TSTP Solution File: LCL448+1 by Twee---2.4.2
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- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : LCL448+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 : n002.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 08:19:02 EDT 2023
% Result : Theorem 237.31s 30.69s
% Output : Proof 241.70s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.07/0.12 % Problem : LCL448+1 : TPTP v8.1.2. Released v3.3.0.
% 0.07/0.13 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.13/0.34 % Computer : n002.cluster.edu
% 0.13/0.34 % Model : x86_64 x86_64
% 0.13/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.34 % Memory : 8042.1875MB
% 0.13/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.34 % CPULimit : 300
% 0.13/0.34 % WCLimit : 300
% 0.13/0.34 % DateTime : Thu Aug 24 17:52:31 EDT 2023
% 0.13/0.34 % CPUTime :
% 237.31/30.69 Command-line arguments: --no-flatten-goal
% 237.31/30.69
% 237.31/30.69 % SZS status Theorem
% 237.31/30.69
% 240.12/31.04 % SZS output start Proof
% 240.12/31.04 Take the following subset of the input axioms:
% 240.12/31.04 fof(and_2, axiom, and_2 <=> ![X, Y]: is_a_theorem(implies(and(X, Y), Y))).
% 240.12/31.04 fof(and_3, axiom, and_3 <=> ![X2, Y2]: is_a_theorem(implies(X2, implies(Y2, and(X2, Y2))))).
% 240.12/31.04 fof(cn2, axiom, cn2 <=> ![P, Q]: is_a_theorem(implies(P, implies(not(P), Q)))).
% 240.12/31.04 fof(implies_1, axiom, implies_1 <=> ![X2, Y2]: is_a_theorem(implies(X2, implies(Y2, X2)))).
% 240.12/31.04 fof(kn1, axiom, kn1 <=> ![P2]: is_a_theorem(implies(P2, and(P2, P2)))).
% 240.12/31.05 fof(modus_ponens, axiom, modus_ponens <=> ![X2, Y2]: ((is_a_theorem(X2) & is_a_theorem(implies(X2, Y2))) => is_a_theorem(Y2))).
% 240.12/31.05 fof(modus_tollens, axiom, modus_tollens <=> ![X2, Y2]: is_a_theorem(implies(implies(not(Y2), not(X2)), implies(X2, Y2)))).
% 240.12/31.05 fof(op_and, axiom, op_and => ![X2, Y2]: and(X2, Y2)=not(or(not(X2), not(Y2)))).
% 240.12/31.05 fof(op_equiv, axiom, op_equiv => ![X2, Y2]: equiv(X2, Y2)=and(implies(X2, Y2), implies(Y2, X2))).
% 240.12/31.05 fof(op_implies_and, axiom, op_implies_and => ![X2, Y2]: implies(X2, Y2)=not(and(X2, not(Y2)))).
% 240.12/31.05 fof(op_implies_or, axiom, op_implies_or => ![X2, Y2]: implies(X2, Y2)=or(not(X2), Y2)).
% 240.12/31.05 fof(or_1, axiom, or_1 <=> ![X2, Y2]: is_a_theorem(implies(X2, or(X2, Y2)))).
% 240.12/31.05 fof(or_2, axiom, or_2 <=> ![X2, Y2]: is_a_theorem(implies(Y2, or(X2, Y2)))).
% 240.12/31.05 fof(principia_modus_ponens, axiom, modus_ponens).
% 240.12/31.05 fof(principia_op_and, axiom, op_and).
% 240.12/31.05 fof(principia_op_equiv, axiom, op_equiv).
% 240.12/31.05 fof(principia_op_implies_or, axiom, op_implies_or).
% 240.12/31.05 fof(principia_r1, axiom, r1).
% 240.12/31.05 fof(principia_r2, axiom, r2).
% 240.12/31.05 fof(principia_r3, axiom, r3).
% 240.12/31.05 fof(principia_r4, conjecture, r4).
% 240.12/31.05 fof(principia_r5, axiom, r5).
% 240.12/31.05 fof(r1, axiom, r1 <=> ![P2]: is_a_theorem(implies(or(P2, P2), P2))).
% 240.12/31.05 fof(r2, axiom, r2 <=> ![P2, Q2]: is_a_theorem(implies(Q2, or(P2, Q2)))).
% 240.12/31.05 fof(r3, axiom, r3 <=> ![P2, Q2]: is_a_theorem(implies(or(P2, Q2), or(Q2, P2)))).
% 240.12/31.05 fof(r4, axiom, r4 <=> ![R, P2, Q2]: is_a_theorem(implies(or(P2, or(Q2, R)), or(Q2, or(P2, R))))).
% 240.12/31.05 fof(r5, axiom, r5 <=> ![P2, Q2, R2]: is_a_theorem(implies(implies(Q2, R2), implies(or(P2, Q2), or(P2, R2))))).
% 240.12/31.05 fof(substitution_of_equivalents, axiom, substitution_of_equivalents <=> ![X2, Y2]: (is_a_theorem(equiv(X2, Y2)) => X2=Y2)).
% 240.12/31.05 fof(substitution_of_equivalents, axiom, substitution_of_equivalents).
% 240.12/31.05
% 240.12/31.05 Now clausify the problem and encode Horn clauses using encoding 3 of
% 240.12/31.05 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 240.12/31.05 We repeatedly replace C & s=t => u=v by the two clauses:
% 240.12/31.05 fresh(y, y, x1...xn) = u
% 240.12/31.05 C => fresh(s, t, x1...xn) = v
% 240.12/31.05 where fresh is a fresh function symbol and x1..xn are the free
% 240.12/31.05 variables of u and v.
% 240.12/31.05 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 240.12/31.05 input problem has no model of domain size 1).
% 240.12/31.05
% 240.12/31.05 The encoding turns the above axioms into the following unit equations and goals:
% 240.12/31.05
% 240.12/31.05 Axiom 1 (principia_op_and): op_and = true.
% 240.12/31.05 Axiom 2 (principia_op_implies_or): op_implies_or = true.
% 240.12/31.05 Axiom 3 (principia_op_equiv): op_equiv = true.
% 240.12/31.05 Axiom 4 (principia_modus_ponens): modus_ponens = true.
% 240.12/31.05 Axiom 5 (substitution_of_equivalents): substitution_of_equivalents = true.
% 240.12/31.05 Axiom 6 (principia_r1): r1 = true.
% 240.12/31.05 Axiom 7 (principia_r2): r2 = true.
% 240.12/31.05 Axiom 8 (principia_r3): r3 = true.
% 240.12/31.05 Axiom 9 (principia_r5): r5 = true.
% 240.12/31.05 Axiom 10 (implies_1): fresh40(X, X) = true.
% 240.12/31.05 Axiom 11 (kn1): fresh34(X, X) = true.
% 240.12/31.05 Axiom 12 (modus_tollens): fresh26(X, X) = true.
% 240.12/31.05 Axiom 13 (or_2): fresh17(X, X) = true.
% 240.12/31.05 Axiom 14 (r4): fresh7(X, X) = true.
% 240.12/31.05 Axiom 15 (modus_ponens_2): fresh60(X, X, Y) = true.
% 240.12/31.05 Axiom 16 (kn1_1): fresh33(X, X, Y) = true.
% 240.12/31.05 Axiom 17 (modus_ponens_2): fresh28(X, X, Y) = is_a_theorem(Y).
% 240.12/31.05 Axiom 18 (r1_1): fresh12(X, X, Y) = true.
% 240.12/31.05 Axiom 19 (substitution_of_equivalents_2): fresh(X, X, Y, Z) = Z.
% 240.12/31.05 Axiom 20 (modus_ponens_2): fresh59(X, X, Y, Z) = fresh60(modus_ponens, true, Z).
% 240.12/31.05 Axiom 21 (op_and): fresh24(X, X, Y, Z) = and(Y, Z).
% 240.12/31.05 Axiom 22 (op_equiv): fresh23(X, X, Y, Z) = equiv(Y, Z).
% 240.12/31.05 Axiom 23 (op_implies_or): fresh21(X, X, Y, Z) = implies(Y, Z).
% 240.12/31.05 Axiom 24 (op_implies_or): fresh21(op_implies_or, true, X, Y) = or(not(X), Y).
% 240.12/31.05 Axiom 25 (r2_1): fresh10(X, X, Y, Z) = true.
% 240.12/31.05 Axiom 26 (r3_1): fresh8(X, X, Y, Z) = true.
% 240.12/31.05 Axiom 27 (substitution_of_equivalents_2): fresh2(X, X, Y, Z) = Y.
% 240.12/31.05 Axiom 28 (op_implies_and): fresh22(op_implies_and, true, X, Y) = not(and(X, not(Y))).
% 240.12/31.05 Axiom 29 (r5_1): fresh4(X, X, Y, Z, W) = true.
% 240.12/31.05 Axiom 30 (op_and): fresh24(op_and, true, X, Y) = not(or(not(X), not(Y))).
% 240.12/31.05 Axiom 31 (kn1_1): fresh33(kn1, true, X) = is_a_theorem(implies(X, and(X, X))).
% 240.12/31.05 Axiom 32 (or_1_1): fresh18(or_1, true, X, Y) = is_a_theorem(implies(X, or(X, Y))).
% 240.12/31.05 Axiom 33 (implies_1_1): fresh39(implies_1, true, X, Y) = is_a_theorem(implies(X, implies(Y, X))).
% 240.12/31.05 Axiom 34 (or_2_1): fresh16(or_2, true, X, Y) = is_a_theorem(implies(Y, or(X, Y))).
% 240.12/31.05 Axiom 35 (r2_1): fresh10(r2, true, X, Y) = is_a_theorem(implies(Y, or(X, Y))).
% 240.12/31.05 Axiom 36 (and_2_1): fresh55(and_2, true, X, Y) = is_a_theorem(implies(and(X, Y), Y)).
% 240.12/31.05 Axiom 37 (r1_1): fresh12(r1, true, X) = is_a_theorem(implies(or(X, X), X)).
% 240.12/31.05 Axiom 38 (op_equiv): fresh23(op_equiv, true, X, Y) = and(implies(X, Y), implies(Y, X)).
% 240.12/31.05 Axiom 39 (cn2_1): fresh49(cn2, true, X, Y) = is_a_theorem(implies(X, implies(not(X), Y))).
% 240.12/31.05 Axiom 40 (substitution_of_equivalents_2): fresh2(substitution_of_equivalents, true, X, Y) = fresh(is_a_theorem(equiv(X, Y)), true, X, Y).
% 240.12/31.05 Axiom 41 (modus_ponens_2): fresh59(is_a_theorem(implies(X, Y)), true, X, Y) = fresh28(is_a_theorem(X), true, Y).
% 240.12/31.05 Axiom 42 (implies_1): fresh40(is_a_theorem(implies(x12, implies(y12, x12))), true) = implies_1.
% 240.12/31.05 Axiom 43 (kn1): fresh34(is_a_theorem(implies(p11, and(p11, p11))), true) = kn1.
% 240.12/31.05 Axiom 44 (or_2): fresh17(is_a_theorem(implies(y5, or(x5, y5))), true) = or_2.
% 240.12/31.05 Axiom 45 (and_3_1): fresh53(and_3, true, X, Y) = is_a_theorem(implies(X, implies(Y, and(X, Y)))).
% 240.12/31.05 Axiom 46 (r3_1): fresh8(r3, true, X, Y) = is_a_theorem(implies(or(X, Y), or(Y, X))).
% 240.12/31.05 Axiom 47 (modus_tollens_1): fresh25(modus_tollens, true, X, Y) = is_a_theorem(implies(implies(not(Y), not(X)), implies(X, Y))).
% 240.12/31.05 Axiom 48 (modus_tollens): fresh26(is_a_theorem(implies(implies(not(y13), not(x13)), implies(x13, y13))), true) = modus_tollens.
% 240.12/31.05 Axiom 49 (r5_1): fresh4(r5, true, X, Y, Z) = is_a_theorem(implies(implies(Y, Z), implies(or(X, Y), or(X, Z)))).
% 240.12/31.05 Axiom 50 (r4_1): fresh6(r4, true, X, Y, Z) = is_a_theorem(implies(or(X, or(Y, Z)), or(Y, or(X, Z)))).
% 240.12/31.05 Axiom 51 (r4): fresh7(is_a_theorem(implies(or(p2, or(q2, r6)), or(q2, or(p2, r6)))), true) = r4.
% 240.12/31.05
% 240.12/31.05 Lemma 52: modus_ponens = op_and.
% 240.12/31.05 Proof:
% 240.12/31.05 modus_ponens
% 240.12/31.05 = { by axiom 4 (principia_modus_ponens) }
% 240.12/31.05 true
% 240.12/31.05 = { by axiom 1 (principia_op_and) R->L }
% 240.12/31.05 op_and
% 240.12/31.05
% 240.12/31.05 Lemma 53: is_a_theorem(implies(X, or(Y, X))) = fresh16(or_2, op_and, Y, X).
% 240.12/31.05 Proof:
% 240.12/31.05 is_a_theorem(implies(X, or(Y, X)))
% 240.12/31.05 = { by axiom 34 (or_2_1) R->L }
% 240.12/31.05 fresh16(or_2, true, Y, X)
% 240.12/31.05 = { by axiom 1 (principia_op_and) R->L }
% 240.12/31.05 fresh16(or_2, op_and, Y, X)
% 240.12/31.05
% 240.12/31.05 Lemma 54: fresh16(or_2, op_and, X, Y) = op_and.
% 240.12/31.05 Proof:
% 240.12/31.05 fresh16(or_2, op_and, X, Y)
% 240.12/31.05 = { by lemma 53 R->L }
% 240.12/31.05 is_a_theorem(implies(Y, or(X, Y)))
% 240.12/31.05 = { by axiom 35 (r2_1) R->L }
% 240.12/31.05 fresh10(r2, true, X, Y)
% 240.12/31.05 = { by axiom 7 (principia_r2) }
% 240.12/31.05 fresh10(true, true, X, Y)
% 240.12/31.05 = { by axiom 1 (principia_op_and) R->L }
% 240.12/31.05 fresh10(op_and, true, X, Y)
% 240.12/31.05 = { by axiom 1 (principia_op_and) R->L }
% 240.12/31.05 fresh10(op_and, op_and, X, Y)
% 240.12/31.05 = { by axiom 25 (r2_1) }
% 240.12/31.05 true
% 240.12/31.05 = { by axiom 1 (principia_op_and) R->L }
% 240.12/31.05 op_and
% 240.12/31.05
% 240.12/31.05 Lemma 55: op_and = or_2.
% 240.12/31.05 Proof:
% 240.12/31.05 op_and
% 240.12/31.05 = { by axiom 1 (principia_op_and) }
% 240.12/31.05 true
% 240.12/31.05 = { by axiom 13 (or_2) R->L }
% 240.12/31.05 fresh17(op_and, op_and)
% 240.12/31.05 = { by axiom 1 (principia_op_and) }
% 240.12/31.05 fresh17(op_and, true)
% 240.12/31.05 = { by lemma 54 R->L }
% 240.12/31.05 fresh17(fresh16(or_2, op_and, x5, y5), true)
% 240.12/31.05 = { by lemma 53 R->L }
% 240.12/31.05 fresh17(is_a_theorem(implies(y5, or(x5, y5))), true)
% 240.12/31.05 = { by axiom 44 (or_2) }
% 240.12/31.05 or_2
% 240.12/31.05
% 240.12/31.05 Lemma 56: is_a_theorem(implies(X, implies(Y, X))) = fresh39(implies_1, op_and, X, Y).
% 240.12/31.05 Proof:
% 240.12/31.05 is_a_theorem(implies(X, implies(Y, X)))
% 240.12/31.05 = { by axiom 33 (implies_1_1) R->L }
% 240.12/31.05 fresh39(implies_1, true, X, Y)
% 240.12/31.05 = { by axiom 1 (principia_op_and) R->L }
% 240.12/31.05 fresh39(implies_1, op_and, X, Y)
% 240.12/31.05
% 240.12/31.05 Lemma 57: or(not(X), Y) = implies(X, Y).
% 240.12/31.05 Proof:
% 240.12/31.05 or(not(X), Y)
% 240.12/31.05 = { by axiom 24 (op_implies_or) R->L }
% 240.12/31.05 fresh21(op_implies_or, true, X, Y)
% 240.12/31.05 = { by axiom 2 (principia_op_implies_or) }
% 240.12/31.05 fresh21(true, true, X, Y)
% 240.12/31.05 = { by axiom 1 (principia_op_and) R->L }
% 240.12/31.05 fresh21(op_and, true, X, Y)
% 240.12/31.05 = { by axiom 1 (principia_op_and) R->L }
% 240.12/31.05 fresh21(op_and, op_and, X, Y)
% 240.12/31.05 = { by axiom 23 (op_implies_or) }
% 240.12/31.05 implies(X, Y)
% 240.12/31.05
% 240.12/31.05 Lemma 58: fresh39(implies_1, or_2, X, Y) = or_2.
% 240.12/31.05 Proof:
% 240.12/31.05 fresh39(implies_1, or_2, X, Y)
% 240.12/31.05 = { by lemma 55 R->L }
% 240.12/31.05 fresh39(implies_1, op_and, X, Y)
% 240.12/31.05 = { by lemma 56 R->L }
% 240.12/31.05 is_a_theorem(implies(X, implies(Y, X)))
% 240.12/31.05 = { by lemma 57 R->L }
% 240.12/31.05 is_a_theorem(implies(X, or(not(Y), X)))
% 240.12/31.05 = { by lemma 53 }
% 240.12/31.05 fresh16(or_2, op_and, not(Y), X)
% 240.12/31.05 = { by lemma 54 }
% 240.12/31.05 op_and
% 240.12/31.05 = { by lemma 55 }
% 240.12/31.05 or_2
% 240.12/31.05
% 240.12/31.05 Lemma 59: or_2 = implies_1.
% 240.12/31.05 Proof:
% 240.12/31.05 or_2
% 240.59/31.05 = { by lemma 55 R->L }
% 240.59/31.05 op_and
% 240.59/31.05 = { by axiom 1 (principia_op_and) }
% 240.59/31.05 true
% 240.59/31.05 = { by axiom 10 (implies_1) R->L }
% 240.59/31.05 fresh40(or_2, or_2)
% 240.59/31.05 = { by lemma 55 R->L }
% 240.59/31.05 fresh40(or_2, op_and)
% 240.59/31.05 = { by lemma 58 R->L }
% 240.59/31.05 fresh40(fresh39(implies_1, or_2, x12, y12), op_and)
% 240.59/31.05 = { by lemma 55 R->L }
% 240.59/31.05 fresh40(fresh39(implies_1, op_and, x12, y12), op_and)
% 240.59/31.05 = { by axiom 1 (principia_op_and) }
% 240.59/31.05 fresh40(fresh39(implies_1, op_and, x12, y12), true)
% 240.59/31.05 = { by lemma 56 R->L }
% 240.59/31.05 fresh40(is_a_theorem(implies(x12, implies(y12, x12))), true)
% 240.59/31.05 = { by axiom 42 (implies_1) }
% 240.59/31.05 implies_1
% 240.59/31.05
% 240.59/31.05 Lemma 60: fresh59(is_a_theorem(implies(X, Y)), op_and, X, Y) = fresh28(is_a_theorem(X), op_and, Y).
% 240.59/31.05 Proof:
% 240.59/31.05 fresh59(is_a_theorem(implies(X, Y)), op_and, X, Y)
% 240.59/31.05 = { by axiom 1 (principia_op_and) }
% 240.59/31.05 fresh59(is_a_theorem(implies(X, Y)), true, X, Y)
% 240.59/31.05 = { by axiom 41 (modus_ponens_2) }
% 240.59/31.05 fresh28(is_a_theorem(X), true, Y)
% 240.59/31.05 = { by axiom 1 (principia_op_and) R->L }
% 240.59/31.05 fresh28(is_a_theorem(X), op_and, Y)
% 240.59/31.05
% 240.59/31.05 Lemma 61: is_a_theorem(implies(or(X, Y), or(Y, X))) = or_2.
% 240.59/31.05 Proof:
% 240.59/31.05 is_a_theorem(implies(or(X, Y), or(Y, X)))
% 240.59/31.05 = { by axiom 46 (r3_1) R->L }
% 240.59/31.05 fresh8(r3, true, X, Y)
% 240.59/31.05 = { by axiom 8 (principia_r3) }
% 240.59/31.05 fresh8(true, true, X, Y)
% 240.59/31.05 = { by axiom 1 (principia_op_and) R->L }
% 240.59/31.05 fresh8(op_and, true, X, Y)
% 240.59/31.05 = { by lemma 55 }
% 240.59/31.05 fresh8(or_2, true, X, Y)
% 240.59/31.05 = { by axiom 1 (principia_op_and) R->L }
% 240.59/31.05 fresh8(or_2, op_and, X, Y)
% 240.59/31.05 = { by lemma 55 }
% 240.59/31.05 fresh8(or_2, or_2, X, Y)
% 240.59/31.05 = { by axiom 26 (r3_1) }
% 240.59/31.05 true
% 240.59/31.05 = { by axiom 1 (principia_op_and) R->L }
% 240.59/31.05 op_and
% 240.59/31.05 = { by lemma 55 }
% 240.59/31.05 or_2
% 240.59/31.05
% 240.59/31.05 Lemma 62: fresh60(X, X, Y) = op_and.
% 240.59/31.05 Proof:
% 240.59/31.05 fresh60(X, X, Y)
% 240.59/31.05 = { by axiom 15 (modus_ponens_2) }
% 240.59/31.05 true
% 240.59/31.05 = { by axiom 1 (principia_op_and) R->L }
% 240.59/31.05 op_and
% 240.59/31.05
% 240.59/31.05 Lemma 63: fresh59(X, X, Y, Z) = op_and.
% 240.59/31.05 Proof:
% 240.59/31.05 fresh59(X, X, Y, Z)
% 240.59/31.05 = { by axiom 20 (modus_ponens_2) }
% 240.59/31.05 fresh60(modus_ponens, true, Z)
% 240.59/31.05 = { by lemma 52 }
% 240.59/31.05 fresh60(op_and, true, Z)
% 240.59/31.05 = { by axiom 1 (principia_op_and) R->L }
% 240.59/31.05 fresh60(op_and, op_and, Z)
% 240.59/31.05 = { by lemma 62 }
% 240.59/31.05 op_and
% 240.59/31.05
% 240.59/31.05 Lemma 64: fresh28(is_a_theorem(or(X, Y)), implies_1, or(Y, X)) = implies_1.
% 240.59/31.05 Proof:
% 240.59/31.05 fresh28(is_a_theorem(or(X, Y)), implies_1, or(Y, X))
% 240.59/31.05 = { by lemma 59 R->L }
% 240.59/31.05 fresh28(is_a_theorem(or(X, Y)), or_2, or(Y, X))
% 240.59/31.05 = { by lemma 55 R->L }
% 240.59/31.05 fresh28(is_a_theorem(or(X, Y)), op_and, or(Y, X))
% 240.59/31.05 = { by lemma 60 R->L }
% 240.59/31.05 fresh59(is_a_theorem(implies(or(X, Y), or(Y, X))), op_and, or(X, Y), or(Y, X))
% 240.59/31.05 = { by lemma 61 }
% 240.59/31.05 fresh59(or_2, op_and, or(X, Y), or(Y, X))
% 240.59/31.05 = { by lemma 59 }
% 240.59/31.05 fresh59(implies_1, op_and, or(X, Y), or(Y, X))
% 240.59/31.05 = { by lemma 55 }
% 240.59/31.05 fresh59(implies_1, or_2, or(X, Y), or(Y, X))
% 240.59/31.05 = { by lemma 59 }
% 240.59/31.05 fresh59(implies_1, implies_1, or(X, Y), or(Y, X))
% 240.59/31.05 = { by lemma 63 }
% 240.59/31.05 op_and
% 240.59/31.05 = { by lemma 55 }
% 240.59/31.05 or_2
% 240.59/31.05 = { by lemma 59 }
% 240.59/31.05 implies_1
% 240.59/31.05
% 240.59/31.05 Lemma 65: fresh28(is_a_theorem(implies(X, Y)), implies_1, or(Y, not(X))) = implies_1.
% 240.59/31.05 Proof:
% 240.59/31.05 fresh28(is_a_theorem(implies(X, Y)), implies_1, or(Y, not(X)))
% 240.59/31.05 = { by lemma 59 R->L }
% 240.59/31.05 fresh28(is_a_theorem(implies(X, Y)), or_2, or(Y, not(X)))
% 240.59/31.05 = { by lemma 55 R->L }
% 240.59/31.05 fresh28(is_a_theorem(implies(X, Y)), op_and, or(Y, not(X)))
% 240.59/31.05 = { by lemma 60 R->L }
% 240.59/31.05 fresh59(is_a_theorem(implies(implies(X, Y), or(Y, not(X)))), op_and, implies(X, Y), or(Y, not(X)))
% 240.59/31.05 = { by lemma 57 R->L }
% 240.59/31.05 fresh59(is_a_theorem(implies(or(not(X), Y), or(Y, not(X)))), op_and, implies(X, Y), or(Y, not(X)))
% 240.59/31.05 = { by lemma 61 }
% 240.59/31.05 fresh59(or_2, op_and, implies(X, Y), or(Y, not(X)))
% 240.59/31.05 = { by lemma 59 }
% 240.59/31.05 fresh59(implies_1, op_and, implies(X, Y), or(Y, not(X)))
% 240.59/31.05 = { by lemma 55 }
% 240.59/31.05 fresh59(implies_1, or_2, implies(X, Y), or(Y, not(X)))
% 240.59/31.05 = { by lemma 59 }
% 240.59/31.05 fresh59(implies_1, implies_1, implies(X, Y), or(Y, not(X)))
% 240.59/31.05 = { by lemma 63 }
% 240.59/31.05 op_and
% 240.59/31.05 = { by lemma 55 }
% 240.59/31.05 or_2
% 240.59/31.05 = { by lemma 59 }
% 240.59/31.05 implies_1
% 240.59/31.05
% 240.59/31.05 Lemma 66: fresh33(X, X, Y) = op_and.
% 240.59/31.05 Proof:
% 240.59/31.05 fresh33(X, X, Y)
% 240.59/31.05 = { by axiom 16 (kn1_1) }
% 240.59/31.05 true
% 240.59/31.05 = { by axiom 1 (principia_op_and) R->L }
% 240.59/31.05 op_and
% 240.59/31.05
% 240.59/31.05 Lemma 67: is_a_theorem(implies(X, and(X, X))) = fresh33(kn1, op_and, X).
% 240.59/31.05 Proof:
% 240.59/31.05 is_a_theorem(implies(X, and(X, X)))
% 240.59/31.05 = { by axiom 31 (kn1_1) R->L }
% 240.59/31.05 fresh33(kn1, true, X)
% 240.59/31.05 = { by axiom 1 (principia_op_and) R->L }
% 240.59/31.05 fresh33(kn1, op_and, X)
% 240.59/31.05
% 240.59/31.05 Lemma 68: r1 = op_and.
% 240.59/31.05 Proof:
% 240.59/31.05 r1
% 240.59/31.05 = { by axiom 6 (principia_r1) }
% 240.59/31.05 true
% 240.59/31.05 = { by axiom 1 (principia_op_and) R->L }
% 240.59/31.05 op_and
% 240.59/31.05
% 240.59/31.05 Lemma 69: fresh12(X, X, Y) = op_and.
% 240.59/31.05 Proof:
% 240.59/31.05 fresh12(X, X, Y)
% 240.59/31.05 = { by axiom 18 (r1_1) }
% 240.59/31.05 true
% 240.59/31.05 = { by axiom 1 (principia_op_and) R->L }
% 240.59/31.05 op_and
% 240.59/31.05
% 240.59/31.05 Lemma 70: is_a_theorem(implies(or(X, X), X)) = op_and.
% 240.59/31.05 Proof:
% 240.59/31.05 is_a_theorem(implies(or(X, X), X))
% 240.59/31.05 = { by axiom 37 (r1_1) R->L }
% 240.59/31.05 fresh12(r1, true, X)
% 240.59/31.05 = { by lemma 68 }
% 240.59/31.05 fresh12(op_and, true, X)
% 240.59/31.05 = { by axiom 1 (principia_op_and) R->L }
% 240.59/31.05 fresh12(op_and, op_and, X)
% 240.59/31.05 = { by lemma 69 }
% 240.59/31.05 op_and
% 240.59/31.05
% 240.59/31.05 Lemma 71: not(implies(X, not(Y))) = and(X, Y).
% 240.59/31.05 Proof:
% 240.59/31.05 not(implies(X, not(Y)))
% 240.59/31.05 = { by lemma 57 R->L }
% 240.59/31.05 not(or(not(X), not(Y)))
% 240.59/31.05 = { by axiom 30 (op_and) R->L }
% 240.59/31.05 fresh24(op_and, true, X, Y)
% 240.59/31.05 = { by axiom 1 (principia_op_and) R->L }
% 240.59/31.05 fresh24(op_and, op_and, X, Y)
% 240.59/31.05 = { by axiom 21 (op_and) }
% 240.59/31.05 and(X, Y)
% 240.59/31.05
% 240.59/31.05 Lemma 72: implies(implies(X, not(Y)), Z) = or(and(X, Y), Z).
% 240.59/31.05 Proof:
% 240.59/31.05 implies(implies(X, not(Y)), Z)
% 240.59/31.05 = { by lemma 57 R->L }
% 240.59/31.05 or(not(implies(X, not(Y))), Z)
% 240.59/31.05 = { by lemma 71 }
% 240.59/31.05 or(and(X, Y), Z)
% 240.59/31.05
% 240.59/31.05 Lemma 73: kn1 = implies_1.
% 240.59/31.05 Proof:
% 240.59/31.05 kn1
% 240.59/31.06 = { by axiom 43 (kn1) R->L }
% 240.59/31.06 fresh34(is_a_theorem(implies(p11, and(p11, p11))), true)
% 240.59/31.06 = { by lemma 67 }
% 240.59/31.06 fresh34(fresh33(kn1, op_and, p11), true)
% 240.59/31.06 = { by axiom 1 (principia_op_and) R->L }
% 240.59/31.06 fresh34(fresh33(kn1, op_and, p11), op_and)
% 240.59/31.06 = { by lemma 55 }
% 240.59/31.06 fresh34(fresh33(kn1, or_2, p11), op_and)
% 240.59/31.06 = { by lemma 55 }
% 240.59/31.06 fresh34(fresh33(kn1, or_2, p11), or_2)
% 240.59/31.06 = { by lemma 59 }
% 240.59/31.06 fresh34(fresh33(kn1, implies_1, p11), or_2)
% 240.59/31.06 = { by lemma 59 }
% 240.59/31.06 fresh34(fresh33(kn1, implies_1, p11), implies_1)
% 240.59/31.06 = { by lemma 59 R->L }
% 240.59/31.06 fresh34(fresh33(kn1, or_2, p11), implies_1)
% 240.59/31.06 = { by lemma 55 R->L }
% 240.59/31.06 fresh34(fresh33(kn1, op_and, p11), implies_1)
% 240.59/31.06 = { by lemma 67 R->L }
% 240.59/31.06 fresh34(is_a_theorem(implies(p11, and(p11, p11))), implies_1)
% 240.59/31.06 = { by lemma 57 R->L }
% 240.59/31.06 fresh34(is_a_theorem(or(not(p11), and(p11, p11))), implies_1)
% 240.59/31.06 = { by axiom 17 (modus_ponens_2) R->L }
% 240.59/31.06 fresh34(fresh28(implies_1, implies_1, or(not(p11), and(p11, p11))), implies_1)
% 240.59/31.06 = { by lemma 59 R->L }
% 240.59/31.06 fresh34(fresh28(or_2, implies_1, or(not(p11), and(p11, p11))), implies_1)
% 240.59/31.06 = { by lemma 55 R->L }
% 240.59/31.06 fresh34(fresh28(op_and, implies_1, or(not(p11), and(p11, p11))), implies_1)
% 240.59/31.06 = { by lemma 70 R->L }
% 240.59/31.06 fresh34(fresh28(is_a_theorem(implies(or(not(p11), not(p11)), not(p11))), implies_1, or(not(p11), and(p11, p11))), implies_1)
% 240.59/31.06 = { by lemma 57 }
% 240.66/31.06 fresh34(fresh28(is_a_theorem(implies(implies(p11, not(p11)), not(p11))), implies_1, or(not(p11), and(p11, p11))), implies_1)
% 240.66/31.06 = { by lemma 72 }
% 240.66/31.06 fresh34(fresh28(is_a_theorem(or(and(p11, p11), not(p11))), implies_1, or(not(p11), and(p11, p11))), implies_1)
% 240.66/31.06 = { by lemma 64 }
% 240.66/31.06 fresh34(implies_1, implies_1)
% 240.66/31.06 = { by axiom 11 (kn1) }
% 240.66/31.06 true
% 240.66/31.06 = { by axiom 1 (principia_op_and) R->L }
% 240.66/31.06 op_and
% 240.66/31.06 = { by lemma 55 }
% 240.66/31.06 or_2
% 240.66/31.06 = { by lemma 59 }
% 240.66/31.06 implies_1
% 240.66/31.06
% 240.66/31.06 Lemma 74: is_a_theorem(implies(implies(X, Y), implies(or(Z, X), or(Z, Y)))) = or_2.
% 240.66/31.06 Proof:
% 240.66/31.06 is_a_theorem(implies(implies(X, Y), implies(or(Z, X), or(Z, Y))))
% 240.66/31.06 = { by axiom 49 (r5_1) R->L }
% 240.66/31.06 fresh4(r5, true, Z, X, Y)
% 240.66/31.06 = { by axiom 9 (principia_r5) }
% 240.66/31.06 fresh4(true, true, Z, X, Y)
% 240.66/31.06 = { by axiom 1 (principia_op_and) R->L }
% 240.66/31.06 fresh4(op_and, true, Z, X, Y)
% 240.66/31.06 = { by lemma 55 }
% 240.66/31.06 fresh4(or_2, true, Z, X, Y)
% 240.66/31.06 = { by axiom 1 (principia_op_and) R->L }
% 240.66/31.06 fresh4(or_2, op_and, Z, X, Y)
% 240.66/31.06 = { by lemma 55 }
% 240.66/31.06 fresh4(or_2, or_2, Z, X, Y)
% 240.66/31.06 = { by axiom 29 (r5_1) }
% 240.66/31.06 true
% 240.66/31.06 = { by axiom 1 (principia_op_and) R->L }
% 240.66/31.06 op_and
% 240.66/31.06 = { by lemma 55 }
% 240.66/31.06 or_2
% 240.66/31.06
% 240.66/31.06 Lemma 75: fresh28(is_a_theorem(implies(X, Y)), implies_1, implies(or(Z, X), or(Z, Y))) = implies_1.
% 240.66/31.06 Proof:
% 240.66/31.06 fresh28(is_a_theorem(implies(X, Y)), implies_1, implies(or(Z, X), or(Z, Y)))
% 240.66/31.06 = { by lemma 59 R->L }
% 240.66/31.06 fresh28(is_a_theorem(implies(X, Y)), or_2, implies(or(Z, X), or(Z, Y)))
% 240.66/31.06 = { by lemma 55 R->L }
% 240.66/31.06 fresh28(is_a_theorem(implies(X, Y)), op_and, implies(or(Z, X), or(Z, Y)))
% 240.66/31.06 = { by lemma 60 R->L }
% 240.66/31.06 fresh59(is_a_theorem(implies(implies(X, Y), implies(or(Z, X), or(Z, Y)))), op_and, implies(X, Y), implies(or(Z, X), or(Z, Y)))
% 240.66/31.06 = { by lemma 74 }
% 240.66/31.06 fresh59(or_2, op_and, implies(X, Y), implies(or(Z, X), or(Z, Y)))
% 240.66/31.06 = { by lemma 59 }
% 240.66/31.06 fresh59(implies_1, op_and, implies(X, Y), implies(or(Z, X), or(Z, Y)))
% 240.66/31.06 = { by lemma 55 }
% 240.66/31.06 fresh59(implies_1, or_2, implies(X, Y), implies(or(Z, X), or(Z, Y)))
% 240.66/31.06 = { by lemma 59 }
% 240.66/31.06 fresh59(implies_1, implies_1, implies(X, Y), implies(or(Z, X), or(Z, Y)))
% 240.66/31.06 = { by lemma 63 }
% 240.66/31.06 op_and
% 240.66/31.06 = { by lemma 55 }
% 240.66/31.06 or_2
% 240.66/31.06 = { by lemma 59 }
% 240.66/31.06 implies_1
% 240.66/31.06
% 240.66/31.06 Lemma 76: fresh(is_a_theorem(equiv(X, Y)), op_and, X, Y) = X.
% 240.66/31.06 Proof:
% 240.66/31.06 fresh(is_a_theorem(equiv(X, Y)), op_and, X, Y)
% 240.66/31.06 = { by axiom 1 (principia_op_and) }
% 240.66/31.06 fresh(is_a_theorem(equiv(X, Y)), true, X, Y)
% 240.66/31.06 = { by axiom 40 (substitution_of_equivalents_2) R->L }
% 240.66/31.06 fresh2(substitution_of_equivalents, true, X, Y)
% 240.66/31.06 = { by axiom 5 (substitution_of_equivalents) }
% 240.66/31.06 fresh2(true, true, X, Y)
% 240.66/31.06 = { by axiom 1 (principia_op_and) R->L }
% 240.66/31.06 fresh2(op_and, true, X, Y)
% 240.66/31.06 = { by axiom 1 (principia_op_and) R->L }
% 240.66/31.06 fresh2(op_and, op_and, X, Y)
% 240.66/31.06 = { by axiom 27 (substitution_of_equivalents_2) }
% 240.66/31.06 X
% 240.66/31.06
% 240.66/31.06 Lemma 77: fresh(is_a_theorem(equiv(X, Y)), implies_1, X, Y) = X.
