))))), X))
% 69.32/9.35 = { by lemma 95 R->L }
% 69.32/9.35 equiv(implies(or(X, Y), or(Y, Z)), implies(implies(fresh(fresh28(is_a_theorem(or(implies(implies(not(Y), X), not(and(not(Y), and(not(not(not(Z))), not(Z))))), and(X, and(not(not(not(Z))), not(Z))))), true, or(X, implies(or(Y, X), or(Y, Z)))), true, or(X, implies(or(Y, X), or(Y, Z))), implies(V, V)), and(X, or(X, implies(or(X, Y), or(Y, Z))))), X))
% 69.32/9.35 = { by lemma 73 R->L }
% 69.32/9.35 equiv(implies(or(X, Y), or(Y, Z)), implies(implies(fresh(fresh28(is_a_theorem(or(implies(implies(not(Y), X), not(and(not(Y), not(implies(not(not(not(Z))), not(not(Z))))))), and(X, and(not(not(not(Z))), not(Z))))), true, or(X, implies(or(Y, X), or(Y, Z)))), true, or(X, implies(or(Y, X), or(Y, Z))), implies(V, V)), and(X, or(X, implies(or(X, Y), or(Y, Z))))), X))
% 69.32/9.35 = { by lemma 74 }
% 69.32/9.35 equiv(implies(or(X, Y), or(Y, Z)), implies(implies(fresh(fresh28(is_a_theorem(or(implies(implies(not(Y), X), implies(not(Y), implies(not(not(not(Z))), not(not(Z))))), and(X, and(not(not(not(Z))), not(Z))))), true, or(X, implies(or(Y, X), or(Y, Z)))), true, or(X, implies(or(Y, X), or(Y, Z))), implies(V, V)), and(X, or(X, implies(or(X, Y), or(Y, Z))))), X))
% 69.32/9.35 = { by lemma 104 }
% 69.32/9.35 equiv(implies(or(X, Y), or(Y, Z)), implies(implies(fresh(fresh28(is_a_theorem(or(and(X, and(not(not(not(Z))), not(Z))), implies(implies(not(Y), X), implies(not(Y), implies(not(not(not(Z))), not(not(Z))))))), true, or(X, implies(or(Y, X), or(Y, Z)))), true, or(X, implies(or(Y, X), or(Y, Z))), implies(V, V)), and(X, or(X, implies(or(X, Y), or(Y, Z))))), X))
% 69.32/9.35 = { by lemma 73 R->L }
% 69.32/9.35 equiv(implies(or(X, Y), or(Y, Z)), implies(implies(fresh(fresh28(is_a_theorem(or(and(X, not(implies(not(not(not(Z))), not(not(Z))))), implies(implies(not(Y), X), implies(not(Y), implies(not(not(not(Z))), not(not(Z))))))), true, or(X, implies(or(Y, X), or(Y, Z)))), true, or(X, implies(or(Y, X), or(Y, Z))), implies(V, V)), and(X, or(X, implies(or(X, Y), or(Y, Z))))), X))
% 69.32/9.35 = { by lemma 76 }
% 69.32/9.35 equiv(implies(or(X, Y), or(Y, Z)), implies(implies(fresh(fresh28(is_a_theorem(implies(implies(X, implies(not(not(not(Z))), not(not(Z)))), implies(implies(not(Y), X), implies(not(Y), implies(not(not(not(Z))), not(not(Z))))))), true, or(X, implies(or(Y, X), or(Y, Z)))), true, or(X, implies(or(Y, X), or(Y, Z))), implies(V, V)), and(X, or(X, implies(or(X, Y), or(Y, Z))))), X))
% 69.32/9.35 = { by lemma 72 R->L }
% 69.32/9.35 equiv(implies(or(X, Y), or(Y, Z)), implies(implies(fresh(fresh28(is_a_theorem(implies(implies(X, implies(not(not(not(Z))), not(not(Z)))), implies(implies(not(Y), X), or(not(not(Y)), implies(not(not(not(Z))), not(not(Z))))))), true, or(X, implies(or(Y, X), or(Y, Z)))), true, or(X, implies(or(Y, X), or(Y, Z))), implies(V, V)), and(X, or(X, implies(or(X, Y), or(Y, Z))))), X))
% 69.32/9.35 = { by lemma 72 R->L }
% 69.32/9.35 equiv(implies(or(X, Y), or(Y, Z)), implies(implies(fresh(fresh28(is_a_theorem(implies(implies(X, implies(not(not(not(Z))), not(not(Z)))), implies(or(not(not(Y)), X), or(not(not(Y)), implies(not(not(not(Z))), not(not(Z))))))), true, or(X, implies(or(Y, X), or(Y, Z)))), true, or(X, implies(or(Y, X), or(Y, Z))), implies(V, V)), and(X, or(X, implies(or(X, Y), or(Y, Z))))), X))
% 69.32/9.35 = { by axiom 56 (r5_1) R->L }
% 69.32/9.35 equiv(implies(or(X, Y), or(Y, Z)), implies(implies(fresh(fresh28(fresh4(r5, true, not(not(Y)), X, implies(not(not(not(Z))), not(not(Z)))), true, or(X, implies(or(Y, X), or(Y, Z)))), true, or(X, implies(or(Y, X), or(Y, Z))), implies(V, V)), and(X, or(X, implies(or(X, Y), or(Y, Z))))), X))
% 69.32/9.35 = { by lemma 74 R->L }
% 69.32/9.35 equiv(implies(or(X, Y), or(Y, Z)), implies(implies(fresh(fresh28(fresh4(r5, true, not(not(Y)), X, not(and(not(not(not(Z))), not(not(not(Z)))))), true, or(X, implies(or(Y, X), or(Y, Z)))), true, or(X, implies(or(Y, X), or(Y, Z))), implies(V, V)), and(X, or(X, implies(or(X, Y), or(Y, Z))))), X))
% 69.32/9.35 = { by lemma 66 }
% 69.32/9.35 equiv(implies(or(X, Y), or(Y, Z)), implies(implies(fresh(fresh28(fresh4(r5, true, not(not(Y)), X, not(not(not(not(Z))))), true, or(X, implies(or(Y, X), or(Y, Z)))), true, or(X, implies(or(Y, X), or(Y, Z))), implies(V, V)), and(X, or(X, implies(or(X, Y), or(Y, Z))))), X))
