TSTP Solution File: LCL269-3 by Twee---2.4.2
View Problem
- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : LCL269-3 : TPTP v8.1.2. Released v2.3.0.
% Transfm : none
% Format : tptp:raw
% Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% Computer : n025.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Thu Aug 31 08:18:21 EDT 2023
% Result : Unsatisfiable 108.06s 14.19s
% Output : Proof 108.98s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : LCL269-3 : TPTP v8.1.2. Released v2.3.0.
% 0.00/0.13 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.13/0.34 % Computer : n025.cluster.edu
% 0.13/0.34 % Model : x86_64 x86_64
% 0.13/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.34 % Memory : 8042.1875MB
% 0.13/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.34 % CPULimit : 300
% 0.13/0.34 % WCLimit : 300
% 0.13/0.34 % DateTime : Thu Aug 24 23:29:52 EDT 2023
% 0.19/0.34 % CPUTime :
% 108.06/14.19 Command-line arguments: --lhs-weight 9 --flip-ordering --complete-subsets --normalise-queue-percent 10 --cp-renormalise-threshold 10
% 108.06/14.19
% 108.06/14.19 % SZS status Unsatisfiable
% 108.06/14.19
% 108.63/14.26 % SZS output start Proof
% 108.63/14.26 Take the following subset of the input axioms:
% 108.63/14.27 fof(and_defn, axiom, ![P, Q]: and(P, Q)=not(or(not(P), not(Q)))).
% 108.63/14.27 fof(axiom_1_2, axiom, ![A]: axiom(implies(or(A, A), A))).
% 108.63/14.27 fof(axiom_1_3, axiom, ![B, A2]: axiom(implies(A2, or(B, A2)))).
% 108.63/14.27 fof(axiom_1_4, axiom, ![A2, B2]: axiom(implies(or(A2, B2), or(B2, A2)))).
% 108.63/14.27 fof(axiom_1_5, axiom, ![C, A2, B2]: axiom(implies(or(A2, or(B2, C)), or(B2, or(A2, C))))).
% 108.63/14.27 fof(axiom_1_6, axiom, ![A2, B2, C2]: axiom(implies(implies(A2, B2), implies(or(C2, A2), or(C2, B2))))).
% 108.63/14.27 fof(equivalent_defn, axiom, ![P2, Q2]: equivalent(P2, Q2)=and(implies(P2, Q2), implies(Q2, P2))).
% 108.63/14.27 fof(implies_definition, axiom, ![X, Y]: implies(X, Y)=or(not(X), Y)).
% 108.63/14.27 fof(prove_this, negated_conjecture, ~theorem(equivalent(equivalent(p, q), equivalent(q, p)))).
% 108.63/14.27 fof(rule_1, axiom, ![X2]: (theorem(X2) | ~axiom(X2))).
% 108.63/14.27 fof(rule_2, axiom, ![X2, Y2]: (theorem(X2) | (~theorem(implies(Y2, X2)) | ~theorem(Y2)))).
% 108.63/14.27
% 108.63/14.27 Now clausify the problem and encode Horn clauses using encoding 3 of
% 108.63/14.27 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 108.63/14.27 We repeatedly replace C & s=t => u=v by the two clauses:
% 108.63/14.27 fresh(y, y, x1...xn) = u
% 108.63/14.27 C => fresh(s, t, x1...xn) = v
% 108.63/14.27 where fresh is a fresh function symbol and x1..xn are the free
% 108.63/14.27 variables of u and v.
% 108.63/14.27 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 108.63/14.27 input problem has no model of domain size 1).
% 108.63/14.27
% 108.63/14.27 The encoding turns the above axioms into the following unit equations and goals:
% 108.63/14.27
% 108.63/14.27 Axiom 1 (implies_definition): implies(X, Y) = or(not(X), Y).
% 108.63/14.27 Axiom 2 (rule_2): fresh(X, X, Y) = true.
% 108.63/14.27 Axiom 3 (rule_1): fresh2(X, X, Y) = true.
% 108.63/14.27 Axiom 4 (rule_2): fresh3(X, X, Y, Z) = theorem(Y).
% 108.63/14.27 Axiom 5 (rule_1): fresh2(axiom(X), true, X) = theorem(X).
% 108.63/14.27 Axiom 6 (axiom_1_3): axiom(implies(X, or(Y, X))) = true.
% 108.63/14.27 Axiom 7 (axiom_1_2): axiom(implies(or(X, X), X)) = true.
% 108.63/14.27 Axiom 8 (and_defn): and(X, Y) = not(or(not(X), not(Y))).
% 108.63/14.27 Axiom 9 (equivalent_defn): equivalent(X, Y) = and(implies(X, Y), implies(Y, X)).
% 108.63/14.27 Axiom 10 (axiom_1_4): axiom(implies(or(X, Y), or(Y, X))) = true.
% 108.63/14.27 Axiom 11 (rule_2): fresh3(theorem(implies(X, Y)), true, Y, X) = fresh(theorem(X), true, Y).
% 108.63/14.27 Axiom 12 (axiom_1_6): axiom(implies(implies(X, Y), implies(or(Z, X), or(Z, Y)))) = true.
% 108.63/14.27 Axiom 13 (axiom_1_5): axiom(implies(or(X, or(Y, Z)), or(Y, or(X, Z)))) = true.
% 108.63/14.27
% 108.63/14.27 Lemma 14: not(implies(X, not(Y))) = and(X, Y).
% 108.63/14.27 Proof:
% 108.63/14.27 not(implies(X, not(Y)))
% 108.63/14.27 = { by axiom 1 (implies_definition) }
% 108.63/14.27 not(or(not(X), not(Y)))
% 108.63/14.27 = { by axiom 8 (and_defn) R->L }
% 108.63/14.27 and(X, Y)
% 108.63/14.27
% 108.63/14.27 Lemma 15: theorem(implies(X, or(Y, X))) = true.
% 108.63/14.27 Proof:
% 108.63/14.27 theorem(implies(X, or(Y, X)))
% 108.63/14.27 = { by axiom 5 (rule_1) R->L }
% 108.63/14.27 fresh2(axiom(implies(X, or(Y, X))), true, implies(X, or(Y, X)))
% 108.63/14.27 = { by axiom 6 (axiom_1_3) }
% 108.63/14.27 fresh2(true, true, implies(X, or(Y, X)))
% 108.63/14.27 = { by axiom 3 (rule_1) }
% 108.63/14.27 true
% 108.63/14.27
% 108.63/14.27 Lemma 16: implies(implies(X, not(Y)), Z) = or(and(X, Y), Z).
% 108.63/14.27 Proof:
% 108.63/14.27 implies(implies(X, not(Y)), Z)
% 108.63/14.27 = { by axiom 1 (implies_definition) }
% 108.63/14.27 or(not(implies(X, not(Y))), Z)
% 108.63/14.27 = { by lemma 14 }
% 108.98/14.27 or(and(X, Y), Z)
% 108.98/14.27
% 108.98/14.27 Lemma 17: theorem(implies(or(X, Y), or(Y, X))) = true.
% 108.98/14.27 Proof:
% 108.98/14.27 theorem(implies(or(X, Y), or(Y, X)))
% 108.98/14.27 = { by axiom 5 (rule_1) R->L }
% 108.98/14.27 fresh2(axiom(implies(or(X, Y), or(Y, X))), true, implies(or(X, Y), or(Y, X)))
% 108.98/14.27 = { by axiom 10 (axiom_1_4) }
% 108.98/14.27 fresh2(true, true, implies(or(X, Y), or(Y, X)))
% 108.98/14.27 = { by axiom 3 (rule_1) }
% 108.98/14.27 true
% 108.98/14.27
% 108.98/14.27 Lemma 18: fresh(theorem(or(X, Y)), true, or(Y, X)) = theorem(or(Y, X)).
% 108.98/14.27 Proof:
% 108.98/14.27 fresh(theorem(or(X, Y)), true, or(Y, X))
% 108.98/14.27 = { by axiom 11 (rule_2) R->L }
% 108.98/14.27 fresh3(theorem(implies(or(X, Y), or(Y, X))), true, or(Y, X), or(X, Y))
% 108.98/14.27 = { by lemma 17 }
% 108.98/14.27 fresh3(true, true, or(Y, X), or(X, Y))
% 108.98/14.27 = { by axiom 4 (rule_2) }
% 108.98/14.27 theorem(or(Y, X))
% 108.98/14.27
% 108.98/14.27 Lemma 19: fresh(theorem(implies(X, Y)), true, implies(or(Z, X), or(Z, Y))) = theorem(implies(or(Z, X), or(Z, Y))).