% 240.66/31.06 Proof:
% 240.66/31.06 fresh(is_a_theorem(equiv(X, Y)), implies_1, X, Y)
% 240.66/31.06 = { by lemma 59 R->L }
% 240.66/31.06 fresh(is_a_theorem(equiv(X, Y)), or_2, X, Y)
% 240.66/31.06 = { by lemma 55 R->L }
% 240.66/31.06 fresh(is_a_theorem(equiv(X, Y)), op_and, X, Y)
% 240.66/31.06 = { by lemma 76 }
% 240.66/31.06 X
% 240.66/31.06
% 240.66/31.06 Lemma 78: is_a_theorem(implies(or(X, or(Y, Y)), or(X, Y))) = implies_1.
% 240.66/31.06 Proof:
% 240.66/31.06 is_a_theorem(implies(or(X, or(Y, Y)), or(X, Y)))
% 240.66/31.06 = { by axiom 17 (modus_ponens_2) R->L }
% 240.66/31.06 fresh28(implies_1, implies_1, implies(or(X, or(Y, Y)), or(X, Y)))
% 240.66/31.06 = { by lemma 59 R->L }
% 240.66/31.06 fresh28(or_2, implies_1, implies(or(X, or(Y, Y)), or(X, Y)))
% 240.66/31.06 = { by lemma 55 R->L }
% 240.66/31.06 fresh28(op_and, implies_1, implies(or(X, or(Y, Y)), or(X, Y)))
% 240.66/31.06 = { by lemma 70 R->L }
% 240.66/31.06 fresh28(is_a_theorem(implies(or(Y, Y), Y)), implies_1, implies(or(X, or(Y, Y)), or(X, Y)))
% 240.66/31.06 = { by lemma 75 }
% 240.66/31.06 implies_1
% 240.66/31.06
% 240.66/31.06 Lemma 79: fresh28(is_a_theorem(or(X, or(Y, Y))), implies_1, or(X, Y)) = implies_1.
% 240.66/31.06 Proof:
% 240.66/31.06 fresh28(is_a_theorem(or(X, or(Y, Y))), implies_1, or(X, Y))
% 240.66/31.06 = { by lemma 59 R->L }
% 240.66/31.06 fresh28(is_a_theorem(or(X, or(Y, Y))), or_2, or(X, Y))
% 240.66/31.06 = { by lemma 55 R->L }
% 240.66/31.06 fresh28(is_a_theorem(or(X, or(Y, Y))), op_and, or(X, Y))
% 240.66/31.06 = { by lemma 60 R->L }
% 240.66/31.06 fresh59(is_a_theorem(implies(or(X, or(Y, Y)), or(X, Y))), op_and, or(X, or(Y, Y)), or(X, Y))
% 240.66/31.06 = { by lemma 78 }
% 240.66/31.06 fresh59(implies_1, op_and, or(X, or(Y, Y)), or(X, Y))
% 240.66/31.06 = { by lemma 55 }
% 240.66/31.06 fresh59(implies_1, or_2, or(X, or(Y, Y)), or(X, Y))
% 240.66/31.06 = { by lemma 59 }
% 240.66/31.06 fresh59(implies_1, implies_1, or(X, or(Y, Y)), or(X, Y))
% 240.66/31.06 = { by lemma 63 }
% 240.66/31.06 op_and
% 240.66/31.06 = { by lemma 55 }
% 240.66/31.06 or_2
% 240.66/31.06 = { by lemma 59 }
% 240.66/31.06 implies_1
% 240.66/31.06
% 240.66/31.06 Lemma 80: fresh28(is_a_theorem(or(X, implies(Y, not(Y)))), implies_1, or(X, not(Y))) = implies_1.
% 240.66/31.06 Proof:
% 240.66/31.06 fresh28(is_a_theorem(or(X, implies(Y, not(Y)))), implies_1, or(X, not(Y)))
% 240.66/31.06 = { by lemma 57 R->L }
% 240.66/31.06 fresh28(is_a_theorem(or(X, or(not(Y), not(Y)))), implies_1, or(X, not(Y)))
% 240.66/31.06 = { by lemma 79 }
% 240.66/31.06 implies_1
% 240.66/31.06
% 240.66/31.06 Lemma 81: is_a_theorem(or(or(X, Y), not(Y))) = implies_1.
% 240.66/31.06 Proof:
% 240.66/31.06 is_a_theorem(or(or(X, Y), not(Y)))
% 240.66/31.06 = { by axiom 17 (modus_ponens_2) R->L }
% 240.66/31.06 fresh28(implies_1, implies_1, or(or(X, Y), not(Y)))
% 240.66/31.06 = { by lemma 59 R->L }
% 240.66/31.06 fresh28(or_2, implies_1, or(or(X, Y), not(Y)))
% 240.66/31.06 = { by lemma 55 R->L }
% 240.66/31.06 fresh28(op_and, implies_1, or(or(X, Y), not(Y)))
% 240.66/31.06 = { by lemma 54 R->L }
% 240.66/31.06 fresh28(fresh16(or_2, op_and, X, Y), implies_1, or(or(X, Y), not(Y)))
% 240.66/31.06 = { by lemma 53 R->L }
% 240.66/31.06 fresh28(is_a_theorem(implies(Y, or(X, Y))), implies_1, or(or(X, Y), not(Y)))
% 240.66/31.06 = { by lemma 65 }
% 240.66/31.06 implies_1
% 240.66/31.06
% 240.66/31.06 Lemma 82: is_a_theorem(implies(or(X, Y), or(X, implies(Z, Y)))) = implies_1.
% 240.66/31.06 Proof:
% 240.66/31.06 is_a_theorem(implies(or(X, Y), or(X, implies(Z, Y))))
% 240.66/31.06 = { by axiom 17 (modus_ponens_2) R->L }
% 240.66/31.06 fresh28(implies_1, implies_1, implies(or(X, Y), or(X, implies(Z, Y))))
% 240.66/31.06 = { by lemma 59 R->L }
% 240.66/31.06 fresh28(or_2, implies_1, implies(or(X, Y), or(X, implies(Z, Y))))
% 240.66/31.06 = { by lemma 58 R->L }
% 240.66/31.06 fresh28(fresh39(implies_1, or_2, Y, Z), implies_1, implies(or(X, Y), or(X, implies(Z, Y))))
% 240.66/31.06 = { by lemma 55 R->L }
% 240.66/31.06 fresh28(fresh39(implies_1, op_and, Y, Z), implies_1, implies(or(X, Y), or(X, implies(Z, Y))))
% 240.66/31.06 = { by lemma 56 R->L }
% 240.66/31.06 fresh28(is_a_theorem(implies(Y, implies(Z, Y))), implies_1, implies(or(X, Y), or(X, implies(Z, Y))))
% 240.66/31.06 = { by lemma 75 }
% 240.66/31.06 implies_1
% 240.66/31.06
% 240.66/31.06 Lemma 83: fresh59(is_a_theorem(implies(X, Y)), implies_1, X, Y) = fresh28(is_a_theorem(X), implies_1, Y).
% 240.66/31.06 Proof:
% 240.66/31.06 fresh59(is_a_theorem(implies(X, Y)), implies_1, X, Y)
% 240.66/31.06 = { by lemma 59 R->L }
% 240.66/31.06 fresh59(is_a_theorem(implies(X, Y)), or_2, X, Y)
% 240.66/31.06 = { by lemma 55 R->L }
% 240.66/31.06 fresh59(is_a_theorem(implies(X, Y)), op_and, X, Y)
% 240.66/31.06 = { by lemma 60 }
% 240.66/31.06 fresh28(is_a_theorem(X), op_and, Y)
% 240.66/31.06 = { by lemma 55 }
% 240.66/31.06 fresh28(is_a_theorem(X), or_2, Y)
% 240.66/31.06 = { by lemma 59 }
% 240.66/31.06 fresh28(is_a_theorem(X), implies_1, Y)
% 240.66/31.06
% 240.66/31.06 Lemma 84: is_a_theorem(implies(X, implies(not(X), Y))) = fresh49(cn2, implies_1, X, Y).
% 240.66/31.06 Proof:
% 240.66/31.06 is_a_theorem(implies(X, implies(not(X), Y)))
% 240.66/31.06 = { by axiom 39 (cn2_1) R->L }
% 240.66/31.06 fresh49(cn2, true, X, Y)
% 240.66/31.06 = { by axiom 1 (principia_op_and) R->L }
% 240.66/31.06 fresh49(cn2, op_and, X, Y)
% 240.66/31.06 = { by lemma 55 }
% 240.66/31.06 fresh49(cn2, or_2, X, Y)
% 240.66/31.06 = { by lemma 59 }
% 240.66/31.06 fresh49(cn2, implies_1, X, Y)
% 240.66/31.06
% 240.66/31.06 Lemma 85: is_a_theorem(implies(or(X, Y), or(X, or(Z, Y)))) = implies_1.
% 240.66/31.06 Proof:
% 240.66/31.06 is_a_theorem(implies(or(X, Y), or(X, or(Z, Y))))
% 240.66/31.06 = { by axiom 17 (modus_ponens_2) R->L }
% 240.66/31.06 fresh28(implies_1, implies_1, implies(or(X, Y), or(X, or(Z, Y))))
% 240.66/31.06 = { by lemma 59 R->L }
% 240.66/31.06 fresh28(or_2, implies_1, implies(or(X, Y), or(X, or(Z, Y))))
% 240.66/31.06 = { by lemma 55 R->L }
% 240.66/31.06 fresh28(op_and, implies_1, implies(or(X, Y), or(X, or(Z, Y))))
% 240.66/31.06 = { by lemma 54 R->L }
% 240.66/31.06 fresh28(fresh16(or_2, op_and, Z, Y), implies_1, implies(or(X, Y), or(X, or(Z, Y))))
% 240.66/31.06 = { by lemma 53 R->L }
% 240.66/31.06 fresh28(is_a_theorem(implies(Y, or(Z, Y))), implies_1, implies(or(X, Y), or(X, or(Z, Y))))
% 240.66/31.06 = { by lemma 75 }
% 240.66/31.06 implies_1
% 240.66/31.06
% 240.66/31.06 Lemma 86: fresh28(is_a_theorem(or(X, Y)), implies_1, or(X, or(Z, Y))) = implies_1.
% 240.66/31.06 Proof:
% 240.66/31.06 fresh28(is_a_theorem(or(X, Y)), implies_1, or(X, or(Z, Y)))
% 240.66/31.06 = { by lemma 59 R->L }
% 240.66/31.06 fresh28(is_a_theorem(or(X, Y)), or_2, or(X, or(Z, Y)))
% 240.66/31.06 = { by lemma 55 R->L }
% 240.66/31.06 fresh28(is_a_theorem(or(X, Y)), op_and, or(X, or(Z, Y)))
% 240.66/31.06 = { by lemma 60 R->L }
% 240.66/31.06 fresh59(is_a_theorem(implies(or(X, Y), or(X, or(Z, Y)))), op_and, or(X, Y), or(X, or(Z, Y)))
% 240.66/31.06 = { by lemma 85 }
% 240.66/31.06 fresh59(implies_1, op_and, or(X, Y), or(X, or(Z, Y)))
% 240.66/31.06 = { by lemma 55 }
% 240.66/31.06 fresh59(implies_1, or_2, or(X, Y), or(X, or(Z, Y)))
% 240.66/31.06 = { by lemma 59 }
% 240.66/31.06 fresh59(implies_1, implies_1, or(X, Y), or(X, or(Z, Y)))
% 240.66/31.06 = { by lemma 63 }
% 240.66/31.06 op_and
% 240.66/31.06 = { by lemma 55 }
% 240.66/31.06 or_2
% 240.66/31.06 = { by lemma 59 }
% 240.66/31.06 implies_1
% 240.66/31.06
% 240.66/31.06 Lemma 87: is_a_theorem(or(or(X, Y), or(Z, not(Y)))) = implies_1.
% 240.66/31.06 Proof:
% 240.66/31.06 is_a_theorem(or(or(X, Y), or(Z, not(Y))))
% 240.66/31.06 = { by axiom 17 (modus_ponens_2) R->L }
% 240.66/31.06 fresh28(implies_1, implies_1, or(or(X, Y), or(Z, not(Y))))
% 240.66/31.06 = { by lemma 81 R->L }
% 240.66/31.06 fresh28(is_a_theorem(or(or(X, Y), not(Y))), implies_1, or(or(X, Y), or(Z, not(Y))))
% 240.66/31.06 = { by lemma 86 }
% 240.66/31.06 implies_1
% 240.66/31.06
% 240.66/31.06 Lemma 88: is_a_theorem(or(X, or(Y, not(X)))) = implies_1.
% 240.66/31.06 Proof:
% 240.66/31.06 is_a_theorem(or(X, or(Y, not(X))))
% 240.66/31.06 = { by axiom 17 (modus_ponens_2) R->L }
% 240.66/31.06 fresh28(implies_1, implies_1, or(X, or(Y, not(X))))
% 240.66/31.06 = { by lemma 79 R->L }
% 240.66/31.06 fresh28(fresh28(is_a_theorem(or(or(Y, not(X)), or(X, X))), implies_1, or(or(Y, not(X)), X)), implies_1, or(X, or(Y, not(X))))
% 240.66/31.06 = { by axiom 17 (modus_ponens_2) R->L }
% 240.66/31.06 fresh28(fresh28(fresh28(implies_1, implies_1, or(or(Y, not(X)), or(X, X))), implies_1, or(or(Y, not(X)), X)), implies_1, or(X, or(Y, not(X))))
% 240.66/31.06 = { by lemma 87 R->L }
% 240.66/31.06 fresh28(fresh28(fresh28(is_a_theorem(or(or(X, X), or(Y, not(X)))), implies_1, or(or(Y, not(X)), or(X, X))), implies_1, or(or(Y, not(X)), X)), implies_1, or(X, or(Y, not(X))))
% 240.66/31.06 = { by lemma 64 }
% 240.66/31.06 fresh28(fresh28(implies_1, implies_1, or(or(Y, not(X)), X)), implies_1, or(X, or(Y, not(X))))
% 240.66/31.06 = { by axiom 17 (modus_ponens_2) }
% 240.66/31.06 fresh28(is_a_theorem(or(or(Y, not(X)), X)), implies_1, or(X, or(Y, not(X))))
% 240.66/31.06 = { by lemma 64 }
% 240.66/31.06 implies_1
% 240.66/31.06
% 240.66/31.06 Lemma 89: is_a_theorem(implies(or(X, or(Y, Z)), or(X, or(Z, Y)))) = implies_1.
% 240.66/31.06 Proof:
% 240.66/31.06 is_a_theorem(implies(or(X, or(Y, Z)), or(X, or(Z, Y))))
% 240.66/31.06 = { by axiom 17 (modus_ponens_2) R->L }
% 240.66/31.06 fresh28(implies_1, implies_1, implies(or(X, or(Y, Z)), or(X, or(Z, Y))))
% 240.66/31.06 = { by lemma 59 R->L }
% 240.66/31.06 fresh28(or_2, implies_1, implies(or(X, or(Y, Z)), or(X, or(Z, Y))))
% 240.66/31.06 = { by lemma 61 R->L }
% 240.66/31.06 fresh28(is_a_theorem(implies(or(Y, Z), or(Z, Y))), implies_1, implies(or(X, or(Y, Z)), or(X, or(Z, Y))))
% 240.66/31.06 = { by lemma 75 }
% 240.66/31.06 implies_1
% 240.66/31.06
% 240.66/31.06 Lemma 90: fresh59(X, X, Y, Z) = implies_1.
% 240.66/31.06 Proof:
% 240.66/31.06 fresh59(X, X, Y, Z)
% 240.66/31.06 = { by axiom 20 (modus_ponens_2) }
% 240.66/31.06 fresh60(modus_ponens, true, Z)
% 240.66/31.06 = { by lemma 52 }
% 240.66/31.06 fresh60(op_and, true, Z)
% 240.66/31.06 = { by axiom 1 (principia_op_and) R->L }
% 240.66/31.06 fresh60(op_and, op_and, Z)
% 240.66/31.06 = { by lemma 62 }
% 240.66/31.06 op_and
% 240.66/31.06 = { by lemma 55 }
% 240.66/31.06 or_2
% 240.66/31.06 = { by lemma 59 }
% 240.66/31.06 implies_1
% 240.66/31.06
% 240.66/31.06 Lemma 91: fresh28(is_a_theorem(or(X, or(Y, Z))), implies_1, or(X, or(Z, Y))) = implies_1.
% 240.66/31.06 Proof:
% 240.66/31.06 fresh28(is_a_theorem(or(X, or(Y, Z))), implies_1, or(X, or(Z, Y)))
% 240.66/31.06 = { by lemma 83 R->L }
% 240.66/31.06 fresh59(is_a_theorem(implies(or(X, or(Y, Z)), or(X, or(Z, Y)))), implies_1, or(X, or(Y, Z)), or(X, or(Z, Y)))
% 240.66/31.06 = { by lemma 89 }
% 240.66/31.06 fresh59(implies_1, implies_1, or(X, or(Y, Z)), or(X, or(Z, Y)))
% 240.66/31.06 = { by lemma 90 }
% 240.66/31.06 implies_1
% 240.66/31.06
% 240.66/31.06 Lemma 92: fresh49(cn2, implies_1, X, Y) = implies_1.
% 240.66/31.06 Proof:
% 240.66/31.06 fresh49(cn2, implies_1, X, Y)
% 240.66/31.06 = { by lemma 84 R->L }
% 240.66/31.06 is_a_theorem(implies(X, implies(not(X), Y)))
% 240.66/31.06 = { by lemma 57 R->L }
% 240.66/31.06 is_a_theorem(or(not(X), implies(not(X), Y)))
% 240.66/31.06 = { by lemma 57 R->L }
% 240.66/31.06 is_a_theorem(or(not(X), or(not(not(X)), Y)))
% 240.66/31.06 = { by axiom 17 (modus_ponens_2) R->L }
% 240.66/31.06 fresh28(implies_1, implies_1, or(not(X), or(not(not(X)), Y)))
% 240.66/31.06 = { by lemma 88 R->L }
% 240.66/31.06 fresh28(is_a_theorem(or(not(X), or(Y, not(not(X))))), implies_1, or(not(X), or(not(not(X)), Y)))
% 240.66/31.06 = { by lemma 91 }
% 240.66/31.06 implies_1
% 240.66/31.06
% 240.66/31.06 Lemma 93: is_a_theorem(implies(or(X, Y), or(X, implies(not(Y), Z)))) = implies_1.
% 240.66/31.06 Proof:
% 240.66/31.06 is_a_theorem(implies(or(X, Y), or(X, implies(not(Y), Z))))
% 240.66/31.06 = { by axiom 17 (modus_ponens_2) R->L }
% 240.66/31.06 fresh28(implies_1, implies_1, implies(or(X, Y), or(X, implies(not(Y), Z))))
% 240.66/31.06 = { by lemma 92 R->L }
% 240.66/31.06 fresh28(fresh49(cn2, implies_1, Y, Z), implies_1, implies(or(X, Y), or(X, implies(not(Y), Z))))
% 240.66/31.06 = { by lemma 59 R->L }
% 240.66/31.06 fresh28(fresh49(cn2, or_2, Y, Z), implies_1, implies(or(X, Y), or(X, implies(not(Y), Z))))
% 240.66/31.07 = { by lemma 55 R->L }
% 240.66/31.07 fresh28(fresh49(cn2, op_and, Y, Z), implies_1, implies(or(X, Y), or(X, implies(not(Y), Z))))
% 240.66/31.07 = { by axiom 1 (principia_op_and) }
% 240.66/31.07 fresh28(fresh49(cn2, true, Y, Z), implies_1, implies(or(X, Y), or(X, implies(not(Y), Z))))
% 240.66/31.07 = { by axiom 39 (cn2_1) }
% 240.66/31.07 fresh28(is_a_theorem(implies(Y, implies(not(Y), Z))), implies_1, implies(or(X, Y), or(X, implies(not(Y), Z))))
% 240.66/31.07 = { by lemma 75 }
% 240.66/31.07 implies_1
% 240.66/31.07
% 240.66/31.07 Lemma 94: fresh55(and_2, implies_1, X, Y) = implies_1.
% 240.66/31.07 Proof:
% 240.66/31.07 fresh55(and_2, implies_1, X, Y)
% 240.66/31.07 = { by lemma 59 R->L }
% 240.66/31.07 fresh55(and_2, or_2, X, Y)
% 240.66/31.07 = { by lemma 55 R->L }
% 240.66/31.07 fresh55(and_2, op_and, X, Y)
% 240.66/31.07 = { by axiom 1 (principia_op_and) }
% 240.66/31.07 fresh55(and_2, true, X, Y)
% 240.66/31.07 = { by axiom 36 (and_2_1) }
% 240.66/31.07 is_a_theorem(implies(and(X, Y), Y))
% 240.66/31.07 = { by lemma 57 R->L }
% 240.66/31.07 is_a_theorem(or(not(and(X, Y)), Y))
% 240.66/31.07 = { by axiom 17 (modus_ponens_2) R->L }
% 240.66/31.07 fresh28(implies_1, implies_1, or(not(and(X, Y)), Y))
% 240.66/31.07 = { by lemma 80 R->L }
% 240.66/31.07 fresh28(fresh28(is_a_theorem(or(Y, implies(and(X, Y), not(and(X, Y))))), implies_1, or(Y, not(and(X, Y)))), implies_1, or(not(and(X, Y)), Y))
% 240.66/31.07 = { by lemma 71 R->L }
% 240.66/31.07 fresh28(fresh28(is_a_theorem(or(Y, implies(not(implies(X, not(Y))), not(and(X, Y))))), implies_1, or(Y, not(and(X, Y)))), implies_1, or(not(and(X, Y)), Y))
% 240.66/31.07 = { by axiom 17 (modus_ponens_2) R->L }
% 240.66/31.07 fresh28(fresh28(fresh28(implies_1, implies_1, or(Y, implies(not(implies(X, not(Y))), not(and(X, Y))))), implies_1, or(Y, not(and(X, Y)))), implies_1, or(not(and(X, Y)), Y))
% 240.66/31.07 = { by lemma 64 R->L }
% 240.66/31.07 fresh28(fresh28(fresh28(fresh28(is_a_theorem(or(implies(X, not(Y)), Y)), implies_1, or(Y, implies(X, not(Y)))), implies_1, or(Y, implies(not(implies(X, not(Y))), not(and(X, Y))))), implies_1, or(Y, not(and(X, Y)))), implies_1, or(not(and(X, Y)), Y))
% 240.66/31.07 = { by axiom 17 (modus_ponens_2) R->L }
% 240.66/31.07 fresh28(fresh28(fresh28(fresh28(fresh28(implies_1, implies_1, or(implies(X, not(Y)), Y)), implies_1, or(Y, implies(X, not(Y)))), implies_1, or(Y, implies(not(implies(X, not(Y))), not(and(X, Y))))), implies_1, or(Y, not(and(X, Y)))), implies_1, or(not(and(X, Y)), Y))
% 240.66/31.07 = { by lemma 64 R->L }
% 240.66/31.07 fresh28(fresh28(fresh28(fresh28(fresh28(fresh28(is_a_theorem(or(or(Y, Y), implies(X, not(Y)))), implies_1, or(implies(X, not(Y)), or(Y, Y))), implies_1, or(implies(X, not(Y)), Y)), implies_1, or(Y, implies(X, not(Y)))), implies_1, or(Y, implies(not(implies(X, not(Y))), not(and(X, Y))))), implies_1, or(Y, not(and(X, Y)))), implies_1, or(not(and(X, Y)), Y))
% 240.66/31.07 = { by axiom 17 (modus_ponens_2) R->L }
% 240.66/31.07 fresh28(fresh28(fresh28(fresh28(fresh28(fresh28(fresh28(implies_1, implies_1, or(or(Y, Y), implies(X, not(Y)))), implies_1, or(implies(X, not(Y)), or(Y, Y))), implies_1, or(implies(X, not(Y)), Y)), implies_1, or(Y, implies(X, not(Y)))), implies_1, or(Y, implies(not(implies(X, not(Y))), not(and(X, Y))))), implies_1, or(Y, not(and(X, Y)))), implies_1, or(not(and(X, Y)), Y))
% 240.66/31.07 = { by lemma 81 R->L }
% 240.66/31.07 fresh28(fresh28(fresh28(fresh28(fresh28(fresh28(fresh28(is_a_theorem(or(or(Y, Y), not(Y))), implies_1, or(or(Y, Y), implies(X, not(Y)))), implies_1, or(implies(X, not(Y)), or(Y, Y))), implies_1, or(implies(X, not(Y)), Y)), implies_1, or(Y, implies(X, not(Y)))), implies_1, or(Y, implies(not(implies(X, not(Y))), not(and(X, Y))))), implies_1, or(Y, not(and(X, Y)))), implies_1, or(not(and(X, Y)), Y))
% 240.66/31.07 = { by lemma 59 R->L }
% 240.66/31.07 fresh28(fresh28(fresh28(fresh28(fresh28(fresh28(fresh28(is_a_theorem(or(or(Y, Y), not(Y))), or_2, or(or(Y, Y), implies(X, not(Y)))), implies_1, or(implies(X, not(Y)), or(Y, Y))), implies_1, or(implies(X, not(Y)), Y)), implies_1, or(Y, implies(X, not(Y)))), implies_1, or(Y, implies(not(implies(X, not(Y))), not(and(X, Y))))), implies_1, or(Y, not(and(X, Y)))), implies_1, or(not(and(X, Y)), Y))
% 240.66/31.07 = { by lemma 55 R->L }
% 240.66/31.07 fresh28(fresh28(fresh28(fresh28(fresh28(fresh28(fresh28(is_a_theorem(or(or(Y, Y), not(Y))), op_and, or(or(Y, Y), implies(X, not(Y)))), implies_1, or(implies(X, not(Y)), or(Y, Y))), implies_1, or(implies(X, not(Y)), Y)), implies_1, or(Y, implies(X, not(Y)))), implies_1, or(Y, implies(not(implies(X, not(Y))), not(and(X, Y))))), implies_1, or(Y, not(and(X, Y)))), implies_1, or(not(and(X, Y)), Y))
% 240.66/31.07 = { by lemma 60 R->L }
% 240.66/31.07 fresh28(fresh28(fresh28(fresh28(fresh28(fresh28(fresh59(is_a_theorem(implies(or(or(Y, Y), not(Y)), or(or(Y, Y), implies(X, not(Y))))), op_and, or(or(Y, Y), not(Y)), or(or(Y, Y), implies(X, not(Y)))), implies_1, or(implies(X, not(Y)), or(Y, Y))), implies_1, or(implies(X, not(Y)), Y)), implies_1, or(Y, implies(X, not(Y)))), implies_1, or(Y, implies(not(implies(X, not(Y))), not(and(X, Y))))), implies_1, or(Y, not(and(X, Y)))), implies_1, or(not(and(X, Y)), Y))
% 240.66/31.07 = { by lemma 82 }
% 240.66/31.07 fresh28(fresh28(fresh28(fresh28(fresh28(fresh28(fresh59(implies_1, op_and, or(or(Y, Y), not(Y)), or(or(Y, Y), implies(X, not(Y)))), implies_1, or(implies(X, not(Y)), or(Y, Y))), implies_1, or(implies(X, not(Y)), Y)), implies_1, or(Y, implies(X, not(Y)))), implies_1, or(Y, implies(not(implies(X, not(Y))), not(and(X, Y))))), implies_1, or(Y, not(and(X, Y)))), implies_1, or(not(and(X, Y)), Y))
% 240.66/31.07 = { by lemma 55 }
% 240.66/31.07 fresh28(fresh28(fresh28(fresh28(fresh28(fresh28(fresh59(implies_1, or_2, or(or(Y, Y), not(Y)), or(or(Y, Y), implies(X, not(Y)))), implies_1, or(implies(X, not(Y)), or(Y, Y))), implies_1, or(implies(X, not(Y)), Y)), implies_1, or(Y, implies(X, not(Y)))), implies_1, or(Y, implies(not(implies(X, not(Y))), not(and(X, Y))))), implies_1, or(Y, not(and(X, Y)))), implies_1, or(not(and(X, Y)), Y))
% 240.66/31.07 = { by lemma 59 }
% 240.66/31.07 fresh28(fresh28(fresh28(fresh28(fresh28(fresh28(fresh59(implies_1, implies_1, or(or(Y, Y), not(Y)), or(or(Y, Y), implies(X, not(Y)))), implies_1, or(implies(X, not(Y)), or(Y, Y))), implies_1, or(implies(X, not(Y)), Y)), implies_1, or(Y, implies(X, not(Y)))), implies_1, or(Y, implies(not(implies(X, not(Y))), not(and(X, Y))))), implies_1, or(Y, not(and(X, Y)))), implies_1, or(not(and(X, Y)), Y))
% 240.66/31.07 = { by lemma 63 }
% 240.66/31.07 fresh28(fresh28(fresh28(fresh28(fresh28(fresh28(op_and, implies_1, or(implies(X, not(Y)), or(Y, Y))), implies_1, or(implies(X, not(Y)), Y)), implies_1, or(Y, implies(X, not(Y)))), implies_1, or(Y, implies(not(implies(X, not(Y))), not(and(X, Y))))), implies_1, or(Y, not(and(X, Y)))), implies_1, or(not(and(X, Y)), Y))
% 240.66/31.07 = { by lemma 55 }
% 240.66/31.07 fresh28(fresh28(fresh28(fresh28(fresh28(fresh28(or_2, implies_1, or(implies(X, not(Y)), or(Y, Y))), implies_1, or(implies(X, not(Y)), Y)), implies_1, or(Y, implies(X, not(Y)))), implies_1, or(Y, implies(not(implies(X, not(Y))), not(and(X, Y))))), implies_1, or(Y, not(and(X, Y)))), implies_1, or(not(and(X, Y)), Y))
% 240.66/31.07 = { by lemma 59 }
% 240.66/31.07 fresh28(fresh28(fresh28(fresh28(fresh28(fresh28(implies_1, implies_1, or(implies(X, not(Y)), or(Y, Y))), implies_1, or(implies(X, not(Y)), Y)), implies_1, or(Y, implies(X, not(Y)))), implies_1, or(Y, implies(not(implies(X, not(Y))), not(and(X, Y))))), implies_1, or(Y, not(and(X, Y)))), implies_1, or(not(and(X, Y)), Y))
% 240.66/31.07 = { by axiom 17 (modus_ponens_2) }
% 240.66/31.07 fresh28(fresh28(fresh28(fresh28(fresh28(is_a_theorem(or(implies(X, not(Y)), or(Y, Y))), implies_1, or(implies(X, not(Y)), Y)), implies_1, or(Y, implies(X, not(Y)))), implies_1, or(Y, implies(not(implies(X, not(Y))), not(and(X, Y))))), implies_1, or(Y, not(and(X, Y)))), implies_1, or(not(and(X, Y)), Y))
% 240.66/31.07 = { by lemma 79 }
% 240.66/31.07 fresh28(fresh28(fresh28(fresh28(implies_1, implies_1, or(Y, implies(X, not(Y)))), implies_1, or(Y, implies(not(implies(X, not(Y))), not(and(X, Y))))), implies_1, or(Y, not(and(X, Y)))), implies_1, or(not(and(X, Y)), Y))
% 240.66/31.07 = { by axiom 17 (modus_ponens_2) }
% 240.66/31.07 fresh28(fresh28(fresh28(is_a_theorem(or(Y, implies(X, not(Y)))), implies_1, or(Y, implies(not(implies(X, not(Y))), not(and(X, Y))))), implies_1, or(Y, not(and(X, Y)))), implies_1, or(not(and(X, Y)), Y))
% 240.66/31.07 = { by lemma 83 R->L }
% 240.66/31.07 fresh28(fresh28(fresh59(is_a_theorem(implies(or(Y, implies(X, not(Y))), or(Y, implies(not(implies(X, not(Y))), not(and(X, Y)))))), implies_1, or(Y, implies(X, not(Y))), or(Y, implies(not(implies(X, not(Y))), not(and(X, Y))))), implies_1, or(Y, not(and(X, Y)))), implies_1, or(not(and(X, Y)), Y))
% 240.66/31.07 = { by lemma 93 }
% 240.66/31.07 fresh28(fresh28(fresh59(implies_1, implies_1, or(Y, implies(X, not(Y))), or(Y, implies(not(implies(X, not(Y))), not(and(X, Y))))), implies_1, or(Y, not(and(X, Y)))), implies_1, or(not(and(X, Y)), Y))
% 240.66/31.07 = { by lemma 90 }
% 240.66/31.07 fresh28(fresh28(implies_1, implies_1, or(Y, not(and(X, Y)))), implies_1, or(not(and(X, Y)), Y))
% 240.66/31.07 = { by axiom 17 (modus_ponens_2) }
% 240.66/31.07 fresh28(is_a_theorem(or(Y, not(and(X, Y)))), implies_1, or(not(and(X, Y)), Y))
% 240.66/31.07 = { by lemma 64 }
% 240.66/31.07 implies_1
% 240.66/31.07
% 240.66/31.07 Lemma 95: and(implies(X, Y), implies(Y, X)) = equiv(X, Y).
% 240.66/31.07 Proof:
% 240.66/31.07 and(implies(X, Y), implies(Y, X))
% 240.66/31.07 = { by axiom 38 (op_equiv) R->L }
% 240.66/31.07 fresh23(op_equiv, true, X, Y)
% 240.66/31.07 = { by axiom 3 (principia_op_equiv) }
% 240.66/31.07 fresh23(true, true, X, Y)
% 240.66/31.07 = { by axiom 1 (principia_op_and) R->L }
% 240.66/31.07 fresh23(op_and, true, X, Y)
% 240.66/31.07 = { by axiom 1 (principia_op_and) R->L }
% 240.66/31.07 fresh23(op_and, op_and, X, Y)
% 240.66/31.07 = { by axiom 22 (op_equiv) }
% 240.66/31.07 equiv(X, Y)
% 240.66/31.07
% 240.66/31.07 Lemma 96: not(implies(X, and(Y, Z))) = and(X, implies(Y, not(Z))).
% 240.66/31.07 Proof:
% 240.66/31.07 not(implies(X, and(Y, Z)))
% 240.66/31.07 = { by lemma 71 R->L }
% 240.66/31.07 not(implies(X, not(implies(Y, not(Z)))))
% 240.66/31.07 = { by lemma 71 }
% 240.66/31.07 and(X, implies(Y, not(Z)))
% 240.66/31.07
% 240.66/31.07 Lemma 97: fresh28(is_a_theorem(X), implies_1, implies(not(X), Y)) = implies_1.
% 240.66/31.07 Proof:
% 240.66/31.07 fresh28(is_a_theorem(X), implies_1, implies(not(X), Y))
% 240.66/31.07 = { by lemma 83 R->L }
% 240.66/31.07 fresh59(is_a_theorem(implies(X, implies(not(X), Y))), implies_1, X, implies(not(X), Y))
% 240.66/31.07 = { by lemma 84 }
% 240.66/31.07 fresh59(fresh49(cn2, implies_1, X, Y), implies_1, X, implies(not(X), Y))
% 240.66/31.07 = { by lemma 92 }
% 240.66/31.07 fresh59(implies_1, implies_1, X, implies(not(X), Y))
% 240.66/31.07 = { by lemma 90 }
% 240.66/31.07 implies_1
% 240.66/31.07
% 240.66/31.07 Lemma 98: fresh28(is_a_theorem(implies(X, Y)), implies_1, implies(implies(Z, X), implies(Z, Y))) = implies_1.
% 240.66/31.07 Proof:
% 240.66/31.07 fresh28(is_a_theorem(implies(X, Y)), implies_1, implies(implies(Z, X), implies(Z, Y)))
% 240.66/31.07 = { by lemma 83 R->L }
% 240.66/31.07 fresh59(is_a_theorem(implies(implies(X, Y), implies(implies(Z, X), implies(Z, Y)))), implies_1, implies(X, Y), implies(implies(Z, X), implies(Z, Y)))
% 240.66/31.07 = { by lemma 57 R->L }
% 240.66/31.07 fresh59(is_a_theorem(implies(implies(X, Y), implies(implies(Z, X), or(not(Z), Y)))), implies_1, implies(X, Y), implies(implies(Z, X), implies(Z, Y)))
% 240.66/31.07 = { by lemma 57 R->L }
% 240.66/31.07 fresh59(is_a_theorem(implies(implies(X, Y), implies(or(not(Z), X), or(not(Z), Y)))), implies_1, implies(X, Y), implies(implies(Z, X), implies(Z, Y)))
% 240.66/31.07 = { by lemma 74 }
% 240.66/31.07 fresh59(or_2, implies_1, implies(X, Y), implies(implies(Z, X), implies(Z, Y)))
% 240.66/31.07 = { by lemma 59 }
% 240.66/31.07 fresh59(implies_1, implies_1, implies(X, Y), implies(implies(Z, X), implies(Z, Y)))
% 240.66/31.07 = { by lemma 90 }
% 240.66/31.07 implies_1
% 240.66/31.07
% 240.66/31.07 Lemma 99: and(X, X) = X.