% 69.32/9.35 = { by lemma 91 }
% 69.32/9.35 equiv(implies(or(X, Y), or(Y, Z)), implies(implies(fresh(fresh28(fresh4(r5, true, not(not(Y)), X, not(not(Z))), true, or(X, implies(or(Y, X), or(Y, Z)))), true, or(X, implies(or(Y, X), or(Y, Z))), implies(V, V)), and(X, or(X, implies(or(X, Y), or(Y, Z))))), X))
% 69.32/9.35 = { by lemma 91 }
% 69.32/9.35 equiv(implies(or(X, Y), or(Y, Z)), implies(implies(fresh(fresh28(fresh4(r5, true, Y, X, not(not(Z))), true, or(X, implies(or(Y, X), or(Y, Z)))), true, or(X, implies(or(Y, X), or(Y, Z))), implies(V, V)), and(X, or(X, implies(or(X, Y), or(Y, Z))))), X))
% 69.32/9.36 = { by lemma 91 }
% 69.32/9.36 equiv(implies(or(X, Y), or(Y, Z)), implies(implies(fresh(fresh28(fresh4(r5, true, Y, X, Z), true, or(X, implies(or(Y, X), or(Y, Z)))), true, or(X, implies(or(Y, X), or(Y, Z))), implies(V, V)), and(X, or(X, implies(or(X, Y), or(Y, Z))))), X))
% 69.32/9.36 = { by axiom 56 (r5_1) }
% 69.32/9.36 equiv(implies(or(X, Y), or(Y, Z)), implies(implies(fresh(fresh28(is_a_theorem(implies(implies(X, Z), implies(or(Y, X), or(Y, Z)))), true, or(X, implies(or(Y, X), or(Y, Z)))), true, or(X, implies(or(Y, X), or(Y, Z))), implies(V, V)), and(X, or(X, implies(or(X, Y), or(Y, Z))))), X))
% 69.32/9.36 = { by lemma 115 R->L }
% 69.32/9.36 equiv(implies(or(X, Y), or(Y, Z)), implies(implies(fresh(fresh28(is_a_theorem(implies(implies(X, Z), implies(or(Y, X), or(Y, Z)))), true, or(implies(implies(X, Z), X), implies(or(Y, X), or(Y, Z)))), true, or(X, implies(or(Y, X), or(Y, Z))), implies(V, V)), and(X, or(X, implies(or(X, Y), or(Y, Z))))), X))
% 69.32/9.36 = { by lemma 128 R->L }
% 69.32/9.36 equiv(implies(or(X, Y), or(Y, Z)), implies(implies(fresh(fresh28(is_a_theorem(implies(implies(X, Z), implies(or(Y, X), or(Y, Z)))), true, or(implies(and(implies(X, Z), or(implies(X, Z), not(implies(implies(X, Z), X)))), X), implies(or(Y, X), or(Y, Z)))), true, or(X, implies(or(Y, X), or(Y, Z))), implies(V, V)), and(X, or(X, implies(or(X, Y), or(Y, Z))))), X))
% 69.32/9.36 = { by lemma 103 }
% 69.32/9.36 equiv(implies(or(X, Y), or(Y, Z)), implies(implies(fresh(fresh28(is_a_theorem(implies(implies(X, Z), implies(or(Y, X), or(Y, Z)))), true, or(implies(and(implies(X, Z), implies(implies(implies(X, Z), X), implies(X, Z))), X), implies(or(Y, X), or(Y, Z)))), true, or(X, implies(or(Y, X), or(Y, Z))), implies(V, V)), and(X, or(X, implies(or(X, Y), or(Y, Z))))), X))
% 69.32/9.36 = { by lemma 118 R->L }
% 69.32/9.36 equiv(implies(or(X, Y), or(Y, Z)), implies(implies(fresh(fresh28(is_a_theorem(implies(implies(X, Z), implies(or(Y, X), or(Y, Z)))), true, or(or(X, implies(implies(X, Z), not(implies(implies(implies(X, Z), X), implies(X, Z))))), implies(or(Y, X), or(Y, Z)))), true, or(X, implies(or(Y, X), or(Y, Z))), implies(V, V)), and(X, or(X, implies(or(X, Y), or(Y, Z))))), X))
% 69.32/9.36 = { by lemma 83 R->L }
% 69.32/9.36 equiv(implies(or(X, Y), or(Y, Z)), implies(implies(fresh(fresh28(is_a_theorem(implies(implies(X, Z), implies(or(Y, X), or(Y, Z)))), true, or(implies(implies(U, U), or(X, implies(implies(X, Z), not(implies(implies(implies(X, Z), X), implies(X, Z)))))), implies(or(Y, X), or(Y, Z)))), true, or(X, implies(or(Y, X), or(Y, Z))), implies(V, V)), and(X, or(X, implies(or(X, Y), or(Y, Z))))), X))
% 69.32/9.36 = { by lemma 133 R->L }
% 69.32/9.36 equiv(implies(or(X, Y), or(Y, Z)), implies(implies(fresh(fresh28(is_a_theorem(implies(implies(X, Z), implies(or(Y, X), or(Y, Z)))), true, or(implies(implies(not(implies(implies(implies(X, Z), X), implies(X, Z))), or(X, implies(implies(X, Z), not(implies(implies(implies(X, Z), X), implies(X, Z)))))), or(X, implies(implies(X, Z), not(implies(implies(implies(X, Z), X), implies(X, Z)))))), implies(or(Y, X), or(Y, Z)))), true, or(X, implies(or(Y, X), or(Y, Z))), implies(V, V)), and(X, or(X, implies(or(X, Y), or(Y, Z))))), X))
% 69.32/9.36 = { by lemma 113 R->L }
% 69.32/9.36 equiv(implies(or(X, Y), or(Y, Z)), implies(implies(fresh(fresh28(is_a_theorem(implies(implies(X, Z), implies(or(Y, X), or(Y, Z)))), true, or(or(not(implies(implies(implies(X, Z), X), implies(X, Z))), implies(implies(not(implies(implies(implies(X, Z), X), implies(X, Z))), or(X, implies(implies(X, Z), not(implies(implies(implies(X, Z), X), implies(X, Z)))))), or(X, implies(implies(X, Z), not(implies(implies(implies(X, Z), X), implies(X, Z))))))), implies(or(Y, X), or(Y, Z)))), true, or(X, implies(or(Y, X), or(Y, Z))), implies(V, V)), and(X, or(X, implies(or(X, Y), or(Y, Z))))), X))