% 108.98/14.27 Proof:
% 108.98/14.27 fresh(theorem(implies(X, Y)), true, implies(or(Z, X), or(Z, Y)))
% 108.98/14.27 = { by axiom 11 (rule_2) R->L }
% 108.98/14.27 fresh3(theorem(implies(implies(X, Y), implies(or(Z, X), or(Z, Y)))), true, implies(or(Z, X), or(Z, Y)), implies(X, Y))
% 108.98/14.27 = { by axiom 5 (rule_1) R->L }
% 108.98/14.27 fresh3(fresh2(axiom(implies(implies(X, Y), implies(or(Z, X), or(Z, Y)))), true, implies(implies(X, Y), implies(or(Z, X), or(Z, Y)))), true, implies(or(Z, X), or(Z, Y)), implies(X, Y))
% 108.98/14.27 = { by axiom 12 (axiom_1_6) }
% 108.98/14.27 fresh3(fresh2(true, true, implies(implies(X, Y), implies(or(Z, X), or(Z, Y)))), true, implies(or(Z, X), or(Z, Y)), implies(X, Y))
% 108.98/14.27 = { by axiom 3 (rule_1) }
% 108.98/14.27 fresh3(true, true, implies(or(Z, X), or(Z, Y)), implies(X, Y))
% 108.98/14.27 = { by axiom 4 (rule_2) }
% 108.98/14.27 theorem(implies(or(Z, X), or(Z, Y)))
% 108.98/14.27
% 108.98/14.27 Lemma 20: theorem(implies(equivalent(X, Y), equivalent(Y, X))) = true.
% 108.98/14.27 Proof:
% 108.98/14.27 theorem(implies(equivalent(X, Y), equivalent(Y, X)))
% 108.98/14.27 = { by axiom 1 (implies_definition) }
% 108.98/14.27 theorem(or(not(equivalent(X, Y)), equivalent(Y, X)))
% 108.98/14.27 = { by lemma 18 R->L }
% 108.98/14.27 fresh(theorem(or(equivalent(Y, X), not(equivalent(X, Y)))), true, or(not(equivalent(X, Y)), equivalent(Y, X)))
% 108.98/14.27 = { by axiom 4 (rule_2) R->L }
% 108.98/14.27 fresh(fresh3(true, true, or(equivalent(Y, X), not(equivalent(X, Y))), or(equivalent(Y, X), or(not(equivalent(X, Y)), not(equivalent(X, Y))))), true, or(not(equivalent(X, Y)), equivalent(Y, X)))
% 108.98/14.27 = { by axiom 2 (rule_2) R->L }
% 108.98/14.27 fresh(fresh3(fresh(true, true, implies(or(equivalent(Y, X), or(not(equivalent(X, Y)), not(equivalent(X, Y)))), or(equivalent(Y, X), not(equivalent(X, Y))))), true, or(equivalent(Y, X), not(equivalent(X, Y))), or(equivalent(Y, X), or(not(equivalent(X, Y)), not(equivalent(X, Y))))), true, or(not(equivalent(X, Y)), equivalent(Y, X)))
% 108.98/14.27 = { by axiom 3 (rule_1) R->L }
% 108.98/14.27 fresh(fresh3(fresh(fresh2(true, true, implies(or(not(equivalent(X, Y)), not(equivalent(X, Y))), not(equivalent(X, Y)))), true, implies(or(equivalent(Y, X), or(not(equivalent(X, Y)), not(equivalent(X, Y)))), or(equivalent(Y, X), not(equivalent(X, Y))))), true, or(equivalent(Y, X), not(equivalent(X, Y))), or(equivalent(Y, X), or(not(equivalent(X, Y)), not(equivalent(X, Y))))), true, or(not(equivalent(X, Y)), equivalent(Y, X)))
% 108.98/14.27 = { by axiom 7 (axiom_1_2) R->L }
% 108.98/14.27 fresh(fresh3(fresh(fresh2(axiom(implies(or(not(equivalent(X, Y)), not(equivalent(X, Y))), not(equivalent(X, Y)))), true, implies(or(not(equivalent(X, Y)), not(equivalent(X, Y))), not(equivalent(X, Y)))), true, implies(or(equivalent(Y, X), or(not(equivalent(X, Y)), not(equivalent(X, Y)))), or(equivalent(Y, X), not(equivalent(X, Y))))), true, or(equivalent(Y, X), not(equivalent(X, Y))), or(equivalent(Y, X), or(not(equivalent(X, Y)), not(equivalent(X, Y))))), true, or(not(equivalent(X, Y)), equivalent(Y, X)))
% 108.98/14.27 = { by axiom 5 (rule_1) }
% 108.98/14.27 fresh(fresh3(fresh(theorem(implies(or(not(equivalent(X, Y)), not(equivalent(X, Y))), not(equivalent(X, Y)))), true, implies(or(equivalent(Y, X), or(not(equivalent(X, Y)), not(equivalent(X, Y)))), or(equivalent(Y, X), not(equivalent(X, Y))))), true, or(equivalent(Y, X), not(equivalent(X, Y))), or(equivalent(Y, X), or(not(equivalent(X, Y)), not(equivalent(X, Y))))), true, or(not(equivalent(X, Y)), equivalent(Y, X)))
% 108.98/14.27 = { by lemma 19 }
% 108.98/14.27 fresh(fresh3(theorem(implies(or(equivalent(Y, X), or(not(equivalent(X, Y)), not(equivalent(X, Y)))), or(equivalent(Y, X), not(equivalent(X, Y))))), true, or(equivalent(Y, X), not(equivalent(X, Y))), or(equivalent(Y, X), or(not(equivalent(X, Y)), not(equivalent(X, Y))))), true, or(not(equivalent(X, Y)), equivalent(Y, X)))
% 108.98/14.27 = { by axiom 11 (rule_2) }
% 108.98/14.27 fresh(fresh(theorem(or(equivalent(Y, X), or(not(equivalent(X, Y)), not(equivalent(X, Y))))), true, or(equivalent(Y, X), not(equivalent(X, Y)))), true, or(not(equivalent(X, Y)), equivalent(Y, X)))
% 108.98/14.27 = { by lemma 18 R->L }
% 108.98/14.27 fresh(fresh(fresh(theorem(or(or(not(equivalent(X, Y)), not(equivalent(X, Y))), equivalent(Y, X))), true, or(equivalent(Y, X), or(not(equivalent(X, Y)), not(equivalent(X, Y))))), true, or(equivalent(Y, X), not(equivalent(X, Y)))), true, or(not(equivalent(X, Y)), equivalent(Y, X)))
% 108.98/14.27 = { by axiom 4 (rule_2) R->L }
% 108.98/14.27 fresh(fresh(fresh(fresh3(true, true, or(or(not(equivalent(X, Y)), not(equivalent(X, Y))), equivalent(Y, X)), or(or(not(equivalent(X, Y)), not(equivalent(X, Y))), or(equivalent(X, Y), equivalent(X, Y)))), true, or(equivalent(Y, X), or(not(equivalent(X, Y)), not(equivalent(X, Y))))), true, or(equivalent(Y, X), not(equivalent(X, Y)))), true, or(not(equivalent(X, Y)), equivalent(Y, X)))
% 108.98/14.28 = { by axiom 2 (rule_2) R->L }
% 108.98/14.28 fresh(fresh(fresh(fresh3(fresh(true, true, implies(or(or(not(equivalent(X, Y)), not(equivalent(X, Y))), or(equivalent(X, Y), equivalent(X, Y))), or(or(not(equivalent(X, Y)), not(equivalent(X, Y))), equivalent(Y, X)))), true, or(or(not(equivalent(X, Y)), not(equivalent(X, Y))), equivalent(Y, X)), or(or(not(equivalent(X, Y)), not(equivalent(X, Y))), or(equivalent(X, Y), equivalent(X, Y)))), true, or(equivalent(Y, X), or(not(equivalent(X, Y)), not(equivalent(X, Y))))), true, or(equivalent(Y, X), not(equivalent(X, Y)))), true, or(not(equivalent(X, Y)), equivalent(Y, X)))
% 108.98/14.28 = { by axiom 2 (rule_2) R->L }
% 108.98/14.28 fresh(fresh(fresh(fresh3(fresh(fresh(true, true, or(not(or(equivalent(X, Y), equivalent(X, Y))), equivalent(Y, X))), true, implies(or(or(not(equivalent(X, Y)), not(equivalent(X, Y))), or(equivalent(X, Y), equivalent(X, Y))), or(or(not(equivalent(X, Y)), not(equivalent(X, Y))), equivalent(Y, X)))), true, or(or(not(equivalent(X, Y)), not(equivalent(X, Y))), equivalent(Y, X)), or(or(not(equivalent(X, Y)), not(equivalent(X, Y))), or(equivalent(X, Y), equivalent(X, Y)))), true, or(equivalent(Y, X), or(not(equivalent(X, Y)), not(equivalent(X, Y))))), true, or(equivalent(Y, X), not(equivalent(X, Y)))), true, or(not(equivalent(X, Y)), equivalent(Y, X)))
% 108.98/14.28 = { by axiom 2 (rule_2) R->L }
% 108.98/14.28 fresh(fresh(fresh(fresh3(fresh(fresh(fresh(true, true, or(equivalent(Y, X), and(implies(implies(X, Y), not(implies(Y, X))), implies(implies(X, Y), not(implies(Y, X)))))), true, or(not(or(equivalent(X, Y), equivalent(X, Y))), equivalent(Y, X))), true, implies(or(or(not(equivalent(X, Y)), not(equivalent(X, Y))), or(equivalent(X, Y), equivalent(X, Y))), or(or(not(equivalent(X, Y)), not(equivalent(X, Y))), equivalent(Y, X)))), true, or(or(not(equivalent(X, Y)), not(equivalent(X, Y))), equivalent(Y, X)), or(or(not(equivalent(X, Y)), not(equivalent(X, Y))), or(equivalent(X, Y), equivalent(X, Y)))), true, or(equivalent(Y, X), or(not(equivalent(X, Y)), not(equivalent(X, Y))))), true, or(equivalent(Y, X), not(equivalent(X, Y)))), true, or(not(equivalent(X, Y)), equivalent(Y, X)))