% 240.66/31.07 Proof:
% 240.66/31.07 and(X, X)
% 240.66/31.07 = { by lemma 77 R->L }
% 240.66/31.07 fresh(is_a_theorem(equiv(and(X, X), X)), implies_1, and(X, X), X)
% 240.66/31.07 = { by axiom 17 (modus_ponens_2) R->L }
% 240.66/31.07 fresh(fresh28(implies_1, implies_1, equiv(and(X, X), X)), implies_1, and(X, X), X)
% 240.66/31.07 = { by lemma 94 R->L }
% 240.66/31.07 fresh(fresh28(fresh55(and_2, implies_1, X, X), implies_1, equiv(and(X, X), X)), implies_1, and(X, X), X)
% 240.66/31.07 = { by lemma 59 R->L }
% 240.66/31.07 fresh(fresh28(fresh55(and_2, or_2, X, X), implies_1, equiv(and(X, X), X)), implies_1, and(X, X), X)
% 240.66/31.07 = { by lemma 55 R->L }
% 240.66/31.07 fresh(fresh28(fresh55(and_2, op_and, X, X), implies_1, equiv(and(X, X), X)), implies_1, and(X, X), X)
% 240.66/31.07 = { by axiom 1 (principia_op_and) }
% 240.66/31.07 fresh(fresh28(fresh55(and_2, true, X, X), implies_1, equiv(and(X, X), X)), implies_1, and(X, X), X)
% 240.66/31.07 = { by axiom 36 (and_2_1) }
% 240.66/31.07 fresh(fresh28(is_a_theorem(implies(and(X, X), X)), implies_1, equiv(and(X, X), X)), implies_1, and(X, X), X)
% 240.66/31.07 = { by lemma 83 R->L }
% 240.66/31.07 fresh(fresh59(is_a_theorem(implies(implies(and(X, X), X), equiv(and(X, X), X))), implies_1, implies(and(X, X), X), equiv(and(X, X), X)), implies_1, and(X, X), X)
% 240.66/31.07 = { by lemma 95 R->L }
% 240.66/31.07 fresh(fresh59(is_a_theorem(implies(implies(and(X, X), X), and(implies(and(X, X), X), implies(X, and(X, X))))), implies_1, implies(and(X, X), X), equiv(and(X, X), X)), implies_1, and(X, X), X)
% 240.66/31.07 = { by lemma 57 R->L }
% 240.66/31.07 fresh(fresh59(is_a_theorem(or(not(implies(and(X, X), X)), and(implies(and(X, X), X), implies(X, and(X, X))))), implies_1, implies(and(X, X), X), equiv(and(X, X), X)), implies_1, and(X, X), X)
% 240.66/31.07 = { by axiom 17 (modus_ponens_2) R->L }
% 240.66/31.07 fresh(fresh59(fresh28(implies_1, implies_1, or(not(implies(and(X, X), X)), and(implies(and(X, X), X), implies(X, and(X, X))))), implies_1, implies(and(X, X), X), equiv(and(X, X), X)), implies_1, and(X, X), X)
% 240.66/31.08 = { by lemma 80 R->L }
% 240.66/31.08 fresh(fresh59(fresh28(fresh28(is_a_theorem(or(and(implies(and(X, X), X), implies(X, and(X, X))), implies(implies(and(X, X), X), not(implies(and(X, X), X))))), implies_1, or(and(implies(and(X, X), X), implies(X, and(X, X))), not(implies(and(X, X), X)))), implies_1, or(not(implies(and(X, X), X)), and(implies(and(X, X), X), implies(X, and(X, X))))), implies_1, implies(and(X, X), X), equiv(and(X, X), X)), implies_1, and(X, X), X)
% 240.66/31.08 = { by lemma 72 R->L }
% 240.66/31.08 fresh(fresh59(fresh28(fresh28(is_a_theorem(implies(implies(implies(and(X, X), X), not(implies(X, and(X, X)))), implies(implies(and(X, X), X), not(implies(and(X, X), X))))), implies_1, or(and(implies(and(X, X), X), implies(X, and(X, X))), not(implies(and(X, X), X)))), implies_1, or(not(implies(and(X, X), X)), and(implies(and(X, X), X), implies(X, and(X, X))))), implies_1, implies(and(X, X), X), equiv(and(X, X), X)), implies_1, and(X, X), X)
% 240.66/31.08 = { by lemma 96 }
% 240.66/31.08 fresh(fresh59(fresh28(fresh28(is_a_theorem(implies(implies(implies(and(X, X), X), and(X, implies(X, not(X)))), implies(implies(and(X, X), X), not(implies(and(X, X), X))))), implies_1, or(and(implies(and(X, X), X), implies(X, and(X, X))), not(implies(and(X, X), X)))), implies_1, or(not(implies(and(X, X), X)), and(implies(and(X, X), X), implies(X, and(X, X))))), implies_1, implies(and(X, X), X), equiv(and(X, X), X)), implies_1, and(X, X), X)
% 240.66/31.08 = { by axiom 17 (modus_ponens_2) R->L }
% 240.66/31.08 fresh(fresh59(fresh28(fresh28(fresh28(implies_1, implies_1, implies(implies(implies(and(X, X), X), and(X, implies(X, not(X)))), implies(implies(and(X, X), X), not(implies(and(X, X), X))))), implies_1, or(and(implies(and(X, X), X), implies(X, and(X, X))), not(implies(and(X, X), X)))), implies_1, or(not(implies(and(X, X), X)), and(implies(and(X, X), X), implies(X, and(X, X))))), implies_1, implies(and(X, X), X), equiv(and(X, X), X)), implies_1, and(X, X), X)
% 240.66/31.08 = { by lemma 97 R->L }
% 240.66/31.08 fresh(fresh59(fresh28(fresh28(fresh28(fresh28(is_a_theorem(implies(X, and(X, X))), implies_1, implies(not(implies(X, and(X, X))), not(implies(and(X, X), X)))), implies_1, implies(implies(implies(and(X, X), X), and(X, implies(X, not(X)))), implies(implies(and(X, X), X), not(implies(and(X, X), X))))), implies_1, or(and(implies(and(X, X), X), implies(X, and(X, X))), not(implies(and(X, X), X)))), implies_1, or(not(implies(and(X, X), X)), and(implies(and(X, X), X), implies(X, and(X, X))))), implies_1, implies(and(X, X), X), equiv(and(X, X), X)), implies_1, and(X, X), X)
% 240.66/31.08 = { by axiom 31 (kn1_1) R->L }
% 240.66/31.08 fresh(fresh59(fresh28(fresh28(fresh28(fresh28(fresh33(kn1, true, X), implies_1, implies(not(implies(X, and(X, X))), not(implies(and(X, X), X)))), implies_1, implies(implies(implies(and(X, X), X), and(X, implies(X, not(X)))), implies(implies(and(X, X), X), not(implies(and(X, X), X))))), implies_1, or(and(implies(and(X, X), X), implies(X, and(X, X))), not(implies(and(X, X), X)))), implies_1, or(not(implies(and(X, X), X)), and(implies(and(X, X), X), implies(X, and(X, X))))), implies_1, implies(and(X, X), X), equiv(and(X, X), X)), implies_1, and(X, X), X)
% 240.66/31.08 = { by axiom 1 (principia_op_and) R->L }
% 240.66/31.08 fresh(fresh59(fresh28(fresh28(fresh28(fresh28(fresh33(kn1, op_and, X), implies_1, implies(not(implies(X, and(X, X))), not(implies(and(X, X), X)))), implies_1, implies(implies(implies(and(X, X), X), and(X, implies(X, not(X)))), implies(implies(and(X, X), X), not(implies(and(X, X), X))))), implies_1, or(and(implies(and(X, X), X), implies(X, and(X, X))), not(implies(and(X, X), X)))), implies_1, or(not(implies(and(X, X), X)), and(implies(and(X, X), X), implies(X, and(X, X))))), implies_1, implies(and(X, X), X), equiv(and(X, X), X)), implies_1, and(X, X), X)
% 240.66/31.08 = { by lemma 73 }
% 240.66/31.08 fresh(fresh59(fresh28(fresh28(fresh28(fresh28(fresh33(implies_1, op_and, X), implies_1, implies(not(implies(X, and(X, X))), not(implies(and(X, X), X)))), implies_1, implies(implies(implies(and(X, X), X), and(X, implies(X, not(X)))), implies(implies(and(X, X), X), not(implies(and(X, X), X))))), implies_1, or(and(implies(and(X, X), X), implies(X, and(X, X))), not(implies(and(X, X), X)))), implies_1, or(not(implies(and(X, X), X)), and(implies(and(X, X), X), implies(X, and(X, X))))), implies_1, implies(and(X, X), X), equiv(and(X, X), X)), implies_1, and(X, X), X)
% 240.66/31.08 = { by lemma 55 }
% 240.66/31.08 fresh(fresh59(fresh28(fresh28(fresh28(fresh28(fresh33(implies_1, or_2, X), implies_1, implies(not(implies(X, and(X, X))), not(implies(and(X, X), X)))), implies_1, implies(implies(implies(and(X, X), X), and(X, implies(X, not(X)))), implies(implies(and(X, X), X), not(implies(and(X, X), X))))), implies_1, or(and(implies(and(X, X), X), implies(X, and(X, X))), not(implies(and(X, X), X)))), implies_1, or(not(implies(and(X, X), X)), and(implies(and(X, X), X), implies(X, and(X, X))))), implies_1, implies(and(X, X), X), equiv(and(X, X), X)), implies_1, and(X, X), X)
% 240.66/31.08 = { by lemma 59 }
% 240.66/31.08 fresh(fresh59(fresh28(fresh28(fresh28(fresh28(fresh33(implies_1, implies_1, X), implies_1, implies(not(implies(X, and(X, X))), not(implies(and(X, X), X)))), implies_1, implies(implies(implies(and(X, X), X), and(X, implies(X, not(X)))), implies(implies(and(X, X), X), not(implies(and(X, X), X))))), implies_1, or(and(implies(and(X, X), X), implies(X, and(X, X))), not(implies(and(X, X), X)))), implies_1, or(not(implies(and(X, X), X)), and(implies(and(X, X), X), implies(X, and(X, X))))), implies_1, implies(and(X, X), X), equiv(and(X, X), X)), implies_1, and(X, X), X)
% 240.66/31.08 = { by axiom 16 (kn1_1) }
% 240.66/31.08 fresh(fresh59(fresh28(fresh28(fresh28(fresh28(true, implies_1, implies(not(implies(X, and(X, X))), not(implies(and(X, X), X)))), implies_1, implies(implies(implies(and(X, X), X), and(X, implies(X, not(X)))), implies(implies(and(X, X), X), not(implies(and(X, X), X))))), implies_1, or(and(implies(and(X, X), X), implies(X, and(X, X))), not(implies(and(X, X), X)))), implies_1, or(not(implies(and(X, X), X)), and(implies(and(X, X), X), implies(X, and(X, X))))), implies_1, implies(and(X, X), X), equiv(and(X, X), X)), implies_1, and(X, X), X)
% 240.66/31.08 = { by axiom 1 (principia_op_and) R->L }
% 240.66/31.08 fresh(fresh59(fresh28(fresh28(fresh28(fresh28(op_and, implies_1, implies(not(implies(X, and(X, X))), not(implies(and(X, X), X)))), implies_1, implies(implies(implies(and(X, X), X), and(X, implies(X, not(X)))), implies(implies(and(X, X), X), not(implies(and(X, X), X))))), implies_1, or(and(implies(and(X, X), X), implies(X, and(X, X))), not(implies(and(X, X), X)))), implies_1, or(not(implies(and(X, X), X)), and(implies(and(X, X), X), implies(X, and(X, X))))), implies_1, implies(and(X, X), X), equiv(and(X, X), X)), implies_1, and(X, X), X)
% 240.66/31.08 = { by lemma 55 }
% 240.66/31.08 fresh(fresh59(fresh28(fresh28(fresh28(fresh28(or_2, implies_1, implies(not(implies(X, and(X, X))), not(implies(and(X, X), X)))), implies_1, implies(implies(implies(and(X, X), X), and(X, implies(X, not(X)))), implies(implies(and(X, X), X), not(implies(and(X, X), X))))), implies_1, or(and(implies(and(X, X), X), implies(X, and(X, X))), not(implies(and(X, X), X)))), implies_1, or(not(implies(and(X, X), X)), and(implies(and(X, X), X), implies(X, and(X, X))))), implies_1, implies(and(X, X), X), equiv(and(X, X), X)), implies_1, and(X, X), X)
% 240.66/31.08 = { by lemma 59 }
% 240.66/31.08 fresh(fresh59(fresh28(fresh28(fresh28(fresh28(implies_1, implies_1, implies(not(implies(X, and(X, X))), not(implies(and(X, X), X)))), implies_1, implies(implies(implies(and(X, X), X), and(X, implies(X, not(X)))), implies(implies(and(X, X), X), not(implies(and(X, X), X))))), implies_1, or(and(implies(and(X, X), X), implies(X, and(X, X))), not(implies(and(X, X), X)))), implies_1, or(not(implies(and(X, X), X)), and(implies(and(X, X), X), implies(X, and(X, X))))), implies_1, implies(and(X, X), X), equiv(and(X, X), X)), implies_1, and(X, X), X)
% 240.66/31.08 = { by axiom 17 (modus_ponens_2) }
% 240.66/31.08 fresh(fresh59(fresh28(fresh28(fresh28(is_a_theorem(implies(not(implies(X, and(X, X))), not(implies(and(X, X), X)))), implies_1, implies(implies(implies(and(X, X), X), and(X, implies(X, not(X)))), implies(implies(and(X, X), X), not(implies(and(X, X), X))))), implies_1, or(and(implies(and(X, X), X), implies(X, and(X, X))), not(implies(and(X, X), X)))), implies_1, or(not(implies(and(X, X), X)), and(implies(and(X, X), X), implies(X, and(X, X))))), implies_1, implies(and(X, X), X), equiv(and(X, X), X)), implies_1, and(X, X), X)
% 240.66/31.08 = { by lemma 96 }
% 240.66/31.08 fresh(fresh59(fresh28(fresh28(fresh28(is_a_theorem(implies(and(X, implies(X, not(X))), not(implies(and(X, X), X)))), implies_1, implies(implies(implies(and(X, X), X), and(X, implies(X, not(X)))), implies(implies(and(X, X), X), not(implies(and(X, X), X))))), implies_1, or(and(implies(and(X, X), X), implies(X, and(X, X))), not(implies(and(X, X), X)))), implies_1, or(not(implies(and(X, X), X)), and(implies(and(X, X), X), implies(X, and(X, X))))), implies_1, implies(and(X, X), X), equiv(and(X, X), X)), implies_1, and(X, X), X)
% 240.66/31.08 = { by lemma 98 }
% 240.66/31.08 fresh(fresh59(fresh28(fresh28(implies_1, implies_1, or(and(implies(and(X, X), X), implies(X, and(X, X))), not(implies(and(X, X), X)))), implies_1, or(not(implies(and(X, X), X)), and(implies(and(X, X), X), implies(X, and(X, X))))), implies_1, implies(and(X, X), X), equiv(and(X, X), X)), implies_1, and(X, X), X)
% 240.66/31.08 = { by axiom 17 (modus_ponens_2) }
% 240.66/31.08 fresh(fresh59(fresh28(is_a_theorem(or(and(implies(and(X, X), X), implies(X, and(X, X))), not(implies(and(X, X), X)))), implies_1, or(not(implies(and(X, X), X)), and(implies(and(X, X), X), implies(X, and(X, X))))), implies_1, implies(and(X, X), X), equiv(and(X, X), X)), implies_1, and(X, X), X)
% 240.66/31.08 = { by lemma 64 }
% 240.66/31.08 fresh(fresh59(implies_1, implies_1, implies(and(X, X), X), equiv(and(X, X), X)), implies_1, and(X, X), X)
% 240.66/31.08 = { by lemma 90 }
% 240.66/31.08 fresh(implies_1, implies_1, and(X, X), X)
% 240.66/31.08 = { by axiom 19 (substitution_of_equivalents_2) }
% 240.66/31.08 X
% 240.66/31.08
% 240.66/31.08 Lemma 100: and(X, implies(Y, not(Y))) = not(implies(X, Y)).
% 240.66/31.08 Proof:
% 240.66/31.08 and(X, implies(Y, not(Y)))
% 240.66/31.08 = { by lemma 96 R->L }
% 240.66/31.08 not(implies(X, and(Y, Y)))
% 240.66/31.08 = { by lemma 99 }
% 240.66/31.08 not(implies(X, Y))
% 240.66/31.08
% 240.66/31.08 Lemma 101: implies(X, not(X)) = not(or(X, X)).
% 240.66/31.08 Proof:
% 240.66/31.08 implies(X, not(X))
% 240.66/31.08 = { by lemma 99 R->L }
% 240.66/31.08 and(implies(X, not(X)), implies(X, not(X)))
% 240.66/31.08 = { by lemma 100 }
% 240.66/31.08 not(implies(implies(X, not(X)), X))
% 240.66/31.08 = { by lemma 72 }
% 240.66/31.08 not(or(and(X, X), X))
% 240.66/31.08 = { by lemma 99 }
% 240.66/31.08 not(or(X, X))
% 240.66/31.08
% 240.66/31.08 Lemma 102: not(or(and(X, Y), not(Z))) = and(implies(X, not(Y)), Z).
% 240.66/31.08 Proof:
% 240.66/31.08 not(or(and(X, Y), not(Z)))
% 240.66/31.08 = { by lemma 72 R->L }
% 240.66/31.08 not(implies(implies(X, not(Y)), not(Z)))
% 240.66/31.08 = { by lemma 71 }
% 240.66/31.08 and(implies(X, not(Y)), Z)
% 240.66/31.08
% 240.66/31.08 Lemma 103: and(not(or(X, X)), Y) = not(or(X, not(Y))).
% 240.66/31.08 Proof:
% 240.66/31.08 and(not(or(X, X)), Y)
% 240.66/31.08 = { by lemma 101 R->L }
% 240.66/31.08 and(implies(X, not(X)), Y)
% 240.66/31.08 = { by lemma 102 R->L }
% 240.66/31.08 not(or(and(X, X), not(Y)))
% 240.66/31.08 = { by lemma 99 }
% 240.66/31.08 not(or(X, not(Y)))
% 240.66/31.08
% 240.66/31.08 Lemma 104: is_a_theorem(implies(X, or(X, Y))) = fresh18(or_1, implies_1, X, Y).
% 240.66/31.08 Proof:
% 240.66/31.08 is_a_theorem(implies(X, or(X, Y)))
% 240.66/31.08 = { by axiom 32 (or_1_1) R->L }
% 240.66/31.08 fresh18(or_1, true, X, Y)
% 240.66/31.08 = { by axiom 1 (principia_op_and) R->L }
% 240.66/31.08 fresh18(or_1, op_and, X, Y)
% 240.66/31.08 = { by lemma 55 }
% 240.66/31.08 fresh18(or_1, or_2, X, Y)
% 240.66/31.08 = { by lemma 59 }
% 240.66/31.08 fresh18(or_1, implies_1, X, Y)
% 240.66/31.08
% 240.66/31.08 Lemma 105: fresh28(is_a_theorem(implies(X, or(Y, Y))), implies_1, implies(X, Y)) = implies_1.
% 240.66/31.08 Proof:
% 240.66/31.08 fresh28(is_a_theorem(implies(X, or(Y, Y))), implies_1, implies(X, Y))
% 240.66/31.08 = { by lemma 59 R->L }
% 240.66/31.08 fresh28(is_a_theorem(implies(X, or(Y, Y))), or_2, implies(X, Y))
% 240.66/31.08 = { by lemma 55 R->L }
% 240.66/31.08 fresh28(is_a_theorem(implies(X, or(Y, Y))), op_and, implies(X, Y))
% 240.66/31.08 = { by lemma 60 R->L }
% 240.66/31.08 fresh59(is_a_theorem(implies(implies(X, or(Y, Y)), implies(X, Y))), op_and, implies(X, or(Y, Y)), implies(X, Y))
% 240.66/31.08 = { by lemma 57 R->L }
% 240.66/31.08 fresh59(is_a_theorem(implies(implies(X, or(Y, Y)), or(not(X), Y))), op_and, implies(X, or(Y, Y)), implies(X, Y))
% 240.66/31.08 = { by lemma 57 R->L }
% 240.66/31.08 fresh59(is_a_theorem(implies(or(not(X), or(Y, Y)), or(not(X), Y))), op_and, implies(X, or(Y, Y)), implies(X, Y))
% 240.66/31.08 = { by lemma 78 }
% 240.66/31.08 fresh59(implies_1, op_and, implies(X, or(Y, Y)), implies(X, Y))
% 240.66/31.08 = { by lemma 55 }
% 240.66/31.08 fresh59(implies_1, or_2, implies(X, or(Y, Y)), implies(X, Y))
% 240.66/31.08 = { by lemma 59 }
% 240.66/31.08 fresh59(implies_1, implies_1, implies(X, or(Y, Y)), implies(X, Y))
% 240.66/31.08 = { by lemma 63 }
% 240.66/31.08 op_and
% 240.66/31.08 = { by lemma 55 }
% 240.66/31.08 or_2
% 240.66/31.08 = { by lemma 59 }
% 240.66/31.08 implies_1
% 240.66/31.08
% 240.66/31.08 Lemma 106: is_a_theorem(implies(or(implies(X, Y), Y), implies(X, Y))) = implies_1.
% 240.66/31.08 Proof:
% 240.66/31.08 is_a_theorem(implies(or(implies(X, Y), Y), implies(X, Y)))
% 240.66/31.08 = { by axiom 17 (modus_ponens_2) R->L }
% 240.66/31.08 fresh28(implies_1, implies_1, implies(or(implies(X, Y), Y), implies(X, Y)))
% 240.66/31.08 = { by lemma 82 R->L }
% 240.66/31.08 fresh28(is_a_theorem(implies(or(implies(X, Y), Y), or(implies(X, Y), implies(X, Y)))), implies_1, implies(or(implies(X, Y), Y), implies(X, Y)))
% 240.66/31.08 = { by lemma 105 }
% 240.66/31.08 implies_1
% 240.66/31.08
% 240.66/31.08 Lemma 107: fresh18(or_1, implies_1, X, Y) = implies_1.
% 240.66/31.08 Proof:
% 240.66/31.08 fresh18(or_1, implies_1, X, Y)
% 240.66/31.08 = { by lemma 104 R->L }
% 240.66/31.08 is_a_theorem(implies(X, or(X, Y)))
% 240.66/31.08 = { by axiom 17 (modus_ponens_2) R->L }
% 240.66/31.08 fresh28(implies_1, implies_1, implies(X, or(X, Y)))
% 240.66/31.08 = { by lemma 91 R->L }
% 240.66/31.08 fresh28(fresh28(is_a_theorem(or(implies(X, or(X, Y)), or(Y, X))), implies_1, or(implies(X, or(X, Y)), or(X, Y))), implies_1, implies(X, or(X, Y)))
% 240.66/31.08 = { by axiom 17 (modus_ponens_2) R->L }
% 240.66/31.08 fresh28(fresh28(fresh28(implies_1, implies_1, or(implies(X, or(X, Y)), or(Y, X))), implies_1, or(implies(X, or(X, Y)), or(X, Y))), implies_1, implies(X, or(X, Y)))
% 240.66/31.09 = { by lemma 91 R->L }
% 240.66/31.09 fresh28(fresh28(fresh28(fresh28(is_a_theorem(or(or(Y, X), or(or(X, Y), not(X)))), implies_1, or(or(Y, X), or(not(X), or(X, Y)))), implies_1, or(implies(X, or(X, Y)), or(Y, X))), implies_1, or(implies(X, or(X, Y)), or(X, Y))), implies_1, implies(X, or(X, Y)))
% 240.66/31.09 = { by lemma 87 }
% 240.66/31.09 fresh28(fresh28(fresh28(fresh28(implies_1, implies_1, or(or(Y, X), or(not(X), or(X, Y)))), implies_1, or(implies(X, or(X, Y)), or(Y, X))), implies_1, or(implies(X, or(X, Y)), or(X, Y))), implies_1, implies(X, or(X, Y)))
% 240.66/31.09 = { by axiom 17 (modus_ponens_2) }
% 240.66/31.09 fresh28(fresh28(fresh28(is_a_theorem(or(or(Y, X), or(not(X), or(X, Y)))), implies_1, or(implies(X, or(X, Y)), or(Y, X))), implies_1, or(implies(X, or(X, Y)), or(X, Y))), implies_1, implies(X, or(X, Y)))
% 240.66/31.09 = { by lemma 57 }
% 240.66/31.09 fresh28(fresh28(fresh28(is_a_theorem(or(or(Y, X), implies(X, or(X, Y)))), implies_1, or(implies(X, or(X, Y)), or(Y, X))), implies_1, or(implies(X, or(X, Y)), or(X, Y))), implies_1, implies(X, or(X, Y)))
% 240.66/31.09 = { by lemma 64 }
% 240.66/31.09 fresh28(fresh28(implies_1, implies_1, or(implies(X, or(X, Y)), or(X, Y))), implies_1, implies(X, or(X, Y)))
% 240.66/31.09 = { by axiom 17 (modus_ponens_2) }
% 240.66/31.09 fresh28(is_a_theorem(or(implies(X, or(X, Y)), or(X, Y))), implies_1, implies(X, or(X, Y)))
% 240.66/31.09 = { by lemma 59 R->L }
% 240.66/31.09 fresh28(is_a_theorem(or(implies(X, or(X, Y)), or(X, Y))), or_2, implies(X, or(X, Y)))
% 240.66/31.09 = { by lemma 55 R->L }
% 240.66/31.09 fresh28(is_a_theorem(or(implies(X, or(X, Y)), or(X, Y))), op_and, implies(X, or(X, Y)))
% 240.66/31.09 = { by lemma 60 R->L }
% 240.66/31.09 fresh59(is_a_theorem(implies(or(implies(X, or(X, Y)), or(X, Y)), implies(X, or(X, Y)))), op_and, or(implies(X, or(X, Y)), or(X, Y)), implies(X, or(X, Y)))
% 240.66/31.09 = { by lemma 106 }
% 240.66/31.09 fresh59(implies_1, op_and, or(implies(X, or(X, Y)), or(X, Y)), implies(X, or(X, Y)))
% 240.66/31.09 = { by lemma 55 }
% 240.66/31.09 fresh59(implies_1, or_2, or(implies(X, or(X, Y)), or(X, Y)), implies(X, or(X, Y)))
% 240.66/31.09 = { by lemma 59 }
% 240.66/31.09 fresh59(implies_1, implies_1, or(implies(X, or(X, Y)), or(X, Y)), implies(X, or(X, Y)))
% 240.66/31.09 = { by lemma 63 }
% 240.66/31.09 op_and
% 240.66/31.09 = { by lemma 55 }
% 240.66/31.09 or_2
% 240.66/31.09 = { by lemma 59 }
% 240.66/31.09 implies_1
% 240.66/31.09
% 240.66/31.09 Lemma 108: is_a_theorem(implies(X, or(X, Y))) = implies_1.
% 240.66/31.09 Proof:
% 240.66/31.09 is_a_theorem(implies(X, or(X, Y)))
% 240.66/31.09 = { by axiom 32 (or_1_1) R->L }
% 240.66/31.09 fresh18(or_1, true, X, Y)
% 240.66/31.09 = { by axiom 1 (principia_op_and) R->L }
% 240.66/31.09 fresh18(or_1, op_and, X, Y)
% 240.66/31.09 = { by lemma 55 }
% 240.66/31.09 fresh18(or_1, or_2, X, Y)
% 240.66/31.09 = { by lemma 59 }
% 240.66/31.09 fresh18(or_1, implies_1, X, Y)
% 240.66/31.09 = { by lemma 107 }
% 240.66/31.09 implies_1
% 240.66/31.09
% 240.66/31.09 Lemma 109: or(X, X) = X.
% 240.66/31.09 Proof:
% 240.66/31.09 or(X, X)
% 240.66/31.09 = { by axiom 19 (substitution_of_equivalents_2) R->L }
% 240.66/31.09 fresh(implies_1, implies_1, X, or(X, X))
% 240.66/31.09 = { by lemma 90 R->L }
% 240.66/31.09 fresh(fresh59(implies_1, implies_1, implies(X, or(X, X)), equiv(X, or(X, X))), implies_1, X, or(X, X))
% 240.66/31.09 = { by lemma 64 R->L }
% 240.66/31.09 fresh(fresh59(fresh28(is_a_theorem(or(and(implies(X, or(X, X)), implies(or(X, X), X)), not(implies(X, or(X, X))))), implies_1, or(not(implies(X, or(X, X))), and(implies(X, or(X, X)), implies(or(X, X), X)))), implies_1, implies(X, or(X, X)), equiv(X, or(X, X))), implies_1, X, or(X, X))
% 240.66/31.09 = { by axiom 17 (modus_ponens_2) R->L }
% 240.66/31.09 fresh(fresh59(fresh28(fresh28(implies_1, implies_1, or(and(implies(X, or(X, X)), implies(or(X, X), X)), not(implies(X, or(X, X))))), implies_1, or(not(implies(X, or(X, X))), and(implies(X, or(X, X)), implies(or(X, X), X)))), implies_1, implies(X, or(X, X)), equiv(X, or(X, X))), implies_1, X, or(X, X))
% 240.66/31.09 = { by lemma 98 R->L }
% 240.66/31.09 fresh(fresh59(fresh28(fresh28(fresh28(is_a_theorem(implies(not(implies(or(X, X), X)), not(implies(X, or(X, X))))), implies_1, implies(implies(implies(X, or(X, X)), not(implies(or(X, X), X))), implies(implies(X, or(X, X)), not(implies(X, or(X, X)))))), implies_1, or(and(implies(X, or(X, X)), implies(or(X, X), X)), not(implies(X, or(X, X))))), implies_1, or(not(implies(X, or(X, X))), and(implies(X, or(X, X)), implies(or(X, X), X)))), implies_1, implies(X, or(X, X)), equiv(X, or(X, X))), implies_1, X, or(X, X))
% 240.66/31.09 = { by axiom 17 (modus_ponens_2) R->L }
% 240.66/31.09 fresh(fresh59(fresh28(fresh28(fresh28(fresh28(implies_1, implies_1, implies(not(implies(or(X, X), X)), not(implies(X, or(X, X))))), implies_1, implies(implies(implies(X, or(X, X)), not(implies(or(X, X), X))), implies(implies(X, or(X, X)), not(implies(X, or(X, X)))))), implies_1, or(and(implies(X, or(X, X)), implies(or(X, X), X)), not(implies(X, or(X, X))))), implies_1, or(not(implies(X, or(X, X))), and(implies(X, or(X, X)), implies(or(X, X), X)))), implies_1, implies(X, or(X, X)), equiv(X, or(X, X))), implies_1, X, or(X, X))
% 240.66/31.09 = { by lemma 59 R->L }
% 240.66/31.09 fresh(fresh59(fresh28(fresh28(fresh28(fresh28(or_2, implies_1, implies(not(implies(or(X, X), X)), not(implies(X, or(X, X))))), implies_1, implies(implies(implies(X, or(X, X)), not(implies(or(X, X), X))), implies(implies(X, or(X, X)), not(implies(X, or(X, X)))))), implies_1, or(and(implies(X, or(X, X)), implies(or(X, X), X)), not(implies(X, or(X, X))))), implies_1, or(not(implies(X, or(X, X))), and(implies(X, or(X, X)), implies(or(X, X), X)))), implies_1, implies(X, or(X, X)), equiv(X, or(X, X))), implies_1, X, or(X, X))
% 240.66/31.09 = { by lemma 55 R->L }
% 240.66/31.09 fresh(fresh59(fresh28(fresh28(fresh28(fresh28(op_and, implies_1, implies(not(implies(or(X, X), X)), not(implies(X, or(X, X))))), implies_1, implies(implies(implies(X, or(X, X)), not(implies(or(X, X), X))), implies(implies(X, or(X, X)), not(implies(X, or(X, X)))))), implies_1, or(and(implies(X, or(X, X)), implies(or(X, X), X)), not(implies(X, or(X, X))))), implies_1, or(not(implies(X, or(X, X))), and(implies(X, or(X, X)), implies(or(X, X), X)))), implies_1, implies(X, or(X, X)), equiv(X, or(X, X))), implies_1, X, or(X, X))
% 240.66/31.09 = { by lemma 69 R->L }
% 240.66/31.09 fresh(fresh59(fresh28(fresh28(fresh28(fresh28(fresh12(op_and, op_and, X), implies_1, implies(not(implies(or(X, X), X)), not(implies(X, or(X, X))))), implies_1, implies(implies(implies(X, or(X, X)), not(implies(or(X, X), X))), implies(implies(X, or(X, X)), not(implies(X, or(X, X)))))), implies_1, or(and(implies(X, or(X, X)), implies(or(X, X), X)), not(implies(X, or(X, X))))), implies_1, or(not(implies(X, or(X, X))), and(implies(X, or(X, X)), implies(or(X, X), X)))), implies_1, implies(X, or(X, X)), equiv(X, or(X, X))), implies_1, X, or(X, X))
% 240.66/31.09 = { by axiom 1 (principia_op_and) }
% 240.66/31.09 fresh(fresh59(fresh28(fresh28(fresh28(fresh28(fresh12(op_and, true, X), implies_1, implies(not(implies(or(X, X), X)), not(implies(X, or(X, X))))), implies_1, implies(implies(implies(X, or(X, X)), not(implies(or(X, X), X))), implies(implies(X, or(X, X)), not(implies(X, or(X, X)))))), implies_1, or(and(implies(X, or(X, X)), implies(or(X, X), X)), not(implies(X, or(X, X))))), implies_1, or(not(implies(X, or(X, X))), and(implies(X, or(X, X)), implies(or(X, X), X)))), implies_1, implies(X, or(X, X)), equiv(X, or(X, X))), implies_1, X, or(X, X))
% 240.66/31.09 = { by lemma 68 R->L }
% 240.66/31.09 fresh(fresh59(fresh28(fresh28(fresh28(fresh28(fresh12(r1, true, X), implies_1, implies(not(implies(or(X, X), X)), not(implies(X, or(X, X))))), implies_1, implies(implies(implies(X, or(X, X)), not(implies(or(X, X), X))), implies(implies(X, or(X, X)), not(implies(X, or(X, X)))))), implies_1, or(and(implies(X, or(X, X)), implies(or(X, X), X)), not(implies(X, or(X, X))))), implies_1, or(not(implies(X, or(X, X))), and(implies(X, or(X, X)), implies(or(X, X), X)))), implies_1, implies(X, or(X, X)), equiv(X, or(X, X))), implies_1, X, or(X, X))
% 240.66/31.09 = { by axiom 37 (r1_1) }
% 240.66/31.09 fresh(fresh59(fresh28(fresh28(fresh28(fresh28(is_a_theorem(implies(or(X, X), X)), implies_1, implies(not(implies(or(X, X), X)), not(implies(X, or(X, X))))), implies_1, implies(implies(implies(X, or(X, X)), not(implies(or(X, X), X))), implies(implies(X, or(X, X)), not(implies(X, or(X, X)))))), implies_1, or(and(implies(X, or(X, X)), implies(or(X, X), X)), not(implies(X, or(X, X))))), implies_1, or(not(implies(X, or(X, X))), and(implies(X, or(X, X)), implies(or(X, X), X)))), implies_1, implies(X, or(X, X)), equiv(X, or(X, X))), implies_1, X, or(X, X))
% 240.66/31.09 = { by lemma 97 }
% 240.66/31.09 fresh(fresh59(fresh28(fresh28(fresh28(implies_1, implies_1, implies(implies(implies(X, or(X, X)), not(implies(or(X, X), X))), implies(implies(X, or(X, X)), not(implies(X, or(X, X)))))), implies_1, or(and(implies(X, or(X, X)), implies(or(X, X), X)), not(implies(X, or(X, X))))), implies_1, or(not(implies(X, or(X, X))), and(implies(X, or(X, X)), implies(or(X, X), X)))), implies_1, implies(X, or(X, X)), equiv(X, or(X, X))), implies_1, X, or(X, X))
% 240.66/31.09 = { by axiom 17 (modus_ponens_2) }
% 240.66/31.09 fresh(fresh59(fresh28(fresh28(is_a_theorem(implies(implies(implies(X, or(X, X)), not(implies(or(X, X), X))), implies(implies(X, or(X, X)), not(implies(X, or(X, X)))))), implies_1, or(and(implies(X, or(X, X)), implies(or(X, X), X)), not(implies(X, or(X, X))))), implies_1, or(not(implies(X, or(X, X))), and(implies(X, or(X, X)), implies(or(X, X), X)))), implies_1, implies(X, or(X, X)), equiv(X, or(X, X))), implies_1, X, or(X, X))
% 240.66/31.09 = { by lemma 72 }
% 240.66/31.09 fresh(fresh59(fresh28(fresh28(is_a_theorem(or(and(implies(X, or(X, X)), implies(or(X, X), X)), implies(implies(X, or(X, X)), not(implies(X, or(X, X)))))), implies_1, or(and(implies(X, or(X, X)), implies(or(X, X), X)), not(implies(X, or(X, X))))), implies_1, or(not(implies(X, or(X, X))), and(implies(X, or(X, X)), implies(or(X, X), X)))), implies_1, implies(X, or(X, X)), equiv(X, or(X, X))), implies_1, X, or(X, X))
% 240.66/31.09 = { by lemma 80 }
% 240.66/31.09 fresh(fresh59(fresh28(implies_1, implies_1, or(not(implies(X, or(X, X))), and(implies(X, or(X, X)), implies(or(X, X), X)))), implies_1, implies(X, or(X, X)), equiv(X, or(X, X))), implies_1, X, or(X, X))
% 240.66/31.09 = { by axiom 17 (modus_ponens_2) }
% 240.66/31.09 fresh(fresh59(is_a_theorem(or(not(implies(X, or(X, X))), and(implies(X, or(X, X)), implies(or(X, X), X)))), implies_1, implies(X, or(X, X)), equiv(X, or(X, X))), implies_1, X, or(X, X))
% 240.66/31.09 = { by lemma 57 }
% 240.66/31.09 fresh(fresh59(is_a_theorem(implies(implies(X, or(X, X)), and(implies(X, or(X, X)), implies(or(X, X), X)))), implies_1, implies(X, or(X, X)), equiv(X, or(X, X))), implies_1, X, or(X, X))
% 240.66/31.09 = { by lemma 95 }
% 240.66/31.09 fresh(fresh59(is_a_theorem(implies(implies(X, or(X, X)), equiv(X, or(X, X)))), implies_1, implies(X, or(X, X)), equiv(X, or(X, X))), implies_1, X, or(X, X))
% 240.66/31.09 = { by lemma 83 }
% 240.66/31.09 fresh(fresh28(is_a_theorem(implies(X, or(X, X))), implies_1, equiv(X, or(X, X))), implies_1, X, or(X, X))
% 240.66/31.09 = { by lemma 108 }
% 240.66/31.09 fresh(fresh28(implies_1, implies_1, equiv(X, or(X, X))), implies_1, X, or(X, X))
% 240.66/31.09 = { by axiom 17 (modus_ponens_2) }
% 240.66/31.09 fresh(is_a_theorem(equiv(X, or(X, X))), implies_1, X, or(X, X))
% 240.66/31.09 = { by lemma 77 }
% 240.66/31.09 X
% 240.66/31.09
% 240.66/31.09 Lemma 110: not(or(X, not(Y))) = and(not(X), Y).