% 69.32/9.36 = { by lemma 133 }
% 69.32/9.36 equiv(implies(or(X, Y), or(Y, Z)), implies(implies(fresh(fresh28(is_a_theorem(implies(implies(X, Z), implies(or(Y, X), or(Y, Z)))), true, or(or(not(implies(implies(implies(X, Z), X), implies(X, Z))), implies(implies(T, T), or(X, implies(implies(X, Z), not(implies(implies(implies(X, Z), X), implies(X, Z))))))), implies(or(Y, X), or(Y, Z)))), true, or(X, implies(or(Y, X), or(Y, Z))), implies(V, V)), and(X, or(X, implies(or(X, Y), or(Y, Z))))), X))
% 69.32/9.36 = { by lemma 83 }
% 69.32/9.36 equiv(implies(or(X, Y), or(Y, Z)), implies(implies(fresh(fresh28(is_a_theorem(implies(implies(X, Z), implies(or(Y, X), or(Y, Z)))), true, or(or(not(implies(implies(implies(X, Z), X), implies(X, Z))), or(X, implies(implies(X, Z), not(implies(implies(implies(X, Z), X), implies(X, Z)))))), implies(or(Y, X), or(Y, Z)))), true, or(X, implies(or(Y, X), or(Y, Z))), implies(V, V)), and(X, or(X, implies(or(X, Y), or(Y, Z))))), X))
% 69.32/9.36 = { by lemma 102 R->L }
% 69.32/9.36 equiv(implies(or(X, Y), or(Y, Z)), implies(implies(fresh(fresh28(is_a_theorem(implies(implies(X, Z), implies(or(Y, X), or(Y, Z)))), true, or(or(not(implies(implies(implies(X, Z), X), implies(X, Z))), or(X, implies(implies(X, Z), and(not(implies(X, Z)), implies(implies(X, Z), X))))), implies(or(Y, X), or(Y, Z)))), true, or(X, implies(or(Y, X), or(Y, Z))), implies(V, V)), and(X, or(X, implies(or(X, Y), or(Y, Z))))), X))
% 69.32/9.36 = { by lemma 72 R->L }
% 69.32/9.36 equiv(implies(or(X, Y), or(Y, Z)), implies(implies(fresh(fresh28(is_a_theorem(implies(implies(X, Z), implies(or(Y, X), or(Y, Z)))), true, or(or(not(implies(implies(implies(X, Z), X), implies(X, Z))), or(X, or(not(implies(X, Z)), and(not(implies(X, Z)), implies(implies(X, Z), X))))), implies(or(Y, X), or(Y, Z)))), true, or(X, implies(or(Y, X), or(Y, Z))), implies(V, V)), and(X, or(X, implies(or(X, Y), or(Y, Z))))), X))
% 69.32/9.36 = { by lemma 114 }
% 69.32/9.36 equiv(implies(or(X, Y), or(Y, Z)), implies(implies(fresh(fresh28(is_a_theorem(implies(implies(X, Z), implies(or(Y, X), or(Y, Z)))), true, or(or(not(implies(implies(implies(X, Z), X), implies(X, Z))), or(X, not(implies(X, Z)))), implies(or(Y, X), or(Y, Z)))), true, or(X, implies(or(Y, X), or(Y, Z))), implies(V, V)), and(X, or(X, implies(or(X, Y), or(Y, Z))))), X))
% 69.32/9.36 = { by lemma 72 }
% 69.32/9.36 equiv(implies(or(X, Y), or(Y, Z)), implies(implies(fresh(fresh28(is_a_theorem(implies(implies(X, Z), implies(or(Y, X), or(Y, Z)))), true, or(implies(implies(implies(implies(X, Z), X), implies(X, Z)), or(X, not(implies(X, Z)))), implies(or(Y, X), or(Y, Z)))), true, or(X, implies(or(Y, X), or(Y, Z))), implies(V, V)), and(X, or(X, implies(or(X, Y), or(Y, Z))))), X))
% 69.32/9.36 = { by lemma 103 }
% 69.32/9.36 equiv(implies(or(X, Y), or(Y, Z)), implies(implies(fresh(fresh28(is_a_theorem(implies(implies(X, Z), implies(or(Y, X), or(Y, Z)))), true, or(implies(implies(implies(implies(X, Z), X), implies(X, Z)), implies(implies(X, Z), X)), implies(or(Y, X), or(Y, Z)))), true, or(X, implies(or(Y, X), or(Y, Z))), implies(V, V)), and(X, or(X, implies(or(X, Y), or(Y, Z))))), X))
% 69.32/9.36 = { by lemma 103 R->L }
% 69.32/9.36 equiv(implies(or(X, Y), or(Y, Z)), implies(implies(fresh(fresh28(is_a_theorem(implies(implies(X, Z), implies(or(Y, X), or(Y, Z)))), true, or(or(implies(implies(X, Z), X), not(implies(implies(implies(X, Z), X), implies(X, Z)))), implies(or(Y, X), or(Y, Z)))), true, or(X, implies(or(Y, X), or(Y, Z))), implies(V, V)), and(X, or(X, implies(or(X, Y), or(Y, Z))))), X))
% 69.32/9.36 = { by lemma 131 R->L }
% 69.32/9.37 equiv(implies(or(X, Y), or(Y, Z)), implies(implies(fresh(fresh28(is_a_theorem(implies(implies(X, Z), implies(or(Y, X), or(Y, Z)))), true, or(implies(implies(implies(implies(X, Z), X), not(implies(implies(implies(X, Z), X), implies(X, Z)))), not(implies(implies(implies(X, Z), X), implies(X, Z)))), implies(or(Y, X), or(Y, Z)))), true, or(X, implies(or(Y, X), or(Y, Z))), implies(V, V)), and(X, or(X, implies(or(X, Y), or(Y, Z))))), X))
% 69.32/9.37 = { by lemma 121 }