% 108.98/14.28 = { by axiom 3 (rule_1) R->L }
% 108.98/14.28 fresh(fresh(fresh(fresh3(fresh(fresh(fresh(fresh2(true, true, or(equivalent(Y, X), implies(implies(X, Y), not(implies(Y, X))))), true, or(equivalent(Y, X), and(implies(implies(X, Y), not(implies(Y, X))), implies(implies(X, Y), not(implies(Y, X)))))), true, or(not(or(equivalent(X, Y), equivalent(X, Y))), equivalent(Y, X))), true, implies(or(or(not(equivalent(X, Y)), not(equivalent(X, Y))), or(equivalent(X, Y), equivalent(X, Y))), or(or(not(equivalent(X, Y)), not(equivalent(X, Y))), equivalent(Y, X)))), true, or(or(not(equivalent(X, Y)), not(equivalent(X, Y))), equivalent(Y, X)), or(or(not(equivalent(X, Y)), not(equivalent(X, Y))), or(equivalent(X, Y), equivalent(X, Y)))), true, or(equivalent(Y, X), or(not(equivalent(X, Y)), not(equivalent(X, Y))))), true, or(equivalent(Y, X), not(equivalent(X, Y)))), true, or(not(equivalent(X, Y)), equivalent(Y, X)))
% 108.98/14.28 = { by axiom 10 (axiom_1_4) R->L }
% 108.98/14.28 fresh(fresh(fresh(fresh3(fresh(fresh(fresh(fresh2(axiom(implies(or(not(implies(Y, X)), not(implies(X, Y))), or(not(implies(X, Y)), not(implies(Y, X))))), true, or(equivalent(Y, X), implies(implies(X, Y), not(implies(Y, X))))), true, or(equivalent(Y, X), and(implies(implies(X, Y), not(implies(Y, X))), implies(implies(X, Y), not(implies(Y, X)))))), true, or(not(or(equivalent(X, Y), equivalent(X, Y))), equivalent(Y, X))), true, implies(or(or(not(equivalent(X, Y)), not(equivalent(X, Y))), or(equivalent(X, Y), equivalent(X, Y))), or(or(not(equivalent(X, Y)), not(equivalent(X, Y))), equivalent(Y, X)))), true, or(or(not(equivalent(X, Y)), not(equivalent(X, Y))), equivalent(Y, X)), or(or(not(equivalent(X, Y)), not(equivalent(X, Y))), or(equivalent(X, Y), equivalent(X, Y)))), true, or(equivalent(Y, X), or(not(equivalent(X, Y)), not(equivalent(X, Y))))), true, or(equivalent(Y, X), not(equivalent(X, Y)))), true, or(not(equivalent(X, Y)), equivalent(Y, X)))
% 108.98/14.28 = { by axiom 1 (implies_definition) R->L }
% 108.98/14.28 fresh(fresh(fresh(fresh3(fresh(fresh(fresh(fresh2(axiom(implies(implies(implies(Y, X), not(implies(X, Y))), or(not(implies(X, Y)), not(implies(Y, X))))), true, or(equivalent(Y, X), implies(implies(X, Y), not(implies(Y, X))))), true, or(equivalent(Y, X), and(implies(implies(X, Y), not(implies(Y, X))), implies(implies(X, Y), not(implies(Y, X)))))), true, or(not(or(equivalent(X, Y), equivalent(X, Y))), equivalent(Y, X))), true, implies(or(or(not(equivalent(X, Y)), not(equivalent(X, Y))), or(equivalent(X, Y), equivalent(X, Y))), or(or(not(equivalent(X, Y)), not(equivalent(X, Y))), equivalent(Y, X)))), true, or(or(not(equivalent(X, Y)), not(equivalent(X, Y))), equivalent(Y, X)), or(or(not(equivalent(X, Y)), not(equivalent(X, Y))), or(equivalent(X, Y), equivalent(X, Y)))), true, or(equivalent(Y, X), or(not(equivalent(X, Y)), not(equivalent(X, Y))))), true, or(equivalent(Y, X), not(equivalent(X, Y)))), true, or(not(equivalent(X, Y)), equivalent(Y, X)))
% 108.98/14.28 = { by axiom 1 (implies_definition) R->L }
% 108.98/14.28 fresh(fresh(fresh(fresh3(fresh(fresh(fresh(fresh2(axiom(implies(implies(implies(Y, X), not(implies(X, Y))), implies(implies(X, Y), not(implies(Y, X))))), true, or(equivalent(Y, X), implies(implies(X, Y), not(implies(Y, X))))), true, or(equivalent(Y, X), and(implies(implies(X, Y), not(implies(Y, X))), implies(implies(X, Y), not(implies(Y, X)))))), true, or(not(or(equivalent(X, Y), equivalent(X, Y))), equivalent(Y, X))), true, implies(or(or(not(equivalent(X, Y)), not(equivalent(X, Y))), or(equivalent(X, Y), equivalent(X, Y))), or(or(not(equivalent(X, Y)), not(equivalent(X, Y))), equivalent(Y, X)))), true, or(or(not(equivalent(X, Y)), not(equivalent(X, Y))), equivalent(Y, X)), or(or(not(equivalent(X, Y)), not(equivalent(X, Y))), or(equivalent(X, Y), equivalent(X, Y)))), true, or(equivalent(Y, X), or(not(equivalent(X, Y)), not(equivalent(X, Y))))), true, or(equivalent(Y, X), not(equivalent(X, Y)))), true, or(not(equivalent(X, Y)), equivalent(Y, X)))
% 108.98/14.28 = { by lemma 16 }
% 108.98/14.28 fresh(fresh(fresh(fresh3(fresh(fresh(fresh(fresh2(axiom(or(and(implies(Y, X), implies(X, Y)), implies(implies(X, Y), not(implies(Y, X))))), true, or(equivalent(Y, X), implies(implies(X, Y), not(implies(Y, X))))), true, or(equivalent(Y, X), and(implies(implies(X, Y), not(implies(Y, X))), implies(implies(X, Y), not(implies(Y, X)))))), true, or(not(or(equivalent(X, Y), equivalent(X, Y))), equivalent(Y, X))), true, implies(or(or(not(equivalent(X, Y)), not(equivalent(X, Y))), or(equivalent(X, Y), equivalent(X, Y))), or(or(not(equivalent(X, Y)), not(equivalent(X, Y))), equivalent(Y, X)))), true, or(or(not(equivalent(X, Y)), not(equivalent(X, Y))), equivalent(Y, X)), or(or(not(equivalent(X, Y)), not(equivalent(X, Y))), or(equivalent(X, Y), equivalent(X, Y)))), true, or(equivalent(Y, X), or(not(equivalent(X, Y)), not(equivalent(X, Y))))), true, or(equivalent(Y, X), not(equivalent(X, Y)))), true, or(not(equivalent(X, Y)), equivalent(Y, X)))
% 108.98/14.28 = { by axiom 9 (equivalent_defn) R->L }
% 108.98/14.28 fresh(fresh(fresh(fresh3(fresh(fresh(fresh(fresh2(axiom(or(equivalent(Y, X), implies(implies(X, Y), not(implies(Y, X))))), true, or(equivalent(Y, X), implies(implies(X, Y), not(implies(Y, X))))), true, or(equivalent(Y, X), and(implies(implies(X, Y), not(implies(Y, X))), implies(implies(X, Y), not(implies(Y, X)))))), true, or(not(or(equivalent(X, Y), equivalent(X, Y))), equivalent(Y, X))), true, implies(or(or(not(equivalent(X, Y)), not(equivalent(X, Y))), or(equivalent(X, Y), equivalent(X, Y))), or(or(not(equivalent(X, Y)), not(equivalent(X, Y))), equivalent(Y, X)))), true, or(or(not(equivalent(X, Y)), not(equivalent(X, Y))), equivalent(Y, X)), or(or(not(equivalent(X, Y)), not(equivalent(X, Y))), or(equivalent(X, Y), equivalent(X, Y)))), true, or(equivalent(Y, X), or(not(equivalent(X, Y)), not(equivalent(X, Y))))), true, or(equivalent(Y, X), not(equivalent(X, Y)))), true, or(not(equivalent(X, Y)), equivalent(Y, X)))
% 108.98/14.28 = { by axiom 5 (rule_1) }
% 108.98/14.28 fresh(fresh(fresh(fresh3(fresh(fresh(fresh(theorem(or(equivalent(Y, X), implies(implies(X, Y), not(implies(Y, X))))), true, or(equivalent(Y, X), and(implies(implies(X, Y), not(implies(Y, X))), implies(implies(X, Y), not(implies(Y, X)))))), true, or(not(or(equivalent(X, Y), equivalent(X, Y))), equivalent(Y, X))), true, implies(or(or(not(equivalent(X, Y)), not(equivalent(X, Y))), or(equivalent(X, Y), equivalent(X, Y))), or(or(not(equivalent(X, Y)), not(equivalent(X, Y))), equivalent(Y, X)))), true, or(or(not(equivalent(X, Y)), not(equivalent(X, Y))), equivalent(Y, X)), or(or(not(equivalent(X, Y)), not(equivalent(X, Y))), or(equivalent(X, Y), equivalent(X, Y)))), true, or(equivalent(Y, X), or(not(equivalent(X, Y)), not(equivalent(X, Y))))), true, or(equivalent(Y, X), not(equivalent(X, Y)))), true, or(not(equivalent(X, Y)), equivalent(Y, X)))