% 240.66/31.09 Proof:
% 240.66/31.09 not(or(X, not(Y)))
% 240.66/31.09 = { by lemma 103 R->L }
% 240.66/31.09 and(not(or(X, X)), Y)
% 240.66/31.09 = { by lemma 109 }
% 240.66/31.09 and(not(X), Y)
% 240.66/31.09
% 240.66/31.09 Lemma 111: is_a_theorem(implies(implies(not(X), not(Y)), implies(Y, X))) = fresh25(modus_tollens, or_2, Y, X).
% 240.66/31.09 Proof:
% 240.66/31.09 is_a_theorem(implies(implies(not(X), not(Y)), implies(Y, X)))
% 240.66/31.09 = { by axiom 47 (modus_tollens_1) R->L }
% 240.66/31.09 fresh25(modus_tollens, true, Y, X)
% 240.66/31.09 = { by axiom 1 (principia_op_and) R->L }
% 240.66/31.09 fresh25(modus_tollens, op_and, Y, X)
% 240.66/31.09 = { by lemma 55 }
% 240.66/31.09 fresh25(modus_tollens, or_2, Y, X)
% 240.66/31.09
% 240.66/31.09 Lemma 112: implies_1 = modus_tollens.
% 240.66/31.09 Proof:
% 240.66/31.09 implies_1
% 240.66/31.09 = { by lemma 59 R->L }
% 240.66/31.09 or_2
% 240.66/31.09 = { by lemma 55 R->L }
% 240.66/31.09 op_and
% 240.66/31.09 = { by axiom 1 (principia_op_and) }
% 240.66/31.09 true
% 240.66/31.09 = { by axiom 12 (modus_tollens) R->L }
% 240.66/31.09 fresh26(implies_1, implies_1)
% 240.66/31.10 = { by lemma 64 R->L }
% 240.66/31.10 fresh26(fresh28(is_a_theorem(or(or(not(x13), y13), and(not(or(y13, not(x13))), not(or(y13, not(x13)))))), implies_1, or(and(not(or(y13, not(x13))), not(or(y13, not(x13)))), or(not(x13), y13))), implies_1)
% 240.66/31.10 = { by axiom 17 (modus_ponens_2) R->L }
% 240.66/31.10 fresh26(fresh28(fresh28(implies_1, implies_1, or(or(not(x13), y13), and(not(or(y13, not(x13))), not(or(y13, not(x13)))))), implies_1, or(and(not(or(y13, not(x13))), not(or(y13, not(x13)))), or(not(x13), y13))), implies_1)
% 240.66/31.10 = { by lemma 65 R->L }
% 240.66/31.10 fresh26(fresh28(fresh28(fresh28(is_a_theorem(implies(or(y13, not(x13)), or(not(x13), y13))), implies_1, or(or(not(x13), y13), not(or(y13, not(x13))))), implies_1, or(or(not(x13), y13), and(not(or(y13, not(x13))), not(or(y13, not(x13)))))), implies_1, or(and(not(or(y13, not(x13))), not(or(y13, not(x13)))), or(not(x13), y13))), implies_1)
% 240.66/31.10 = { by lemma 61 }
% 240.66/31.10 fresh26(fresh28(fresh28(fresh28(or_2, implies_1, or(or(not(x13), y13), not(or(y13, not(x13))))), implies_1, or(or(not(x13), y13), and(not(or(y13, not(x13))), not(or(y13, not(x13)))))), implies_1, or(and(not(or(y13, not(x13))), not(or(y13, not(x13)))), or(not(x13), y13))), implies_1)
% 240.66/31.10 = { by lemma 59 }
% 240.66/31.10 fresh26(fresh28(fresh28(fresh28(implies_1, implies_1, or(or(not(x13), y13), not(or(y13, not(x13))))), implies_1, or(or(not(x13), y13), and(not(or(y13, not(x13))), not(or(y13, not(x13)))))), implies_1, or(and(not(or(y13, not(x13))), not(or(y13, not(x13)))), or(not(x13), y13))), implies_1)
% 240.66/31.10 = { by axiom 17 (modus_ponens_2) }
% 240.66/31.10 fresh26(fresh28(fresh28(is_a_theorem(or(or(not(x13), y13), not(or(y13, not(x13))))), implies_1, or(or(not(x13), y13), and(not(or(y13, not(x13))), not(or(y13, not(x13)))))), implies_1, or(and(not(or(y13, not(x13))), not(or(y13, not(x13)))), or(not(x13), y13))), implies_1)
% 240.66/31.10 = { by lemma 59 R->L }
% 240.66/31.10 fresh26(fresh28(fresh28(is_a_theorem(or(or(not(x13), y13), not(or(y13, not(x13))))), or_2, or(or(not(x13), y13), and(not(or(y13, not(x13))), not(or(y13, not(x13)))))), implies_1, or(and(not(or(y13, not(x13))), not(or(y13, not(x13)))), or(not(x13), y13))), implies_1)
% 240.66/31.10 = { by lemma 55 R->L }
% 240.66/31.10 fresh26(fresh28(fresh28(is_a_theorem(or(or(not(x13), y13), not(or(y13, not(x13))))), op_and, or(or(not(x13), y13), and(not(or(y13, not(x13))), not(or(y13, not(x13)))))), implies_1, or(and(not(or(y13, not(x13))), not(or(y13, not(x13)))), or(not(x13), y13))), implies_1)
% 240.66/31.10 = { by lemma 60 R->L }
% 240.66/31.10 fresh26(fresh28(fresh59(is_a_theorem(implies(or(or(not(x13), y13), not(or(y13, not(x13)))), or(or(not(x13), y13), and(not(or(y13, not(x13))), not(or(y13, not(x13))))))), op_and, or(or(not(x13), y13), not(or(y13, not(x13)))), or(or(not(x13), y13), and(not(or(y13, not(x13))), not(or(y13, not(x13)))))), implies_1, or(and(not(or(y13, not(x13))), not(or(y13, not(x13)))), or(not(x13), y13))), implies_1)
% 240.66/31.10 = { by axiom 17 (modus_ponens_2) R->L }
% 240.66/31.10 fresh26(fresh28(fresh59(fresh28(implies_1, implies_1, implies(or(or(not(x13), y13), not(or(y13, not(x13)))), or(or(not(x13), y13), and(not(or(y13, not(x13))), not(or(y13, not(x13))))))), op_and, or(or(not(x13), y13), not(or(y13, not(x13)))), or(or(not(x13), y13), and(not(or(y13, not(x13))), not(or(y13, not(x13)))))), implies_1, or(and(not(or(y13, not(x13))), not(or(y13, not(x13)))), or(not(x13), y13))), implies_1)
% 240.66/31.10 = { by lemma 59 R->L }
% 240.66/31.10 fresh26(fresh28(fresh59(fresh28(or_2, implies_1, implies(or(or(not(x13), y13), not(or(y13, not(x13)))), or(or(not(x13), y13), and(not(or(y13, not(x13))), not(or(y13, not(x13))))))), op_and, or(or(not(x13), y13), not(or(y13, not(x13)))), or(or(not(x13), y13), and(not(or(y13, not(x13))), not(or(y13, not(x13)))))), implies_1, or(and(not(or(y13, not(x13))), not(or(y13, not(x13)))), or(not(x13), y13))), implies_1)
% 240.66/31.10 = { by lemma 55 R->L }
% 240.66/31.10 fresh26(fresh28(fresh59(fresh28(op_and, implies_1, implies(or(or(not(x13), y13), not(or(y13, not(x13)))), or(or(not(x13), y13), and(not(or(y13, not(x13))), not(or(y13, not(x13))))))), op_and, or(or(not(x13), y13), not(or(y13, not(x13)))), or(or(not(x13), y13), and(not(or(y13, not(x13))), not(or(y13, not(x13)))))), implies_1, or(and(not(or(y13, not(x13))), not(or(y13, not(x13)))), or(not(x13), y13))), implies_1)
% 240.66/31.10 = { by lemma 66 R->L }
% 240.66/31.10 fresh26(fresh28(fresh59(fresh28(fresh33(implies_1, implies_1, not(or(y13, not(x13)))), implies_1, implies(or(or(not(x13), y13), not(or(y13, not(x13)))), or(or(not(x13), y13), and(not(or(y13, not(x13))), not(or(y13, not(x13))))))), op_and, or(or(not(x13), y13), not(or(y13, not(x13)))), or(or(not(x13), y13), and(not(or(y13, not(x13))), not(or(y13, not(x13)))))), implies_1, or(and(not(or(y13, not(x13))), not(or(y13, not(x13)))), or(not(x13), y13))), implies_1)
% 240.66/31.10 = { by lemma 59 R->L }
% 240.66/31.10 fresh26(fresh28(fresh59(fresh28(fresh33(implies_1, or_2, not(or(y13, not(x13)))), implies_1, implies(or(or(not(x13), y13), not(or(y13, not(x13)))), or(or(not(x13), y13), and(not(or(y13, not(x13))), not(or(y13, not(x13))))))), op_and, or(or(not(x13), y13), not(or(y13, not(x13)))), or(or(not(x13), y13), and(not(or(y13, not(x13))), not(or(y13, not(x13)))))), implies_1, or(and(not(or(y13, not(x13))), not(or(y13, not(x13)))), or(not(x13), y13))), implies_1)
% 240.66/31.10 = { by lemma 55 R->L }
% 240.66/31.10 fresh26(fresh28(fresh59(fresh28(fresh33(implies_1, op_and, not(or(y13, not(x13)))), implies_1, implies(or(or(not(x13), y13), not(or(y13, not(x13)))), or(or(not(x13), y13), and(not(or(y13, not(x13))), not(or(y13, not(x13))))))), op_and, or(or(not(x13), y13), not(or(y13, not(x13)))), or(or(not(x13), y13), and(not(or(y13, not(x13))), not(or(y13, not(x13)))))), implies_1, or(and(not(or(y13, not(x13))), not(or(y13, not(x13)))), or(not(x13), y13))), implies_1)
% 240.66/31.10 = { by lemma 73 R->L }
% 240.66/31.10 fresh26(fresh28(fresh59(fresh28(fresh33(kn1, op_and, not(or(y13, not(x13)))), implies_1, implies(or(or(not(x13), y13), not(or(y13, not(x13)))), or(or(not(x13), y13), and(not(or(y13, not(x13))), not(or(y13, not(x13))))))), op_and, or(or(not(x13), y13), not(or(y13, not(x13)))), or(or(not(x13), y13), and(not(or(y13, not(x13))), not(or(y13, not(x13)))))), implies_1, or(and(not(or(y13, not(x13))), not(or(y13, not(x13)))), or(not(x13), y13))), implies_1)
% 240.66/31.10 = { by lemma 67 R->L }
% 240.66/31.10 fresh26(fresh28(fresh59(fresh28(is_a_theorem(implies(not(or(y13, not(x13))), and(not(or(y13, not(x13))), not(or(y13, not(x13)))))), implies_1, implies(or(or(not(x13), y13), not(or(y13, not(x13)))), or(or(not(x13), y13), and(not(or(y13, not(x13))), not(or(y13, not(x13))))))), op_and, or(or(not(x13), y13), not(or(y13, not(x13)))), or(or(not(x13), y13), and(not(or(y13, not(x13))), not(or(y13, not(x13)))))), implies_1, or(and(not(or(y13, not(x13))), not(or(y13, not(x13)))), or(not(x13), y13))), implies_1)
% 240.66/31.10 = { by lemma 75 }
% 240.66/31.10 fresh26(fresh28(fresh59(implies_1, op_and, or(or(not(x13), y13), not(or(y13, not(x13)))), or(or(not(x13), y13), and(not(or(y13, not(x13))), not(or(y13, not(x13)))))), implies_1, or(and(not(or(y13, not(x13))), not(or(y13, not(x13)))), or(not(x13), y13))), implies_1)
% 240.66/31.10 = { by lemma 55 }
% 240.66/31.10 fresh26(fresh28(fresh59(implies_1, or_2, or(or(not(x13), y13), not(or(y13, not(x13)))), or(or(not(x13), y13), and(not(or(y13, not(x13))), not(or(y13, not(x13)))))), implies_1, or(and(not(or(y13, not(x13))), not(or(y13, not(x13)))), or(not(x13), y13))), implies_1)
% 240.66/31.10 = { by lemma 59 }
% 240.66/31.10 fresh26(fresh28(fresh59(implies_1, implies_1, or(or(not(x13), y13), not(or(y13, not(x13)))), or(or(not(x13), y13), and(not(or(y13, not(x13))), not(or(y13, not(x13)))))), implies_1, or(and(not(or(y13, not(x13))), not(or(y13, not(x13)))), or(not(x13), y13))), implies_1)
% 240.66/31.10 = { by lemma 63 }
% 240.66/31.10 fresh26(fresh28(op_and, implies_1, or(and(not(or(y13, not(x13))), not(or(y13, not(x13)))), or(not(x13), y13))), implies_1)
% 240.66/31.10 = { by lemma 55 }
% 240.66/31.10 fresh26(fresh28(or_2, implies_1, or(and(not(or(y13, not(x13))), not(or(y13, not(x13)))), or(not(x13), y13))), implies_1)
% 240.66/31.10 = { by lemma 59 }
% 240.66/31.10 fresh26(fresh28(implies_1, implies_1, or(and(not(or(y13, not(x13))), not(or(y13, not(x13)))), or(not(x13), y13))), implies_1)
% 240.66/31.10 = { by axiom 17 (modus_ponens_2) }
% 240.66/31.10 fresh26(is_a_theorem(or(and(not(or(y13, not(x13))), not(or(y13, not(x13)))), or(not(x13), y13))), implies_1)
% 240.66/31.10 = { by lemma 110 }
% 240.66/31.10 fresh26(is_a_theorem(or(and(and(not(y13), x13), not(or(y13, not(x13)))), or(not(x13), y13))), implies_1)
% 240.66/31.10 = { by lemma 110 }
% 240.66/31.10 fresh26(is_a_theorem(or(and(and(not(y13), x13), and(not(y13), x13)), or(not(x13), y13))), implies_1)
% 240.66/31.10 = { by lemma 99 }
% 240.66/31.10 fresh26(is_a_theorem(or(and(not(y13), x13), or(not(x13), y13))), implies_1)
% 240.66/31.10 = { by lemma 57 }
% 240.66/31.10 fresh26(is_a_theorem(or(and(not(y13), x13), implies(x13, y13))), implies_1)
% 240.66/31.10 = { by lemma 72 R->L }
% 240.66/31.10 fresh26(is_a_theorem(implies(implies(not(y13), not(x13)), implies(x13, y13))), implies_1)
% 240.66/31.10 = { by lemma 111 }
% 240.66/31.10 fresh26(fresh25(modus_tollens, or_2, x13, y13), implies_1)
% 240.66/31.10 = { by lemma 59 }
% 240.66/31.10 fresh26(fresh25(modus_tollens, implies_1, x13, y13), implies_1)
% 240.66/31.10 = { by lemma 59 R->L }
% 240.66/31.10 fresh26(fresh25(modus_tollens, implies_1, x13, y13), or_2)
% 240.66/31.10 = { by lemma 59 R->L }
% 240.66/31.10 fresh26(fresh25(modus_tollens, or_2, x13, y13), or_2)
% 240.66/31.10 = { by lemma 55 R->L }
% 240.66/31.10 fresh26(fresh25(modus_tollens, or_2, x13, y13), op_and)
% 240.66/31.10 = { by axiom 1 (principia_op_and) }
% 240.66/31.10 fresh26(fresh25(modus_tollens, or_2, x13, y13), true)
% 240.66/31.10 = { by lemma 111 R->L }
% 240.66/31.10 fresh26(is_a_theorem(implies(implies(not(y13), not(x13)), implies(x13, y13))), true)
% 240.66/31.10 = { by axiom 48 (modus_tollens) }
% 240.66/31.10 modus_tollens
% 240.66/31.10
% 240.66/31.10 Lemma 113: implies(X, X) = equiv(X, X).
% 240.66/31.10 Proof:
% 240.66/31.10 implies(X, X)
% 240.66/31.10 = { by lemma 99 R->L }
% 240.66/31.10 and(implies(X, X), implies(X, X))
% 240.66/31.10 = { by lemma 95 }
% 240.66/31.10 equiv(X, X)
% 240.66/31.10
% 240.66/31.10 Lemma 114: fresh59(is_a_theorem(implies(X, Y)), modus_tollens, X, Y) = fresh28(is_a_theorem(X), modus_tollens, Y).
% 240.66/31.10 Proof:
% 240.66/31.10 fresh59(is_a_theorem(implies(X, Y)), modus_tollens, X, Y)
% 240.66/31.10 = { by lemma 112 R->L }
% 240.66/31.10 fresh59(is_a_theorem(implies(X, Y)), implies_1, X, Y)
% 240.66/31.10 = { by lemma 59 R->L }
% 240.66/31.10 fresh59(is_a_theorem(implies(X, Y)), or_2, X, Y)
% 240.66/31.10 = { by lemma 55 R->L }
% 240.66/31.10 fresh59(is_a_theorem(implies(X, Y)), op_and, X, Y)
% 240.66/31.10 = { by lemma 60 }
% 240.66/31.10 fresh28(is_a_theorem(X), op_and, Y)
% 240.66/31.10 = { by lemma 55 }
% 240.66/31.10 fresh28(is_a_theorem(X), or_2, Y)
% 240.66/31.10 = { by lemma 59 }
% 240.66/31.10 fresh28(is_a_theorem(X), implies_1, Y)
% 240.66/31.10 = { by lemma 112 }
% 240.66/31.10 fresh28(is_a_theorem(X), modus_tollens, Y)
% 240.66/31.10
% 240.66/31.10 Lemma 115: is_a_theorem(implies(X, implies(Y, and(X, Y)))) = fresh53(and_3, modus_tollens, X, Y).
% 240.66/31.10 Proof:
% 240.66/31.10 is_a_theorem(implies(X, implies(Y, and(X, Y))))
% 240.66/31.10 = { by axiom 45 (and_3_1) R->L }
% 240.66/31.10 fresh53(and_3, true, X, Y)
% 240.66/31.10 = { by axiom 1 (principia_op_and) R->L }
% 240.66/31.10 fresh53(and_3, op_and, X, Y)
% 240.66/31.10 = { by lemma 55 }
% 240.66/31.10 fresh53(and_3, or_2, X, Y)
% 240.66/31.10 = { by lemma 59 }
% 240.66/31.10 fresh53(and_3, implies_1, X, Y)
% 240.66/31.10 = { by lemma 112 }
% 240.66/31.10 fresh53(and_3, modus_tollens, X, Y)
% 240.66/31.10
% 240.66/31.10 Lemma 116: is_a_theorem(implies(not(X), implies(X, Y))) = implies_1.
% 240.66/31.10 Proof:
% 240.66/31.10 is_a_theorem(implies(not(X), implies(X, Y)))
% 240.66/31.10 = { by lemma 57 R->L }
% 240.66/31.10 is_a_theorem(implies(not(X), or(not(X), Y)))
% 240.66/31.10 = { by axiom 32 (or_1_1) R->L }
% 240.66/31.10 fresh18(or_1, true, not(X), Y)
% 240.66/31.10 = { by axiom 1 (principia_op_and) R->L }
% 240.66/31.10 fresh18(or_1, op_and, not(X), Y)
% 240.66/31.10 = { by lemma 55 }
% 240.66/31.10 fresh18(or_1, or_2, not(X), Y)
% 240.66/31.10 = { by lemma 59 }
% 240.66/31.10 fresh18(or_1, implies_1, not(X), Y)
% 240.66/31.10 = { by lemma 107 }
% 240.66/31.10 implies_1
% 240.66/31.10
% 240.66/31.10 Lemma 117: fresh53(and_3, modus_tollens, X, Y) = modus_tollens.
% 240.66/31.10 Proof:
% 240.66/31.10 fresh53(and_3, modus_tollens, X, Y)
% 240.66/31.10 = { by lemma 115 R->L }
% 240.66/31.10 is_a_theorem(implies(X, implies(Y, and(X, Y))))
% 240.66/31.10 = { by axiom 17 (modus_ponens_2) R->L }
% 240.66/31.10 fresh28(modus_tollens, modus_tollens, implies(X, implies(Y, and(X, Y))))
% 240.66/31.10 = { by lemma 112 R->L }
% 240.66/31.10 fresh28(implies_1, modus_tollens, implies(X, implies(Y, and(X, Y))))
% 240.66/31.10 = { by lemma 59 R->L }
% 240.66/31.10 fresh28(or_2, modus_tollens, implies(X, implies(Y, and(X, Y))))
% 240.66/31.10 = { by lemma 55 R->L }
% 240.66/31.10 fresh28(op_and, modus_tollens, implies(X, implies(Y, and(X, Y))))
% 240.66/31.10 = { by lemma 63 R->L }
% 240.66/31.10 fresh28(fresh59(implies_1, implies_1, or(implies(X, implies(Y, and(X, Y))), and(X, Y)), or(implies(X, implies(Y, and(X, Y))), implies(Y, and(X, Y)))), modus_tollens, implies(X, implies(Y, and(X, Y))))
% 240.66/31.10 = { by lemma 59 R->L }
% 240.66/31.10 fresh28(fresh59(implies_1, or_2, or(implies(X, implies(Y, and(X, Y))), and(X, Y)), or(implies(X, implies(Y, and(X, Y))), implies(Y, and(X, Y)))), modus_tollens, implies(X, implies(Y, and(X, Y))))
% 240.66/31.10 = { by lemma 55 R->L }
% 240.66/31.10 fresh28(fresh59(implies_1, op_and, or(implies(X, implies(Y, and(X, Y))), and(X, Y)), or(implies(X, implies(Y, and(X, Y))), implies(Y, and(X, Y)))), modus_tollens, implies(X, implies(Y, and(X, Y))))
% 240.66/31.10 = { by lemma 82 R->L }
% 240.66/31.10 fresh28(fresh59(is_a_theorem(implies(or(implies(X, implies(Y, and(X, Y))), and(X, Y)), or(implies(X, implies(Y, and(X, Y))), implies(Y, and(X, Y))))), op_and, or(implies(X, implies(Y, and(X, Y))), and(X, Y)), or(implies(X, implies(Y, and(X, Y))), implies(Y, and(X, Y)))), modus_tollens, implies(X, implies(Y, and(X, Y))))
% 240.66/31.10 = { by lemma 60 }
% 240.66/31.10 fresh28(fresh28(is_a_theorem(or(implies(X, implies(Y, and(X, Y))), and(X, Y))), op_and, or(implies(X, implies(Y, and(X, Y))), implies(Y, and(X, Y)))), modus_tollens, implies(X, implies(Y, and(X, Y))))
% 240.66/31.10 = { by lemma 55 }
% 240.66/31.11 fresh28(fresh28(is_a_theorem(or(implies(X, implies(Y, and(X, Y))), and(X, Y))), or_2, or(implies(X, implies(Y, and(X, Y))), implies(Y, and(X, Y)))), modus_tollens, implies(X, implies(Y, and(X, Y))))
% 240.66/31.11 = { by lemma 59 }
% 240.66/31.11 fresh28(fresh28(is_a_theorem(or(implies(X, implies(Y, and(X, Y))), and(X, Y))), implies_1, or(implies(X, implies(Y, and(X, Y))), implies(Y, and(X, Y)))), modus_tollens, implies(X, implies(Y, and(X, Y))))
% 240.66/31.11 = { by lemma 112 }
% 240.66/31.11 fresh28(fresh28(is_a_theorem(or(implies(X, implies(Y, and(X, Y))), and(X, Y))), modus_tollens, or(implies(X, implies(Y, and(X, Y))), implies(Y, and(X, Y)))), modus_tollens, implies(X, implies(Y, and(X, Y))))
% 240.66/31.11 = { by axiom 17 (modus_ponens_2) R->L }
% 240.66/31.11 fresh28(fresh28(fresh28(implies_1, implies_1, or(implies(X, implies(Y, and(X, Y))), and(X, Y))), modus_tollens, or(implies(X, implies(Y, and(X, Y))), implies(Y, and(X, Y)))), modus_tollens, implies(X, implies(Y, and(X, Y))))
% 240.66/31.11 = { by lemma 98 R->L }
% 240.66/31.11 fresh28(fresh28(fresh28(fresh28(is_a_theorem(implies(not(Y), implies(Y, and(X, Y)))), implies_1, implies(implies(X, not(Y)), implies(X, implies(Y, and(X, Y))))), implies_1, or(implies(X, implies(Y, and(X, Y))), and(X, Y))), modus_tollens, or(implies(X, implies(Y, and(X, Y))), implies(Y, and(X, Y)))), modus_tollens, implies(X, implies(Y, and(X, Y))))
% 240.66/31.11 = { by lemma 116 }
% 240.66/31.11 fresh28(fresh28(fresh28(fresh28(implies_1, implies_1, implies(implies(X, not(Y)), implies(X, implies(Y, and(X, Y))))), implies_1, or(implies(X, implies(Y, and(X, Y))), and(X, Y))), modus_tollens, or(implies(X, implies(Y, and(X, Y))), implies(Y, and(X, Y)))), modus_tollens, implies(X, implies(Y, and(X, Y))))
% 240.66/31.11 = { by axiom 17 (modus_ponens_2) }
% 240.66/31.11 fresh28(fresh28(fresh28(is_a_theorem(implies(implies(X, not(Y)), implies(X, implies(Y, and(X, Y))))), implies_1, or(implies(X, implies(Y, and(X, Y))), and(X, Y))), modus_tollens, or(implies(X, implies(Y, and(X, Y))), implies(Y, and(X, Y)))), modus_tollens, implies(X, implies(Y, and(X, Y))))
% 240.66/31.11 = { by lemma 72 }
% 240.66/31.11 fresh28(fresh28(fresh28(is_a_theorem(or(and(X, Y), implies(X, implies(Y, and(X, Y))))), implies_1, or(implies(X, implies(Y, and(X, Y))), and(X, Y))), modus_tollens, or(implies(X, implies(Y, and(X, Y))), implies(Y, and(X, Y)))), modus_tollens, implies(X, implies(Y, and(X, Y))))
% 240.66/31.11 = { by lemma 64 }
% 240.66/31.11 fresh28(fresh28(implies_1, modus_tollens, or(implies(X, implies(Y, and(X, Y))), implies(Y, and(X, Y)))), modus_tollens, implies(X, implies(Y, and(X, Y))))
% 240.66/31.11 = { by lemma 112 }
% 240.66/31.11 fresh28(fresh28(modus_tollens, modus_tollens, or(implies(X, implies(Y, and(X, Y))), implies(Y, and(X, Y)))), modus_tollens, implies(X, implies(Y, and(X, Y))))
% 240.66/31.11 = { by axiom 17 (modus_ponens_2) }
% 240.66/31.11 fresh28(is_a_theorem(or(implies(X, implies(Y, and(X, Y))), implies(Y, and(X, Y)))), modus_tollens, implies(X, implies(Y, and(X, Y))))
% 240.66/31.11 = { by lemma 112 R->L }
% 240.66/31.11 fresh28(is_a_theorem(or(implies(X, implies(Y, and(X, Y))), implies(Y, and(X, Y)))), implies_1, implies(X, implies(Y, and(X, Y))))
% 240.66/31.11 = { by lemma 59 R->L }
% 240.66/31.11 fresh28(is_a_theorem(or(implies(X, implies(Y, and(X, Y))), implies(Y, and(X, Y)))), or_2, implies(X, implies(Y, and(X, Y))))
% 240.66/31.11 = { by lemma 55 R->L }
% 240.66/31.11 fresh28(is_a_theorem(or(implies(X, implies(Y, and(X, Y))), implies(Y, and(X, Y)))), op_and, implies(X, implies(Y, and(X, Y))))
% 240.66/31.11 = { by lemma 60 R->L }
% 240.66/31.11 fresh59(is_a_theorem(implies(or(implies(X, implies(Y, and(X, Y))), implies(Y, and(X, Y))), implies(X, implies(Y, and(X, Y))))), op_and, or(implies(X, implies(Y, and(X, Y))), implies(Y, and(X, Y))), implies(X, implies(Y, and(X, Y))))
% 240.66/31.11 = { by lemma 106 }
% 240.66/31.11 fresh59(implies_1, op_and, or(implies(X, implies(Y, and(X, Y))), implies(Y, and(X, Y))), implies(X, implies(Y, and(X, Y))))
% 240.66/31.11 = { by lemma 55 }
% 240.66/31.11 fresh59(implies_1, or_2, or(implies(X, implies(Y, and(X, Y))), implies(Y, and(X, Y))), implies(X, implies(Y, and(X, Y))))
% 240.66/31.11 = { by lemma 59 }
% 240.66/31.11 fresh59(implies_1, implies_1, or(implies(X, implies(Y, and(X, Y))), implies(Y, and(X, Y))), implies(X, implies(Y, and(X, Y))))
% 240.66/31.11 = { by lemma 63 }
% 240.66/31.11 op_and
% 240.66/31.11 = { by lemma 55 }
% 240.66/31.11 or_2
% 240.66/31.11 = { by lemma 59 }
% 240.66/31.11 implies_1
% 240.66/31.11 = { by lemma 112 }
% 240.66/31.11 modus_tollens
% 240.66/31.11
% 240.66/31.11 Lemma 118: fresh59(X, X, Y, Z) = modus_tollens.
% 240.66/31.11 Proof:
% 240.66/31.11 fresh59(X, X, Y, Z)
% 240.66/31.11 = { by axiom 20 (modus_ponens_2) }
% 240.66/31.11 fresh60(modus_ponens, true, Z)
% 240.66/31.11 = { by lemma 52 }
% 240.66/31.11 fresh60(op_and, true, Z)
% 240.66/31.11 = { by axiom 1 (principia_op_and) R->L }
% 240.66/31.11 fresh60(op_and, op_and, Z)
% 240.66/31.11 = { by lemma 62 }
% 240.66/31.11 op_and
% 240.66/31.11 = { by lemma 55 }
% 240.66/31.11 or_2
% 240.66/31.11 = { by lemma 59 }
% 240.66/31.11 implies_1
% 240.66/31.11 = { by lemma 112 }
% 240.66/31.11 modus_tollens
% 240.66/31.11
% 240.66/31.11 Lemma 119: fresh28(is_a_theorem(implies(X, Y)), modus_tollens, implies(implies(Y, X), equiv(X, Y))) = modus_tollens.
% 240.66/31.11 Proof:
% 240.66/31.11 fresh28(is_a_theorem(implies(X, Y)), modus_tollens, implies(implies(Y, X), equiv(X, Y)))
% 240.66/31.11 = { by lemma 114 R->L }
% 240.66/31.11 fresh59(is_a_theorem(implies(implies(X, Y), implies(implies(Y, X), equiv(X, Y)))), modus_tollens, implies(X, Y), implies(implies(Y, X), equiv(X, Y)))
% 240.66/31.11 = { by lemma 95 R->L }
% 240.66/31.11 fresh59(is_a_theorem(implies(implies(X, Y), implies(implies(Y, X), and(implies(X, Y), implies(Y, X))))), modus_tollens, implies(X, Y), implies(implies(Y, X), equiv(X, Y)))
% 240.66/31.11 = { by axiom 45 (and_3_1) R->L }
% 240.66/31.11 fresh59(fresh53(and_3, true, implies(X, Y), implies(Y, X)), modus_tollens, implies(X, Y), implies(implies(Y, X), equiv(X, Y)))
% 240.66/31.11 = { by axiom 1 (principia_op_and) R->L }
% 240.66/31.11 fresh59(fresh53(and_3, op_and, implies(X, Y), implies(Y, X)), modus_tollens, implies(X, Y), implies(implies(Y, X), equiv(X, Y)))
% 240.66/31.11 = { by lemma 55 }
% 240.66/31.11 fresh59(fresh53(and_3, or_2, implies(X, Y), implies(Y, X)), modus_tollens, implies(X, Y), implies(implies(Y, X), equiv(X, Y)))
% 240.66/31.11 = { by lemma 59 }
% 240.66/31.11 fresh59(fresh53(and_3, implies_1, implies(X, Y), implies(Y, X)), modus_tollens, implies(X, Y), implies(implies(Y, X), equiv(X, Y)))
% 240.66/31.11 = { by lemma 112 }
% 240.66/31.11 fresh59(fresh53(and_3, modus_tollens, implies(X, Y), implies(Y, X)), modus_tollens, implies(X, Y), implies(implies(Y, X), equiv(X, Y)))
% 240.66/31.11 = { by lemma 117 }
% 240.66/31.11 fresh59(modus_tollens, modus_tollens, implies(X, Y), implies(implies(Y, X), equiv(X, Y)))
% 240.66/31.11 = { by lemma 118 }
% 240.66/31.11 modus_tollens
% 240.66/31.11
% 240.66/31.11 Lemma 120: fresh28(is_a_theorem(implies(or(X, Y), X)), modus_tollens, equiv(X, or(X, Y))) = modus_tollens.
% 240.66/31.11 Proof:
% 240.66/31.11 fresh28(is_a_theorem(implies(or(X, Y), X)), modus_tollens, equiv(X, or(X, Y)))
% 240.66/31.11 = { by lemma 114 R->L }
% 240.66/31.11 fresh59(is_a_theorem(implies(implies(or(X, Y), X), equiv(X, or(X, Y)))), modus_tollens, implies(or(X, Y), X), equiv(X, or(X, Y)))
% 240.66/31.11 = { by axiom 17 (modus_ponens_2) R->L }
% 240.66/31.11 fresh59(fresh28(modus_tollens, modus_tollens, implies(implies(or(X, Y), X), equiv(X, or(X, Y)))), modus_tollens, implies(or(X, Y), X), equiv(X, or(X, Y)))
% 240.66/31.11 = { by lemma 112 R->L }
% 240.66/31.11 fresh59(fresh28(implies_1, modus_tollens, implies(implies(or(X, Y), X), equiv(X, or(X, Y)))), modus_tollens, implies(or(X, Y), X), equiv(X, or(X, Y)))
% 240.66/31.11 = { by lemma 107 R->L }
% 240.66/31.11 fresh59(fresh28(fresh18(or_1, implies_1, X, Y), modus_tollens, implies(implies(or(X, Y), X), equiv(X, or(X, Y)))), modus_tollens, implies(or(X, Y), X), equiv(X, or(X, Y)))
% 240.66/31.11 = { by lemma 59 R->L }
% 240.66/31.11 fresh59(fresh28(fresh18(or_1, or_2, X, Y), modus_tollens, implies(implies(or(X, Y), X), equiv(X, or(X, Y)))), modus_tollens, implies(or(X, Y), X), equiv(X, or(X, Y)))
% 240.66/31.11 = { by lemma 55 R->L }
% 240.66/31.11 fresh59(fresh28(fresh18(or_1, op_and, X, Y), modus_tollens, implies(implies(or(X, Y), X), equiv(X, or(X, Y)))), modus_tollens, implies(or(X, Y), X), equiv(X, or(X, Y)))
% 240.66/31.11 = { by axiom 1 (principia_op_and) }
% 240.66/31.11 fresh59(fresh28(fresh18(or_1, true, X, Y), modus_tollens, implies(implies(or(X, Y), X), equiv(X, or(X, Y)))), modus_tollens, implies(or(X, Y), X), equiv(X, or(X, Y)))
% 240.66/31.11 = { by axiom 32 (or_1_1) }
% 240.66/31.11 fresh59(fresh28(is_a_theorem(implies(X, or(X, Y))), modus_tollens, implies(implies(or(X, Y), X), equiv(X, or(X, Y)))), modus_tollens, implies(or(X, Y), X), equiv(X, or(X, Y)))
% 240.66/31.11 = { by lemma 119 }
% 240.66/31.11 fresh59(modus_tollens, modus_tollens, implies(or(X, Y), X), equiv(X, or(X, Y)))
% 240.66/31.11 = { by lemma 118 }
% 240.66/31.11 modus_tollens
% 240.66/31.11
% 240.66/31.11 Lemma 121: fresh(is_a_theorem(equiv(X, Y)), modus_tollens, X, Y) = X.
% 240.66/31.11 Proof:
% 240.66/31.11 fresh(is_a_theorem(equiv(X, Y)), modus_tollens, X, Y)
% 240.66/31.11 = { by lemma 112 R->L }
% 240.66/31.11 fresh(is_a_theorem(equiv(X, Y)), implies_1, X, Y)
% 240.66/31.11 = { by lemma 59 R->L }
% 240.66/31.11 fresh(is_a_theorem(equiv(X, Y)), or_2, X, Y)
% 240.66/31.11 = { by lemma 55 R->L }
% 240.66/31.11 fresh(is_a_theorem(equiv(X, Y)), op_and, X, Y)
% 240.66/31.11 = { by lemma 76 }
% 240.66/31.11 X
% 240.66/31.11
% 240.66/31.11 Lemma 122: or(or(X, Y), or(Y, X)) = or(X, Y).