% 69.32/9.37 equiv(implies(or(X, Y), or(Y, Z)), implies(implies(fresh(fresh28(is_a_theorem(implies(implies(X, Z), implies(or(Y, X), or(Y, Z)))), true, or(implies(not(equiv(implies(X, Z), implies(implies(X, Z), X))), not(implies(implies(implies(X, Z), X), implies(X, Z)))), implies(or(Y, X), or(Y, Z)))), true, or(X, implies(or(Y, X), or(Y, Z))), implies(V, V)), and(X, or(X, implies(or(X, Y), or(Y, Z))))), X))
% 69.32/9.37 = { by lemma 75 }
% 69.32/9.37 equiv(implies(or(X, Y), or(Y, Z)), implies(implies(fresh(fresh28(is_a_theorem(implies(implies(X, Z), implies(or(Y, X), or(Y, Z)))), true, or(or(equiv(implies(X, Z), implies(implies(X, Z), X)), not(implies(implies(implies(X, Z), X), implies(X, Z)))), implies(or(Y, X), or(Y, Z)))), true, or(X, implies(or(Y, X), or(Y, Z))), implies(V, V)), and(X, or(X, implies(or(X, Y), or(Y, Z))))), X))
% 69.32/9.37 = { by lemma 103 }
% 69.32/9.37 equiv(implies(or(X, Y), or(Y, Z)), implies(implies(fresh(fresh28(is_a_theorem(implies(implies(X, Z), implies(or(Y, X), or(Y, Z)))), true, or(implies(implies(implies(implies(X, Z), X), implies(X, Z)), equiv(implies(X, Z), implies(implies(X, Z), X))), implies(or(Y, X), or(Y, Z)))), true, or(X, implies(or(Y, X), or(Y, Z))), implies(V, V)), and(X, or(X, implies(or(X, Y), or(Y, Z))))), X))
% 69.32/9.37 = { by lemma 115 }
% 69.32/9.37 equiv(implies(or(X, Y), or(Y, Z)), implies(implies(fresh(fresh28(is_a_theorem(implies(implies(X, Z), implies(or(Y, X), or(Y, Z)))), true, or(implies(implies(X, Z), equiv(implies(X, Z), implies(implies(X, Z), X))), implies(or(Y, X), or(Y, Z)))), true, or(X, implies(or(Y, X), or(Y, Z))), implies(V, V)), and(X, or(X, implies(or(X, Y), or(Y, Z))))), X))
% 69.32/9.37 = { by lemma 117 }
% 69.32/9.37 equiv(implies(or(X, Y), or(Y, Z)), implies(implies(fresh(fresh28(is_a_theorem(implies(implies(X, Z), implies(or(Y, X), or(Y, Z)))), true, or(implies(implies(X, Z), and(implies(X, Z), implies(implies(X, Z), X))), implies(or(Y, X), or(Y, Z)))), true, or(X, implies(or(Y, X), or(Y, Z))), implies(V, V)), and(X, or(X, implies(or(X, Y), or(Y, Z))))), X))
% 69.32/9.37 = { by lemma 127 }
% 69.32/9.37 equiv(implies(or(X, Y), or(Y, Z)), implies(implies(fresh(fresh28(is_a_theorem(implies(implies(X, Z), implies(or(Y, X), or(Y, Z)))), true, or(implies(implies(X, Z), and(implies(X, Z), X)), implies(or(Y, X), or(Y, Z)))), true, or(X, implies(or(Y, X), or(Y, Z))), implies(V, V)), and(X, or(X, implies(or(X, Y), or(Y, Z))))), X))
% 69.32/9.37 = { by lemma 75 R->L }
% 69.32/9.37 equiv(implies(or(X, Y), or(Y, Z)), implies(implies(fresh(fresh28(is_a_theorem(implies(implies(X, Z), implies(or(Y, X), or(Y, Z)))), true, implies(not(implies(implies(X, Z), and(implies(X, Z), X))), implies(or(Y, X), or(Y, Z)))), true, or(X, implies(or(Y, X), or(Y, Z))), implies(V, V)), and(X, or(X, implies(or(X, Y), or(Y, Z))))), X))
% 69.32/9.37 = { by lemma 119 R->L }
% 69.32/9.37 equiv(implies(or(X, Y), or(Y, Z)), implies(implies(fresh(fresh28(is_a_theorem(implies(implies(X, Z), implies(or(Y, X), or(Y, Z)))), true, implies(and(implies(X, Z), implies(implies(X, Z), not(X))), implies(or(Y, X), or(Y, Z)))), true, or(X, implies(or(Y, X), or(Y, Z))), implies(V, V)), and(X, or(X, implies(or(X, Y), or(Y, Z))))), X))
% 69.32/9.37 = { by lemma 117 R->L }
% 69.32/9.37 equiv(implies(or(X, Y), or(Y, Z)), implies(implies(fresh(fresh28(is_a_theorem(implies(implies(X, Z), implies(or(Y, X), or(Y, Z)))), true, implies(equiv(implies(X, Z), implies(implies(X, Z), not(X))), implies(or(Y, X), or(Y, Z)))), true, or(X, implies(or(Y, X), or(Y, Z))), implies(V, V)), and(X, or(X, implies(or(X, Y), or(Y, Z))))), X))
% 69.32/9.37 = { by lemma 108 R->L }
% 69.32/9.37 equiv(implies(or(X, Y), or(Y, Z)), implies(implies(fresh(fresh28(is_a_theorem(implies(implies(X, Z), implies(or(Y, X), or(Y, Z)))), true, implies(equiv(implies(X, Z), implies(X, not(implies(X, Z)))), implies(or(Y, X), or(Y, Z)))), true, or(X, implies(or(Y, X), or(Y, Z))), implies(V, V)), and(X, or(X, implies(or(X, Y), or(Y, Z))))), X))
% 69.32/9.37 = { by lemma 100 R->L }
% 69.32/9.37 equiv(implies(or(X, Y), or(Y, Z)), implies(implies(fresh(fresh28(is_a_theorem(implies(implies(X, Z), implies(or(Y, X), or(Y, Z)))), true, implies(equiv(implies(X, Z), not(and(X, implies(X, Z)))), implies(or(Y, X), or(Y, Z)))), true, or(X, implies(or(Y, X), or(Y, Z))), implies(V, V)), and(X, or(X, implies(or(X, Y), or(Y, Z))))), X))