% 108.98/14.28 = { by axiom 11 (rule_2) R->L }
% 108.98/14.29 fresh(fresh(fresh(fresh3(fresh(fresh(fresh3(theorem(implies(or(equivalent(Y, X), implies(implies(X, Y), not(implies(Y, X)))), or(equivalent(Y, X), and(implies(implies(X, Y), not(implies(Y, X))), implies(implies(X, Y), not(implies(Y, X))))))), true, or(equivalent(Y, X), and(implies(implies(X, Y), not(implies(Y, X))), implies(implies(X, Y), not(implies(Y, X))))), or(equivalent(Y, X), implies(implies(X, Y), not(implies(Y, X))))), true, or(not(or(equivalent(X, Y), equivalent(X, Y))), equivalent(Y, X))), true, implies(or(or(not(equivalent(X, Y)), not(equivalent(X, Y))), or(equivalent(X, Y), equivalent(X, Y))), or(or(not(equivalent(X, Y)), not(equivalent(X, Y))), equivalent(Y, X)))), true, or(or(not(equivalent(X, Y)), not(equivalent(X, Y))), equivalent(Y, X)), or(or(not(equivalent(X, Y)), not(equivalent(X, Y))), or(equivalent(X, Y), equivalent(X, Y)))), true, or(equivalent(Y, X), or(not(equivalent(X, Y)), not(equivalent(X, Y))))), true, or(equivalent(Y, X), not(equivalent(X, Y)))), true, or(not(equivalent(X, Y)), equivalent(Y, X)))
% 108.98/14.29 = { by lemma 19 R->L }
% 108.98/14.29 fresh(fresh(fresh(fresh3(fresh(fresh(fresh3(fresh(theorem(implies(implies(implies(X, Y), not(implies(Y, X))), and(implies(implies(X, Y), not(implies(Y, X))), implies(implies(X, Y), not(implies(Y, X)))))), true, implies(or(equivalent(Y, X), implies(implies(X, Y), not(implies(Y, X)))), or(equivalent(Y, X), and(implies(implies(X, Y), not(implies(Y, X))), implies(implies(X, Y), not(implies(Y, X))))))), true, or(equivalent(Y, X), and(implies(implies(X, Y), not(implies(Y, X))), implies(implies(X, Y), not(implies(Y, X))))), or(equivalent(Y, X), implies(implies(X, Y), not(implies(Y, X))))), true, or(not(or(equivalent(X, Y), equivalent(X, Y))), equivalent(Y, X))), true, implies(or(or(not(equivalent(X, Y)), not(equivalent(X, Y))), or(equivalent(X, Y), equivalent(X, Y))), or(or(not(equivalent(X, Y)), not(equivalent(X, Y))), equivalent(Y, X)))), true, or(or(not(equivalent(X, Y)), not(equivalent(X, Y))), equivalent(Y, X)), or(or(not(equivalent(X, Y)), not(equivalent(X, Y))), or(equivalent(X, Y), equivalent(X, Y)))), true, or(equivalent(Y, X), or(not(equivalent(X, Y)), not(equivalent(X, Y))))), true, or(equivalent(Y, X), not(equivalent(X, Y)))), true, or(not(equivalent(X, Y)), equivalent(Y, X)))
% 108.98/14.29 = { by axiom 1 (implies_definition) }
% 108.98/14.29 fresh(fresh(fresh(fresh3(fresh(fresh(fresh3(fresh(theorem(or(not(implies(implies(X, Y), not(implies(Y, X)))), and(implies(implies(X, Y), not(implies(Y, X))), implies(implies(X, Y), not(implies(Y, X)))))), true, implies(or(equivalent(Y, X), implies(implies(X, Y), not(implies(Y, X)))), or(equivalent(Y, X), and(implies(implies(X, Y), not(implies(Y, X))), implies(implies(X, Y), not(implies(Y, X))))))), true, or(equivalent(Y, X), and(implies(implies(X, Y), not(implies(Y, X))), implies(implies(X, Y), not(implies(Y, X))))), or(equivalent(Y, X), implies(implies(X, Y), not(implies(Y, X))))), true, or(not(or(equivalent(X, Y), equivalent(X, Y))), equivalent(Y, X))), true, implies(or(or(not(equivalent(X, Y)), not(equivalent(X, Y))), or(equivalent(X, Y), equivalent(X, Y))), or(or(not(equivalent(X, Y)), not(equivalent(X, Y))), equivalent(Y, X)))), true, or(or(not(equivalent(X, Y)), not(equivalent(X, Y))), equivalent(Y, X)), or(or(not(equivalent(X, Y)), not(equivalent(X, Y))), or(equivalent(X, Y), equivalent(X, Y)))), true, or(equivalent(Y, X), or(not(equivalent(X, Y)), not(equivalent(X, Y))))), true, or(equivalent(Y, X), not(equivalent(X, Y)))), true, or(not(equivalent(X, Y)), equivalent(Y, X)))
% 108.98/14.29 = { by lemma 18 R->L }
% 108.98/14.29 fresh(fresh(fresh(fresh3(fresh(fresh(fresh3(fresh(fresh(theorem(or(and(implies(implies(X, Y), not(implies(Y, X))), implies(implies(X, Y), not(implies(Y, X)))), not(implies(implies(X, Y), not(implies(Y, X)))))), true, or(not(implies(implies(X, Y), not(implies(Y, X)))), and(implies(implies(X, Y), not(implies(Y, X))), implies(implies(X, Y), not(implies(Y, X)))))), true, implies(or(equivalent(Y, X), implies(implies(X, Y), not(implies(Y, X)))), or(equivalent(Y, X), and(implies(implies(X, Y), not(implies(Y, X))), implies(implies(X, Y), not(implies(Y, X))))))), true, or(equivalent(Y, X), and(implies(implies(X, Y), not(implies(Y, X))), implies(implies(X, Y), not(implies(Y, X))))), or(equivalent(Y, X), implies(implies(X, Y), not(implies(Y, X))))), true, or(not(or(equivalent(X, Y), equivalent(X, Y))), equivalent(Y, X))), true, implies(or(or(not(equivalent(X, Y)), not(equivalent(X, Y))), or(equivalent(X, Y), equivalent(X, Y))), or(or(not(equivalent(X, Y)), not(equivalent(X, Y))), equivalent(Y, X)))), true, or(or(not(equivalent(X, Y)), not(equivalent(X, Y))), equivalent(Y, X)), or(or(not(equivalent(X, Y)), not(equivalent(X, Y))), or(equivalent(X, Y), equivalent(X, Y)))), true, or(equivalent(Y, X), or(not(equivalent(X, Y)), not(equivalent(X, Y))))), true, or(equivalent(Y, X), not(equivalent(X, Y)))), true, or(not(equivalent(X, Y)), equivalent(Y, X)))
% 108.98/14.29 = { by axiom 5 (rule_1) R->L }
% 108.98/14.29 fresh(fresh(fresh(fresh3(fresh(fresh(fresh3(fresh(fresh(fresh2(axiom(or(and(implies(implies(X, Y), not(implies(Y, X))), implies(implies(X, Y), not(implies(Y, X)))), not(implies(implies(X, Y), not(implies(Y, X)))))), true, or(and(implies(implies(X, Y), not(implies(Y, X))), implies(implies(X, Y), not(implies(Y, X)))), not(implies(implies(X, Y), not(implies(Y, X)))))), true, or(not(implies(implies(X, Y), not(implies(Y, X)))), and(implies(implies(X, Y), not(implies(Y, X))), implies(implies(X, Y), not(implies(Y, X)))))), true, implies(or(equivalent(Y, X), implies(implies(X, Y), not(implies(Y, X)))), or(equivalent(Y, X), and(implies(implies(X, Y), not(implies(Y, X))), implies(implies(X, Y), not(implies(Y, X))))))), true, or(equivalent(Y, X), and(implies(implies(X, Y), not(implies(Y, X))), implies(implies(X, Y), not(implies(Y, X))))), or(equivalent(Y, X), implies(implies(X, Y), not(implies(Y, X))))), true, or(not(or(equivalent(X, Y), equivalent(X, Y))), equivalent(Y, X))), true, implies(or(or(not(equivalent(X, Y)), not(equivalent(X, Y))), or(equivalent(X, Y), equivalent(X, Y))), or(or(not(equivalent(X, Y)), not(equivalent(X, Y))), equivalent(Y, X)))), true, or(or(not(equivalent(X, Y)), not(equivalent(X, Y))), equivalent(Y, X)), or(or(not(equivalent(X, Y)), not(equivalent(X, Y))), or(equivalent(X, Y), equivalent(X, Y)))), true, or(equivalent(Y, X), or(not(equivalent(X, Y)), not(equivalent(X, Y))))), true, or(equivalent(Y, X), not(equivalent(X, Y)))), true, or(not(equivalent(X, Y)), equivalent(Y, X)))