% 240.66/31.11 Proof:
% 240.66/31.11 or(or(X, Y), or(Y, X))
% 240.66/31.11 = { by axiom 19 (substitution_of_equivalents_2) R->L }
% 240.66/31.11 fresh(modus_tollens, modus_tollens, or(X, Y), or(or(X, Y), or(Y, X)))
% 240.66/31.11 = { by lemma 120 R->L }
% 240.66/31.11 fresh(fresh28(is_a_theorem(implies(or(or(X, Y), or(Y, X)), or(X, Y))), modus_tollens, equiv(or(X, Y), or(or(X, Y), or(Y, X)))), modus_tollens, or(X, Y), or(or(X, Y), or(Y, X)))
% 240.66/31.11 = { by axiom 17 (modus_ponens_2) R->L }
% 240.66/31.11 fresh(fresh28(fresh28(implies_1, implies_1, implies(or(or(X, Y), or(Y, X)), or(X, Y))), modus_tollens, equiv(or(X, Y), or(or(X, Y), or(Y, X)))), modus_tollens, or(X, Y), or(or(X, Y), or(Y, X)))
% 240.66/31.11 = { by lemma 89 R->L }
% 240.66/31.11 fresh(fresh28(fresh28(is_a_theorem(implies(or(or(X, Y), or(Y, X)), or(or(X, Y), or(X, Y)))), implies_1, implies(or(or(X, Y), or(Y, X)), or(X, Y))), modus_tollens, equiv(or(X, Y), or(or(X, Y), or(Y, X)))), modus_tollens, or(X, Y), or(or(X, Y), or(Y, X)))
% 240.66/31.11 = { by lemma 105 }
% 240.66/31.11 fresh(fresh28(implies_1, modus_tollens, equiv(or(X, Y), or(or(X, Y), or(Y, X)))), modus_tollens, or(X, Y), or(or(X, Y), or(Y, X)))
% 240.66/31.11 = { by lemma 112 }
% 240.66/31.11 fresh(fresh28(modus_tollens, modus_tollens, equiv(or(X, Y), or(or(X, Y), or(Y, X)))), modus_tollens, or(X, Y), or(or(X, Y), or(Y, X)))
% 240.66/31.11 = { by axiom 17 (modus_ponens_2) }
% 240.66/31.11 fresh(is_a_theorem(equiv(or(X, Y), or(or(X, Y), or(Y, X)))), modus_tollens, or(X, Y), or(or(X, Y), or(Y, X)))
% 240.66/31.11 = { by lemma 121 }
% 240.66/31.11 or(X, Y)
% 240.66/31.11
% 240.66/31.11 Lemma 123: not(and(X, not(Y))) = fresh22(op_implies_and, implies_1, X, Y).
% 240.66/31.11 Proof:
% 240.66/31.11 not(and(X, not(Y)))
% 240.66/31.11 = { by axiom 28 (op_implies_and) R->L }
% 240.66/31.11 fresh22(op_implies_and, true, X, Y)
% 240.66/31.11 = { by axiom 1 (principia_op_and) R->L }
% 240.66/31.11 fresh22(op_implies_and, op_and, X, Y)
% 240.66/31.11 = { by lemma 55 }
% 240.66/31.11 fresh22(op_implies_and, or_2, X, Y)
% 240.66/31.11 = { by lemma 59 }
% 240.66/31.11 fresh22(op_implies_and, implies_1, X, Y)
% 240.66/31.11
% 240.66/31.11 Lemma 124: and(implies(X, not(Y)), implies(Z, not(W))) = not(or(and(X, Y), and(Z, W))).
% 240.66/31.11 Proof:
% 240.66/31.11 and(implies(X, not(Y)), implies(Z, not(W)))
% 240.66/31.11 = { by lemma 96 R->L }
% 240.66/31.11 not(implies(implies(X, not(Y)), and(Z, W)))
% 240.66/31.11 = { by lemma 72 }
% 240.66/31.11 not(or(and(X, Y), and(Z, W)))
% 240.66/31.11
% 240.66/31.11 Lemma 125: implies(X, not(Y)) = not(and(X, Y)).
% 240.66/31.11 Proof:
% 240.66/31.11 implies(X, not(Y))
% 240.66/31.11 = { by lemma 99 R->L }
% 240.66/31.11 and(implies(X, not(Y)), implies(X, not(Y)))
% 240.66/31.11 = { by lemma 124 }
% 240.66/31.11 not(or(and(X, Y), and(X, Y)))
% 240.66/31.11 = { by lemma 109 }
% 240.66/31.11 not(and(X, Y))
% 240.66/31.11
% 240.66/31.11 Lemma 126: not(not(or(X, X))) = X.
% 240.66/31.11 Proof:
% 240.66/31.11 not(not(or(X, X)))
% 240.66/31.11 = { by lemma 101 R->L }
% 240.66/31.11 not(implies(X, not(X)))
% 240.66/31.11 = { by lemma 71 }
% 240.66/31.11 and(X, X)
% 240.66/31.11 = { by lemma 99 }
% 240.66/31.11 X
% 240.66/31.11
% 240.66/31.11 Lemma 127: fresh22(op_implies_and, modus_tollens, X, Y) = implies(X, Y).
% 240.66/31.11 Proof:
% 240.66/31.11 fresh22(op_implies_and, modus_tollens, X, Y)
% 240.66/31.11 = { by lemma 109 R->L }
% 240.66/31.11 fresh22(op_implies_and, modus_tollens, X, or(Y, Y))
% 240.66/31.11 = { by lemma 112 R->L }
% 240.66/31.11 fresh22(op_implies_and, implies_1, X, or(Y, Y))
% 240.66/31.11 = { by lemma 123 R->L }
% 240.66/31.11 not(and(X, not(or(Y, Y))))
% 240.66/31.11 = { by lemma 125 R->L }
% 240.66/31.11 implies(X, not(not(or(Y, Y))))
% 240.66/31.11 = { by lemma 126 }
% 240.66/31.11 implies(X, Y)
% 240.66/31.11
% 240.66/31.11 Lemma 128: fresh28(is_a_theorem(or(implies(X, Y), not(X))), modus_tollens, implies(X, Y)) = modus_tollens.
% 240.66/31.11 Proof:
% 240.66/31.11 fresh28(is_a_theorem(or(implies(X, Y), not(X))), modus_tollens, implies(X, Y))
% 240.66/31.11 = { by lemma 112 R->L }
% 240.66/31.11 fresh28(is_a_theorem(or(implies(X, Y), not(X))), implies_1, implies(X, Y))
% 240.66/31.11 = { by lemma 83 R->L }
% 240.66/31.11 fresh59(is_a_theorem(implies(or(implies(X, Y), not(X)), implies(X, Y))), implies_1, or(implies(X, Y), not(X)), implies(X, Y))
% 240.66/31.11 = { by axiom 17 (modus_ponens_2) R->L }
% 240.66/31.11 fresh59(fresh28(implies_1, implies_1, implies(or(implies(X, Y), not(X)), implies(X, Y))), implies_1, or(implies(X, Y), not(X)), implies(X, Y))
% 240.66/31.11 = { by lemma 75 R->L }
% 240.66/31.11 fresh59(fresh28(fresh28(is_a_theorem(implies(not(X), implies(X, Y))), implies_1, implies(or(implies(X, Y), not(X)), or(implies(X, Y), implies(X, Y)))), implies_1, implies(or(implies(X, Y), not(X)), implies(X, Y))), implies_1, or(implies(X, Y), not(X)), implies(X, Y))
% 240.66/31.11 = { by lemma 116 }
% 240.66/31.11 fresh59(fresh28(fresh28(implies_1, implies_1, implies(or(implies(X, Y), not(X)), or(implies(X, Y), implies(X, Y)))), implies_1, implies(or(implies(X, Y), not(X)), implies(X, Y))), implies_1, or(implies(X, Y), not(X)), implies(X, Y))
% 240.66/31.11 = { by axiom 17 (modus_ponens_2) }
% 240.66/31.11 fresh59(fresh28(is_a_theorem(implies(or(implies(X, Y), not(X)), or(implies(X, Y), implies(X, Y)))), implies_1, implies(or(implies(X, Y), not(X)), implies(X, Y))), implies_1, or(implies(X, Y), not(X)), implies(X, Y))
% 240.66/31.11 = { by lemma 105 }
% 240.66/31.11 fresh59(implies_1, implies_1, or(implies(X, Y), not(X)), implies(X, Y))
% 240.66/31.11 = { by lemma 90 }
% 240.66/31.11 implies_1
% 240.66/31.11 = { by lemma 112 }
% 240.66/31.11 modus_tollens
% 240.66/31.11
% 240.66/31.11 Lemma 129: not(and(X, not(Y))) = fresh22(op_implies_and, op_and, X, Y).
% 240.66/31.11 Proof:
% 240.66/31.11 not(and(X, not(Y)))
% 240.66/31.11 = { by axiom 28 (op_implies_and) R->L }
% 240.66/31.11 fresh22(op_implies_and, true, X, Y)
% 240.66/31.11 = { by axiom 1 (principia_op_and) R->L }
% 240.66/31.11 fresh22(op_implies_and, op_and, X, Y)
% 240.66/31.11
% 240.66/31.11 Lemma 130: not(implies(X, Y)) = and(X, not(Y)).
% 240.66/31.11 Proof:
% 240.66/31.11 not(implies(X, Y))
% 240.66/31.11 = { by lemma 100 R->L }
% 240.66/31.11 and(X, implies(Y, not(Y)))
% 240.66/31.11 = { by lemma 125 }
% 240.66/31.11 and(X, not(and(Y, Y)))
% 240.66/31.11 = { by lemma 99 }
% 240.66/31.11 and(X, not(Y))
% 240.66/31.11
% 240.66/31.11 Lemma 131: not(not(X)) = X.
% 240.66/31.11 Proof:
% 240.66/31.11 not(not(X))
% 240.66/31.11 = { by lemma 109 R->L }
% 240.66/31.11 not(not(or(X, X)))
% 240.66/31.11 = { by lemma 126 }
% 240.66/31.11 X
% 240.66/31.11
% 240.66/31.11 Lemma 132: fresh28(is_a_theorem(or(X, Y)), modus_tollens, or(Y, X)) = modus_tollens.
% 240.66/31.11 Proof:
% 240.66/31.11 fresh28(is_a_theorem(or(X, Y)), modus_tollens, or(Y, X))
% 240.66/31.11 = { by lemma 112 R->L }
% 240.66/31.11 fresh28(is_a_theorem(or(X, Y)), implies_1, or(Y, X))
% 240.66/31.11 = { by lemma 59 R->L }
% 240.66/31.11 fresh28(is_a_theorem(or(X, Y)), or_2, or(Y, X))
% 240.66/31.11 = { by lemma 55 R->L }
% 240.66/31.11 fresh28(is_a_theorem(or(X, Y)), op_and, or(Y, X))
% 240.66/31.12 = { by lemma 60 R->L }
% 240.66/31.12 fresh59(is_a_theorem(implies(or(X, Y), or(Y, X))), op_and, or(X, Y), or(Y, X))
% 240.66/31.12 = { by lemma 61 }
% 240.66/31.12 fresh59(or_2, op_and, or(X, Y), or(Y, X))
% 240.66/31.12 = { by lemma 59 }
% 240.66/31.12 fresh59(implies_1, op_and, or(X, Y), or(Y, X))
% 240.66/31.12 = { by lemma 55 }
% 240.66/31.12 fresh59(implies_1, or_2, or(X, Y), or(Y, X))
% 240.66/31.12 = { by lemma 59 }
% 240.66/31.12 fresh59(implies_1, implies_1, or(X, Y), or(Y, X))
% 240.66/31.12 = { by lemma 63 }
% 240.66/31.12 op_and
% 240.66/31.12 = { by lemma 55 }
% 240.66/31.12 or_2
% 240.66/31.12 = { by lemma 59 }
% 240.66/31.12 implies_1
% 240.66/31.12 = { by lemma 112 }
% 240.66/31.12 modus_tollens
% 240.66/31.12
% 240.66/31.12 Lemma 133: is_a_theorem(or(implies(X, and(implies(Y, Z), X)), and(Y, not(Z)))) = modus_tollens.
% 240.66/31.12 Proof:
% 240.66/31.12 is_a_theorem(or(implies(X, and(implies(Y, Z), X)), and(Y, not(Z))))
% 240.66/31.12 = { by axiom 17 (modus_ponens_2) R->L }
% 240.66/31.12 fresh28(modus_tollens, modus_tollens, or(implies(X, and(implies(Y, Z), X)), and(Y, not(Z))))
% 240.66/31.12 = { by lemma 117 R->L }
% 240.66/31.12 fresh28(fresh53(and_3, modus_tollens, implies(Y, Z), X), modus_tollens, or(implies(X, and(implies(Y, Z), X)), and(Y, not(Z))))
% 240.66/31.12 = { by lemma 115 R->L }
% 240.66/31.12 fresh28(is_a_theorem(implies(implies(Y, Z), implies(X, and(implies(Y, Z), X)))), modus_tollens, or(implies(X, and(implies(Y, Z), X)), and(Y, not(Z))))
% 240.66/31.12 = { by lemma 127 R->L }
% 240.66/31.12 fresh28(is_a_theorem(implies(fresh22(op_implies_and, modus_tollens, Y, Z), implies(X, and(implies(Y, Z), X)))), modus_tollens, or(implies(X, and(implies(Y, Z), X)), and(Y, not(Z))))
% 240.66/31.12 = { by lemma 127 R->L }
% 240.66/31.12 fresh28(is_a_theorem(fresh22(op_implies_and, modus_tollens, fresh22(op_implies_and, modus_tollens, Y, Z), implies(X, and(implies(Y, Z), X)))), modus_tollens, or(implies(X, and(implies(Y, Z), X)), and(Y, not(Z))))
% 240.66/31.12 = { by lemma 112 R->L }
% 240.66/31.12 fresh28(is_a_theorem(fresh22(op_implies_and, modus_tollens, fresh22(op_implies_and, implies_1, Y, Z), implies(X, and(implies(Y, Z), X)))), modus_tollens, or(implies(X, and(implies(Y, Z), X)), and(Y, not(Z))))
% 240.66/31.12 = { by lemma 112 R->L }
% 240.66/31.12 fresh28(is_a_theorem(fresh22(op_implies_and, implies_1, fresh22(op_implies_and, implies_1, Y, Z), implies(X, and(implies(Y, Z), X)))), modus_tollens, or(implies(X, and(implies(Y, Z), X)), and(Y, not(Z))))
% 240.66/31.12 = { by lemma 59 R->L }
% 240.66/31.12 fresh28(is_a_theorem(fresh22(op_implies_and, implies_1, fresh22(op_implies_and, or_2, Y, Z), implies(X, and(implies(Y, Z), X)))), modus_tollens, or(implies(X, and(implies(Y, Z), X)), and(Y, not(Z))))
% 240.66/31.12 = { by lemma 59 R->L }
% 240.66/31.12 fresh28(is_a_theorem(fresh22(op_implies_and, or_2, fresh22(op_implies_and, or_2, Y, Z), implies(X, and(implies(Y, Z), X)))), modus_tollens, or(implies(X, and(implies(Y, Z), X)), and(Y, not(Z))))
% 240.66/31.12 = { by lemma 55 R->L }
% 240.66/31.12 fresh28(is_a_theorem(fresh22(op_implies_and, or_2, fresh22(op_implies_and, op_and, Y, Z), implies(X, and(implies(Y, Z), X)))), modus_tollens, or(implies(X, and(implies(Y, Z), X)), and(Y, not(Z))))
% 240.66/31.12 = { by lemma 55 R->L }
% 240.66/31.12 fresh28(is_a_theorem(fresh22(op_implies_and, op_and, fresh22(op_implies_and, op_and, Y, Z), implies(X, and(implies(Y, Z), X)))), modus_tollens, or(implies(X, and(implies(Y, Z), X)), and(Y, not(Z))))
% 240.66/31.12 = { by lemma 129 R->L }
% 240.66/31.12 fresh28(is_a_theorem(fresh22(op_implies_and, op_and, not(and(Y, not(Z))), implies(X, and(implies(Y, Z), X)))), modus_tollens, or(implies(X, and(implies(Y, Z), X)), and(Y, not(Z))))
% 240.66/31.12 = { by lemma 55 }
% 240.66/31.12 fresh28(is_a_theorem(fresh22(op_implies_and, or_2, not(and(Y, not(Z))), implies(X, and(implies(Y, Z), X)))), modus_tollens, or(implies(X, and(implies(Y, Z), X)), and(Y, not(Z))))
% 240.66/31.12 = { by lemma 59 }
% 240.66/31.12 fresh28(is_a_theorem(fresh22(op_implies_and, implies_1, not(and(Y, not(Z))), implies(X, and(implies(Y, Z), X)))), modus_tollens, or(implies(X, and(implies(Y, Z), X)), and(Y, not(Z))))
% 240.66/31.12 = { by lemma 123 R->L }
% 240.66/31.12 fresh28(is_a_theorem(not(and(not(and(Y, not(Z))), not(implies(X, and(implies(Y, Z), X)))))), modus_tollens, or(implies(X, and(implies(Y, Z), X)), and(Y, not(Z))))
% 240.66/31.12 = { by lemma 130 R->L }
% 240.66/31.12 fresh28(is_a_theorem(not(not(implies(not(and(Y, not(Z))), implies(X, and(implies(Y, Z), X)))))), modus_tollens, or(implies(X, and(implies(Y, Z), X)), and(Y, not(Z))))
% 240.66/31.12 = { by lemma 100 R->L }
% 240.66/31.12 fresh28(is_a_theorem(not(and(not(and(Y, not(Z))), implies(implies(X, and(implies(Y, Z), X)), not(implies(X, and(implies(Y, Z), X))))))), modus_tollens, or(implies(X, and(implies(Y, Z), X)), and(Y, not(Z))))
% 240.66/31.12 = { by lemma 101 }
% 240.66/31.12 fresh28(is_a_theorem(not(and(not(and(Y, not(Z))), not(or(implies(X, and(implies(Y, Z), X)), implies(X, and(implies(Y, Z), X))))))), modus_tollens, or(implies(X, and(implies(Y, Z), X)), and(Y, not(Z))))
% 240.66/31.12 = { by lemma 110 R->L }
% 240.66/31.12 fresh28(is_a_theorem(not(not(or(and(Y, not(Z)), not(not(or(implies(X, and(implies(Y, Z), X)), implies(X, and(implies(Y, Z), X))))))))), modus_tollens, or(implies(X, and(implies(Y, Z), X)), and(Y, not(Z))))
% 240.66/31.12 = { by lemma 126 }
% 240.66/31.12 fresh28(is_a_theorem(not(not(or(and(Y, not(Z)), implies(X, and(implies(Y, Z), X)))))), modus_tollens, or(implies(X, and(implies(Y, Z), X)), and(Y, not(Z))))
% 240.66/31.12 = { by lemma 131 }
% 240.66/31.12 fresh28(is_a_theorem(or(and(Y, not(Z)), implies(X, and(implies(Y, Z), X)))), modus_tollens, or(implies(X, and(implies(Y, Z), X)), and(Y, not(Z))))
% 240.66/31.12 = { by lemma 132 }
% 240.66/31.12 modus_tollens
% 240.66/31.12
% 240.66/31.12 Lemma 134: fresh28(is_a_theorem(implies(X, Y)), modus_tollens, implies(or(Z, X), or(Z, Y))) = modus_tollens.
% 240.66/31.12 Proof:
% 240.66/31.12 fresh28(is_a_theorem(implies(X, Y)), modus_tollens, implies(or(Z, X), or(Z, Y)))
% 240.66/31.12 = { by lemma 112 R->L }
% 240.66/31.12 fresh28(is_a_theorem(implies(X, Y)), implies_1, implies(or(Z, X), or(Z, Y)))
% 240.66/31.12 = { by lemma 59 R->L }
% 240.66/31.12 fresh28(is_a_theorem(implies(X, Y)), or_2, implies(or(Z, X), or(Z, Y)))
% 240.66/31.12 = { by lemma 55 R->L }
% 240.66/31.12 fresh28(is_a_theorem(implies(X, Y)), op_and, implies(or(Z, X), or(Z, Y)))
% 240.66/31.12 = { by lemma 60 R->L }
% 240.66/31.12 fresh59(is_a_theorem(implies(implies(X, Y), implies(or(Z, X), or(Z, Y)))), op_and, implies(X, Y), implies(or(Z, X), or(Z, Y)))
% 240.66/31.12 = { by lemma 74 }
% 240.66/31.12 fresh59(or_2, op_and, implies(X, Y), implies(or(Z, X), or(Z, Y)))
% 240.66/31.12 = { by lemma 59 }
% 240.66/31.12 fresh59(implies_1, op_and, implies(X, Y), implies(or(Z, X), or(Z, Y)))
% 240.66/31.12 = { by lemma 55 }
% 240.66/31.12 fresh59(implies_1, or_2, implies(X, Y), implies(or(Z, X), or(Z, Y)))
% 240.66/31.12 = { by lemma 59 }
% 240.66/31.12 fresh59(implies_1, implies_1, implies(X, Y), implies(or(Z, X), or(Z, Y)))
% 240.66/31.12 = { by lemma 63 }
% 240.66/31.12 op_and
% 240.66/31.12 = { by lemma 55 }
% 240.66/31.12 or_2
% 240.66/31.12 = { by lemma 59 }
% 240.66/31.12 implies_1
% 240.66/31.12 = { by lemma 112 }
% 240.66/31.12 modus_tollens
% 240.66/31.12
% 240.66/31.12 Lemma 135: implies(not(X), Y) = or(X, Y).
% 240.66/31.12 Proof:
% 240.66/31.12 implies(not(X), Y)
% 240.66/31.12 = { by lemma 109 R->L }
% 240.66/31.12 implies(not(or(X, X)), Y)
% 240.66/31.12 = { by lemma 101 R->L }
% 240.66/31.12 implies(implies(X, not(X)), Y)
% 240.66/31.12 = { by lemma 72 }
% 240.66/31.12 or(and(X, X), Y)
% 240.66/31.12 = { by lemma 99 }
% 240.66/31.12 or(X, Y)
% 240.66/31.12
% 240.66/31.12 Lemma 136: is_a_theorem(implies(and(X, Y), Y)) = modus_tollens.
% 240.66/31.12 Proof:
% 240.66/31.12 is_a_theorem(implies(and(X, Y), Y))
% 240.66/31.12 = { by axiom 36 (and_2_1) R->L }
% 240.66/31.12 fresh55(and_2, true, X, Y)
% 240.66/31.12 = { by axiom 1 (principia_op_and) R->L }
% 240.66/31.12 fresh55(and_2, op_and, X, Y)
% 240.66/31.12 = { by lemma 55 }
% 240.66/31.12 fresh55(and_2, or_2, X, Y)
% 240.66/31.12 = { by lemma 59 }
% 240.66/31.12 fresh55(and_2, implies_1, X, Y)
% 240.66/31.12 = { by lemma 94 }
% 240.66/31.12 implies_1
% 240.66/31.12 = { by lemma 112 }
% 240.66/31.12 modus_tollens
% 240.66/31.12
% 240.66/31.12 Lemma 137: fresh28(is_a_theorem(implies(X, and(Y, X))), modus_tollens, equiv(and(Y, X), X)) = modus_tollens.
% 240.66/31.12 Proof:
% 240.66/31.12 fresh28(is_a_theorem(implies(X, and(Y, X))), modus_tollens, equiv(and(Y, X), X))
% 240.66/31.12 = { by lemma 114 R->L }
% 240.66/31.12 fresh59(is_a_theorem(implies(implies(X, and(Y, X)), equiv(and(Y, X), X))), modus_tollens, implies(X, and(Y, X)), equiv(and(Y, X), X))
% 240.66/31.12 = { by axiom 17 (modus_ponens_2) R->L }
% 240.66/31.12 fresh59(fresh28(modus_tollens, modus_tollens, implies(implies(X, and(Y, X)), equiv(and(Y, X), X))), modus_tollens, implies(X, and(Y, X)), equiv(and(Y, X), X))
% 240.66/31.12 = { by lemma 136 R->L }
% 240.66/31.12 fresh59(fresh28(is_a_theorem(implies(and(Y, X), X)), modus_tollens, implies(implies(X, and(Y, X)), equiv(and(Y, X), X))), modus_tollens, implies(X, and(Y, X)), equiv(and(Y, X), X))
% 240.66/31.12 = { by lemma 119 }
% 240.66/31.12 fresh59(modus_tollens, modus_tollens, implies(X, and(Y, X)), equiv(and(Y, X), X))
% 240.66/31.12 = { by lemma 118 }
% 240.66/31.12 modus_tollens
% 240.66/31.12
% 240.66/31.12 Lemma 138: implies(implies(X, Y), X) = X.
% 240.66/31.12 Proof:
% 240.66/31.12 implies(implies(X, Y), X)
% 240.66/31.12 = { by lemma 57 R->L }
% 240.66/31.12 implies(or(not(X), Y), X)
% 240.66/31.12 = { by lemma 127 R->L }
% 240.66/31.12 fresh22(op_implies_and, modus_tollens, or(not(X), Y), X)
% 240.66/31.12 = { by lemma 112 R->L }
% 240.66/31.12 fresh22(op_implies_and, implies_1, or(not(X), Y), X)
% 240.66/31.12 = { by lemma 59 R->L }
% 240.66/31.12 fresh22(op_implies_and, or_2, or(not(X), Y), X)
% 240.66/31.12 = { by lemma 55 R->L }
% 240.66/31.12 fresh22(op_implies_and, op_and, or(not(X), Y), X)
% 240.66/31.12 = { by axiom 1 (principia_op_and) }
% 240.66/31.12 fresh22(op_implies_and, true, or(not(X), Y), X)
% 240.66/31.12 = { by axiom 28 (op_implies_and) }
% 240.66/31.12 not(and(or(not(X), Y), not(X)))
% 240.66/31.12 = { by lemma 121 R->L }
% 240.66/31.12 not(fresh(is_a_theorem(equiv(and(or(not(X), Y), not(X)), not(X))), modus_tollens, and(or(not(X), Y), not(X)), not(X)))
% 240.66/31.12 = { by axiom 17 (modus_ponens_2) R->L }
% 240.66/31.12 not(fresh(fresh28(modus_tollens, modus_tollens, equiv(and(or(not(X), Y), not(X)), not(X))), modus_tollens, and(or(not(X), Y), not(X)), not(X)))
% 240.66/31.12 = { by lemma 128 R->L }
% 240.66/31.12 not(fresh(fresh28(fresh28(is_a_theorem(or(implies(not(X), and(implies(not(not(X)), Y), not(X))), not(not(X)))), modus_tollens, implies(not(X), and(implies(not(not(X)), Y), not(X)))), modus_tollens, equiv(and(or(not(X), Y), not(X)), not(X))), modus_tollens, and(or(not(X), Y), not(X)), not(X)))
% 240.66/31.12 = { by axiom 17 (modus_ponens_2) R->L }
% 240.66/31.12 not(fresh(fresh28(fresh28(fresh28(modus_tollens, modus_tollens, or(implies(not(X), and(implies(not(not(X)), Y), not(X))), not(not(X)))), modus_tollens, implies(not(X), and(implies(not(not(X)), Y), not(X)))), modus_tollens, equiv(and(or(not(X), Y), not(X)), not(X))), modus_tollens, and(or(not(X), Y), not(X)), not(X)))
% 240.66/31.12 = { by lemma 133 R->L }
% 240.66/31.12 not(fresh(fresh28(fresh28(fresh28(is_a_theorem(or(implies(not(X), and(implies(not(not(X)), Y), not(X))), and(not(not(X)), not(Y)))), modus_tollens, or(implies(not(X), and(implies(not(not(X)), Y), not(X))), not(not(X)))), modus_tollens, implies(not(X), and(implies(not(not(X)), Y), not(X)))), modus_tollens, equiv(and(or(not(X), Y), not(X)), not(X))), modus_tollens, and(or(not(X), Y), not(X)), not(X)))
% 240.66/31.12 = { by lemma 114 R->L }
% 240.66/31.12 not(fresh(fresh28(fresh28(fresh59(is_a_theorem(implies(or(implies(not(X), and(implies(not(not(X)), Y), not(X))), and(not(not(X)), not(Y))), or(implies(not(X), and(implies(not(not(X)), Y), not(X))), not(not(X))))), modus_tollens, or(implies(not(X), and(implies(not(not(X)), Y), not(X))), and(not(not(X)), not(Y))), or(implies(not(X), and(implies(not(not(X)), Y), not(X))), not(not(X)))), modus_tollens, implies(not(X), and(implies(not(not(X)), Y), not(X)))), modus_tollens, equiv(and(or(not(X), Y), not(X)), not(X))), modus_tollens, and(or(not(X), Y), not(X)), not(X)))
% 240.66/31.13 = { by axiom 17 (modus_ponens_2) R->L }
% 240.66/31.13 not(fresh(fresh28(fresh28(fresh59(fresh28(modus_tollens, modus_tollens, implies(or(implies(not(X), and(implies(not(not(X)), Y), not(X))), and(not(not(X)), not(Y))), or(implies(not(X), and(implies(not(not(X)), Y), not(X))), not(not(X))))), modus_tollens, or(implies(not(X), and(implies(not(not(X)), Y), not(X))), and(not(not(X)), not(Y))), or(implies(not(X), and(implies(not(not(X)), Y), not(X))), not(not(X)))), modus_tollens, implies(not(X), and(implies(not(not(X)), Y), not(X)))), modus_tollens, equiv(and(or(not(X), Y), not(X)), not(X))), modus_tollens, and(or(not(X), Y), not(X)), not(X)))
% 240.66/31.13 = { by lemma 112 R->L }
% 240.66/31.13 not(fresh(fresh28(fresh28(fresh59(fresh28(implies_1, modus_tollens, implies(or(implies(not(X), and(implies(not(not(X)), Y), not(X))), and(not(not(X)), not(Y))), or(implies(not(X), and(implies(not(not(X)), Y), not(X))), not(not(X))))), modus_tollens, or(implies(not(X), and(implies(not(not(X)), Y), not(X))), and(not(not(X)), not(Y))), or(implies(not(X), and(implies(not(not(X)), Y), not(X))), not(not(X)))), modus_tollens, implies(not(X), and(implies(not(not(X)), Y), not(X)))), modus_tollens, equiv(and(or(not(X), Y), not(X)), not(X))), modus_tollens, and(or(not(X), Y), not(X)), not(X)))
% 240.66/31.13 = { by lemma 90 R->L }
% 240.66/31.13 not(fresh(fresh28(fresh28(fresh59(fresh28(fresh59(implies_1, implies_1, or(implies(not(implies(not(not(X)), not(not(Y)))), not(not(X))), implies(not(not(X)), not(not(Y)))), implies(not(implies(not(not(X)), not(not(Y)))), not(not(X)))), modus_tollens, implies(or(implies(not(X), and(implies(not(not(X)), Y), not(X))), and(not(not(X)), not(Y))), or(implies(not(X), and(implies(not(not(X)), Y), not(X))), not(not(X))))), modus_tollens, or(implies(not(X), and(implies(not(not(X)), Y), not(X))), and(not(not(X)), not(Y))), or(implies(not(X), and(implies(not(not(X)), Y), not(X))), not(not(X)))), modus_tollens, implies(not(X), and(implies(not(not(X)), Y), not(X)))), modus_tollens, equiv(and(or(not(X), Y), not(X)), not(X))), modus_tollens, and(or(not(X), Y), not(X)), not(X)))
% 240.66/31.13 = { by lemma 105 R->L }
% 240.66/31.13 not(fresh(fresh28(fresh28(fresh59(fresh28(fresh59(fresh28(is_a_theorem(implies(or(implies(not(implies(not(not(X)), not(not(Y)))), not(not(X))), implies(not(not(X)), not(not(Y)))), or(implies(not(implies(not(not(X)), not(not(Y)))), not(not(X))), implies(not(implies(not(not(X)), not(not(Y)))), not(not(X)))))), implies_1, implies(or(implies(not(implies(not(not(X)), not(not(Y)))), not(not(X))), implies(not(not(X)), not(not(Y)))), implies(not(implies(not(not(X)), not(not(Y)))), not(not(X))))), implies_1, or(implies(not(implies(not(not(X)), not(not(Y)))), not(not(X))), implies(not(not(X)), not(not(Y)))), implies(not(implies(not(not(X)), not(not(Y)))), not(not(X)))), modus_tollens, implies(or(implies(not(X), and(implies(not(not(X)), Y), not(X))), and(not(not(X)), not(Y))), or(implies(not(X), and(implies(not(not(X)), Y), not(X))), not(not(X))))), modus_tollens, or(implies(not(X), and(implies(not(not(X)), Y), not(X))), and(not(not(X)), not(Y))), or(implies(not(X), and(implies(not(not(X)), Y), not(X))), not(not(X)))), modus_tollens, implies(not(X), and(implies(not(not(X)), Y), not(X)))), modus_tollens, equiv(and(or(not(X), Y), not(X)), not(X))), modus_tollens, and(or(not(X), Y), not(X)), not(X)))
% 240.66/31.13 = { by lemma 93 }
% 240.66/31.13 not(fresh(fresh28(fresh28(fresh59(fresh28(fresh59(fresh28(implies_1, implies_1, implies(or(implies(not(implies(not(not(X)), not(not(Y)))), not(not(X))), implies(not(not(X)), not(not(Y)))), implies(not(implies(not(not(X)), not(not(Y)))), not(not(X))))), implies_1, or(implies(not(implies(not(not(X)), not(not(Y)))), not(not(X))), implies(not(not(X)), not(not(Y)))), implies(not(implies(not(not(X)), not(not(Y)))), not(not(X)))), modus_tollens, implies(or(implies(not(X), and(implies(not(not(X)), Y), not(X))), and(not(not(X)), not(Y))), or(implies(not(X), and(implies(not(not(X)), Y), not(X))), not(not(X))))), modus_tollens, or(implies(not(X), and(implies(not(not(X)), Y), not(X))), and(not(not(X)), not(Y))), or(implies(not(X), and(implies(not(not(X)), Y), not(X))), not(not(X)))), modus_tollens, implies(not(X), and(implies(not(not(X)), Y), not(X)))), modus_tollens, equiv(and(or(not(X), Y), not(X)), not(X))), modus_tollens, and(or(not(X), Y), not(X)), not(X)))
% 240.66/31.13 = { by axiom 17 (modus_ponens_2) }