% 69.32/9.37 = { by lemma 103 R->L }
% 69.32/9.37 equiv(implies(or(X, Y), or(Y, Z)), implies(implies(fresh(fresh28(is_a_theorem(implies(implies(X, Z), implies(or(Y, X), or(Y, Z)))), true, or(implies(or(Y, X), or(Y, Z)), not(equiv(implies(X, Z), not(and(X, implies(X, Z))))))), true, or(X, implies(or(Y, X), or(Y, Z))), implies(V, V)), and(X, or(X, implies(or(X, Y), or(Y, Z))))), X))
% 69.32/9.37 = { by lemma 114 R->L }
% 69.32/9.37 equiv(implies(or(X, Y), or(Y, Z)), implies(implies(fresh(fresh28(is_a_theorem(implies(or(implies(X, Z), and(implies(X, Z), X)), implies(or(Y, X), or(Y, Z)))), true, or(implies(or(Y, X), or(Y, Z)), not(equiv(implies(X, Z), not(and(X, implies(X, Z))))))), true, or(X, implies(or(Y, X), or(Y, Z))), implies(V, V)), and(X, or(X, implies(or(X, Y), or(Y, Z))))), X))
% 69.32/9.37 = { by lemma 97 R->L }
% 69.32/9.37 equiv(implies(or(X, Y), or(Y, Z)), implies(implies(fresh(fresh28(is_a_theorem(implies(or(implies(X, Z), and(X, implies(X, Z))), implies(or(Y, X), or(Y, Z)))), true, or(implies(or(Y, X), or(Y, Z)), not(equiv(implies(X, Z), not(and(X, implies(X, Z))))))), true, or(X, implies(or(Y, X), or(Y, Z))), implies(V, V)), and(X, or(X, implies(or(X, Y), or(Y, Z))))), X))
% 69.32/9.37 = { by lemma 117 R->L }
% 69.32/9.37 equiv(implies(or(X, Y), or(Y, Z)), implies(implies(fresh(fresh28(is_a_theorem(implies(or(implies(X, Z), equiv(X, implies(X, Z))), implies(or(Y, X), or(Y, Z)))), true, or(implies(or(Y, X), or(Y, Z)), not(equiv(implies(X, Z), not(and(X, implies(X, Z))))))), true, or(X, implies(or(Y, X), or(Y, Z))), implies(V, V)), and(X, or(X, implies(or(X, Y), or(Y, Z))))), X))
% 69.32/9.37 = { by lemma 117 R->L }
% 69.32/9.37 equiv(implies(or(X, Y), or(Y, Z)), implies(implies(fresh(fresh28(is_a_theorem(implies(or(implies(X, Z), equiv(X, implies(X, Z))), implies(or(Y, X), or(Y, Z)))), true, or(implies(or(Y, X), or(Y, Z)), not(equiv(implies(X, Z), not(equiv(X, implies(X, Z))))))), true, or(X, implies(or(Y, X), or(Y, Z))), implies(V, V)), and(X, or(X, implies(or(X, Y), or(Y, Z))))), X))
% 69.32/9.37 = { by lemma 106 R->L }
% 69.32/9.37 equiv(implies(or(X, Y), or(Y, Z)), implies(implies(fresh(fresh28(is_a_theorem(implies(or(implies(X, Z), equiv(X, implies(X, Z))), implies(or(Y, X), or(Y, Z)))), true, or(implies(or(Y, X), or(Y, Z)), not(equiv(equiv(X, implies(X, Z)), not(implies(X, Z)))))), true, or(X, implies(or(Y, X), or(Y, Z))), implies(V, V)), and(X, or(X, implies(or(X, Y), or(Y, Z))))), X))
% 69.32/9.37 = { by lemma 135 R->L }
% 69.32/9.37 equiv(implies(or(X, Y), or(Y, Z)), implies(implies(fresh(fresh28(is_a_theorem(implies(or(implies(X, Z), equiv(X, implies(X, Z))), implies(or(Y, X), or(Y, Z)))), true, or(implies(or(Y, X), or(Y, Z)), not(equiv(and(implies(X, Z), implies(implies(X, Z), X)), not(implies(X, Z)))))), true, or(X, implies(or(Y, X), or(Y, Z))), implies(V, V)), and(X, or(X, implies(or(X, Y), or(Y, Z))))), X))
% 69.32/9.37 = { by lemma 107 R->L }
% 69.32/9.37 equiv(implies(or(X, Y), or(Y, Z)), implies(implies(fresh(fresh28(is_a_theorem(implies(or(implies(X, Z), equiv(X, implies(X, Z))), implies(or(Y, X), or(Y, Z)))), true, or(implies(or(Y, X), or(Y, Z)), not(equiv(not(and(implies(X, Z), implies(implies(X, Z), X))), implies(X, Z))))), true, or(X, implies(or(Y, X), or(Y, Z))), implies(V, V)), and(X, or(X, implies(or(X, Y), or(Y, Z))))), X))
% 69.32/9.37 = { by lemma 100 }
% 69.32/9.37 equiv(implies(or(X, Y), or(Y, Z)), implies(implies(fresh(fresh28(is_a_theorem(implies(or(implies(X, Z), equiv(X, implies(X, Z))), implies(or(Y, X), or(Y, Z)))), true, or(implies(or(Y, X), or(Y, Z)), not(equiv(implies(implies(X, Z), not(implies(implies(X, Z), X))), implies(X, Z))))), true, or(X, implies(or(Y, X), or(Y, Z))), implies(V, V)), and(X, or(X, implies(or(X, Y), or(Y, Z))))), X))
% 69.32/9.38 = { by lemma 120 R->L }
% 69.32/9.38 equiv(implies(or(X, Y), or(Y, Z)), implies(implies(fresh(fresh28(is_a_theorem(implies(or(implies(X, Z), equiv(X, implies(X, Z))), implies(or(Y, X), or(Y, Z)))), true, or(implies(or(Y, X), or(Y, Z)), implies(implies(implies(implies(X, Z), not(implies(implies(X, Z), X))), implies(X, Z)), not(implies(implies(X, Z), implies(implies(X, Z), not(implies(implies(X, Z), X)))))))), true, or(X, implies(or(Y, X), or(Y, Z))), implies(V, V)), and(X, or(X, implies(or(X, Y), or(Y, Z))))), X))