% 108.98/14.29 = { by lemma 16 R->L }
% 108.98/14.29 fresh(fresh(fresh(fresh3(fresh(fresh(fresh3(fresh(fresh(fresh2(axiom(implies(implies(implies(implies(X, Y), not(implies(Y, X))), not(implies(implies(X, Y), not(implies(Y, X))))), not(implies(implies(X, Y), not(implies(Y, X)))))), true, or(and(implies(implies(X, Y), not(implies(Y, X))), implies(implies(X, Y), not(implies(Y, X)))), not(implies(implies(X, Y), not(implies(Y, X)))))), true, or(not(implies(implies(X, Y), not(implies(Y, X)))), and(implies(implies(X, Y), not(implies(Y, X))), implies(implies(X, Y), not(implies(Y, X)))))), true, implies(or(equivalent(Y, X), implies(implies(X, Y), not(implies(Y, X)))), or(equivalent(Y, X), and(implies(implies(X, Y), not(implies(Y, X))), implies(implies(X, Y), not(implies(Y, X))))))), true, or(equivalent(Y, X), and(implies(implies(X, Y), not(implies(Y, X))), implies(implies(X, Y), not(implies(Y, X))))), or(equivalent(Y, X), implies(implies(X, Y), not(implies(Y, X))))), true, or(not(or(equivalent(X, Y), equivalent(X, Y))), equivalent(Y, X))), true, implies(or(or(not(equivalent(X, Y)), not(equivalent(X, Y))), or(equivalent(X, Y), equivalent(X, Y))), or(or(not(equivalent(X, Y)), not(equivalent(X, Y))), equivalent(Y, X)))), true, or(or(not(equivalent(X, Y)), not(equivalent(X, Y))), equivalent(Y, X)), or(or(not(equivalent(X, Y)), not(equivalent(X, Y))), or(equivalent(X, Y), equivalent(X, Y)))), true, or(equivalent(Y, X), or(not(equivalent(X, Y)), not(equivalent(X, Y))))), true, or(equivalent(Y, X), not(equivalent(X, Y)))), true, or(not(equivalent(X, Y)), equivalent(Y, X)))
% 108.98/14.29 = { by axiom 1 (implies_definition) }
% 108.98/14.30 fresh(fresh(fresh(fresh3(fresh(fresh(fresh3(fresh(fresh(fresh2(axiom(implies(or(not(implies(implies(X, Y), not(implies(Y, X)))), not(implies(implies(X, Y), not(implies(Y, X))))), not(implies(implies(X, Y), not(implies(Y, X)))))), true, or(and(implies(implies(X, Y), not(implies(Y, X))), implies(implies(X, Y), not(implies(Y, X)))), not(implies(implies(X, Y), not(implies(Y, X)))))), true, or(not(implies(implies(X, Y), not(implies(Y, X)))), and(implies(implies(X, Y), not(implies(Y, X))), implies(implies(X, Y), not(implies(Y, X)))))), true, implies(or(equivalent(Y, X), implies(implies(X, Y), not(implies(Y, X)))), or(equivalent(Y, X), and(implies(implies(X, Y), not(implies(Y, X))), implies(implies(X, Y), not(implies(Y, X))))))), true, or(equivalent(Y, X), and(implies(implies(X, Y), not(implies(Y, X))), implies(implies(X, Y), not(implies(Y, X))))), or(equivalent(Y, X), implies(implies(X, Y), not(implies(Y, X))))), true, or(not(or(equivalent(X, Y), equivalent(X, Y))), equivalent(Y, X))), true, implies(or(or(not(equivalent(X, Y)), not(equivalent(X, Y))), or(equivalent(X, Y), equivalent(X, Y))), or(or(not(equivalent(X, Y)), not(equivalent(X, Y))), equivalent(Y, X)))), true, or(or(not(equivalent(X, Y)), not(equivalent(X, Y))), equivalent(Y, X)), or(or(not(equivalent(X, Y)), not(equivalent(X, Y))), or(equivalent(X, Y), equivalent(X, Y)))), true, or(equivalent(Y, X), or(not(equivalent(X, Y)), not(equivalent(X, Y))))), true, or(equivalent(Y, X), not(equivalent(X, Y)))), true, or(not(equivalent(X, Y)), equivalent(Y, X)))
% 108.98/14.30 = { by axiom 7 (axiom_1_2) }
% 108.98/14.30 fresh(fresh(fresh(fresh3(fresh(fresh(fresh3(fresh(fresh(fresh2(true, true, or(and(implies(implies(X, Y), not(implies(Y, X))), implies(implies(X, Y), not(implies(Y, X)))), not(implies(implies(X, Y), not(implies(Y, X)))))), true, or(not(implies(implies(X, Y), not(implies(Y, X)))), and(implies(implies(X, Y), not(implies(Y, X))), implies(implies(X, Y), not(implies(Y, X)))))), true, implies(or(equivalent(Y, X), implies(implies(X, Y), not(implies(Y, X)))), or(equivalent(Y, X), and(implies(implies(X, Y), not(implies(Y, X))), implies(implies(X, Y), not(implies(Y, X))))))), true, or(equivalent(Y, X), and(implies(implies(X, Y), not(implies(Y, X))), implies(implies(X, Y), not(implies(Y, X))))), or(equivalent(Y, X), implies(implies(X, Y), not(implies(Y, X))))), true, or(not(or(equivalent(X, Y), equivalent(X, Y))), equivalent(Y, X))), true, implies(or(or(not(equivalent(X, Y)), not(equivalent(X, Y))), or(equivalent(X, Y), equivalent(X, Y))), or(or(not(equivalent(X, Y)), not(equivalent(X, Y))), equivalent(Y, X)))), true, or(or(not(equivalent(X, Y)), not(equivalent(X, Y))), equivalent(Y, X)), or(or(not(equivalent(X, Y)), not(equivalent(X, Y))), or(equivalent(X, Y), equivalent(X, Y)))), true, or(equivalent(Y, X), or(not(equivalent(X, Y)), not(equivalent(X, Y))))), true, or(equivalent(Y, X), not(equivalent(X, Y)))), true, or(not(equivalent(X, Y)), equivalent(Y, X)))
% 108.98/14.30 = { by axiom 3 (rule_1) }
% 108.98/14.30 fresh(fresh(fresh(fresh3(fresh(fresh(fresh3(fresh(fresh(true, true, or(not(implies(implies(X, Y), not(implies(Y, X)))), and(implies(implies(X, Y), not(implies(Y, X))), implies(implies(X, Y), not(implies(Y, X)))))), true, implies(or(equivalent(Y, X), implies(implies(X, Y), not(implies(Y, X)))), or(equivalent(Y, X), and(implies(implies(X, Y), not(implies(Y, X))), implies(implies(X, Y), not(implies(Y, X))))))), true, or(equivalent(Y, X), and(implies(implies(X, Y), not(implies(Y, X))), implies(implies(X, Y), not(implies(Y, X))))), or(equivalent(Y, X), implies(implies(X, Y), not(implies(Y, X))))), true, or(not(or(equivalent(X, Y), equivalent(X, Y))), equivalent(Y, X))), true, implies(or(or(not(equivalent(X, Y)), not(equivalent(X, Y))), or(equivalent(X, Y), equivalent(X, Y))), or(or(not(equivalent(X, Y)), not(equivalent(X, Y))), equivalent(Y, X)))), true, or(or(not(equivalent(X, Y)), not(equivalent(X, Y))), equivalent(Y, X)), or(or(not(equivalent(X, Y)), not(equivalent(X, Y))), or(equivalent(X, Y), equivalent(X, Y)))), true, or(equivalent(Y, X), or(not(equivalent(X, Y)), not(equivalent(X, Y))))), true, or(equivalent(Y, X), not(equivalent(X, Y)))), true, or(not(equivalent(X, Y)), equivalent(Y, X)))
% 108.98/14.30 = { by axiom 2 (rule_2) }
% 108.98/14.30 fresh(fresh(fresh(fresh3(fresh(fresh(fresh3(fresh(true, true, implies(or(equivalent(Y, X), implies(implies(X, Y), not(implies(Y, X)))), or(equivalent(Y, X), and(implies(implies(X, Y), not(implies(Y, X))), implies(implies(X, Y), not(implies(Y, X))))))), true, or(equivalent(Y, X), and(implies(implies(X, Y), not(implies(Y, X))), implies(implies(X, Y), not(implies(Y, X))))), or(equivalent(Y, X), implies(implies(X, Y), not(implies(Y, X))))), true, or(not(or(equivalent(X, Y), equivalent(X, Y))), equivalent(Y, X))), true, implies(or(or(not(equivalent(X, Y)), not(equivalent(X, Y))), or(equivalent(X, Y), equivalent(X, Y))), or(or(not(equivalent(X, Y)), not(equivalent(X, Y))), equivalent(Y, X)))), true, or(or(not(equivalent(X, Y)), not(equivalent(X, Y))), equivalent(Y, X)), or(or(not(equivalent(X, Y)), not(equivalent(X, Y))), or(equivalent(X, Y), equivalent(X, Y)))), true, or(equivalent(Y, X), or(not(equivalent(X, Y)), not(equivalent(X, Y))))), true, or(equivalent(Y, X), not(equivalent(X, Y)))), true, or(not(equivalent(X, Y)), equivalent(Y, X)))