% 240.66/31.13 not(fresh(fresh28(fresh28(fresh59(fresh28(fresh59(is_a_theorem(implies(or(implies(not(implies(not(not(X)), not(not(Y)))), not(not(X))), implies(not(not(X)), not(not(Y)))), implies(not(implies(not(not(X)), not(not(Y)))), not(not(X))))), implies_1, or(implies(not(implies(not(not(X)), not(not(Y)))), not(not(X))), implies(not(not(X)), not(not(Y)))), implies(not(implies(not(not(X)), not(not(Y)))), not(not(X)))), modus_tollens, implies(or(implies(not(X), and(implies(not(not(X)), Y), not(X))), and(not(not(X)), not(Y))), or(implies(not(X), and(implies(not(not(X)), Y), not(X))), not(not(X))))), modus_tollens, or(implies(not(X), and(implies(not(not(X)), Y), not(X))), and(not(not(X)), not(Y))), or(implies(not(X), and(implies(not(not(X)), Y), not(X))), not(not(X)))), modus_tollens, implies(not(X), and(implies(not(not(X)), Y), not(X)))), modus_tollens, equiv(and(or(not(X), Y), not(X)), not(X))), modus_tollens, and(or(not(X), Y), not(X)), not(X)))
% 240.66/31.13 = { by lemma 83 }
% 240.66/31.13 not(fresh(fresh28(fresh28(fresh59(fresh28(fresh28(is_a_theorem(or(implies(not(implies(not(not(X)), not(not(Y)))), not(not(X))), implies(not(not(X)), not(not(Y))))), implies_1, implies(not(implies(not(not(X)), not(not(Y)))), not(not(X)))), modus_tollens, implies(or(implies(not(X), and(implies(not(not(X)), Y), not(X))), and(not(not(X)), not(Y))), or(implies(not(X), and(implies(not(not(X)), Y), not(X))), not(not(X))))), modus_tollens, or(implies(not(X), and(implies(not(not(X)), Y), not(X))), and(not(not(X)), not(Y))), or(implies(not(X), and(implies(not(not(X)), Y), not(X))), not(not(X)))), modus_tollens, implies(not(X), and(implies(not(not(X)), Y), not(X)))), modus_tollens, equiv(and(or(not(X), Y), not(X)), not(X))), modus_tollens, and(or(not(X), Y), not(X)), not(X)))
% 240.66/31.13 = { by lemma 57 R->L }
% 240.66/31.13 not(fresh(fresh28(fresh28(fresh59(fresh28(fresh28(is_a_theorem(or(implies(not(implies(not(not(X)), not(not(Y)))), not(not(X))), or(not(not(not(X))), not(not(Y))))), implies_1, implies(not(implies(not(not(X)), not(not(Y)))), not(not(X)))), modus_tollens, implies(or(implies(not(X), and(implies(not(not(X)), Y), not(X))), and(not(not(X)), not(Y))), or(implies(not(X), and(implies(not(not(X)), Y), not(X))), not(not(X))))), modus_tollens, or(implies(not(X), and(implies(not(not(X)), Y), not(X))), and(not(not(X)), not(Y))), or(implies(not(X), and(implies(not(not(X)), Y), not(X))), not(not(X)))), modus_tollens, implies(not(X), and(implies(not(not(X)), Y), not(X)))), modus_tollens, equiv(and(or(not(X), Y), not(X)), not(X))), modus_tollens, and(or(not(X), Y), not(X)), not(X)))
% 240.66/31.13 = { by axiom 17 (modus_ponens_2) R->L }
% 240.66/31.13 not(fresh(fresh28(fresh28(fresh59(fresh28(fresh28(fresh28(implies_1, implies_1, or(implies(not(implies(not(not(X)), not(not(Y)))), not(not(X))), or(not(not(not(X))), not(not(Y))))), implies_1, implies(not(implies(not(not(X)), not(not(Y)))), not(not(X)))), modus_tollens, implies(or(implies(not(X), and(implies(not(not(X)), Y), not(X))), and(not(not(X)), not(Y))), or(implies(not(X), and(implies(not(not(X)), Y), not(X))), not(not(X))))), modus_tollens, or(implies(not(X), and(implies(not(not(X)), Y), not(X))), and(not(not(X)), not(Y))), or(implies(not(X), and(implies(not(not(X)), Y), not(X))), not(not(X)))), modus_tollens, implies(not(X), and(implies(not(not(X)), Y), not(X)))), modus_tollens, equiv(and(or(not(X), Y), not(X)), not(X))), modus_tollens, and(or(not(X), Y), not(X)), not(X)))
% 240.66/31.13 = { by lemma 86 R->L }
% 240.66/31.13 not(fresh(fresh28(fresh28(fresh59(fresh28(fresh28(fresh28(fresh28(is_a_theorem(or(implies(not(implies(not(not(X)), not(not(Y)))), not(not(X))), not(not(not(X))))), implies_1, or(implies(not(implies(not(not(X)), not(not(Y)))), not(not(X))), or(not(not(Y)), not(not(not(X)))))), implies_1, or(implies(not(implies(not(not(X)), not(not(Y)))), not(not(X))), or(not(not(not(X))), not(not(Y))))), implies_1, implies(not(implies(not(not(X)), not(not(Y)))), not(not(X)))), modus_tollens, implies(or(implies(not(X), and(implies(not(not(X)), Y), not(X))), and(not(not(X)), not(Y))), or(implies(not(X), and(implies(not(not(X)), Y), not(X))), not(not(X))))), modus_tollens, or(implies(not(X), and(implies(not(not(X)), Y), not(X))), and(not(not(X)), not(Y))), or(implies(not(X), and(implies(not(not(X)), Y), not(X))), not(not(X)))), modus_tollens, implies(not(X), and(implies(not(not(X)), Y), not(X)))), modus_tollens, equiv(and(or(not(X), Y), not(X)), not(X))), modus_tollens, and(or(not(X), Y), not(X)), not(X)))
% 240.66/31.14 = { by axiom 17 (modus_ponens_2) R->L }
% 240.66/31.14 not(fresh(fresh28(fresh28(fresh59(fresh28(fresh28(fresh28(fresh28(fresh28(implies_1, implies_1, or(implies(not(implies(not(not(X)), not(not(Y)))), not(not(X))), not(not(not(X))))), implies_1, or(implies(not(implies(not(not(X)), not(not(Y)))), not(not(X))), or(not(not(Y)), not(not(not(X)))))), implies_1, or(implies(not(implies(not(not(X)), not(not(Y)))), not(not(X))), or(not(not(not(X))), not(not(Y))))), implies_1, implies(not(implies(not(not(X)), not(not(Y)))), not(not(X)))), modus_tollens, implies(or(implies(not(X), and(implies(not(not(X)), Y), not(X))), and(not(not(X)), not(Y))), or(implies(not(X), and(implies(not(not(X)), Y), not(X))), not(not(X))))), modus_tollens, or(implies(not(X), and(implies(not(not(X)), Y), not(X))), and(not(not(X)), not(Y))), or(implies(not(X), and(implies(not(not(X)), Y), not(X))), not(not(X)))), modus_tollens, implies(not(X), and(implies(not(not(X)), Y), not(X)))), modus_tollens, equiv(and(or(not(X), Y), not(X)), not(X))), modus_tollens, and(or(not(X), Y), not(X)), not(X)))
% 240.66/31.14 = { by lemma 59 R->L }
% 240.66/31.14 not(fresh(fresh28(fresh28(fresh59(fresh28(fresh28(fresh28(fresh28(fresh28(or_2, implies_1, or(implies(not(implies(not(not(X)), not(not(Y)))), not(not(X))), not(not(not(X))))), implies_1, or(implies(not(implies(not(not(X)), not(not(Y)))), not(not(X))), or(not(not(Y)), not(not(not(X)))))), implies_1, or(implies(not(implies(not(not(X)), not(not(Y)))), not(not(X))), or(not(not(not(X))), not(not(Y))))), implies_1, implies(not(implies(not(not(X)), not(not(Y)))), not(not(X)))), modus_tollens, implies(or(implies(not(X), and(implies(not(not(X)), Y), not(X))), and(not(not(X)), not(Y))), or(implies(not(X), and(implies(not(not(X)), Y), not(X))), not(not(X))))), modus_tollens, or(implies(not(X), and(implies(not(not(X)), Y), not(X))), and(not(not(X)), not(Y))), or(implies(not(X), and(implies(not(not(X)), Y), not(X))), not(not(X)))), modus_tollens, implies(not(X), and(implies(not(not(X)), Y), not(X)))), modus_tollens, equiv(and(or(not(X), Y), not(X)), not(X))), modus_tollens, and(or(not(X), Y), not(X)), not(X)))
% 240.66/31.14 = { by lemma 58 R->L }
% 240.66/31.14 not(fresh(fresh28(fresh28(fresh59(fresh28(fresh28(fresh28(fresh28(fresh28(fresh39(implies_1, or_2, not(not(X)), not(implies(not(not(X)), not(not(Y))))), implies_1, or(implies(not(implies(not(not(X)), not(not(Y)))), not(not(X))), not(not(not(X))))), implies_1, or(implies(not(implies(not(not(X)), not(not(Y)))), not(not(X))), or(not(not(Y)), not(not(not(X)))))), implies_1, or(implies(not(implies(not(not(X)), not(not(Y)))), not(not(X))), or(not(not(not(X))), not(not(Y))))), implies_1, implies(not(implies(not(not(X)), not(not(Y)))), not(not(X)))), modus_tollens, implies(or(implies(not(X), and(implies(not(not(X)), Y), not(X))), and(not(not(X)), not(Y))), or(implies(not(X), and(implies(not(not(X)), Y), not(X))), not(not(X))))), modus_tollens, or(implies(not(X), and(implies(not(not(X)), Y), not(X))), and(not(not(X)), not(Y))), or(implies(not(X), and(implies(not(not(X)), Y), not(X))), not(not(X)))), modus_tollens, implies(not(X), and(implies(not(not(X)), Y), not(X)))), modus_tollens, equiv(and(or(not(X), Y), not(X)), not(X))), modus_tollens, and(or(not(X), Y), not(X)), not(X)))
% 240.66/31.14 = { by lemma 55 R->L }
% 240.66/31.15 not(fresh(fresh28(fresh28(fresh59(fresh28(fresh28(fresh28(fresh28(fresh28(fresh39(implies_1, op_and, not(not(X)), not(implies(not(not(X)), not(not(Y))))), implies_1, or(implies(not(implies(not(not(X)), not(not(Y)))), not(not(X))), not(not(not(X))))), implies_1, or(implies(not(implies(not(not(X)), not(not(Y)))), not(not(X))), or(not(not(Y)), not(not(not(X)))))), implies_1, or(implies(not(implies(not(not(X)), not(not(Y)))), not(not(X))), or(not(not(not(X))), not(not(Y))))), implies_1, implies(not(implies(not(not(X)), not(not(Y)))), not(not(X)))), modus_tollens, implies(or(implies(not(X), and(implies(not(not(X)), Y), not(X))), and(not(not(X)), not(Y))), or(implies(not(X), and(implies(not(not(X)), Y), not(X))), not(not(X))))), modus_tollens, or(implies(not(X), and(implies(not(not(X)), Y), not(X))), and(not(not(X)), not(Y))), or(implies(not(X), and(implies(not(not(X)), Y), not(X))), not(not(X)))), modus_tollens, implies(not(X), and(implies(not(not(X)), Y), not(X)))), modus_tollens, equiv(and(or(not(X), Y), not(X)), not(X))), modus_tollens, and(or(not(X), Y), not(X)), not(X)))
% 240.66/31.15 = { by lemma 56 R->L }
% 240.66/31.15 not(fresh(fresh28(fresh28(fresh59(fresh28(fresh28(fresh28(fresh28(fresh28(is_a_theorem(implies(not(not(X)), implies(not(implies(not(not(X)), not(not(Y)))), not(not(X))))), implies_1, or(implies(not(implies(not(not(X)), not(not(Y)))), not(not(X))), not(not(not(X))))), implies_1, or(implies(not(implies(not(not(X)), not(not(Y)))), not(not(X))), or(not(not(Y)), not(not(not(X)))))), implies_1, or(implies(not(implies(not(not(X)), not(not(Y)))), not(not(X))), or(not(not(not(X))), not(not(Y))))), implies_1, implies(not(implies(not(not(X)), not(not(Y)))), not(not(X)))), modus_tollens, implies(or(implies(not(X), and(implies(not(not(X)), Y), not(X))), and(not(not(X)), not(Y))), or(implies(not(X), and(implies(not(not(X)), Y), not(X))), not(not(X))))), modus_tollens, or(implies(not(X), and(implies(not(not(X)), Y), not(X))), and(not(not(X)), not(Y))), or(implies(not(X), and(implies(not(not(X)), Y), not(X))), not(not(X)))), modus_tollens, implies(not(X), and(implies(not(not(X)), Y), not(X)))), modus_tollens, equiv(and(or(not(X), Y), not(X)), not(X))), modus_tollens, and(or(not(X), Y), not(X)), not(X)))
% 240.66/31.15 = { by lemma 65 }
% 240.66/31.15 not(fresh(fresh28(fresh28(fresh59(fresh28(fresh28(fresh28(fresh28(implies_1, implies_1, or(implies(not(implies(not(not(X)), not(not(Y)))), not(not(X))), or(not(not(Y)), not(not(not(X)))))), implies_1, or(implies(not(implies(not(not(X)), not(not(Y)))), not(not(X))), or(not(not(not(X))), not(not(Y))))), implies_1, implies(not(implies(not(not(X)), not(not(Y)))), not(not(X)))), modus_tollens, implies(or(implies(not(X), and(implies(not(not(X)), Y), not(X))), and(not(not(X)), not(Y))), or(implies(not(X), and(implies(not(not(X)), Y), not(X))), not(not(X))))), modus_tollens, or(implies(not(X), and(implies(not(not(X)), Y), not(X))), and(not(not(X)), not(Y))), or(implies(not(X), and(implies(not(not(X)), Y), not(X))), not(not(X)))), modus_tollens, implies(not(X), and(implies(not(not(X)), Y), not(X)))), modus_tollens, equiv(and(or(not(X), Y), not(X)), not(X))), modus_tollens, and(or(not(X), Y), not(X)), not(X)))
% 240.66/31.15 = { by axiom 17 (modus_ponens_2) }
% 240.66/31.15 not(fresh(fresh28(fresh28(fresh59(fresh28(fresh28(fresh28(is_a_theorem(or(implies(not(implies(not(not(X)), not(not(Y)))), not(not(X))), or(not(not(Y)), not(not(not(X)))))), implies_1, or(implies(not(implies(not(not(X)), not(not(Y)))), not(not(X))), or(not(not(not(X))), not(not(Y))))), implies_1, implies(not(implies(not(not(X)), not(not(Y)))), not(not(X)))), modus_tollens, implies(or(implies(not(X), and(implies(not(not(X)), Y), not(X))), and(not(not(X)), not(Y))), or(implies(not(X), and(implies(not(not(X)), Y), not(X))), not(not(X))))), modus_tollens, or(implies(not(X), and(implies(not(not(X)), Y), not(X))), and(not(not(X)), not(Y))), or(implies(not(X), and(implies(not(not(X)), Y), not(X))), not(not(X)))), modus_tollens, implies(not(X), and(implies(not(not(X)), Y), not(X)))), modus_tollens, equiv(and(or(not(X), Y), not(X)), not(X))), modus_tollens, and(or(not(X), Y), not(X)), not(X)))
% 240.66/31.15 = { by lemma 91 }
% 240.66/31.15 not(fresh(fresh28(fresh28(fresh59(fresh28(fresh28(implies_1, implies_1, implies(not(implies(not(not(X)), not(not(Y)))), not(not(X)))), modus_tollens, implies(or(implies(not(X), and(implies(not(not(X)), Y), not(X))), and(not(not(X)), not(Y))), or(implies(not(X), and(implies(not(not(X)), Y), not(X))), not(not(X))))), modus_tollens, or(implies(not(X), and(implies(not(not(X)), Y), not(X))), and(not(not(X)), not(Y))), or(implies(not(X), and(implies(not(not(X)), Y), not(X))), not(not(X)))), modus_tollens, implies(not(X), and(implies(not(not(X)), Y), not(X)))), modus_tollens, equiv(and(or(not(X), Y), not(X)), not(X))), modus_tollens, and(or(not(X), Y), not(X)), not(X)))
% 240.66/31.15 = { by axiom 17 (modus_ponens_2) }
% 240.66/31.15 not(fresh(fresh28(fresh28(fresh59(fresh28(is_a_theorem(implies(not(implies(not(not(X)), not(not(Y)))), not(not(X)))), modus_tollens, implies(or(implies(not(X), and(implies(not(not(X)), Y), not(X))), and(not(not(X)), not(Y))), or(implies(not(X), and(implies(not(not(X)), Y), not(X))), not(not(X))))), modus_tollens, or(implies(not(X), and(implies(not(not(X)), Y), not(X))), and(not(not(X)), not(Y))), or(implies(not(X), and(implies(not(not(X)), Y), not(X))), not(not(X)))), modus_tollens, implies(not(X), and(implies(not(not(X)), Y), not(X)))), modus_tollens, equiv(and(or(not(X), Y), not(X)), not(X))), modus_tollens, and(or(not(X), Y), not(X)), not(X)))
% 240.66/31.15 = { by lemma 71 }
% 240.66/31.15 not(fresh(fresh28(fresh28(fresh59(fresh28(is_a_theorem(implies(and(not(not(X)), not(Y)), not(not(X)))), modus_tollens, implies(or(implies(not(X), and(implies(not(not(X)), Y), not(X))), and(not(not(X)), not(Y))), or(implies(not(X), and(implies(not(not(X)), Y), not(X))), not(not(X))))), modus_tollens, or(implies(not(X), and(implies(not(not(X)), Y), not(X))), and(not(not(X)), not(Y))), or(implies(not(X), and(implies(not(not(X)), Y), not(X))), not(not(X)))), modus_tollens, implies(not(X), and(implies(not(not(X)), Y), not(X)))), modus_tollens, equiv(and(or(not(X), Y), not(X)), not(X))), modus_tollens, and(or(not(X), Y), not(X)), not(X)))
% 240.66/31.15 = { by lemma 134 }
% 240.66/31.15 not(fresh(fresh28(fresh28(fresh59(modus_tollens, modus_tollens, or(implies(not(X), and(implies(not(not(X)), Y), not(X))), and(not(not(X)), not(Y))), or(implies(not(X), and(implies(not(not(X)), Y), not(X))), not(not(X)))), modus_tollens, implies(not(X), and(implies(not(not(X)), Y), not(X)))), modus_tollens, equiv(and(or(not(X), Y), not(X)), not(X))), modus_tollens, and(or(not(X), Y), not(X)), not(X)))
% 240.66/31.15 = { by lemma 118 }
% 240.66/31.15 not(fresh(fresh28(fresh28(modus_tollens, modus_tollens, implies(not(X), and(implies(not(not(X)), Y), not(X)))), modus_tollens, equiv(and(or(not(X), Y), not(X)), not(X))), modus_tollens, and(or(not(X), Y), not(X)), not(X)))
% 240.66/31.15 = { by axiom 17 (modus_ponens_2) }
% 240.66/31.15 not(fresh(fresh28(is_a_theorem(implies(not(X), and(implies(not(not(X)), Y), not(X)))), modus_tollens, equiv(and(or(not(X), Y), not(X)), not(X))), modus_tollens, and(or(not(X), Y), not(X)), not(X)))
% 240.66/31.15 = { by lemma 135 }
% 240.66/31.15 not(fresh(fresh28(is_a_theorem(implies(not(X), and(or(not(X), Y), not(X)))), modus_tollens, equiv(and(or(not(X), Y), not(X)), not(X))), modus_tollens, and(or(not(X), Y), not(X)), not(X)))
% 240.66/31.15 = { by lemma 137 }
% 240.66/31.15 not(fresh(modus_tollens, modus_tollens, and(or(not(X), Y), not(X)), not(X)))
% 240.66/31.15 = { by axiom 19 (substitution_of_equivalents_2) }
% 240.66/31.15 not(not(X))
% 240.66/31.15 = { by lemma 131 }
% 240.66/31.15 X
% 240.66/31.15
% 240.66/31.15 Lemma 139: fresh28(is_a_theorem(or(X, and(Y, Z))), modus_tollens, or(X, Z)) = modus_tollens.
% 240.66/31.15 Proof:
% 240.66/31.15 fresh28(is_a_theorem(or(X, and(Y, Z))), modus_tollens, or(X, Z))
% 240.66/31.16 = { by lemma 114 R->L }
% 240.66/31.16 fresh59(is_a_theorem(implies(or(X, and(Y, Z)), or(X, Z))), modus_tollens, or(X, and(Y, Z)), or(X, Z))
% 240.66/31.16 = { by axiom 17 (modus_ponens_2) R->L }
% 240.66/31.16 fresh59(fresh28(modus_tollens, modus_tollens, implies(or(X, and(Y, Z)), or(X, Z))), modus_tollens, or(X, and(Y, Z)), or(X, Z))
% 240.66/31.16 = { by lemma 136 R->L }
% 240.66/31.16 fresh59(fresh28(is_a_theorem(implies(and(Y, Z), Z)), modus_tollens, implies(or(X, and(Y, Z)), or(X, Z))), modus_tollens, or(X, and(Y, Z)), or(X, Z))
% 240.66/31.16 = { by lemma 134 }
% 240.66/31.16 fresh59(modus_tollens, modus_tollens, or(X, and(Y, Z)), or(X, Z))
% 240.66/31.16 = { by lemma 118 }
% 240.66/31.16 modus_tollens
% 240.66/31.16
% 240.66/31.16 Lemma 140: and(implies(X, Y), Y) = Y.
% 240.66/31.16 Proof:
% 240.66/31.16 and(implies(X, Y), Y)
% 240.66/31.16 = { by lemma 121 R->L }
% 240.66/31.16 fresh(is_a_theorem(equiv(and(implies(X, Y), Y), Y)), modus_tollens, and(implies(X, Y), Y), Y)
% 240.66/31.16 = { by axiom 17 (modus_ponens_2) R->L }
% 240.66/31.16 fresh(fresh28(modus_tollens, modus_tollens, equiv(and(implies(X, Y), Y), Y)), modus_tollens, and(implies(X, Y), Y), Y)
% 240.66/31.16 = { by lemma 128 R->L }
% 240.66/31.16 fresh(fresh28(fresh28(is_a_theorem(or(implies(Y, and(implies(X, Y), Y)), not(Y))), modus_tollens, implies(Y, and(implies(X, Y), Y))), modus_tollens, equiv(and(implies(X, Y), Y), Y)), modus_tollens, and(implies(X, Y), Y), Y)
% 240.66/31.16 = { by axiom 17 (modus_ponens_2) R->L }
% 240.66/31.16 fresh(fresh28(fresh28(fresh28(modus_tollens, modus_tollens, or(implies(Y, and(implies(X, Y), Y)), not(Y))), modus_tollens, implies(Y, and(implies(X, Y), Y))), modus_tollens, equiv(and(implies(X, Y), Y), Y)), modus_tollens, and(implies(X, Y), Y), Y)
% 240.66/31.16 = { by lemma 133 R->L }
% 240.66/31.16 fresh(fresh28(fresh28(fresh28(is_a_theorem(or(implies(Y, and(implies(X, Y), Y)), and(X, not(Y)))), modus_tollens, or(implies(Y, and(implies(X, Y), Y)), not(Y))), modus_tollens, implies(Y, and(implies(X, Y), Y))), modus_tollens, equiv(and(implies(X, Y), Y), Y)), modus_tollens, and(implies(X, Y), Y), Y)
% 240.66/31.16 = { by lemma 139 }
% 240.66/31.16 fresh(fresh28(fresh28(modus_tollens, modus_tollens, implies(Y, and(implies(X, Y), Y))), modus_tollens, equiv(and(implies(X, Y), Y), Y)), modus_tollens, and(implies(X, Y), Y), Y)
% 240.66/31.16 = { by axiom 17 (modus_ponens_2) }
% 240.66/31.16 fresh(fresh28(is_a_theorem(implies(Y, and(implies(X, Y), Y))), modus_tollens, equiv(and(implies(X, Y), Y), Y)), modus_tollens, and(implies(X, Y), Y), Y)
% 240.66/31.16 = { by lemma 137 }
% 240.66/31.16 fresh(modus_tollens, modus_tollens, and(implies(X, Y), Y), Y)
% 240.66/31.16 = { by axiom 19 (substitution_of_equivalents_2) }
% 240.66/31.16 Y
% 240.66/31.16
% 240.66/31.16 Lemma 141: or(Y, X) = or(X, Y).
% 240.66/31.16 Proof:
% 240.66/31.16 or(Y, X)
% 240.66/31.16 = { by lemma 122 R->L }
% 240.66/31.16 or(or(Y, X), or(X, Y))
% 240.66/31.16 = { by lemma 138 R->L }
% 240.66/31.16 implies(implies(or(or(Y, X), or(X, Y)), not(or(X, Y))), or(or(Y, X), or(X, Y)))
% 240.66/31.16 = { by lemma 125 }
% 240.66/31.16 implies(not(and(or(or(Y, X), or(X, Y)), or(X, Y))), or(or(Y, X), or(X, Y)))
% 240.66/31.16 = { by lemma 135 }
% 240.66/31.16 or(and(or(or(Y, X), or(X, Y)), or(X, Y)), or(or(Y, X), or(X, Y)))
% 240.66/31.16 = { by lemma 138 R->L }
% 240.66/31.16 or(and(or(implies(implies(or(Y, X), or(Y, X)), or(Y, X)), or(X, Y)), or(X, Y)), or(or(Y, X), or(X, Y)))
% 240.66/31.16 = { by lemma 113 }
% 240.66/31.16 or(and(or(implies(equiv(or(Y, X), or(Y, X)), or(Y, X)), or(X, Y)), or(X, Y)), or(or(Y, X), or(X, Y)))
% 240.66/31.16 = { by lemma 127 R->L }
% 240.66/31.16 or(and(or(fresh22(op_implies_and, modus_tollens, equiv(or(Y, X), or(Y, X)), or(Y, X)), or(X, Y)), or(X, Y)), or(or(Y, X), or(X, Y)))
% 240.66/31.16 = { by lemma 112 R->L }
% 240.66/31.16 or(and(or(fresh22(op_implies_and, implies_1, equiv(or(Y, X), or(Y, X)), or(Y, X)), or(X, Y)), or(X, Y)), or(or(Y, X), or(X, Y)))
% 240.66/31.16 = { by lemma 59 R->L }
% 240.66/31.16 or(and(or(fresh22(op_implies_and, or_2, equiv(or(Y, X), or(Y, X)), or(Y, X)), or(X, Y)), or(X, Y)), or(or(Y, X), or(X, Y)))
% 240.66/31.16 = { by lemma 55 R->L }
% 240.66/31.16 or(and(or(fresh22(op_implies_and, op_and, equiv(or(Y, X), or(Y, X)), or(Y, X)), or(X, Y)), or(X, Y)), or(or(Y, X), or(X, Y)))
% 240.66/31.16 = { by lemma 129 R->L }
% 240.66/31.16 or(and(or(not(and(equiv(or(Y, X), or(Y, X)), not(or(Y, X)))), or(X, Y)), or(X, Y)), or(or(Y, X), or(X, Y)))
% 240.66/31.16 = { by lemma 57 }
% 240.66/31.16 or(and(implies(and(equiv(or(Y, X), or(Y, X)), not(or(Y, X))), or(X, Y)), or(X, Y)), or(or(Y, X), or(X, Y)))
% 240.66/31.16 = { by lemma 140 }
% 240.66/31.16 or(or(X, Y), or(or(Y, X), or(X, Y)))
% 240.66/31.16 = { by lemma 122 R->L }
% 240.66/31.16 or(or(or(X, Y), or(Y, X)), or(or(Y, X), or(X, Y)))
% 240.66/31.16 = { by lemma 122 }
% 240.66/31.16 or(or(X, Y), or(Y, X))
% 240.66/31.16 = { by lemma 122 }
% 240.66/31.16 or(X, Y)
% 240.66/31.16
% 240.66/31.16 Lemma 142: is_a_theorem(equiv(X, X)) = modus_tollens.
% 240.66/31.16 Proof:
% 240.66/31.16 is_a_theorem(equiv(X, X))
% 240.66/31.16 = { by axiom 17 (modus_ponens_2) R->L }
% 240.66/31.16 fresh28(implies_1, implies_1, equiv(X, X))
% 240.66/31.16 = { by lemma 64 R->L }
% 240.66/31.16 fresh28(fresh28(is_a_theorem(or(X, not(X))), implies_1, or(not(X), X)), implies_1, equiv(X, X))
% 240.66/31.16 = { by axiom 17 (modus_ponens_2) R->L }
% 240.66/31.16 fresh28(fresh28(fresh28(implies_1, implies_1, or(X, not(X))), implies_1, or(not(X), X)), implies_1, equiv(X, X))
% 240.66/31.16 = { by lemma 88 R->L }
% 240.66/31.16 fresh28(fresh28(fresh28(is_a_theorem(or(X, or(not(X), not(X)))), implies_1, or(X, not(X))), implies_1, or(not(X), X)), implies_1, equiv(X, X))
% 240.66/31.16 = { by lemma 79 }
% 240.66/31.16 fresh28(fresh28(implies_1, implies_1, or(not(X), X)), implies_1, equiv(X, X))
% 240.66/31.16 = { by axiom 17 (modus_ponens_2) }
% 240.66/31.16 fresh28(is_a_theorem(or(not(X), X)), implies_1, equiv(X, X))
% 240.66/31.16 = { by lemma 57 }
% 240.66/31.16 fresh28(is_a_theorem(implies(X, X)), implies_1, equiv(X, X))
% 240.66/31.16 = { by lemma 59 R->L }
% 240.66/31.16 fresh28(is_a_theorem(implies(X, X)), or_2, equiv(X, X))
% 240.66/31.16 = { by lemma 55 R->L }
% 240.66/31.16 fresh28(is_a_theorem(implies(X, X)), op_and, equiv(X, X))
% 240.66/31.16 = { by lemma 60 R->L }
% 240.66/31.16 fresh59(is_a_theorem(implies(implies(X, X), equiv(X, X))), op_and, implies(X, X), equiv(X, X))
% 240.66/31.16 = { by lemma 95 R->L }
% 240.66/31.16 fresh59(is_a_theorem(implies(implies(X, X), and(implies(X, X), implies(X, X)))), op_and, implies(X, X), equiv(X, X))
% 240.66/31.16 = { by lemma 67 }
% 240.66/31.16 fresh59(fresh33(kn1, op_and, implies(X, X)), op_and, implies(X, X), equiv(X, X))
% 240.66/31.16 = { by lemma 73 }
% 240.66/31.16 fresh59(fresh33(implies_1, op_and, implies(X, X)), op_and, implies(X, X), equiv(X, X))
% 240.66/31.16 = { by lemma 55 }
% 240.66/31.16 fresh59(fresh33(implies_1, or_2, implies(X, X)), op_and, implies(X, X), equiv(X, X))
% 240.66/31.16 = { by lemma 59 }
% 240.66/31.16 fresh59(fresh33(implies_1, implies_1, implies(X, X)), op_and, implies(X, X), equiv(X, X))
% 240.66/31.16 = { by lemma 66 }
% 240.66/31.16 fresh59(op_and, op_and, implies(X, X), equiv(X, X))
% 240.66/31.16 = { by lemma 55 }
% 240.66/31.16 fresh59(or_2, op_and, implies(X, X), equiv(X, X))
% 240.66/31.16 = { by lemma 59 }
% 240.66/31.16 fresh59(implies_1, op_and, implies(X, X), equiv(X, X))
% 240.66/31.16 = { by lemma 55 }
% 240.66/31.16 fresh59(implies_1, or_2, implies(X, X), equiv(X, X))
% 240.66/31.16 = { by lemma 59 }
% 240.66/31.16 fresh59(implies_1, implies_1, implies(X, X), equiv(X, X))
% 240.66/31.16 = { by lemma 63 }
% 240.66/31.16 op_and
% 240.66/31.16 = { by lemma 55 }
% 240.66/31.16 or_2
% 240.66/31.16 = { by lemma 59 }
% 240.66/31.16 implies_1
% 240.66/31.16 = { by lemma 112 }
% 240.66/31.16 modus_tollens
% 240.66/31.16
% 240.66/31.16 Lemma 143: not(or(and(not(X), Y), and(not(Y), X))) = equiv(not(X), not(Y)).
% 240.66/31.16 Proof:
% 240.66/31.16 not(or(and(not(X), Y), and(not(Y), X)))
% 240.66/31.16 = { by lemma 124 R->L }
% 240.66/31.16 and(implies(not(X), not(Y)), implies(not(Y), not(X)))
% 240.66/31.16 = { by lemma 95 }
% 240.66/31.16 equiv(not(X), not(Y))
% 240.66/31.16
% 240.66/31.16 Lemma 144: equiv(not(X), not(X)) = or(X, not(X)).
% 240.66/31.16 Proof:
% 240.66/31.16 equiv(not(X), not(X))
% 240.66/31.16 = { by lemma 109 R->L }
% 240.66/31.16 equiv(not(or(X, X)), not(X))
% 240.66/31.16 = { by lemma 109 R->L }
% 240.66/31.16 equiv(not(or(X, X)), not(or(X, X)))
% 240.66/31.16 = { by lemma 143 R->L }
% 240.66/31.16 not(or(and(not(or(X, X)), or(X, X)), and(not(or(X, X)), or(X, X))))
% 240.66/31.16 = { by lemma 109 }
% 240.66/31.16 not(and(not(or(X, X)), or(X, X)))
% 240.66/31.16 = { by lemma 103 }
% 240.66/31.16 not(not(or(X, not(or(X, X)))))
% 240.66/31.16 = { by lemma 131 }
% 240.66/31.16 or(X, not(or(X, X)))
% 240.66/31.16 = { by lemma 109 }
% 240.66/31.16 or(X, not(X))
% 240.66/31.16
% 240.66/31.16 Lemma 145: implies(X, equiv(Y, Y)) = equiv(Y, Y).