% 69.32/9.38 = { by lemma 116 }
% 69.32/9.38 equiv(implies(or(X, Y), or(Y, Z)), implies(implies(fresh(fresh28(is_a_theorem(implies(or(implies(X, Z), equiv(X, implies(X, Z))), implies(or(Y, X), or(Y, Z)))), true, or(implies(or(Y, X), or(Y, Z)), implies(implies(implies(implies(X, Z), not(implies(implies(X, Z), X))), implies(X, Z)), not(implies(implies(X, Z), not(implies(implies(X, Z), X))))))), true, or(X, implies(or(Y, X), or(Y, Z))), implies(V, V)), and(X, or(X, implies(or(X, Y), or(Y, Z))))), X))
% 69.32/9.38 = { by lemma 121 }
% 69.32/9.38 equiv(implies(or(X, Y), or(Y, Z)), implies(implies(fresh(fresh28(is_a_theorem(implies(or(implies(X, Z), equiv(X, implies(X, Z))), implies(or(Y, X), or(Y, Z)))), true, or(implies(or(Y, X), or(Y, Z)), implies(implies(not(equiv(X, implies(X, Z))), implies(X, Z)), not(implies(implies(X, Z), not(implies(implies(X, Z), X))))))), true, or(X, implies(or(Y, X), or(Y, Z))), implies(V, V)), and(X, or(X, implies(or(X, Y), or(Y, Z))))), X))
% 69.32/9.38 = { by lemma 75 }
% 69.32/9.38 equiv(implies(or(X, Y), or(Y, Z)), implies(implies(fresh(fresh28(is_a_theorem(implies(or(implies(X, Z), equiv(X, implies(X, Z))), implies(or(Y, X), or(Y, Z)))), true, or(implies(or(Y, X), or(Y, Z)), implies(or(equiv(X, implies(X, Z)), implies(X, Z)), not(implies(implies(X, Z), not(implies(implies(X, Z), X))))))), true, or(X, implies(or(Y, X), or(Y, Z))), implies(V, V)), and(X, or(X, implies(or(X, Y), or(Y, Z))))), X))
% 69.32/9.38 = { by lemma 73 }
% 69.32/9.38 equiv(implies(or(X, Y), or(Y, Z)), implies(implies(fresh(fresh28(is_a_theorem(implies(or(implies(X, Z), equiv(X, implies(X, Z))), implies(or(Y, X), or(Y, Z)))), true, or(implies(or(Y, X), or(Y, Z)), implies(or(equiv(X, implies(X, Z)), implies(X, Z)), and(implies(X, Z), implies(implies(X, Z), X))))), true, or(X, implies(or(Y, X), or(Y, Z))), implies(V, V)), and(X, or(X, implies(or(X, Y), or(Y, Z))))), X))
% 69.32/9.38 = { by lemma 135 }
% 69.32/9.38 equiv(implies(or(X, Y), or(Y, Z)), implies(implies(fresh(fresh28(is_a_theorem(implies(or(implies(X, Z), equiv(X, implies(X, Z))), implies(or(Y, X), or(Y, Z)))), true, or(implies(or(Y, X), or(Y, Z)), implies(or(equiv(X, implies(X, Z)), implies(X, Z)), equiv(X, implies(X, Z))))), true, or(X, implies(or(Y, X), or(Y, Z))), implies(V, V)), and(X, or(X, implies(or(X, Y), or(Y, Z))))), X))
% 69.32/9.38 = { by lemma 104 R->L }
% 69.32/9.38 equiv(implies(or(X, Y), or(Y, Z)), implies(implies(fresh(fresh28(is_a_theorem(implies(or(implies(X, Z), equiv(X, implies(X, Z))), implies(or(Y, X), or(Y, Z)))), true, or(implies(or(Y, X), or(Y, Z)), implies(or(implies(X, Z), equiv(X, implies(X, Z))), equiv(X, implies(X, Z))))), true, or(X, implies(or(Y, X), or(Y, Z))), implies(V, V)), and(X, or(X, implies(or(X, Y), or(Y, Z))))), X))
% 69.32/9.38 = { by lemma 131 R->L }
% 69.32/9.38 equiv(implies(or(X, Y), or(Y, Z)), implies(implies(fresh(fresh28(is_a_theorem(implies(or(implies(X, Z), equiv(X, implies(X, Z))), implies(or(Y, X), or(Y, Z)))), true, implies(implies(implies(or(Y, X), or(Y, Z)), implies(or(implies(X, Z), equiv(X, implies(X, Z))), equiv(X, implies(X, Z)))), implies(or(implies(X, Z), equiv(X, implies(X, Z))), equiv(X, implies(X, Z))))), true, or(X, implies(or(Y, X), or(Y, Z))), implies(V, V)), and(X, or(X, implies(or(X, Y), or(Y, Z))))), X))
% 69.32/9.38 = { by lemma 116 R->L }
% 69.32/9.38 equiv(implies(or(X, Y), or(Y, Z)), implies(implies(fresh(fresh28(is_a_theorem(implies(or(implies(X, Z), equiv(X, implies(X, Z))), implies(or(Y, X), or(Y, Z)))), true, implies(implies(implies(or(Y, X), or(Y, Z)), implies(or(implies(X, Z), equiv(X, implies(X, Z))), equiv(X, implies(X, Z)))), implies(or(implies(X, Z), equiv(X, implies(X, Z))), implies(or(implies(X, Z), equiv(X, implies(X, Z))), equiv(X, implies(X, Z)))))), true, or(X, implies(or(Y, X), or(Y, Z))), implies(V, V)), and(X, or(X, implies(or(X, Y), or(Y, Z))))), X))
% 69.32/9.38 = { by axiom 52 (modus_ponens_2) R->L }