% 108.98/14.30 = { by axiom 2 (rule_2) }
% 108.98/14.30 fresh(fresh(fresh(fresh3(fresh(fresh(fresh3(true, true, or(equivalent(Y, X), and(implies(implies(X, Y), not(implies(Y, X))), implies(implies(X, Y), not(implies(Y, X))))), or(equivalent(Y, X), implies(implies(X, Y), not(implies(Y, X))))), true, or(not(or(equivalent(X, Y), equivalent(X, Y))), equivalent(Y, X))), true, implies(or(or(not(equivalent(X, Y)), not(equivalent(X, Y))), or(equivalent(X, Y), equivalent(X, Y))), or(or(not(equivalent(X, Y)), not(equivalent(X, Y))), equivalent(Y, X)))), true, or(or(not(equivalent(X, Y)), not(equivalent(X, Y))), equivalent(Y, X)), or(or(not(equivalent(X, Y)), not(equivalent(X, Y))), or(equivalent(X, Y), equivalent(X, Y)))), true, or(equivalent(Y, X), or(not(equivalent(X, Y)), not(equivalent(X, Y))))), true, or(equivalent(Y, X), not(equivalent(X, Y)))), true, or(not(equivalent(X, Y)), equivalent(Y, X)))
% 108.98/14.30 = { by axiom 4 (rule_2) }
% 108.98/14.30 fresh(fresh(fresh(fresh3(fresh(fresh(theorem(or(equivalent(Y, X), and(implies(implies(X, Y), not(implies(Y, X))), implies(implies(X, Y), not(implies(Y, X)))))), true, or(not(or(equivalent(X, Y), equivalent(X, Y))), equivalent(Y, X))), true, implies(or(or(not(equivalent(X, Y)), not(equivalent(X, Y))), or(equivalent(X, Y), equivalent(X, Y))), or(or(not(equivalent(X, Y)), not(equivalent(X, Y))), equivalent(Y, X)))), true, or(or(not(equivalent(X, Y)), not(equivalent(X, Y))), equivalent(Y, X)), or(or(not(equivalent(X, Y)), not(equivalent(X, Y))), or(equivalent(X, Y), equivalent(X, Y)))), true, or(equivalent(Y, X), or(not(equivalent(X, Y)), not(equivalent(X, Y))))), true, or(equivalent(Y, X), not(equivalent(X, Y)))), true, or(not(equivalent(X, Y)), equivalent(Y, X)))
% 108.98/14.30 = { by lemma 14 R->L }
% 108.98/14.30 fresh(fresh(fresh(fresh3(fresh(fresh(theorem(or(equivalent(Y, X), not(implies(implies(implies(X, Y), not(implies(Y, X))), not(implies(implies(X, Y), not(implies(Y, X)))))))), true, or(not(or(equivalent(X, Y), equivalent(X, Y))), equivalent(Y, X))), true, implies(or(or(not(equivalent(X, Y)), not(equivalent(X, Y))), or(equivalent(X, Y), equivalent(X, Y))), or(or(not(equivalent(X, Y)), not(equivalent(X, Y))), equivalent(Y, X)))), true, or(or(not(equivalent(X, Y)), not(equivalent(X, Y))), equivalent(Y, X)), or(or(not(equivalent(X, Y)), not(equivalent(X, Y))), or(equivalent(X, Y), equivalent(X, Y)))), true, or(equivalent(Y, X), or(not(equivalent(X, Y)), not(equivalent(X, Y))))), true, or(equivalent(Y, X), not(equivalent(X, Y)))), true, or(not(equivalent(X, Y)), equivalent(Y, X)))
% 108.98/14.30 = { by lemma 16 }
% 108.98/14.30 fresh(fresh(fresh(fresh3(fresh(fresh(theorem(or(equivalent(Y, X), not(or(and(implies(X, Y), implies(Y, X)), not(implies(implies(X, Y), not(implies(Y, X)))))))), true, or(not(or(equivalent(X, Y), equivalent(X, Y))), equivalent(Y, X))), true, implies(or(or(not(equivalent(X, Y)), not(equivalent(X, Y))), or(equivalent(X, Y), equivalent(X, Y))), or(or(not(equivalent(X, Y)), not(equivalent(X, Y))), equivalent(Y, X)))), true, or(or(not(equivalent(X, Y)), not(equivalent(X, Y))), equivalent(Y, X)), or(or(not(equivalent(X, Y)), not(equivalent(X, Y))), or(equivalent(X, Y), equivalent(X, Y)))), true, or(equivalent(Y, X), or(not(equivalent(X, Y)), not(equivalent(X, Y))))), true, or(equivalent(Y, X), not(equivalent(X, Y)))), true, or(not(equivalent(X, Y)), equivalent(Y, X)))
% 108.98/14.30 = { by axiom 9 (equivalent_defn) R->L }
% 108.98/14.30 fresh(fresh(fresh(fresh3(fresh(fresh(theorem(or(equivalent(Y, X), not(or(equivalent(X, Y), not(implies(implies(X, Y), not(implies(Y, X)))))))), true, or(not(or(equivalent(X, Y), equivalent(X, Y))), equivalent(Y, X))), true, implies(or(or(not(equivalent(X, Y)), not(equivalent(X, Y))), or(equivalent(X, Y), equivalent(X, Y))), or(or(not(equivalent(X, Y)), not(equivalent(X, Y))), equivalent(Y, X)))), true, or(or(not(equivalent(X, Y)), not(equivalent(X, Y))), equivalent(Y, X)), or(or(not(equivalent(X, Y)), not(equivalent(X, Y))), or(equivalent(X, Y), equivalent(X, Y)))), true, or(equivalent(Y, X), or(not(equivalent(X, Y)), not(equivalent(X, Y))))), true, or(equivalent(Y, X), not(equivalent(X, Y)))), true, or(not(equivalent(X, Y)), equivalent(Y, X)))
% 108.98/14.30 = { by lemma 14 }
% 108.98/14.30 fresh(fresh(fresh(fresh3(fresh(fresh(theorem(or(equivalent(Y, X), not(or(equivalent(X, Y), and(implies(X, Y), implies(Y, X)))))), true, or(not(or(equivalent(X, Y), equivalent(X, Y))), equivalent(Y, X))), true, implies(or(or(not(equivalent(X, Y)), not(equivalent(X, Y))), or(equivalent(X, Y), equivalent(X, Y))), or(or(not(equivalent(X, Y)), not(equivalent(X, Y))), equivalent(Y, X)))), true, or(or(not(equivalent(X, Y)), not(equivalent(X, Y))), equivalent(Y, X)), or(or(not(equivalent(X, Y)), not(equivalent(X, Y))), or(equivalent(X, Y), equivalent(X, Y)))), true, or(equivalent(Y, X), or(not(equivalent(X, Y)), not(equivalent(X, Y))))), true, or(equivalent(Y, X), not(equivalent(X, Y)))), true, or(not(equivalent(X, Y)), equivalent(Y, X)))
% 108.98/14.30 = { by axiom 9 (equivalent_defn) R->L }
% 108.98/14.30 fresh(fresh(fresh(fresh3(fresh(fresh(theorem(or(equivalent(Y, X), not(or(equivalent(X, Y), equivalent(X, Y))))), true, or(not(or(equivalent(X, Y), equivalent(X, Y))), equivalent(Y, X))), true, implies(or(or(not(equivalent(X, Y)), not(equivalent(X, Y))), or(equivalent(X, Y), equivalent(X, Y))), or(or(not(equivalent(X, Y)), not(equivalent(X, Y))), equivalent(Y, X)))), true, or(or(not(equivalent(X, Y)), not(equivalent(X, Y))), equivalent(Y, X)), or(or(not(equivalent(X, Y)), not(equivalent(X, Y))), or(equivalent(X, Y), equivalent(X, Y)))), true, or(equivalent(Y, X), or(not(equivalent(X, Y)), not(equivalent(X, Y))))), true, or(equivalent(Y, X), not(equivalent(X, Y)))), true, or(not(equivalent(X, Y)), equivalent(Y, X)))
% 108.98/14.30 = { by lemma 18 }
% 108.98/14.31 fresh(fresh(fresh(fresh3(fresh(theorem(or(not(or(equivalent(X, Y), equivalent(X, Y))), equivalent(Y, X))), true, implies(or(or(not(equivalent(X, Y)), not(equivalent(X, Y))), or(equivalent(X, Y), equivalent(X, Y))), or(or(not(equivalent(X, Y)), not(equivalent(X, Y))), equivalent(Y, X)))), true, or(or(not(equivalent(X, Y)), not(equivalent(X, Y))), equivalent(Y, X)), or(or(not(equivalent(X, Y)), not(equivalent(X, Y))), or(equivalent(X, Y), equivalent(X, Y)))), true, or(equivalent(Y, X), or(not(equivalent(X, Y)), not(equivalent(X, Y))))), true, or(equivalent(Y, X), not(equivalent(X, Y)))), true, or(not(equivalent(X, Y)), equivalent(Y, X)))
% 108.98/14.31 = { by axiom 1 (implies_definition) R->L }