% 240.66/31.16 Proof:
% 240.66/31.16 implies(X, equiv(Y, Y))
% 240.66/31.16 = { by lemma 121 R->L }
% 240.66/31.16 fresh(is_a_theorem(equiv(implies(X, equiv(Y, Y)), equiv(Y, Y))), modus_tollens, implies(X, equiv(Y, Y)), equiv(Y, Y))
% 240.66/31.16 = { by axiom 17 (modus_ponens_2) R->L }
% 240.66/31.16 fresh(fresh28(modus_tollens, modus_tollens, equiv(implies(X, equiv(Y, Y)), equiv(Y, Y))), modus_tollens, implies(X, equiv(Y, Y)), equiv(Y, Y))
% 240.66/31.16 = { by lemma 142 R->L }
% 240.66/31.16 fresh(fresh28(is_a_theorem(equiv(Y, Y)), modus_tollens, equiv(implies(X, equiv(Y, Y)), equiv(Y, Y))), modus_tollens, implies(X, equiv(Y, Y)), equiv(Y, Y))
% 240.66/31.16 = { by lemma 114 R->L }
% 240.66/31.16 fresh(fresh59(is_a_theorem(implies(equiv(Y, Y), equiv(implies(X, equiv(Y, Y)), equiv(Y, Y)))), modus_tollens, equiv(Y, Y), equiv(implies(X, equiv(Y, Y)), equiv(Y, Y))), modus_tollens, implies(X, equiv(Y, Y)), equiv(Y, Y))
% 240.66/31.16 = { by lemma 57 R->L }
% 240.66/31.16 fresh(fresh59(is_a_theorem(or(not(equiv(Y, Y)), equiv(implies(X, equiv(Y, Y)), equiv(Y, Y)))), modus_tollens, equiv(Y, Y), equiv(implies(X, equiv(Y, Y)), equiv(Y, Y))), modus_tollens, implies(X, equiv(Y, Y)), equiv(Y, Y))
% 240.66/31.16 = { by axiom 17 (modus_ponens_2) R->L }
% 240.66/31.16 fresh(fresh59(fresh28(modus_tollens, modus_tollens, or(not(equiv(Y, Y)), equiv(implies(X, equiv(Y, Y)), equiv(Y, Y)))), modus_tollens, equiv(Y, Y), equiv(implies(X, equiv(Y, Y)), equiv(Y, Y))), modus_tollens, implies(X, equiv(Y, Y)), equiv(Y, Y))
% 240.66/31.16 = { by lemma 139 R->L }
% 240.66/31.16 fresh(fresh59(fresh28(fresh28(is_a_theorem(or(equiv(implies(X, equiv(Y, Y)), equiv(Y, Y)), and(implies(X, equiv(Y, Y)), not(equiv(Y, Y))))), modus_tollens, or(equiv(implies(X, equiv(Y, Y)), equiv(Y, Y)), not(equiv(Y, Y)))), modus_tollens, or(not(equiv(Y, Y)), equiv(implies(X, equiv(Y, Y)), equiv(Y, Y)))), modus_tollens, equiv(Y, Y), equiv(implies(X, equiv(Y, Y)), equiv(Y, Y))), modus_tollens, implies(X, equiv(Y, Y)), equiv(Y, Y))
% 240.66/31.16 = { by lemma 130 R->L }
% 240.66/31.16 fresh(fresh59(fresh28(fresh28(is_a_theorem(or(equiv(implies(X, equiv(Y, Y)), equiv(Y, Y)), not(implies(implies(X, equiv(Y, Y)), equiv(Y, Y))))), modus_tollens, or(equiv(implies(X, equiv(Y, Y)), equiv(Y, Y)), not(equiv(Y, Y)))), modus_tollens, or(not(equiv(Y, Y)), equiv(implies(X, equiv(Y, Y)), equiv(Y, Y)))), modus_tollens, equiv(Y, Y), equiv(implies(X, equiv(Y, Y)), equiv(Y, Y))), modus_tollens, implies(X, equiv(Y, Y)), equiv(Y, Y))
% 240.66/31.16 = { by lemma 95 R->L }
% 240.66/31.16 fresh(fresh59(fresh28(fresh28(is_a_theorem(or(and(implies(implies(X, equiv(Y, Y)), equiv(Y, Y)), implies(equiv(Y, Y), implies(X, equiv(Y, Y)))), not(implies(implies(X, equiv(Y, Y)), equiv(Y, Y))))), modus_tollens, or(equiv(implies(X, equiv(Y, Y)), equiv(Y, Y)), not(equiv(Y, Y)))), modus_tollens, or(not(equiv(Y, Y)), equiv(implies(X, equiv(Y, Y)), equiv(Y, Y)))), modus_tollens, equiv(Y, Y), equiv(implies(X, equiv(Y, Y)), equiv(Y, Y))), modus_tollens, implies(X, equiv(Y, Y)), equiv(Y, Y))
% 240.66/31.16 = { by axiom 17 (modus_ponens_2) R->L }
% 240.66/31.16 fresh(fresh59(fresh28(fresh28(fresh28(implies_1, implies_1, or(and(implies(implies(X, equiv(Y, Y)), equiv(Y, Y)), implies(equiv(Y, Y), implies(X, equiv(Y, Y)))), not(implies(implies(X, equiv(Y, Y)), equiv(Y, Y))))), modus_tollens, or(equiv(implies(X, equiv(Y, Y)), equiv(Y, Y)), not(equiv(Y, Y)))), modus_tollens, or(not(equiv(Y, Y)), equiv(implies(X, equiv(Y, Y)), equiv(Y, Y)))), modus_tollens, equiv(Y, Y), equiv(implies(X, equiv(Y, Y)), equiv(Y, Y))), modus_tollens, implies(X, equiv(Y, Y)), equiv(Y, Y))
% 240.66/31.16 = { by lemma 98 R->L }
% 240.66/31.16 fresh(fresh59(fresh28(fresh28(fresh28(fresh28(is_a_theorem(implies(not(implies(equiv(Y, Y), implies(X, equiv(Y, Y)))), not(implies(implies(X, equiv(Y, Y)), equiv(Y, Y))))), implies_1, implies(implies(implies(implies(X, equiv(Y, Y)), equiv(Y, Y)), not(implies(equiv(Y, Y), implies(X, equiv(Y, Y))))), implies(implies(implies(X, equiv(Y, Y)), equiv(Y, Y)), not(implies(implies(X, equiv(Y, Y)), equiv(Y, Y)))))), implies_1, or(and(implies(implies(X, equiv(Y, Y)), equiv(Y, Y)), implies(equiv(Y, Y), implies(X, equiv(Y, Y)))), not(implies(implies(X, equiv(Y, Y)), equiv(Y, Y))))), modus_tollens, or(equiv(implies(X, equiv(Y, Y)), equiv(Y, Y)), not(equiv(Y, Y)))), modus_tollens, or(not(equiv(Y, Y)), equiv(implies(X, equiv(Y, Y)), equiv(Y, Y)))), modus_tollens, equiv(Y, Y), equiv(implies(X, equiv(Y, Y)), equiv(Y, Y))), modus_tollens, implies(X, equiv(Y, Y)), equiv(Y, Y))
% 240.66/31.17 = { by axiom 17 (modus_ponens_2) R->L }
% 240.66/31.17 fresh(fresh59(fresh28(fresh28(fresh28(fresh28(fresh28(implies_1, implies_1, implies(not(implies(equiv(Y, Y), implies(X, equiv(Y, Y)))), not(implies(implies(X, equiv(Y, Y)), equiv(Y, Y))))), implies_1, implies(implies(implies(implies(X, equiv(Y, Y)), equiv(Y, Y)), not(implies(equiv(Y, Y), implies(X, equiv(Y, Y))))), implies(implies(implies(X, equiv(Y, Y)), equiv(Y, Y)), not(implies(implies(X, equiv(Y, Y)), equiv(Y, Y)))))), implies_1, or(and(implies(implies(X, equiv(Y, Y)), equiv(Y, Y)), implies(equiv(Y, Y), implies(X, equiv(Y, Y)))), not(implies(implies(X, equiv(Y, Y)), equiv(Y, Y))))), modus_tollens, or(equiv(implies(X, equiv(Y, Y)), equiv(Y, Y)), not(equiv(Y, Y)))), modus_tollens, or(not(equiv(Y, Y)), equiv(implies(X, equiv(Y, Y)), equiv(Y, Y)))), modus_tollens, equiv(Y, Y), equiv(implies(X, equiv(Y, Y)), equiv(Y, Y))), modus_tollens, implies(X, equiv(Y, Y)), equiv(Y, Y))
% 240.66/31.17 = { by lemma 59 R->L }
% 240.66/31.17 fresh(fresh59(fresh28(fresh28(fresh28(fresh28(fresh28(or_2, implies_1, implies(not(implies(equiv(Y, Y), implies(X, equiv(Y, Y)))), not(implies(implies(X, equiv(Y, Y)), equiv(Y, Y))))), implies_1, implies(implies(implies(implies(X, equiv(Y, Y)), equiv(Y, Y)), not(implies(equiv(Y, Y), implies(X, equiv(Y, Y))))), implies(implies(implies(X, equiv(Y, Y)), equiv(Y, Y)), not(implies(implies(X, equiv(Y, Y)), equiv(Y, Y)))))), implies_1, or(and(implies(implies(X, equiv(Y, Y)), equiv(Y, Y)), implies(equiv(Y, Y), implies(X, equiv(Y, Y)))), not(implies(implies(X, equiv(Y, Y)), equiv(Y, Y))))), modus_tollens, or(equiv(implies(X, equiv(Y, Y)), equiv(Y, Y)), not(equiv(Y, Y)))), modus_tollens, or(not(equiv(Y, Y)), equiv(implies(X, equiv(Y, Y)), equiv(Y, Y)))), modus_tollens, equiv(Y, Y), equiv(implies(X, equiv(Y, Y)), equiv(Y, Y))), modus_tollens, implies(X, equiv(Y, Y)), equiv(Y, Y))
% 240.66/31.17 = { by lemma 58 R->L }
% 240.66/31.17 fresh(fresh59(fresh28(fresh28(fresh28(fresh28(fresh28(fresh39(implies_1, or_2, equiv(Y, Y), X), implies_1, implies(not(implies(equiv(Y, Y), implies(X, equiv(Y, Y)))), not(implies(implies(X, equiv(Y, Y)), equiv(Y, Y))))), implies_1, implies(implies(implies(implies(X, equiv(Y, Y)), equiv(Y, Y)), not(implies(equiv(Y, Y), implies(X, equiv(Y, Y))))), implies(implies(implies(X, equiv(Y, Y)), equiv(Y, Y)), not(implies(implies(X, equiv(Y, Y)), equiv(Y, Y)))))), implies_1, or(and(implies(implies(X, equiv(Y, Y)), equiv(Y, Y)), implies(equiv(Y, Y), implies(X, equiv(Y, Y)))), not(implies(implies(X, equiv(Y, Y)), equiv(Y, Y))))), modus_tollens, or(equiv(implies(X, equiv(Y, Y)), equiv(Y, Y)), not(equiv(Y, Y)))), modus_tollens, or(not(equiv(Y, Y)), equiv(implies(X, equiv(Y, Y)), equiv(Y, Y)))), modus_tollens, equiv(Y, Y), equiv(implies(X, equiv(Y, Y)), equiv(Y, Y))), modus_tollens, implies(X, equiv(Y, Y)), equiv(Y, Y))
% 240.66/31.17 = { by lemma 55 R->L }
% 240.66/31.17 fresh(fresh59(fresh28(fresh28(fresh28(fresh28(fresh28(fresh39(implies_1, op_and, equiv(Y, Y), X), implies_1, implies(not(implies(equiv(Y, Y), implies(X, equiv(Y, Y)))), not(implies(implies(X, equiv(Y, Y)), equiv(Y, Y))))), implies_1, implies(implies(implies(implies(X, equiv(Y, Y)), equiv(Y, Y)), not(implies(equiv(Y, Y), implies(X, equiv(Y, Y))))), implies(implies(implies(X, equiv(Y, Y)), equiv(Y, Y)), not(implies(implies(X, equiv(Y, Y)), equiv(Y, Y)))))), implies_1, or(and(implies(implies(X, equiv(Y, Y)), equiv(Y, Y)), implies(equiv(Y, Y), implies(X, equiv(Y, Y)))), not(implies(implies(X, equiv(Y, Y)), equiv(Y, Y))))), modus_tollens, or(equiv(implies(X, equiv(Y, Y)), equiv(Y, Y)), not(equiv(Y, Y)))), modus_tollens, or(not(equiv(Y, Y)), equiv(implies(X, equiv(Y, Y)), equiv(Y, Y)))), modus_tollens, equiv(Y, Y), equiv(implies(X, equiv(Y, Y)), equiv(Y, Y))), modus_tollens, implies(X, equiv(Y, Y)), equiv(Y, Y))
% 240.66/31.17 = { by axiom 1 (principia_op_and) }
% 240.66/31.17 fresh(fresh59(fresh28(fresh28(fresh28(fresh28(fresh28(fresh39(implies_1, true, equiv(Y, Y), X), implies_1, implies(not(implies(equiv(Y, Y), implies(X, equiv(Y, Y)))), not(implies(implies(X, equiv(Y, Y)), equiv(Y, Y))))), implies_1, implies(implies(implies(implies(X, equiv(Y, Y)), equiv(Y, Y)), not(implies(equiv(Y, Y), implies(X, equiv(Y, Y))))), implies(implies(implies(X, equiv(Y, Y)), equiv(Y, Y)), not(implies(implies(X, equiv(Y, Y)), equiv(Y, Y)))))), implies_1, or(and(implies(implies(X, equiv(Y, Y)), equiv(Y, Y)), implies(equiv(Y, Y), implies(X, equiv(Y, Y)))), not(implies(implies(X, equiv(Y, Y)), equiv(Y, Y))))), modus_tollens, or(equiv(implies(X, equiv(Y, Y)), equiv(Y, Y)), not(equiv(Y, Y)))), modus_tollens, or(not(equiv(Y, Y)), equiv(implies(X, equiv(Y, Y)), equiv(Y, Y)))), modus_tollens, equiv(Y, Y), equiv(implies(X, equiv(Y, Y)), equiv(Y, Y))), modus_tollens, implies(X, equiv(Y, Y)), equiv(Y, Y))
% 240.66/31.17 = { by axiom 33 (implies_1_1) }
% 240.66/31.17 fresh(fresh59(fresh28(fresh28(fresh28(fresh28(fresh28(is_a_theorem(implies(equiv(Y, Y), implies(X, equiv(Y, Y)))), implies_1, implies(not(implies(equiv(Y, Y), implies(X, equiv(Y, Y)))), not(implies(implies(X, equiv(Y, Y)), equiv(Y, Y))))), implies_1, implies(implies(implies(implies(X, equiv(Y, Y)), equiv(Y, Y)), not(implies(equiv(Y, Y), implies(X, equiv(Y, Y))))), implies(implies(implies(X, equiv(Y, Y)), equiv(Y, Y)), not(implies(implies(X, equiv(Y, Y)), equiv(Y, Y)))))), implies_1, or(and(implies(implies(X, equiv(Y, Y)), equiv(Y, Y)), implies(equiv(Y, Y), implies(X, equiv(Y, Y)))), not(implies(implies(X, equiv(Y, Y)), equiv(Y, Y))))), modus_tollens, or(equiv(implies(X, equiv(Y, Y)), equiv(Y, Y)), not(equiv(Y, Y)))), modus_tollens, or(not(equiv(Y, Y)), equiv(implies(X, equiv(Y, Y)), equiv(Y, Y)))), modus_tollens, equiv(Y, Y), equiv(implies(X, equiv(Y, Y)), equiv(Y, Y))), modus_tollens, implies(X, equiv(Y, Y)), equiv(Y, Y))
% 240.66/31.17 = { by lemma 97 }
% 240.66/31.17 fresh(fresh59(fresh28(fresh28(fresh28(fresh28(implies_1, implies_1, implies(implies(implies(implies(X, equiv(Y, Y)), equiv(Y, Y)), not(implies(equiv(Y, Y), implies(X, equiv(Y, Y))))), implies(implies(implies(X, equiv(Y, Y)), equiv(Y, Y)), not(implies(implies(X, equiv(Y, Y)), equiv(Y, Y)))))), implies_1, or(and(implies(implies(X, equiv(Y, Y)), equiv(Y, Y)), implies(equiv(Y, Y), implies(X, equiv(Y, Y)))), not(implies(implies(X, equiv(Y, Y)), equiv(Y, Y))))), modus_tollens, or(equiv(implies(X, equiv(Y, Y)), equiv(Y, Y)), not(equiv(Y, Y)))), modus_tollens, or(not(equiv(Y, Y)), equiv(implies(X, equiv(Y, Y)), equiv(Y, Y)))), modus_tollens, equiv(Y, Y), equiv(implies(X, equiv(Y, Y)), equiv(Y, Y))), modus_tollens, implies(X, equiv(Y, Y)), equiv(Y, Y))
% 240.66/31.17 = { by axiom 17 (modus_ponens_2) }
% 240.66/31.17 fresh(fresh59(fresh28(fresh28(fresh28(is_a_theorem(implies(implies(implies(implies(X, equiv(Y, Y)), equiv(Y, Y)), not(implies(equiv(Y, Y), implies(X, equiv(Y, Y))))), implies(implies(implies(X, equiv(Y, Y)), equiv(Y, Y)), not(implies(implies(X, equiv(Y, Y)), equiv(Y, Y)))))), implies_1, or(and(implies(implies(X, equiv(Y, Y)), equiv(Y, Y)), implies(equiv(Y, Y), implies(X, equiv(Y, Y)))), not(implies(implies(X, equiv(Y, Y)), equiv(Y, Y))))), modus_tollens, or(equiv(implies(X, equiv(Y, Y)), equiv(Y, Y)), not(equiv(Y, Y)))), modus_tollens, or(not(equiv(Y, Y)), equiv(implies(X, equiv(Y, Y)), equiv(Y, Y)))), modus_tollens, equiv(Y, Y), equiv(implies(X, equiv(Y, Y)), equiv(Y, Y))), modus_tollens, implies(X, equiv(Y, Y)), equiv(Y, Y))
% 240.66/31.17 = { by lemma 72 }
% 240.66/31.17 fresh(fresh59(fresh28(fresh28(fresh28(is_a_theorem(or(and(implies(implies(X, equiv(Y, Y)), equiv(Y, Y)), implies(equiv(Y, Y), implies(X, equiv(Y, Y)))), implies(implies(implies(X, equiv(Y, Y)), equiv(Y, Y)), not(implies(implies(X, equiv(Y, Y)), equiv(Y, Y)))))), implies_1, or(and(implies(implies(X, equiv(Y, Y)), equiv(Y, Y)), implies(equiv(Y, Y), implies(X, equiv(Y, Y)))), not(implies(implies(X, equiv(Y, Y)), equiv(Y, Y))))), modus_tollens, or(equiv(implies(X, equiv(Y, Y)), equiv(Y, Y)), not(equiv(Y, Y)))), modus_tollens, or(not(equiv(Y, Y)), equiv(implies(X, equiv(Y, Y)), equiv(Y, Y)))), modus_tollens, equiv(Y, Y), equiv(implies(X, equiv(Y, Y)), equiv(Y, Y))), modus_tollens, implies(X, equiv(Y, Y)), equiv(Y, Y))
% 240.66/31.17 = { by lemma 80 }
% 240.66/31.17 fresh(fresh59(fresh28(fresh28(implies_1, modus_tollens, or(equiv(implies(X, equiv(Y, Y)), equiv(Y, Y)), not(equiv(Y, Y)))), modus_tollens, or(not(equiv(Y, Y)), equiv(implies(X, equiv(Y, Y)), equiv(Y, Y)))), modus_tollens, equiv(Y, Y), equiv(implies(X, equiv(Y, Y)), equiv(Y, Y))), modus_tollens, implies(X, equiv(Y, Y)), equiv(Y, Y))
% 240.66/31.17 = { by lemma 112 }
% 240.66/31.17 fresh(fresh59(fresh28(fresh28(modus_tollens, modus_tollens, or(equiv(implies(X, equiv(Y, Y)), equiv(Y, Y)), not(equiv(Y, Y)))), modus_tollens, or(not(equiv(Y, Y)), equiv(implies(X, equiv(Y, Y)), equiv(Y, Y)))), modus_tollens, equiv(Y, Y), equiv(implies(X, equiv(Y, Y)), equiv(Y, Y))), modus_tollens, implies(X, equiv(Y, Y)), equiv(Y, Y))
% 240.66/31.17 = { by axiom 17 (modus_ponens_2) }
% 240.66/31.17 fresh(fresh59(fresh28(is_a_theorem(or(equiv(implies(X, equiv(Y, Y)), equiv(Y, Y)), not(equiv(Y, Y)))), modus_tollens, or(not(equiv(Y, Y)), equiv(implies(X, equiv(Y, Y)), equiv(Y, Y)))), modus_tollens, equiv(Y, Y), equiv(implies(X, equiv(Y, Y)), equiv(Y, Y))), modus_tollens, implies(X, equiv(Y, Y)), equiv(Y, Y))
% 240.66/31.17 = { by lemma 132 }
% 240.66/31.17 fresh(fresh59(modus_tollens, modus_tollens, equiv(Y, Y), equiv(implies(X, equiv(Y, Y)), equiv(Y, Y))), modus_tollens, implies(X, equiv(Y, Y)), equiv(Y, Y))
% 240.66/31.17 = { by lemma 118 }
% 240.66/31.17 fresh(modus_tollens, modus_tollens, implies(X, equiv(Y, Y)), equiv(Y, Y))
% 240.66/31.17 = { by axiom 19 (substitution_of_equivalents_2) }
% 240.66/31.17 equiv(Y, Y)
% 240.66/31.17
% 240.66/31.17 Lemma 146: implies(equiv(X, not(X)), Y) = or(or(X, not(X)), Y).
% 240.66/31.17 Proof:
% 240.66/31.17 implies(equiv(X, not(X)), Y)
% 240.66/31.17 = { by lemma 131 R->L }
% 240.66/31.17 implies(equiv(not(not(X)), not(X)), Y)
% 240.66/31.17 = { by lemma 143 R->L }
% 240.66/31.17 implies(not(or(and(not(not(X)), X), and(not(X), not(X)))), Y)
% 240.66/31.17 = { by lemma 99 }
% 240.66/31.17 implies(not(or(and(not(not(X)), X), not(X))), Y)
% 240.66/31.17 = { by lemma 102 }
% 240.66/31.17 implies(and(implies(not(not(X)), not(X)), X), Y)
% 240.66/31.17 = { by lemma 125 }
% 240.66/31.17 implies(and(not(and(not(not(X)), X)), X), Y)
% 240.66/31.17 = { by lemma 131 }
% 240.66/31.17 implies(and(not(and(X, X)), X), Y)
% 240.66/31.17 = { by lemma 99 }
% 240.66/31.17 implies(and(not(X), X), Y)
% 240.66/31.17 = { by lemma 109 R->L }
% 240.66/31.17 implies(or(and(not(X), X), and(not(X), X)), Y)
% 240.66/31.17 = { by lemma 57 R->L }
% 240.66/31.17 or(not(or(and(not(X), X), and(not(X), X))), Y)
% 240.66/31.17 = { by lemma 143 }
% 240.66/31.17 or(equiv(not(X), not(X)), Y)
% 240.66/31.17 = { by lemma 144 }
% 240.66/31.17 or(or(X, not(X)), Y)
% 240.66/31.17
% 240.66/31.17 Lemma 147: or(or(X, not(X)), Y) = or(X, not(X)).
% 240.66/31.17 Proof:
% 240.66/31.17 or(or(X, not(X)), Y)
% 240.66/31.17 = { by lemma 144 R->L }
% 240.66/31.17 or(equiv(not(X), not(X)), Y)
% 240.66/31.17 = { by lemma 121 R->L }
% 240.66/31.17 fresh(is_a_theorem(equiv(or(equiv(not(X), not(X)), Y), equiv(not(X), not(X)))), modus_tollens, or(equiv(not(X), not(X)), Y), equiv(not(X), not(X)))
% 240.66/31.17 = { by axiom 17 (modus_ponens_2) R->L }
% 240.66/31.17 fresh(fresh28(modus_tollens, modus_tollens, equiv(or(equiv(not(X), not(X)), Y), equiv(not(X), not(X)))), modus_tollens, or(equiv(not(X), not(X)), Y), equiv(not(X), not(X)))
% 240.66/31.17 = { by lemma 142 R->L }
% 240.66/31.17 fresh(fresh28(is_a_theorem(equiv(not(X), not(X))), modus_tollens, equiv(or(equiv(not(X), not(X)), Y), equiv(not(X), not(X)))), modus_tollens, or(equiv(not(X), not(X)), Y), equiv(not(X), not(X)))
% 240.66/31.17 = { by lemma 114 R->L }
% 240.66/31.17 fresh(fresh59(is_a_theorem(implies(equiv(not(X), not(X)), equiv(or(equiv(not(X), not(X)), Y), equiv(not(X), not(X))))), modus_tollens, equiv(not(X), not(X)), equiv(or(equiv(not(X), not(X)), Y), equiv(not(X), not(X)))), modus_tollens, or(equiv(not(X), not(X)), Y), equiv(not(X), not(X)))
% 240.66/31.17 = { by lemma 57 R->L }
% 240.66/31.17 fresh(fresh59(is_a_theorem(or(not(equiv(not(X), not(X))), equiv(or(equiv(not(X), not(X)), Y), equiv(not(X), not(X))))), modus_tollens, equiv(not(X), not(X)), equiv(or(equiv(not(X), not(X)), Y), equiv(not(X), not(X)))), modus_tollens, or(equiv(not(X), not(X)), Y), equiv(not(X), not(X)))
% 240.66/31.17 = { by axiom 17 (modus_ponens_2) R->L }
% 240.66/31.17 fresh(fresh59(fresh28(modus_tollens, modus_tollens, or(not(equiv(not(X), not(X))), equiv(or(equiv(not(X), not(X)), Y), equiv(not(X), not(X))))), modus_tollens, equiv(not(X), not(X)), equiv(or(equiv(not(X), not(X)), Y), equiv(not(X), not(X)))), modus_tollens, or(equiv(not(X), not(X)), Y), equiv(not(X), not(X)))
% 240.66/31.17 = { by lemma 139 R->L }
% 240.66/31.18 fresh(fresh59(fresh28(fresh28(is_a_theorem(or(equiv(or(equiv(not(X), not(X)), Y), equiv(not(X), not(X))), and(or(equiv(not(X), not(X)), Y), not(equiv(not(X), not(X)))))), modus_tollens, or(equiv(or(equiv(not(X), not(X)), Y), equiv(not(X), not(X))), not(equiv(not(X), not(X))))), modus_tollens, or(not(equiv(not(X), not(X))), equiv(or(equiv(not(X), not(X)), Y), equiv(not(X), not(X))))), modus_tollens, equiv(not(X), not(X)), equiv(or(equiv(not(X), not(X)), Y), equiv(not(X), not(X)))), modus_tollens, or(equiv(not(X), not(X)), Y), equiv(not(X), not(X)))
% 240.66/31.18 = { by lemma 130 R->L }
% 240.66/31.18 fresh(fresh59(fresh28(fresh28(is_a_theorem(or(equiv(or(equiv(not(X), not(X)), Y), equiv(not(X), not(X))), not(implies(or(equiv(not(X), not(X)), Y), equiv(not(X), not(X)))))), modus_tollens, or(equiv(or(equiv(not(X), not(X)), Y), equiv(not(X), not(X))), not(equiv(not(X), not(X))))), modus_tollens, or(not(equiv(not(X), not(X))), equiv(or(equiv(not(X), not(X)), Y), equiv(not(X), not(X))))), modus_tollens, equiv(not(X), not(X)), equiv(or(equiv(not(X), not(X)), Y), equiv(not(X), not(X)))), modus_tollens, or(equiv(not(X), not(X)), Y), equiv(not(X), not(X)))
% 240.66/31.18 = { by lemma 95 R->L }
% 240.66/31.18 fresh(fresh59(fresh28(fresh28(is_a_theorem(or(and(implies(or(equiv(not(X), not(X)), Y), equiv(not(X), not(X))), implies(equiv(not(X), not(X)), or(equiv(not(X), not(X)), Y))), not(implies(or(equiv(not(X), not(X)), Y), equiv(not(X), not(X)))))), modus_tollens, or(equiv(or(equiv(not(X), not(X)), Y), equiv(not(X), not(X))), not(equiv(not(X), not(X))))), modus_tollens, or(not(equiv(not(X), not(X))), equiv(or(equiv(not(X), not(X)), Y), equiv(not(X), not(X))))), modus_tollens, equiv(not(X), not(X)), equiv(or(equiv(not(X), not(X)), Y), equiv(not(X), not(X)))), modus_tollens, or(equiv(not(X), not(X)), Y), equiv(not(X), not(X)))
% 240.66/31.18 = { by axiom 17 (modus_ponens_2) R->L }
% 240.66/31.18 fresh(fresh59(fresh28(fresh28(fresh28(implies_1, implies_1, or(and(implies(or(equiv(not(X), not(X)), Y), equiv(not(X), not(X))), implies(equiv(not(X), not(X)), or(equiv(not(X), not(X)), Y))), not(implies(or(equiv(not(X), not(X)), Y), equiv(not(X), not(X)))))), modus_tollens, or(equiv(or(equiv(not(X), not(X)), Y), equiv(not(X), not(X))), not(equiv(not(X), not(X))))), modus_tollens, or(not(equiv(not(X), not(X))), equiv(or(equiv(not(X), not(X)), Y), equiv(not(X), not(X))))), modus_tollens, equiv(not(X), not(X)), equiv(or(equiv(not(X), not(X)), Y), equiv(not(X), not(X)))), modus_tollens, or(equiv(not(X), not(X)), Y), equiv(not(X), not(X)))
% 240.66/31.18 = { by lemma 98 R->L }
% 240.66/31.18 fresh(fresh59(fresh28(fresh28(fresh28(fresh28(is_a_theorem(implies(not(implies(equiv(not(X), not(X)), or(equiv(not(X), not(X)), Y))), not(implies(or(equiv(not(X), not(X)), Y), equiv(not(X), not(X)))))), implies_1, implies(implies(implies(or(equiv(not(X), not(X)), Y), equiv(not(X), not(X))), not(implies(equiv(not(X), not(X)), or(equiv(not(X), not(X)), Y)))), implies(implies(or(equiv(not(X), not(X)), Y), equiv(not(X), not(X))), not(implies(or(equiv(not(X), not(X)), Y), equiv(not(X), not(X))))))), implies_1, or(and(implies(or(equiv(not(X), not(X)), Y), equiv(not(X), not(X))), implies(equiv(not(X), not(X)), or(equiv(not(X), not(X)), Y))), not(implies(or(equiv(not(X), not(X)), Y), equiv(not(X), not(X)))))), modus_tollens, or(equiv(or(equiv(not(X), not(X)), Y), equiv(not(X), not(X))), not(equiv(not(X), not(X))))), modus_tollens, or(not(equiv(not(X), not(X))), equiv(or(equiv(not(X), not(X)), Y), equiv(not(X), not(X))))), modus_tollens, equiv(not(X), not(X)), equiv(or(equiv(not(X), not(X)), Y), equiv(not(X), not(X)))), modus_tollens, or(equiv(not(X), not(X)), Y), equiv(not(X), not(X)))
% 240.66/31.18 = { by axiom 17 (modus_ponens_2) R->L }
% 240.66/31.18 fresh(fresh59(fresh28(fresh28(fresh28(fresh28(fresh28(implies_1, implies_1, implies(not(implies(equiv(not(X), not(X)), or(equiv(not(X), not(X)), Y))), not(implies(or(equiv(not(X), not(X)), Y), equiv(not(X), not(X)))))), implies_1, implies(implies(implies(or(equiv(not(X), not(X)), Y), equiv(not(X), not(X))), not(implies(equiv(not(X), not(X)), or(equiv(not(X), not(X)), Y)))), implies(implies(or(equiv(not(X), not(X)), Y), equiv(not(X), not(X))), not(implies(or(equiv(not(X), not(X)), Y), equiv(not(X), not(X))))))), implies_1, or(and(implies(or(equiv(not(X), not(X)), Y), equiv(not(X), not(X))), implies(equiv(not(X), not(X)), or(equiv(not(X), not(X)), Y))), not(implies(or(equiv(not(X), not(X)), Y), equiv(not(X), not(X)))))), modus_tollens, or(equiv(or(equiv(not(X), not(X)), Y), equiv(not(X), not(X))), not(equiv(not(X), not(X))))), modus_tollens, or(not(equiv(not(X), not(X))), equiv(or(equiv(not(X), not(X)), Y), equiv(not(X), not(X))))), modus_tollens, equiv(not(X), not(X)), equiv(or(equiv(not(X), not(X)), Y), equiv(not(X), not(X)))), modus_tollens, or(equiv(not(X), not(X)), Y), equiv(not(X), not(X)))
% 240.66/31.18 = { by lemma 107 R->L }
% 240.66/31.18 fresh(fresh59(fresh28(fresh28(fresh28(fresh28(fresh28(fresh18(or_1, implies_1, equiv(not(X), not(X)), Y), implies_1, implies(not(implies(equiv(not(X), not(X)), or(equiv(not(X), not(X)), Y))), not(implies(or(equiv(not(X), not(X)), Y), equiv(not(X), not(X)))))), implies_1, implies(implies(implies(or(equiv(not(X), not(X)), Y), equiv(not(X), not(X))), not(implies(equiv(not(X), not(X)), or(equiv(not(X), not(X)), Y)))), implies(implies(or(equiv(not(X), not(X)), Y), equiv(not(X), not(X))), not(implies(or(equiv(not(X), not(X)), Y), equiv(not(X), not(X))))))), implies_1, or(and(implies(or(equiv(not(X), not(X)), Y), equiv(not(X), not(X))), implies(equiv(not(X), not(X)), or(equiv(not(X), not(X)), Y))), not(implies(or(equiv(not(X), not(X)), Y), equiv(not(X), not(X)))))), modus_tollens, or(equiv(or(equiv(not(X), not(X)), Y), equiv(not(X), not(X))), not(equiv(not(X), not(X))))), modus_tollens, or(not(equiv(not(X), not(X))), equiv(or(equiv(not(X), not(X)), Y), equiv(not(X), not(X))))), modus_tollens, equiv(not(X), not(X)), equiv(or(equiv(not(X), not(X)), Y), equiv(not(X), not(X)))), modus_tollens, or(equiv(not(X), not(X)), Y), equiv(not(X), not(X)))
% 240.66/31.18 = { by lemma 104 R->L }
% 240.66/31.18 fresh(fresh59(fresh28(fresh28(fresh28(fresh28(fresh28(is_a_theorem(implies(equiv(not(X), not(X)), or(equiv(not(X), not(X)), Y))), implies_1, implies(not(implies(equiv(not(X), not(X)), or(equiv(not(X), not(X)), Y))), not(implies(or(equiv(not(X), not(X)), Y), equiv(not(X), not(X)))))), implies_1, implies(implies(implies(or(equiv(not(X), not(X)), Y), equiv(not(X), not(X))), not(implies(equiv(not(X), not(X)), or(equiv(not(X), not(X)), Y)))), implies(implies(or(equiv(not(X), not(X)), Y), equiv(not(X), not(X))), not(implies(or(equiv(not(X), not(X)), Y), equiv(not(X), not(X))))))), implies_1, or(and(implies(or(equiv(not(X), not(X)), Y), equiv(not(X), not(X))), implies(equiv(not(X), not(X)), or(equiv(not(X), not(X)), Y))), not(implies(or(equiv(not(X), not(X)), Y), equiv(not(X), not(X)))))), modus_tollens, or(equiv(or(equiv(not(X), not(X)), Y), equiv(not(X), not(X))), not(equiv(not(X), not(X))))), modus_tollens, or(not(equiv(not(X), not(X))), equiv(or(equiv(not(X), not(X)), Y), equiv(not(X), not(X))))), modus_tollens, equiv(not(X), not(X)), equiv(or(equiv(not(X), not(X)), Y), equiv(not(X), not(X)))), modus_tollens, or(equiv(not(X), not(X)), Y), equiv(not(X), not(X)))
% 240.66/31.18 = { by lemma 97 }
% 240.66/31.18 fresh(fresh59(fresh28(fresh28(fresh28(fresh28(implies_1, implies_1, implies(implies(implies(or(equiv(not(X), not(X)), Y), equiv(not(X), not(X))), not(implies(equiv(not(X), not(X)), or(equiv(not(X), not(X)), Y)))), implies(implies(or(equiv(not(X), not(X)), Y), equiv(not(X), not(X))), not(implies(or(equiv(not(X), not(X)), Y), equiv(not(X), not(X))))))), implies_1, or(and(implies(or(equiv(not(X), not(X)), Y), equiv(not(X), not(X))), implies(equiv(not(X), not(X)), or(equiv(not(X), not(X)), Y))), not(implies(or(equiv(not(X), not(X)), Y), equiv(not(X), not(X)))))), modus_tollens, or(equiv(or(equiv(not(X), not(X)), Y), equiv(not(X), not(X))), not(equiv(not(X), not(X))))), modus_tollens, or(not(equiv(not(X), not(X))), equiv(or(equiv(not(X), not(X)), Y), equiv(not(X), not(X))))), modus_tollens, equiv(not(X), not(X)), equiv(or(equiv(not(X), not(X)), Y), equiv(not(X), not(X)))), modus_tollens, or(equiv(not(X), not(X)), Y), equiv(not(X), not(X)))
% 240.66/31.18 = { by axiom 17 (modus_ponens_2) }
% 240.66/31.18 fresh(fresh59(fresh28(fresh28(fresh28(is_a_theorem(implies(implies(implies(or(equiv(not(X), not(X)), Y), equiv(not(X), not(X))), not(implies(equiv(not(X), not(X)), or(equiv(not(X), not(X)), Y)))), implies(implies(or(equiv(not(X), not(X)), Y), equiv(not(X), not(X))), not(implies(or(equiv(not(X), not(X)), Y), equiv(not(X), not(X))))))), implies_1, or(and(implies(or(equiv(not(X), not(X)), Y), equiv(not(X), not(X))), implies(equiv(not(X), not(X)), or(equiv(not(X), not(X)), Y))), not(implies(or(equiv(not(X), not(X)), Y), equiv(not(X), not(X)))))), modus_tollens, or(equiv(or(equiv(not(X), not(X)), Y), equiv(not(X), not(X))), not(equiv(not(X), not(X))))), modus_tollens, or(not(equiv(not(X), not(X))), equiv(or(equiv(not(X), not(X)), Y), equiv(not(X), not(X))))), modus_tollens, equiv(not(X), not(X)), equiv(or(equiv(not(X), not(X)), Y), equiv(not(X), not(X)))), modus_tollens, or(equiv(not(X), not(X)), Y), equiv(not(X), not(X)))
% 240.66/31.18 = { by lemma 72 }
% 240.66/31.18 fresh(fresh59(fresh28(fresh28(fresh28(is_a_theorem(or(and(implies(or(equiv(not(X), not(X)), Y), equiv(not(X), not(X))), implies(equiv(not(X), not(X)), or(equiv(not(X), not(X)), Y))), implies(implies(or(equiv(not(X), not(X)), Y), equiv(not(X), not(X))), not(implies(or(equiv(not(X), not(X)), Y), equiv(not(X), not(X))))))), implies_1, or(and(implies(or(equiv(not(X), not(X)), Y), equiv(not(X), not(X))), implies(equiv(not(X), not(X)), or(equiv(not(X), not(X)), Y))), not(implies(or(equiv(not(X), not(X)), Y), equiv(not(X), not(X)))))), modus_tollens, or(equiv(or(equiv(not(X), not(X)), Y), equiv(not(X), not(X))), not(equiv(not(X), not(X))))), modus_tollens, or(not(equiv(not(X), not(X))), equiv(or(equiv(not(X), not(X)), Y), equiv(not(X), not(X))))), modus_tollens, equiv(not(X), not(X)), equiv(or(equiv(not(X), not(X)), Y), equiv(not(X), not(X)))), modus_tollens, or(equiv(not(X), not(X)), Y), equiv(not(X), not(X)))
% 240.66/31.18 = { by lemma 80 }
% 240.66/31.18 fresh(fresh59(fresh28(fresh28(implies_1, modus_tollens, or(equiv(or(equiv(not(X), not(X)), Y), equiv(not(X), not(X))), not(equiv(not(X), not(X))))), modus_tollens, or(not(equiv(not(X), not(X))), equiv(or(equiv(not(X), not(X)), Y), equiv(not(X), not(X))))), modus_tollens, equiv(not(X), not(X)), equiv(or(equiv(not(X), not(X)), Y), equiv(not(X), not(X)))), modus_tollens, or(equiv(not(X), not(X)), Y), equiv(not(X), not(X)))
% 240.66/31.18 = { by lemma 112 }
% 240.66/31.18 fresh(fresh59(fresh28(fresh28(modus_tollens, modus_tollens, or(equiv(or(equiv(not(X), not(X)), Y), equiv(not(X), not(X))), not(equiv(not(X), not(X))))), modus_tollens, or(not(equiv(not(X), not(X))), equiv(or(equiv(not(X), not(X)), Y), equiv(not(X), not(X))))), modus_tollens, equiv(not(X), not(X)), equiv(or(equiv(not(X), not(X)), Y), equiv(not(X), not(X)))), modus_tollens, or(equiv(not(X), not(X)), Y), equiv(not(X), not(X)))
% 240.66/31.18 = { by axiom 17 (modus_ponens_2) }
% 240.66/31.18 fresh(fresh59(fresh28(is_a_theorem(or(equiv(or(equiv(not(X), not(X)), Y), equiv(not(X), not(X))), not(equiv(not(X), not(X))))), modus_tollens, or(not(equiv(not(X), not(X))), equiv(or(equiv(not(X), not(X)), Y), equiv(not(X), not(X))))), modus_tollens, equiv(not(X), not(X)), equiv(or(equiv(not(X), not(X)), Y), equiv(not(X), not(X)))), modus_tollens, or(equiv(not(X), not(X)), Y), equiv(not(X), not(X)))
% 240.66/31.18 = { by lemma 132 }
% 240.66/31.18 fresh(fresh59(modus_tollens, modus_tollens, equiv(not(X), not(X)), equiv(or(equiv(not(X), not(X)), Y), equiv(not(X), not(X)))), modus_tollens, or(equiv(not(X), not(X)), Y), equiv(not(X), not(X)))
% 240.66/31.18 = { by lemma 118 }
% 240.66/31.18 fresh(modus_tollens, modus_tollens, or(equiv(not(X), not(X)), Y), equiv(not(X), not(X)))
% 240.66/31.18 = { by axiom 19 (substitution_of_equivalents_2) }
% 240.66/31.18 equiv(not(X), not(X))
% 240.66/31.18 = { by lemma 144 }
% 240.66/31.18 or(X, not(X))
% 240.66/31.18
% 240.66/31.18 Lemma 148: fresh28(is_a_theorem(X), implies_1, implies(Y, X)) = implies_1.
% 240.66/31.18 Proof:
% 240.66/31.18 fresh28(is_a_theorem(X), implies_1, implies(Y, X))
% 240.66/31.18 = { by lemma 59 R->L }
% 240.66/31.18 fresh28(is_a_theorem(X), or_2, implies(Y, X))
% 240.66/31.18 = { by lemma 57 R->L }
% 240.66/31.18 fresh28(is_a_theorem(X), or_2, or(not(Y), X))
% 240.66/31.18 = { by lemma 55 R->L }
% 240.66/31.18 fresh28(is_a_theorem(X), op_and, or(not(Y), X))
% 240.66/31.18 = { by lemma 60 R->L }
% 240.66/31.18 fresh59(is_a_theorem(implies(X, or(not(Y), X))), op_and, X, or(not(Y), X))
% 240.66/31.18 = { by lemma 53 }
% 240.66/31.18 fresh59(fresh16(or_2, op_and, not(Y), X), op_and, X, or(not(Y), X))
% 240.66/31.18 = { by lemma 54 }
% 240.66/31.18 fresh59(op_and, op_and, X, or(not(Y), X))
% 240.66/31.18 = { by lemma 63 }
% 240.66/31.18 op_and
% 240.66/31.18 = { by lemma 55 }
% 240.66/31.18 or_2
% 240.66/31.18 = { by lemma 59 }
% 240.66/31.18 implies_1
% 240.66/31.18
% 240.66/31.18 Lemma 149: fresh28(is_a_theorem(implies(implies(X, Y), Y)), modus_tollens, equiv(Y, implies(X, Y))) = modus_tollens.
% 240.66/31.18 Proof:
% 240.66/31.18 fresh28(is_a_theorem(implies(implies(X, Y), Y)), modus_tollens, equiv(Y, implies(X, Y)))
% 240.66/31.18 = { by lemma 114 R->L }
% 240.66/31.18 fresh59(is_a_theorem(implies(implies(implies(X, Y), Y), equiv(Y, implies(X, Y)))), modus_tollens, implies(implies(X, Y), Y), equiv(Y, implies(X, Y)))
% 240.66/31.18 = { by axiom 17 (modus_ponens_2) R->L }
% 240.66/31.18 fresh59(fresh28(modus_tollens, modus_tollens, implies(implies(implies(X, Y), Y), equiv(Y, implies(X, Y)))), modus_tollens, implies(implies(X, Y), Y), equiv(Y, implies(X, Y)))
% 240.66/31.18 = { by lemma 112 R->L }
% 240.66/31.18 fresh59(fresh28(implies_1, modus_tollens, implies(implies(implies(X, Y), Y), equiv(Y, implies(X, Y)))), modus_tollens, implies(implies(X, Y), Y), equiv(Y, implies(X, Y)))
% 240.66/31.18 = { by lemma 59 R->L }
% 240.66/31.18 fresh59(fresh28(or_2, modus_tollens, implies(implies(implies(X, Y), Y), equiv(Y, implies(X, Y)))), modus_tollens, implies(implies(X, Y), Y), equiv(Y, implies(X, Y)))
% 240.66/31.19 = { by lemma 58 R->L }
% 240.66/31.19 fresh59(fresh28(fresh39(implies_1, or_2, Y, X), modus_tollens, implies(implies(implies(X, Y), Y), equiv(Y, implies(X, Y)))), modus_tollens, implies(implies(X, Y), Y), equiv(Y, implies(X, Y)))
% 240.66/31.19 = { by lemma 55 R->L }
% 240.66/31.19 fresh59(fresh28(fresh39(implies_1, op_and, Y, X), modus_tollens, implies(implies(implies(X, Y), Y), equiv(Y, implies(X, Y)))), modus_tollens, implies(implies(X, Y), Y), equiv(Y, implies(X, Y)))
% 240.66/31.19 = { by axiom 1 (principia_op_and) }
% 240.66/31.19 fresh59(fresh28(fresh39(implies_1, true, Y, X), modus_tollens, implies(implies(implies(X, Y), Y), equiv(Y, implies(X, Y)))), modus_tollens, implies(implies(X, Y), Y), equiv(Y, implies(X, Y)))
% 240.66/31.19 = { by axiom 33 (implies_1_1) }
% 240.66/31.19 fresh59(fresh28(is_a_theorem(implies(Y, implies(X, Y))), modus_tollens, implies(implies(implies(X, Y), Y), equiv(Y, implies(X, Y)))), modus_tollens, implies(implies(X, Y), Y), equiv(Y, implies(X, Y)))
% 240.66/31.19 = { by lemma 119 }
% 240.66/31.19 fresh59(modus_tollens, modus_tollens, implies(implies(X, Y), Y), equiv(Y, implies(X, Y)))
% 240.66/31.19 = { by lemma 118 }
% 240.66/31.19 modus_tollens
% 240.66/31.19
% 240.66/31.19 Lemma 150: implies(X, implies(Y, or(Z, Y))) = implies(Y, or(Z, Y)).