% 69.32/9.38 equiv(implies(or(X, Y), or(Y, Z)), implies(implies(fresh(fresh59(is_a_theorem(implies(implies(or(implies(X, Z), equiv(X, implies(X, Z))), implies(or(Y, X), or(Y, Z))), implies(implies(implies(or(Y, X), or(Y, Z)), implies(or(implies(X, Z), equiv(X, implies(X, Z))), equiv(X, implies(X, Z)))), implies(or(implies(X, Z), equiv(X, implies(X, Z))), implies(or(implies(X, Z), equiv(X, implies(X, Z))), equiv(X, implies(X, Z))))))), true, implies(or(implies(X, Z), equiv(X, implies(X, Z))), implies(or(Y, X), or(Y, Z))), implies(implies(implies(or(Y, X), or(Y, Z)), implies(or(implies(X, Z), equiv(X, implies(X, Z))), equiv(X, implies(X, Z)))), implies(or(implies(X, Z), equiv(X, implies(X, Z))), implies(or(implies(X, Z), equiv(X, implies(X, Z))), equiv(X, implies(X, Z)))))), true, or(X, implies(or(Y, X), or(Y, Z))), implies(V, V)), and(X, or(X, implies(or(X, Y), or(Y, Z))))), X))
% 69.32/9.38 = { by lemma 111 }
% 69.32/9.38 equiv(implies(or(X, Y), or(Y, Z)), implies(implies(fresh(fresh59(true, true, implies(or(implies(X, Z), equiv(X, implies(X, Z))), implies(or(Y, X), or(Y, Z))), implies(implies(implies(or(Y, X), or(Y, Z)), implies(or(implies(X, Z), equiv(X, implies(X, Z))), equiv(X, implies(X, Z)))), implies(or(implies(X, Z), equiv(X, implies(X, Z))), implies(or(implies(X, Z), equiv(X, implies(X, Z))), equiv(X, implies(X, Z)))))), true, or(X, implies(or(Y, X), or(Y, Z))), implies(V, V)), and(X, or(X, implies(or(X, Y), or(Y, Z))))), X))
% 69.32/9.38 = { by lemma 59 }
% 69.32/9.38 equiv(implies(or(X, Y), or(Y, Z)), implies(implies(fresh(true, true, or(X, implies(or(Y, X), or(Y, Z))), implies(V, V)), and(X, or(X, implies(or(X, Y), or(Y, Z))))), X))
% 69.32/9.38 = { by axiom 21 (substitution_of_equivalents_2) }
% 69.32/9.38 equiv(implies(or(X, Y), or(Y, Z)), implies(implies(implies(V, V), and(X, or(X, implies(or(X, Y), or(Y, Z))))), X))
% 69.32/9.38 = { by lemma 83 }
% 69.32/9.38 equiv(implies(or(X, Y), or(Y, Z)), implies(and(X, or(X, implies(or(X, Y), or(Y, Z)))), X))
% 69.32/9.38 = { by lemma 128 }
% 69.32/9.38 equiv(implies(or(X, Y), or(Y, Z)), implies(X, X))
% 69.32/9.38 = { by lemma 94 R->L }
% 69.32/9.38 equiv(implies(or(X, Y), or(Y, Z)), fresh(is_a_theorem(implies(X, X)), true, implies(X, X), implies(S, S)))
% 69.32/9.38 = { by lemma 68 }
% 69.32/9.38 equiv(implies(or(X, Y), or(Y, Z)), fresh(true, true, implies(X, X), implies(S, S)))
% 69.32/9.38 = { by axiom 21 (substitution_of_equivalents_2) }
% 69.32/9.38 equiv(implies(or(X, Y), or(Y, Z)), implies(S, S))
% 69.32/9.38 = { by lemma 93 }
% 69.32/9.38 implies(or(X, Y), or(Y, Z))
% 69.32/9.38
% 69.32/9.38 Lemma 137: or(or(X, Y), or(X, Z)) = or(Z, or(X, Y)).
% 69.32/9.38 Proof:
% 69.32/9.38 or(or(X, Y), or(X, Z))
% 69.32/9.38 = { by lemma 104 }
% 69.32/9.38 or(or(X, Y), or(Z, X))
% 69.32/9.38 = { by lemma 132 R->L }
% 69.32/9.38 equiv(or(X, Y), implies(or(Z, X), or(X, Y)))
% 69.32/9.38 = { by lemma 136 R->L }
% 69.32/9.38 equiv(or(X, Y), implies(Z, implies(or(Z, X), or(X, Y))))
% 69.32/9.38 = { by lemma 93 R->L }
% 69.32/9.38 equiv(or(X, Y), equiv(implies(Z, implies(or(Z, X), or(X, Y))), implies(W, W)))
% 69.32/9.38 = { by axiom 21 (substitution_of_equivalents_2) R->L }
% 69.32/9.38 equiv(or(X, Y), equiv(implies(Z, implies(or(Z, X), or(X, Y))), fresh(true, true, implies(implies(Z, or(X, Y)), implies(Z, implies(or(Z, X), or(X, Y)))), implies(W, W))))
% 69.32/9.38 = { by lemma 111 R->L }
% 69.32/9.38 equiv(or(X, Y), equiv(implies(Z, implies(or(Z, X), or(X, Y))), fresh(is_a_theorem(implies(implies(Z, or(X, Y)), implies(implies(or(X, Y), implies(or(Z, X), or(X, Y))), implies(Z, implies(or(Z, X), or(X, Y)))))), true, implies(implies(Z, or(X, Y)), implies(Z, implies(or(Z, X), or(X, Y)))), implies(W, W))))
% 69.32/9.38 = { by lemma 122 }
% 69.32/9.38 equiv(or(X, Y), equiv(implies(Z, implies(or(Z, X), or(X, Y))), fresh(is_a_theorem(implies(implies(Z, or(X, Y)), implies(implies(V, V), implies(Z, implies(or(Z, X), or(X, Y)))))), true, implies(implies(Z, or(X, Y)), implies(Z, implies(or(Z, X), or(X, Y)))), implies(W, W))))
% 69.32/9.38 = { by lemma 83 }
% 69.32/9.38 equiv(or(X, Y), equiv(implies(Z, implies(or(Z, X), or(X, Y))), fresh(is_a_theorem(implies(implies(Z, or(X, Y)), implies(Z, implies(or(Z, X), or(X, Y))))), true, implies(implies(Z, or(X, Y)), implies(Z, implies(or(Z, X), or(X, Y)))), implies(W, W))))
% 69.32/9.38 = { by lemma 94 }
% 69.32/9.38 equiv(or(X, Y), equiv(implies(Z, implies(or(Z, X), or(X, Y))), implies(implies(Z, or(X, Y)), implies(Z, implies(or(Z, X), or(X, Y))))))
% 69.32/9.38 = { by lemma 64 R->L }