% 108.98/14.31 fresh(fresh(fresh(fresh3(fresh(theorem(implies(or(equivalent(X, Y), equivalent(X, Y)), equivalent(Y, X))), true, implies(or(or(not(equivalent(X, Y)), not(equivalent(X, Y))), or(equivalent(X, Y), equivalent(X, Y))), or(or(not(equivalent(X, Y)), not(equivalent(X, Y))), equivalent(Y, X)))), true, or(or(not(equivalent(X, Y)), not(equivalent(X, Y))), equivalent(Y, X)), or(or(not(equivalent(X, Y)), not(equivalent(X, Y))), or(equivalent(X, Y), equivalent(X, Y)))), true, or(equivalent(Y, X), or(not(equivalent(X, Y)), not(equivalent(X, Y))))), true, or(equivalent(Y, X), not(equivalent(X, Y)))), true, or(not(equivalent(X, Y)), equivalent(Y, X)))
% 108.98/14.31 = { by lemma 19 }
% 108.98/14.31 fresh(fresh(fresh(fresh3(theorem(implies(or(or(not(equivalent(X, Y)), not(equivalent(X, Y))), or(equivalent(X, Y), equivalent(X, Y))), or(or(not(equivalent(X, Y)), not(equivalent(X, Y))), equivalent(Y, X)))), true, or(or(not(equivalent(X, Y)), not(equivalent(X, Y))), equivalent(Y, X)), or(or(not(equivalent(X, Y)), not(equivalent(X, Y))), or(equivalent(X, Y), equivalent(X, Y)))), true, or(equivalent(Y, X), or(not(equivalent(X, Y)), not(equivalent(X, Y))))), true, or(equivalent(Y, X), not(equivalent(X, Y)))), true, or(not(equivalent(X, Y)), equivalent(Y, X)))
% 108.98/14.31 = { by axiom 11 (rule_2) }
% 108.98/14.31 fresh(fresh(fresh(fresh(theorem(or(or(not(equivalent(X, Y)), not(equivalent(X, Y))), or(equivalent(X, Y), equivalent(X, Y)))), true, or(or(not(equivalent(X, Y)), not(equivalent(X, Y))), equivalent(Y, X))), true, or(equivalent(Y, X), or(not(equivalent(X, Y)), not(equivalent(X, Y))))), true, or(equivalent(Y, X), not(equivalent(X, Y)))), true, or(not(equivalent(X, Y)), equivalent(Y, X)))
% 108.98/14.31 = { by lemma 18 R->L }
% 108.98/14.31 fresh(fresh(fresh(fresh(fresh(theorem(or(or(equivalent(X, Y), equivalent(X, Y)), or(not(equivalent(X, Y)), not(equivalent(X, Y))))), true, or(or(not(equivalent(X, Y)), not(equivalent(X, Y))), or(equivalent(X, Y), equivalent(X, Y)))), true, or(or(not(equivalent(X, Y)), not(equivalent(X, Y))), equivalent(Y, X))), true, or(equivalent(Y, X), or(not(equivalent(X, Y)), not(equivalent(X, Y))))), true, or(equivalent(Y, X), not(equivalent(X, Y)))), true, or(not(equivalent(X, Y)), equivalent(Y, X)))
% 108.98/14.31 = { by axiom 4 (rule_2) R->L }
% 108.98/14.31 fresh(fresh(fresh(fresh(fresh(fresh3(true, true, or(or(equivalent(X, Y), equivalent(X, Y)), or(not(equivalent(X, Y)), not(equivalent(X, Y)))), or(not(equivalent(X, Y)), or(or(equivalent(X, Y), equivalent(X, Y)), not(equivalent(X, Y))))), true, or(or(not(equivalent(X, Y)), not(equivalent(X, Y))), or(equivalent(X, Y), equivalent(X, Y)))), true, or(or(not(equivalent(X, Y)), not(equivalent(X, Y))), equivalent(Y, X))), true, or(equivalent(Y, X), or(not(equivalent(X, Y)), not(equivalent(X, Y))))), true, or(equivalent(Y, X), not(equivalent(X, Y)))), true, or(not(equivalent(X, Y)), equivalent(Y, X)))
% 108.98/14.31 = { by axiom 3 (rule_1) R->L }
% 108.98/14.31 fresh(fresh(fresh(fresh(fresh(fresh3(fresh2(true, true, implies(or(not(equivalent(X, Y)), or(or(equivalent(X, Y), equivalent(X, Y)), not(equivalent(X, Y)))), or(or(equivalent(X, Y), equivalent(X, Y)), or(not(equivalent(X, Y)), not(equivalent(X, Y)))))), true, or(or(equivalent(X, Y), equivalent(X, Y)), or(not(equivalent(X, Y)), not(equivalent(X, Y)))), or(not(equivalent(X, Y)), or(or(equivalent(X, Y), equivalent(X, Y)), not(equivalent(X, Y))))), true, or(or(not(equivalent(X, Y)), not(equivalent(X, Y))), or(equivalent(X, Y), equivalent(X, Y)))), true, or(or(not(equivalent(X, Y)), not(equivalent(X, Y))), equivalent(Y, X))), true, or(equivalent(Y, X), or(not(equivalent(X, Y)), not(equivalent(X, Y))))), true, or(equivalent(Y, X), not(equivalent(X, Y)))), true, or(not(equivalent(X, Y)), equivalent(Y, X)))
% 108.98/14.31 = { by axiom 13 (axiom_1_5) R->L }
% 108.98/14.31 fresh(fresh(fresh(fresh(fresh(fresh3(fresh2(axiom(implies(or(not(equivalent(X, Y)), or(or(equivalent(X, Y), equivalent(X, Y)), not(equivalent(X, Y)))), or(or(equivalent(X, Y), equivalent(X, Y)), or(not(equivalent(X, Y)), not(equivalent(X, Y)))))), true, implies(or(not(equivalent(X, Y)), or(or(equivalent(X, Y), equivalent(X, Y)), not(equivalent(X, Y)))), or(or(equivalent(X, Y), equivalent(X, Y)), or(not(equivalent(X, Y)), not(equivalent(X, Y)))))), true, or(or(equivalent(X, Y), equivalent(X, Y)), or(not(equivalent(X, Y)), not(equivalent(X, Y)))), or(not(equivalent(X, Y)), or(or(equivalent(X, Y), equivalent(X, Y)), not(equivalent(X, Y))))), true, or(or(not(equivalent(X, Y)), not(equivalent(X, Y))), or(equivalent(X, Y), equivalent(X, Y)))), true, or(or(not(equivalent(X, Y)), not(equivalent(X, Y))), equivalent(Y, X))), true, or(equivalent(Y, X), or(not(equivalent(X, Y)), not(equivalent(X, Y))))), true, or(equivalent(Y, X), not(equivalent(X, Y)))), true, or(not(equivalent(X, Y)), equivalent(Y, X)))
% 108.98/14.31 = { by axiom 5 (rule_1) }
% 108.98/14.31 fresh(fresh(fresh(fresh(fresh(fresh3(theorem(implies(or(not(equivalent(X, Y)), or(or(equivalent(X, Y), equivalent(X, Y)), not(equivalent(X, Y)))), or(or(equivalent(X, Y), equivalent(X, Y)), or(not(equivalent(X, Y)), not(equivalent(X, Y)))))), true, or(or(equivalent(X, Y), equivalent(X, Y)), or(not(equivalent(X, Y)), not(equivalent(X, Y)))), or(not(equivalent(X, Y)), or(or(equivalent(X, Y), equivalent(X, Y)), not(equivalent(X, Y))))), true, or(or(not(equivalent(X, Y)), not(equivalent(X, Y))), or(equivalent(X, Y), equivalent(X, Y)))), true, or(or(not(equivalent(X, Y)), not(equivalent(X, Y))), equivalent(Y, X))), true, or(equivalent(Y, X), or(not(equivalent(X, Y)), not(equivalent(X, Y))))), true, or(equivalent(Y, X), not(equivalent(X, Y)))), true, or(not(equivalent(X, Y)), equivalent(Y, X)))
% 108.98/14.31 = { by axiom 11 (rule_2) }
% 108.98/14.31 fresh(fresh(fresh(fresh(fresh(fresh(theorem(or(not(equivalent(X, Y)), or(or(equivalent(X, Y), equivalent(X, Y)), not(equivalent(X, Y))))), true, or(or(equivalent(X, Y), equivalent(X, Y)), or(not(equivalent(X, Y)), not(equivalent(X, Y))))), true, or(or(not(equivalent(X, Y)), not(equivalent(X, Y))), or(equivalent(X, Y), equivalent(X, Y)))), true, or(or(not(equivalent(X, Y)), not(equivalent(X, Y))), equivalent(Y, X))), true, or(equivalent(Y, X), or(not(equivalent(X, Y)), not(equivalent(X, Y))))), true, or(equivalent(Y, X), not(equivalent(X, Y)))), true, or(not(equivalent(X, Y)), equivalent(Y, X)))
% 108.98/14.31 = { by axiom 4 (rule_2) R->L }
% 108.98/14.31 fresh(fresh(fresh(fresh(fresh(fresh(fresh3(true, true, or(not(equivalent(X, Y)), or(or(equivalent(X, Y), equivalent(X, Y)), not(equivalent(X, Y)))), or(or(equivalent(X, Y), equivalent(X, Y)), not(equivalent(X, Y)))), true, or(or(equivalent(X, Y), equivalent(X, Y)), or(not(equivalent(X, Y)), not(equivalent(X, Y))))), true, or(or(not(equivalent(X, Y)), not(equivalent(X, Y))), or(equivalent(X, Y), equivalent(X, Y)))), true, or(or(not(equivalent(X, Y)), not(equivalent(X, Y))), equivalent(Y, X))), true, or(equivalent(Y, X), or(not(equivalent(X, Y)), not(equivalent(X, Y))))), true, or(equivalent(Y, X), not(equivalent(X, Y)))), true, or(not(equivalent(X, Y)), equivalent(Y, X)))