% 240.66/31.19 Proof:
% 240.66/31.19 implies(X, implies(Y, or(Z, Y)))
% 240.66/31.19 = { by axiom 19 (substitution_of_equivalents_2) R->L }
% 240.66/31.19 fresh(modus_tollens, modus_tollens, implies(Y, or(Z, Y)), implies(X, implies(Y, or(Z, Y))))
% 240.66/31.19 = { by lemma 149 R->L }
% 240.66/31.19 fresh(fresh28(is_a_theorem(implies(implies(X, implies(Y, or(Z, Y))), implies(Y, or(Z, Y)))), modus_tollens, equiv(implies(Y, or(Z, Y)), implies(X, implies(Y, or(Z, Y))))), modus_tollens, implies(Y, or(Z, Y)), implies(X, implies(Y, or(Z, Y))))
% 240.66/31.19 = { by axiom 17 (modus_ponens_2) R->L }
% 240.66/31.19 fresh(fresh28(fresh28(implies_1, implies_1, implies(implies(X, implies(Y, or(Z, Y))), implies(Y, or(Z, Y)))), modus_tollens, equiv(implies(Y, or(Z, Y)), implies(X, implies(Y, or(Z, Y))))), modus_tollens, implies(Y, or(Z, Y)), implies(X, implies(Y, or(Z, Y))))
% 240.66/31.19 = { by lemma 59 R->L }
% 240.66/31.19 fresh(fresh28(fresh28(or_2, implies_1, implies(implies(X, implies(Y, or(Z, Y))), implies(Y, or(Z, Y)))), modus_tollens, equiv(implies(Y, or(Z, Y)), implies(X, implies(Y, or(Z, Y))))), modus_tollens, implies(Y, or(Z, Y)), implies(X, implies(Y, or(Z, Y))))
% 240.66/31.19 = { by lemma 55 R->L }
% 240.66/31.19 fresh(fresh28(fresh28(op_and, implies_1, implies(implies(X, implies(Y, or(Z, Y))), implies(Y, or(Z, Y)))), modus_tollens, equiv(implies(Y, or(Z, Y)), implies(X, implies(Y, or(Z, Y))))), modus_tollens, implies(Y, or(Z, Y)), implies(X, implies(Y, or(Z, Y))))
% 240.66/31.19 = { by lemma 54 R->L }
% 240.66/31.19 fresh(fresh28(fresh28(fresh16(or_2, op_and, Z, Y), implies_1, implies(implies(X, implies(Y, or(Z, Y))), implies(Y, or(Z, Y)))), modus_tollens, equiv(implies(Y, or(Z, Y)), implies(X, implies(Y, or(Z, Y))))), modus_tollens, implies(Y, or(Z, Y)), implies(X, implies(Y, or(Z, Y))))
% 240.66/31.19 = { by lemma 53 R->L }
% 240.66/31.19 fresh(fresh28(fresh28(is_a_theorem(implies(Y, or(Z, Y))), implies_1, implies(implies(X, implies(Y, or(Z, Y))), implies(Y, or(Z, Y)))), modus_tollens, equiv(implies(Y, or(Z, Y)), implies(X, implies(Y, or(Z, Y))))), modus_tollens, implies(Y, or(Z, Y)), implies(X, implies(Y, or(Z, Y))))
% 240.66/31.19 = { by lemma 148 }
% 240.66/31.19 fresh(fresh28(implies_1, modus_tollens, equiv(implies(Y, or(Z, Y)), implies(X, implies(Y, or(Z, Y))))), modus_tollens, implies(Y, or(Z, Y)), implies(X, implies(Y, or(Z, Y))))
% 240.66/31.19 = { by lemma 112 }
% 240.66/31.19 fresh(fresh28(modus_tollens, modus_tollens, equiv(implies(Y, or(Z, Y)), implies(X, implies(Y, or(Z, Y))))), modus_tollens, implies(Y, or(Z, Y)), implies(X, implies(Y, or(Z, Y))))
% 240.66/31.19 = { by axiom 17 (modus_ponens_2) }
% 240.66/31.19 fresh(is_a_theorem(equiv(implies(Y, or(Z, Y)), implies(X, implies(Y, or(Z, Y))))), modus_tollens, implies(Y, or(Z, Y)), implies(X, implies(Y, or(Z, Y))))
% 240.66/31.19 = { by lemma 121 }
% 240.66/31.19 implies(Y, or(Z, Y))
% 240.66/31.19
% 240.66/31.19 Lemma 151: is_a_theorem(implies(or(X, Y), or(X, or(Z, Y)))) = modus_tollens.
% 240.66/31.19 Proof:
% 240.66/31.19 is_a_theorem(implies(or(X, Y), or(X, or(Z, Y))))
% 240.66/31.19 = { by axiom 17 (modus_ponens_2) R->L }
% 240.66/31.19 fresh28(implies_1, implies_1, implies(or(X, Y), or(X, or(Z, Y))))
% 240.66/31.19 = { by lemma 59 R->L }
% 240.66/31.19 fresh28(or_2, implies_1, implies(or(X, Y), or(X, or(Z, Y))))
% 240.66/31.19 = { by lemma 55 R->L }
% 240.66/31.19 fresh28(op_and, implies_1, implies(or(X, Y), or(X, or(Z, Y))))
% 240.66/31.19 = { by lemma 54 R->L }
% 240.66/31.19 fresh28(fresh16(or_2, op_and, Z, Y), implies_1, implies(or(X, Y), or(X, or(Z, Y))))
% 240.66/31.19 = { by lemma 53 R->L }
% 240.66/31.19 fresh28(is_a_theorem(implies(Y, or(Z, Y))), implies_1, implies(or(X, Y), or(X, or(Z, Y))))
% 240.66/31.19 = { by lemma 75 }
% 240.66/31.19 implies_1
% 240.66/31.19 = { by lemma 112 }
% 240.66/31.19 modus_tollens
% 240.66/31.19
% 240.66/31.19 Goal 1 (principia_r4): r4 = true.
% 240.66/31.19 Proof:
% 240.66/31.19 r4
% 240.66/31.19 = { by axiom 51 (r4) R->L }
% 240.66/31.19 fresh7(is_a_theorem(implies(or(p2, or(q2, r6)), or(q2, or(p2, r6)))), true)
% 240.66/31.19 = { by axiom 50 (r4_1) R->L }
% 240.66/31.19 fresh7(fresh6(r4, true, p2, q2, r6), true)
% 240.66/31.19 = { by axiom 1 (principia_op_and) R->L }
% 240.66/31.19 fresh7(fresh6(r4, op_and, p2, q2, r6), true)
% 240.66/31.19 = { by lemma 55 }
% 240.66/31.19 fresh7(fresh6(r4, or_2, p2, q2, r6), true)
% 240.66/31.19 = { by axiom 1 (principia_op_and) R->L }
% 240.66/31.19 fresh7(fresh6(r4, or_2, p2, q2, r6), op_and)
% 240.66/31.19 = { by lemma 55 }
% 240.66/31.19 fresh7(fresh6(r4, or_2, p2, q2, r6), or_2)
% 240.66/31.19 = { by lemma 59 }
% 240.66/31.19 fresh7(fresh6(r4, implies_1, p2, q2, r6), or_2)
% 240.66/31.19 = { by lemma 59 }
% 240.66/31.19 fresh7(fresh6(r4, implies_1, p2, q2, r6), implies_1)
% 240.66/31.19 = { by lemma 112 }
% 240.66/31.19 fresh7(fresh6(r4, implies_1, p2, q2, r6), modus_tollens)
% 240.66/31.19 = { by lemma 59 R->L }
% 240.66/31.19 fresh7(fresh6(r4, or_2, p2, q2, r6), modus_tollens)
% 240.66/31.19 = { by lemma 55 R->L }
% 240.66/31.19 fresh7(fresh6(r4, op_and, p2, q2, r6), modus_tollens)
% 240.66/31.19 = { by axiom 1 (principia_op_and) }
% 240.66/31.19 fresh7(fresh6(r4, true, p2, q2, r6), modus_tollens)
% 240.66/31.19 = { by axiom 50 (r4_1) }
% 240.66/31.19 fresh7(is_a_theorem(implies(or(p2, or(q2, r6)), or(q2, or(p2, r6)))), modus_tollens)
% 240.66/31.19 = { by lemma 141 }
% 240.66/31.19 fresh7(is_a_theorem(implies(or(p2, or(q2, r6)), or(q2, or(r6, p2)))), modus_tollens)
% 240.66/31.19 = { by lemma 121 R->L }
% 240.66/31.19 fresh7(is_a_theorem(implies(or(p2, or(q2, r6)), fresh(is_a_theorem(equiv(or(q2, or(r6, p2)), or(p2, or(q2, or(r6, p2))))), modus_tollens, or(q2, or(r6, p2)), or(p2, or(q2, or(r6, p2)))))), modus_tollens)
% 240.66/31.19 = { by lemma 141 }
% 240.66/31.19 fresh7(is_a_theorem(implies(or(p2, or(q2, r6)), fresh(is_a_theorem(equiv(or(q2, or(r6, p2)), or(or(q2, or(r6, p2)), p2))), modus_tollens, or(q2, or(r6, p2)), or(p2, or(q2, or(r6, p2)))))), modus_tollens)
% 240.66/31.19 = { by lemma 95 R->L }
% 240.66/31.19 fresh7(is_a_theorem(implies(or(p2, or(q2, r6)), fresh(is_a_theorem(and(implies(or(q2, or(r6, p2)), or(or(q2, or(r6, p2)), p2)), implies(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2))))), modus_tollens, or(q2, or(r6, p2)), or(p2, or(q2, or(r6, p2)))))), modus_tollens)
% 240.66/31.19 = { by lemma 121 R->L }
% 240.66/31.19 fresh7(is_a_theorem(implies(or(p2, or(q2, r6)), fresh(is_a_theorem(and(implies(or(q2, or(r6, p2)), fresh(is_a_theorem(equiv(or(or(q2, or(r6, p2)), p2), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2))))), modus_tollens, or(or(q2, or(r6, p2)), p2), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2))))), implies(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2))))), modus_tollens, or(q2, or(r6, p2)), or(p2, or(q2, or(r6, p2)))))), modus_tollens)
% 240.66/31.19 = { by axiom 17 (modus_ponens_2) R->L }
% 240.66/31.19 fresh7(is_a_theorem(implies(or(p2, or(q2, r6)), fresh(is_a_theorem(and(implies(or(q2, or(r6, p2)), fresh(fresh28(modus_tollens, modus_tollens, equiv(or(or(q2, or(r6, p2)), p2), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2))))), modus_tollens, or(or(q2, or(r6, p2)), p2), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2))))), implies(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2))))), modus_tollens, or(q2, or(r6, p2)), or(p2, or(q2, or(r6, p2)))))), modus_tollens)
% 240.66/31.20 = { by lemma 112 R->L }
% 240.66/31.20 fresh7(is_a_theorem(implies(or(p2, or(q2, r6)), fresh(is_a_theorem(and(implies(or(q2, or(r6, p2)), fresh(fresh28(implies_1, modus_tollens, equiv(or(or(q2, or(r6, p2)), p2), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2))))), modus_tollens, or(or(q2, or(r6, p2)), p2), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2))))), implies(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2))))), modus_tollens, or(q2, or(r6, p2)), or(p2, or(q2, or(r6, p2)))))), modus_tollens)
% 240.66/31.20 = { by lemma 105 R->L }
% 240.66/31.20 fresh7(is_a_theorem(implies(or(p2, or(q2, r6)), fresh(is_a_theorem(and(implies(or(q2, or(r6, p2)), fresh(fresh28(fresh28(is_a_theorem(implies(or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2))), or(or(or(q2, or(r6, p2)), p2), or(or(q2, or(r6, p2)), p2)))), implies_1, implies(or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2))), or(or(q2, or(r6, p2)), p2))), modus_tollens, equiv(or(or(q2, or(r6, p2)), p2), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2))))), modus_tollens, or(or(q2, or(r6, p2)), p2), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2))))), implies(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2))))), modus_tollens, or(q2, or(r6, p2)), or(p2, or(q2, or(r6, p2)))))), modus_tollens)
% 240.66/31.20 = { by axiom 17 (modus_ponens_2) R->L }
% 240.66/31.20 fresh7(is_a_theorem(implies(or(p2, or(q2, r6)), fresh(is_a_theorem(and(implies(or(q2, or(r6, p2)), fresh(fresh28(fresh28(fresh28(implies_1, implies_1, implies(or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2))), or(or(or(q2, or(r6, p2)), p2), or(or(q2, or(r6, p2)), p2)))), implies_1, implies(or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2))), or(or(q2, or(r6, p2)), p2))), modus_tollens, equiv(or(or(q2, or(r6, p2)), p2), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2))))), modus_tollens, or(or(q2, or(r6, p2)), p2), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2))))), implies(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2))))), modus_tollens, or(q2, or(r6, p2)), or(p2, or(q2, or(r6, p2)))))), modus_tollens)
% 240.66/31.20 = { by lemma 108 R->L }
% 240.66/31.20 fresh7(is_a_theorem(implies(or(p2, or(q2, r6)), fresh(is_a_theorem(and(implies(or(q2, or(r6, p2)), fresh(fresh28(fresh28(fresh28(is_a_theorem(implies(or(q2, or(r6, p2)), or(or(q2, or(r6, p2)), p2))), implies_1, implies(or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2))), or(or(or(q2, or(r6, p2)), p2), or(or(q2, or(r6, p2)), p2)))), implies_1, implies(or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2))), or(or(q2, or(r6, p2)), p2))), modus_tollens, equiv(or(or(q2, or(r6, p2)), p2), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2))))), modus_tollens, or(or(q2, or(r6, p2)), p2), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2))))), implies(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2))))), modus_tollens, or(q2, or(r6, p2)), or(p2, or(q2, or(r6, p2)))))), modus_tollens)
% 240.66/31.20 = { by lemma 75 }
% 240.66/31.20 fresh7(is_a_theorem(implies(or(p2, or(q2, r6)), fresh(is_a_theorem(and(implies(or(q2, or(r6, p2)), fresh(fresh28(fresh28(implies_1, implies_1, implies(or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2))), or(or(q2, or(r6, p2)), p2))), modus_tollens, equiv(or(or(q2, or(r6, p2)), p2), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2))))), modus_tollens, or(or(q2, or(r6, p2)), p2), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2))))), implies(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2))))), modus_tollens, or(q2, or(r6, p2)), or(p2, or(q2, or(r6, p2)))))), modus_tollens)
% 240.66/31.20 = { by axiom 17 (modus_ponens_2) }
% 240.66/31.20 fresh7(is_a_theorem(implies(or(p2, or(q2, r6)), fresh(is_a_theorem(and(implies(or(q2, or(r6, p2)), fresh(fresh28(is_a_theorem(implies(or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2))), or(or(q2, or(r6, p2)), p2))), modus_tollens, equiv(or(or(q2, or(r6, p2)), p2), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2))))), modus_tollens, or(or(q2, or(r6, p2)), p2), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2))))), implies(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2))))), modus_tollens, or(q2, or(r6, p2)), or(p2, or(q2, or(r6, p2)))))), modus_tollens)
% 240.66/31.20 = { by lemma 120 }
% 240.66/31.20 fresh7(is_a_theorem(implies(or(p2, or(q2, r6)), fresh(is_a_theorem(and(implies(or(q2, or(r6, p2)), fresh(modus_tollens, modus_tollens, or(or(q2, or(r6, p2)), p2), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2))))), implies(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2))))), modus_tollens, or(q2, or(r6, p2)), or(p2, or(q2, or(r6, p2)))))), modus_tollens)
% 240.66/31.20 = { by axiom 19 (substitution_of_equivalents_2) }
% 240.66/31.20 fresh7(is_a_theorem(implies(or(p2, or(q2, r6)), fresh(is_a_theorem(and(implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2)))), implies(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2))))), modus_tollens, or(q2, or(r6, p2)), or(p2, or(q2, or(r6, p2)))))), modus_tollens)
% 240.66/31.20 = { by lemma 150 R->L }
% 240.66/31.21 fresh7(is_a_theorem(implies(or(p2, or(q2, r6)), fresh(is_a_theorem(and(implies(implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2)))), implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2))))), implies(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2))))), modus_tollens, or(q2, or(r6, p2)), or(p2, or(q2, or(r6, p2)))))), modus_tollens)
% 240.66/31.21 = { by lemma 150 R->L }
% 240.66/31.21 fresh7(is_a_theorem(implies(or(p2, or(q2, r6)), fresh(is_a_theorem(and(implies(implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2)))), implies(X, implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2)))))), implies(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2))))), modus_tollens, or(q2, or(r6, p2)), or(p2, or(q2, or(r6, p2)))))), modus_tollens)
% 241.70/31.21 = { by lemma 121 R->L }
% 241.70/31.21 fresh7(is_a_theorem(implies(or(p2, or(q2, r6)), fresh(is_a_theorem(and(fresh(is_a_theorem(equiv(implies(implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2)))), implies(X, implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2)))))), implies(implies(implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2)))), implies(X, implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2)))))), implies(implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2)))), implies(X, implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2))))))))), modus_tollens, implies(implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2)))), implies(X, implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2)))))), implies(implies(implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2)))), implies(X, implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2)))))), implies(implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2)))), implies(X, implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2)))))))), implies(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2))))), modus_tollens, or(q2, or(r6, p2)), or(p2, or(q2, or(r6, p2)))))), modus_tollens)
% 241.70/31.21 = { by axiom 17 (modus_ponens_2) R->L }
% 241.70/31.21 fresh7(is_a_theorem(implies(or(p2, or(q2, r6)), fresh(is_a_theorem(and(fresh(fresh28(modus_tollens, modus_tollens, equiv(implies(implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2)))), implies(X, implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2)))))), implies(implies(implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2)))), implies(X, implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2)))))), implies(implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2)))), implies(X, implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2))))))))), modus_tollens, implies(implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2)))), implies(X, implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2)))))), implies(implies(implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2)))), implies(X, implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2)))))), implies(implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2)))), implies(X, implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2)))))))), implies(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2))))), modus_tollens, or(q2, or(r6, p2)), or(p2, or(q2, or(r6, p2)))))), modus_tollens)
% 241.70/31.21 = { by lemma 112 R->L }
% 241.70/31.21 fresh7(is_a_theorem(implies(or(p2, or(q2, r6)), fresh(is_a_theorem(and(fresh(fresh28(implies_1, modus_tollens, equiv(implies(implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2)))), implies(X, implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2)))))), implies(implies(implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2)))), implies(X, implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2)))))), implies(implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2)))), implies(X, implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2))))))))), modus_tollens, implies(implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2)))), implies(X, implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2)))))), implies(implies(implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2)))), implies(X, implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2)))))), implies(implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2)))), implies(X, implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2)))))))), implies(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2))))), modus_tollens, or(q2, or(r6, p2)), or(p2, or(q2, or(r6, p2)))))), modus_tollens)
% 241.70/31.21 = { by lemma 148 R->L }
% 241.70/31.22 fresh7(is_a_theorem(implies(or(p2, or(q2, r6)), fresh(is_a_theorem(and(fresh(fresh28(fresh28(is_a_theorem(implies(implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2)))), implies(X, implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2))))))), implies_1, implies(implies(implies(implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2)))), implies(X, implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2)))))), implies(implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2)))), implies(X, implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2))))))), implies(implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2)))), implies(X, implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2)))))))), modus_tollens, equiv(implies(implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2)))), implies(X, implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2)))))), implies(implies(implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2)))), implies(X, implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2)))))), implies(implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2)))), implies(X, implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2))))))))), modus_tollens, implies(implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2)))), implies(X, implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2)))))), implies(implies(implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2)))), implies(X, implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2)))))), implies(implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2)))), implies(X, implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2)))))))), implies(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2))))), modus_tollens, or(q2, or(r6, p2)), or(p2, or(q2, or(r6, p2)))))), modus_tollens)
% 241.70/31.22 = { by lemma 56 }
% 241.70/31.22 fresh7(is_a_theorem(implies(or(p2, or(q2, r6)), fresh(is_a_theorem(and(fresh(fresh28(fresh28(fresh39(implies_1, op_and, implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2)))), X), implies_1, implies(implies(implies(implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2)))), implies(X, implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2)))))), implies(implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2)))), implies(X, implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2))))))), implies(implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2)))), implies(X, implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2)))))))), modus_tollens, equiv(implies(implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2)))), implies(X, implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2)))))), implies(implies(implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2)))), implies(X, implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2)))))), implies(implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2)))), implies(X, implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2))))))))), modus_tollens, implies(implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2)))), implies(X, implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2)))))), implies(implies(implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2)))), implies(X, implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2)))))), implies(implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2)))), implies(X, implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2)))))))), implies(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2))))), modus_tollens, or(q2, or(r6, p2)), or(p2, or(q2, or(r6, p2)))))), modus_tollens)
% 241.70/31.22 = { by lemma 55 }
% 241.70/31.22 fresh7(is_a_theorem(implies(or(p2, or(q2, r6)), fresh(is_a_theorem(and(fresh(fresh28(fresh28(fresh39(implies_1, or_2, implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2)))), X), implies_1, implies(implies(implies(implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2)))), implies(X, implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2)))))), implies(implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2)))), implies(X, implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2))))))), implies(implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2)))), implies(X, implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2)))))))), modus_tollens, equiv(implies(implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2)))), implies(X, implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2)))))), implies(implies(implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2)))), implies(X, implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2)))))), implies(implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2)))), implies(X, implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2))))))))), modus_tollens, implies(implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2)))), implies(X, implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2)))))), implies(implies(implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2)))), implies(X, implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2)))))), implies(implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2)))), implies(X, implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2)))))))), implies(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2))))), modus_tollens, or(q2, or(r6, p2)), or(p2, or(q2, or(r6, p2)))))), modus_tollens)
% 241.70/31.22 = { by lemma 58 }
% 241.70/31.22 fresh7(is_a_theorem(implies(or(p2, or(q2, r6)), fresh(is_a_theorem(and(fresh(fresh28(fresh28(or_2, implies_1, implies(implies(implies(implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2)))), implies(X, implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2)))))), implies(implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2)))), implies(X, implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2))))))), implies(implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2)))), implies(X, implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2)))))))), modus_tollens, equiv(implies(implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2)))), implies(X, implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2)))))), implies(implies(implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2)))), implies(X, implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2)))))), implies(implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2)))), implies(X, implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2))))))))), modus_tollens, implies(implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2)))), implies(X, implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2)))))), implies(implies(implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2)))), implies(X, implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2)))))), implies(implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2)))), implies(X, implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2)))))))), implies(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2))))), modus_tollens, or(q2, or(r6, p2)), or(p2, or(q2, or(r6, p2)))))), modus_tollens)
% 241.70/31.23 = { by lemma 59 }
% 241.70/31.23 fresh7(is_a_theorem(implies(or(p2, or(q2, r6)), fresh(is_a_theorem(and(fresh(fresh28(fresh28(implies_1, implies_1, implies(implies(implies(implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2)))), implies(X, implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2)))))), implies(implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2)))), implies(X, implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2))))))), implies(implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2)))), implies(X, implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2)))))))), modus_tollens, equiv(implies(implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2)))), implies(X, implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2)))))), implies(implies(implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2)))), implies(X, implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2)))))), implies(implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2)))), implies(X, implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2))))))))), modus_tollens, implies(implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2)))), implies(X, implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2)))))), implies(implies(implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2)))), implies(X, implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2)))))), implies(implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2)))), implies(X, implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2)))))))), implies(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2))))), modus_tollens, or(q2, or(r6, p2)), or(p2, or(q2, or(r6, p2)))))), modus_tollens)
% 241.70/31.23 = { by axiom 17 (modus_ponens_2) }
% 241.70/31.23 fresh7(is_a_theorem(implies(or(p2, or(q2, r6)), fresh(is_a_theorem(and(fresh(fresh28(is_a_theorem(implies(implies(implies(implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2)))), implies(X, implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2)))))), implies(implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2)))), implies(X, implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2))))))), implies(implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2)))), implies(X, implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2)))))))), modus_tollens, equiv(implies(implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2)))), implies(X, implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2)))))), implies(implies(implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2)))), implies(X, implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2)))))), implies(implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2)))), implies(X, implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2))))))))), modus_tollens, implies(implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2)))), implies(X, implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2)))))), implies(implies(implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2)))), implies(X, implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2)))))), implies(implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2)))), implies(X, implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2)))))))), implies(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2))))), modus_tollens, or(q2, or(r6, p2)), or(p2, or(q2, or(r6, p2)))))), modus_tollens)
% 241.70/31.23 = { by lemma 149 }
% 241.70/31.23 fresh7(is_a_theorem(implies(or(p2, or(q2, r6)), fresh(is_a_theorem(and(fresh(modus_tollens, modus_tollens, implies(implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2)))), implies(X, implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2)))))), implies(implies(implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2)))), implies(X, implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2)))))), implies(implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2)))), implies(X, implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2)))))))), implies(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2))))), modus_tollens, or(q2, or(r6, p2)), or(p2, or(q2, or(r6, p2)))))), modus_tollens)
% 241.70/31.23 = { by axiom 19 (substitution_of_equivalents_2) }
% 241.70/31.23 fresh7(is_a_theorem(implies(or(p2, or(q2, r6)), fresh(is_a_theorem(and(implies(implies(implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2)))), implies(X, implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2)))))), implies(implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2)))), implies(X, implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2))))))), implies(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2))))), modus_tollens, or(q2, or(r6, p2)), or(p2, or(q2, or(r6, p2)))))), modus_tollens)
% 241.70/31.23 = { by lemma 113 }
% 241.70/31.23 fresh7(is_a_theorem(implies(or(p2, or(q2, r6)), fresh(is_a_theorem(and(equiv(implies(implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2)))), implies(X, implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2)))))), implies(implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2)))), implies(X, implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2))))))), implies(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2))))), modus_tollens, or(q2, or(r6, p2)), or(p2, or(q2, or(r6, p2)))))), modus_tollens)
% 241.70/31.23 = { by lemma 145 R->L }
% 241.70/31.23 fresh7(is_a_theorem(implies(or(p2, or(q2, r6)), fresh(is_a_theorem(and(implies(equiv(Y, not(Y)), equiv(implies(implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2)))), implies(X, implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2)))))), implies(implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2)))), implies(X, implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2)))))))), implies(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2))))), modus_tollens, or(q2, or(r6, p2)), or(p2, or(q2, or(r6, p2)))))), modus_tollens)
% 241.70/31.23 = { by lemma 146 }
% 241.70/31.23 fresh7(is_a_theorem(implies(or(p2, or(q2, r6)), fresh(is_a_theorem(and(or(or(Y, not(Y)), equiv(implies(implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2)))), implies(X, implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2)))))), implies(implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2)))), implies(X, implies(or(q2, or(r6, p2)), or(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2)))))))), implies(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2))))), modus_tollens, or(q2, or(r6, p2)), or(p2, or(q2, or(r6, p2)))))), modus_tollens)
% 241.70/31.23 = { by lemma 147 }
% 241.70/31.23 fresh7(is_a_theorem(implies(or(p2, or(q2, r6)), fresh(is_a_theorem(and(or(Y, not(Y)), implies(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2))))), modus_tollens, or(q2, or(r6, p2)), or(p2, or(q2, or(r6, p2)))))), modus_tollens)
% 241.70/31.23 = { by lemma 147 R->L }
% 241.70/31.23 fresh7(is_a_theorem(implies(or(p2, or(q2, r6)), fresh(is_a_theorem(and(or(or(Y, not(Y)), equiv(implies(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2))), implies(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2))))), implies(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2))))), modus_tollens, or(q2, or(r6, p2)), or(p2, or(q2, or(r6, p2)))))), modus_tollens)
% 241.70/31.23 = { by lemma 146 R->L }
% 241.70/31.23 fresh7(is_a_theorem(implies(or(p2, or(q2, r6)), fresh(is_a_theorem(and(implies(equiv(Y, not(Y)), equiv(implies(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2))), implies(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2))))), implies(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2))))), modus_tollens, or(q2, or(r6, p2)), or(p2, or(q2, or(r6, p2)))))), modus_tollens)
% 241.70/31.23 = { by lemma 145 }
% 241.70/31.23 fresh7(is_a_theorem(implies(or(p2, or(q2, r6)), fresh(is_a_theorem(and(equiv(implies(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2))), implies(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2)))), implies(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2))))), modus_tollens, or(q2, or(r6, p2)), or(p2, or(q2, or(r6, p2)))))), modus_tollens)
% 241.70/31.23 = { by lemma 113 R->L }
% 241.70/31.23 fresh7(is_a_theorem(implies(or(p2, or(q2, r6)), fresh(is_a_theorem(and(implies(implies(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2))), implies(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2)))), implies(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2))))), modus_tollens, or(q2, or(r6, p2)), or(p2, or(q2, or(r6, p2)))))), modus_tollens)
% 241.70/31.23 = { by lemma 140 }
% 241.70/31.23 fresh7(is_a_theorem(implies(or(p2, or(q2, r6)), fresh(is_a_theorem(implies(or(or(q2, or(r6, p2)), p2), or(q2, or(r6, p2)))), modus_tollens, or(q2, or(r6, p2)), or(p2, or(q2, or(r6, p2)))))), modus_tollens)
% 241.70/31.23 = { by lemma 121 R->L }
% 241.70/31.23 fresh7(is_a_theorem(implies(or(p2, or(q2, r6)), fresh(is_a_theorem(implies(or(or(q2, or(r6, p2)), p2), fresh(is_a_theorem(equiv(or(q2, or(r6, p2)), or(or(q2, or(r6, p2)), or(r6, p2)))), modus_tollens, or(q2, or(r6, p2)), or(or(q2, or(r6, p2)), or(r6, p2))))), modus_tollens, or(q2, or(r6, p2)), or(p2, or(q2, or(r6, p2)))))), modus_tollens)
% 241.70/31.23 = { by axiom 17 (modus_ponens_2) R->L }
% 241.70/31.23 fresh7(is_a_theorem(implies(or(p2, or(q2, r6)), fresh(is_a_theorem(implies(or(or(q2, or(r6, p2)), p2), fresh(fresh28(modus_tollens, modus_tollens, equiv(or(q2, or(r6, p2)), or(or(q2, or(r6, p2)), or(r6, p2)))), modus_tollens, or(q2, or(r6, p2)), or(or(q2, or(r6, p2)), or(r6, p2))))), modus_tollens, or(q2, or(r6, p2)), or(p2, or(q2, or(r6, p2)))))), modus_tollens)
% 241.70/31.24 = { by lemma 112 R->L }
% 241.70/31.24 fresh7(is_a_theorem(implies(or(p2, or(q2, r6)), fresh(is_a_theorem(implies(or(or(q2, or(r6, p2)), p2), fresh(fresh28(implies_1, modus_tollens, equiv(or(q2, or(r6, p2)), or(or(q2, or(r6, p2)), or(r6, p2)))), modus_tollens, or(q2, or(r6, p2)), or(or(q2, or(r6, p2)), or(r6, p2))))), modus_tollens, or(q2, or(r6, p2)), or(p2, or(q2, or(r6, p2)))))), modus_tollens)
% 241.70/31.24 = { by lemma 105 R->L }
% 241.70/31.24 fresh7(is_a_theorem(implies(or(p2, or(q2, r6)), fresh(is_a_theorem(implies(or(or(q2, or(r6, p2)), p2), fresh(fresh28(fresh28(is_a_theorem(implies(or(or(q2, or(r6, p2)), or(r6, p2)), or(or(q2, or(r6, p2)), or(q2, or(r6, p2))))), implies_1, implies(or(or(q2, or(r6, p2)), or(r6, p2)), or(q2, or(r6, p2)))), modus_tollens, equiv(or(q2, or(r6, p2)), or(or(q2, or(r6, p2)), or(r6, p2)))), modus_tollens, or(q2, or(r6, p2)), or(or(q2, or(r6, p2)), or(r6, p2))))), modus_tollens, or(q2, or(r6, p2)), or(p2, or(q2, or(r6, p2)))))), modus_tollens)
% 241.70/31.24 = { by lemma 85 }
% 241.70/31.24 fresh7(is_a_theorem(implies(or(p2, or(q2, r6)), fresh(is_a_theorem(implies(or(or(q2, or(r6, p2)), p2), fresh(fresh28(fresh28(implies_1, implies_1, implies(or(or(q2, or(r6, p2)), or(r6, p2)), or(q2, or(r6, p2)))), modus_tollens, equiv(or(q2, or(r6, p2)), or(or(q2, or(r6, p2)), or(r6, p2)))), modus_tollens, or(q2, or(r6, p2)), or(or(q2, or(r6, p2)), or(r6, p2))))), modus_tollens, or(q2, or(r6, p2)), or(p2, or(q2, or(r6, p2)))))), modus_tollens)
% 241.70/31.24 = { by axiom 17 (modus_ponens_2) }
% 241.70/31.24 fresh7(is_a_theorem(implies(or(p2, or(q2, r6)), fresh(is_a_theorem(implies(or(or(q2, or(r6, p2)), p2), fresh(fresh28(is_a_theorem(implies(or(or(q2, or(r6, p2)), or(r6, p2)), or(q2, or(r6, p2)))), modus_tollens, equiv(or(q2, or(r6, p2)), or(or(q2, or(r6, p2)), or(r6, p2)))), modus_tollens, or(q2, or(r6, p2)), or(or(q2, or(r6, p2)), or(r6, p2))))), modus_tollens, or(q2, or(r6, p2)), or(p2, or(q2, or(r6, p2)))))), modus_tollens)
% 241.70/31.24 = { by lemma 120 }
% 241.70/31.24 fresh7(is_a_theorem(implies(or(p2, or(q2, r6)), fresh(is_a_theorem(implies(or(or(q2, or(r6, p2)), p2), fresh(modus_tollens, modus_tollens, or(q2, or(r6, p2)), or(or(q2, or(r6, p2)), or(r6, p2))))), modus_tollens, or(q2, or(r6, p2)), or(p2, or(q2, or(r6, p2)))))), modus_tollens)
% 241.70/31.24 = { by axiom 19 (substitution_of_equivalents_2) }
% 241.70/31.24 fresh7(is_a_theorem(implies(or(p2, or(q2, r6)), fresh(is_a_theorem(implies(or(or(q2, or(r6, p2)), p2), or(or(q2, or(r6, p2)), or(r6, p2)))), modus_tollens, or(q2, or(r6, p2)), or(p2, or(q2, or(r6, p2)))))), modus_tollens)
% 241.70/31.24 = { by lemma 151 }
% 241.70/31.24 fresh7(is_a_theorem(implies(or(p2, or(q2, r6)), fresh(modus_tollens, modus_tollens, or(q2, or(r6, p2)), or(p2, or(q2, or(r6, p2)))))), modus_tollens)
% 241.70/31.24 = { by axiom 19 (substitution_of_equivalents_2) }
% 241.70/31.24 fresh7(is_a_theorem(implies(or(p2, or(q2, r6)), or(p2, or(q2, or(r6, p2))))), modus_tollens)
% 241.70/31.24 = { by lemma 141 }
% 241.70/31.24 fresh7(is_a_theorem(implies(or(p2, or(q2, r6)), or(p2, or(q2, or(p2, r6))))), modus_tollens)
% 241.70/31.24 = { by axiom 17 (modus_ponens_2) R->L }
% 241.70/31.24 fresh7(fresh28(modus_tollens, modus_tollens, implies(or(p2, or(q2, r6)), or(p2, or(q2, or(p2, r6))))), modus_tollens)
% 241.70/31.24 = { by lemma 151 R->L }
% 241.70/31.24 fresh7(fresh28(is_a_theorem(implies(or(q2, r6), or(q2, or(p2, r6)))), modus_tollens, implies(or(p2, or(q2, r6)), or(p2, or(q2, or(p2, r6))))), modus_tollens)
% 241.70/31.24 = { by lemma 134 }
% 241.70/31.24 fresh7(modus_tollens, modus_tollens)
% 241.70/31.24 = { by axiom 14 (r4) }
% 241.70/31.24 true
% 241.70/31.24 % SZS output end Proof
% 241.70/31.24
% 241.70/31.24 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------