% 69.32/9.38 equiv(or(X, Y), and(implies(implies(Z, implies(or(Z, X), or(X, Y))), implies(implies(Z, or(X, Y)), implies(Z, implies(or(Z, X), or(X, Y))))), implies(implies(implies(Z, or(X, Y)), implies(Z, implies(or(Z, X), or(X, Y)))), implies(Z, implies(or(Z, X), or(X, Y))))))
% 69.32/9.38 = { by lemma 129 }
% 69.32/9.38 equiv(or(X, Y), implies(implies(implies(Z, or(X, Y)), implies(Z, implies(or(Z, X), or(X, Y)))), implies(Z, implies(or(Z, X), or(X, Y)))))
% 69.32/9.38 = { by lemma 131 }
% 69.32/9.38 equiv(or(X, Y), or(implies(Z, or(X, Y)), implies(Z, implies(or(Z, X), or(X, Y)))))
% 69.32/9.38 = { by lemma 136 }
% 69.32/9.38 equiv(or(X, Y), or(implies(Z, or(X, Y)), implies(or(Z, X), or(X, Y))))
% 69.32/9.38 = { by lemma 104 }
% 69.32/9.38 equiv(or(X, Y), or(implies(Z, or(X, Y)), implies(or(X, Z), or(X, Y))))
% 69.32/9.38 = { by lemma 93 R->L }
% 69.32/9.38 equiv(or(X, Y), or(implies(Z, or(X, Y)), equiv(implies(or(X, Z), or(X, Y)), implies(U, U))))
% 69.32/9.38 = { by lemma 99 R->L }
% 69.32/9.38 equiv(or(X, Y), or(implies(Z, or(X, Y)), equiv(implies(U, U), implies(or(X, Z), or(X, Y)))))
% 69.32/9.38 = { by lemma 64 R->L }
% 69.32/9.38 equiv(or(X, Y), or(implies(Z, or(X, Y)), and(implies(implies(U, U), implies(or(X, Z), or(X, Y))), implies(implies(or(X, Z), or(X, Y)), implies(U, U)))))
% 69.32/9.39 = { by lemma 83 }
% 69.32/9.39 equiv(or(X, Y), or(implies(Z, or(X, Y)), and(implies(or(X, Z), or(X, Y)), implies(implies(or(X, Z), or(X, Y)), implies(U, U)))))
% 69.32/9.39 = { by lemma 70 }
% 69.32/9.39 equiv(or(X, Y), or(implies(Z, or(X, Y)), and(implies(or(X, Z), or(X, Y)), implies(U, U))))
% 69.32/9.39 = { by lemma 124 R->L }
% 69.32/9.39 equiv(or(X, Y), or(implies(Z, or(X, Y)), and(implies(or(X, Z), or(X, Y)), implies(implies(implies(not(X), Z), or(X, Y)), implies(Z, or(X, Y))))))
% 69.32/9.39 = { by lemma 75 }
% 69.32/9.39 equiv(or(X, Y), or(implies(Z, or(X, Y)), and(implies(or(X, Z), or(X, Y)), implies(implies(or(X, Z), or(X, Y)), implies(Z, or(X, Y))))))
% 69.32/9.39 = { by lemma 104 }
% 69.32/9.39 equiv(or(X, Y), or(and(implies(or(X, Z), or(X, Y)), implies(implies(or(X, Z), or(X, Y)), implies(Z, or(X, Y)))), implies(Z, or(X, Y))))
% 69.32/9.39 = { by lemma 83 R->L }
% 69.32/9.39 equiv(or(X, Y), or(and(implies(or(X, Z), or(X, Y)), implies(implies(or(X, Z), or(X, Y)), implies(Z, or(X, Y)))), implies(implies(T, T), implies(Z, or(X, Y)))))
% 69.32/9.39 = { by lemma 134 R->L }
% 69.32/9.39 equiv(or(X, Y), or(and(implies(or(X, Z), or(X, Y)), implies(implies(or(X, Z), or(X, Y)), implies(Z, or(X, Y)))), implies(implies(and(implies(or(X, Z), or(X, Y)), implies(implies(or(X, Z), or(X, Y)), implies(Z, or(X, Y)))), implies(Z, or(X, Y))), implies(Z, or(X, Y)))))
% 69.32/9.39 = { by lemma 113 }
% 69.32/9.39 equiv(or(X, Y), implies(implies(and(implies(or(X, Z), or(X, Y)), implies(implies(or(X, Z), or(X, Y)), implies(Z, or(X, Y)))), implies(Z, or(X, Y))), implies(Z, or(X, Y))))
% 69.32/9.39 = { by lemma 134 }
% 69.32/9.39 equiv(or(X, Y), implies(implies(S, S), implies(Z, or(X, Y))))
% 69.32/9.39 = { by lemma 83 }
% 69.32/9.39 equiv(or(X, Y), implies(Z, or(X, Y)))
% 69.32/9.39 = { by lemma 132 }
% 69.32/9.39 or(or(X, Y), Z)
% 69.32/9.39 = { by lemma 104 R->L }
% 69.32/9.39 or(Z, or(X, Y))
% 69.32/9.39
% 69.32/9.39 Goal 1 (principia_r4): r4 = true.
% 69.32/9.39 Proof:
% 69.32/9.39 r4
% 69.32/9.39 = { by axiom 58 (r4) R->L }
% 69.32/9.39 fresh7(is_a_theorem(implies(or(p2, or(q2, r6)), or(q2, or(p2, r6)))), true)
% 69.32/9.39 = { by lemma 104 }
% 69.32/9.39 fresh7(is_a_theorem(implies(or(p2, or(r6, q2)), or(q2, or(p2, r6)))), true)
% 69.32/9.39 = { by lemma 137 R->L }
% 69.32/9.39 fresh7(is_a_theorem(implies(or(or(r6, q2), or(r6, p2)), or(q2, or(p2, r6)))), true)
% 69.32/9.39 = { by lemma 104 R->L }
% 69.32/9.39 fresh7(is_a_theorem(implies(or(or(r6, p2), or(r6, q2)), or(q2, or(p2, r6)))), true)
% 69.32/9.39 = { by lemma 137 }
% 69.32/9.39 fresh7(is_a_theorem(implies(or(q2, or(r6, p2)), or(q2, or(p2, r6)))), true)
% 69.32/9.39 = { by lemma 104 R->L }
% 69.32/9.39 fresh7(is_a_theorem(implies(or(q2, or(p2, r6)), or(q2, or(p2, r6)))), true)
% 69.32/9.39 = { by lemma 68 }
% 69.32/9.39 fresh7(true, true)
% 69.32/9.39 = { by axiom 18 (r4) }
% 69.32/9.39 true
% 69.32/9.39 % SZS output end Proof
% 69.32/9.39
% 69.32/9.39 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------