% 108.98/14.31 = { by lemma 15 R->L }
% 108.98/14.31 fresh(fresh(fresh(fresh(fresh(fresh(fresh3(theorem(implies(or(or(equivalent(X, Y), equivalent(X, Y)), not(equivalent(X, Y))), or(not(equivalent(X, Y)), or(or(equivalent(X, Y), equivalent(X, Y)), not(equivalent(X, Y)))))), true, or(not(equivalent(X, Y)), or(or(equivalent(X, Y), equivalent(X, Y)), not(equivalent(X, Y)))), or(or(equivalent(X, Y), equivalent(X, Y)), not(equivalent(X, Y)))), true, or(or(equivalent(X, Y), equivalent(X, Y)), or(not(equivalent(X, Y)), not(equivalent(X, Y))))), true, or(or(not(equivalent(X, Y)), not(equivalent(X, Y))), or(equivalent(X, Y), equivalent(X, Y)))), true, or(or(not(equivalent(X, Y)), not(equivalent(X, Y))), equivalent(Y, X))), true, or(equivalent(Y, X), or(not(equivalent(X, Y)), not(equivalent(X, Y))))), true, or(equivalent(Y, X), not(equivalent(X, Y)))), true, or(not(equivalent(X, Y)), equivalent(Y, X)))
% 108.98/14.31 = { by axiom 11 (rule_2) }
% 108.98/14.31 fresh(fresh(fresh(fresh(fresh(fresh(fresh(theorem(or(or(equivalent(X, Y), equivalent(X, Y)), not(equivalent(X, Y)))), true, or(not(equivalent(X, Y)), or(or(equivalent(X, Y), equivalent(X, Y)), not(equivalent(X, Y))))), true, or(or(equivalent(X, Y), equivalent(X, Y)), or(not(equivalent(X, Y)), not(equivalent(X, Y))))), true, or(or(not(equivalent(X, Y)), not(equivalent(X, Y))), or(equivalent(X, Y), equivalent(X, Y)))), true, or(or(not(equivalent(X, Y)), not(equivalent(X, Y))), equivalent(Y, X))), true, or(equivalent(Y, X), or(not(equivalent(X, Y)), not(equivalent(X, Y))))), true, or(equivalent(Y, X), not(equivalent(X, Y)))), true, or(not(equivalent(X, Y)), equivalent(Y, X)))
% 108.98/14.31 = { by axiom 4 (rule_2) R->L }
% 108.98/14.31 fresh(fresh(fresh(fresh(fresh(fresh(fresh(fresh3(true, true, or(or(equivalent(X, Y), equivalent(X, Y)), not(equivalent(X, Y))), implies(equivalent(X, Y), or(equivalent(X, Y), equivalent(X, Y)))), true, or(not(equivalent(X, Y)), or(or(equivalent(X, Y), equivalent(X, Y)), not(equivalent(X, Y))))), true, or(or(equivalent(X, Y), equivalent(X, Y)), or(not(equivalent(X, Y)), not(equivalent(X, Y))))), true, or(or(not(equivalent(X, Y)), not(equivalent(X, Y))), or(equivalent(X, Y), equivalent(X, Y)))), true, or(or(not(equivalent(X, Y)), not(equivalent(X, Y))), equivalent(Y, X))), true, or(equivalent(Y, X), or(not(equivalent(X, Y)), not(equivalent(X, Y))))), true, or(equivalent(Y, X), not(equivalent(X, Y)))), true, or(not(equivalent(X, Y)), equivalent(Y, X)))
% 108.98/14.31 = { by lemma 17 R->L }
% 108.98/14.31 fresh(fresh(fresh(fresh(fresh(fresh(fresh(fresh3(theorem(implies(or(not(equivalent(X, Y)), or(equivalent(X, Y), equivalent(X, Y))), or(or(equivalent(X, Y), equivalent(X, Y)), not(equivalent(X, Y))))), true, or(or(equivalent(X, Y), equivalent(X, Y)), not(equivalent(X, Y))), implies(equivalent(X, Y), or(equivalent(X, Y), equivalent(X, Y)))), true, or(not(equivalent(X, Y)), or(or(equivalent(X, Y), equivalent(X, Y)), not(equivalent(X, Y))))), true, or(or(equivalent(X, Y), equivalent(X, Y)), or(not(equivalent(X, Y)), not(equivalent(X, Y))))), true, or(or(not(equivalent(X, Y)), not(equivalent(X, Y))), or(equivalent(X, Y), equivalent(X, Y)))), true, or(or(not(equivalent(X, Y)), not(equivalent(X, Y))), equivalent(Y, X))), true, or(equivalent(Y, X), or(not(equivalent(X, Y)), not(equivalent(X, Y))))), true, or(equivalent(Y, X), not(equivalent(X, Y)))), true, or(not(equivalent(X, Y)), equivalent(Y, X)))
% 108.98/14.31 = { by axiom 1 (implies_definition) R->L }
% 108.98/14.31 fresh(fresh(fresh(fresh(fresh(fresh(fresh(fresh3(theorem(implies(implies(equivalent(X, Y), or(equivalent(X, Y), equivalent(X, Y))), or(or(equivalent(X, Y), equivalent(X, Y)), not(equivalent(X, Y))))), true, or(or(equivalent(X, Y), equivalent(X, Y)), not(equivalent(X, Y))), implies(equivalent(X, Y), or(equivalent(X, Y), equivalent(X, Y)))), true, or(not(equivalent(X, Y)), or(or(equivalent(X, Y), equivalent(X, Y)), not(equivalent(X, Y))))), true, or(or(equivalent(X, Y), equivalent(X, Y)), or(not(equivalent(X, Y)), not(equivalent(X, Y))))), true, or(or(not(equivalent(X, Y)), not(equivalent(X, Y))), or(equivalent(X, Y), equivalent(X, Y)))), true, or(or(not(equivalent(X, Y)), not(equivalent(X, Y))), equivalent(Y, X))), true, or(equivalent(Y, X), or(not(equivalent(X, Y)), not(equivalent(X, Y))))), true, or(equivalent(Y, X), not(equivalent(X, Y)))), true, or(not(equivalent(X, Y)), equivalent(Y, X)))
% 108.98/14.32 = { by axiom 11 (rule_2) }
% 108.98/14.32 fresh(fresh(fresh(fresh(fresh(fresh(fresh(fresh(theorem(implies(equivalent(X, Y), or(equivalent(X, Y), equivalent(X, Y)))), true, or(or(equivalent(X, Y), equivalent(X, Y)), not(equivalent(X, Y)))), true, or(not(equivalent(X, Y)), or(or(equivalent(X, Y), equivalent(X, Y)), not(equivalent(X, Y))))), true, or(or(equivalent(X, Y), equivalent(X, Y)), or(not(equivalent(X, Y)), not(equivalent(X, Y))))), true, or(or(not(equivalent(X, Y)), not(equivalent(X, Y))), or(equivalent(X, Y), equivalent(X, Y)))), true, or(or(not(equivalent(X, Y)), not(equivalent(X, Y))), equivalent(Y, X))), true, or(equivalent(Y, X), or(not(equivalent(X, Y)), not(equivalent(X, Y))))), true, or(equivalent(Y, X), not(equivalent(X, Y)))), true, or(not(equivalent(X, Y)), equivalent(Y, X)))
% 108.98/14.32 = { by lemma 15 }
% 108.98/14.32 fresh(fresh(fresh(fresh(fresh(fresh(fresh(fresh(true, true, or(or(equivalent(X, Y), equivalent(X, Y)), not(equivalent(X, Y)))), true, or(not(equivalent(X, Y)), or(or(equivalent(X, Y), equivalent(X, Y)), not(equivalent(X, Y))))), true, or(or(equivalent(X, Y), equivalent(X, Y)), or(not(equivalent(X, Y)), not(equivalent(X, Y))))), true, or(or(not(equivalent(X, Y)), not(equivalent(X, Y))), or(equivalent(X, Y), equivalent(X, Y)))), true, or(or(not(equivalent(X, Y)), not(equivalent(X, Y))), equivalent(Y, X))), true, or(equivalent(Y, X), or(not(equivalent(X, Y)), not(equivalent(X, Y))))), true, or(equivalent(Y, X), not(equivalent(X, Y)))), true, or(not(equivalent(X, Y)), equivalent(Y, X)))
% 108.98/14.32 = { by axiom 2 (rule_2) }
% 108.98/14.32 fresh(fresh(fresh(fresh(fresh(fresh(fresh(true, true, or(not(equivalent(X, Y)), or(or(equivalent(X, Y), equivalent(X, Y)), not(equivalent(X, Y))))), true, or(or(equivalent(X, Y), equivalent(X, Y)), or(not(equivalent(X, Y)), not(equivalent(X, Y))))), true, or(or(not(equivalent(X, Y)), not(equivalent(X, Y))), or(equivalent(X, Y), equivalent(X, Y)))), true, or(or(not(equivalent(X, Y)), not(equivalent(X, Y))), equivalent(Y, X))), true, or(equivalent(Y, X), or(not(equivalent(X, Y)), not(equivalent(X, Y))))), true, or(equivalent(Y, X), not(equivalent(X, Y)))), true, or(not(equivalent(X, Y)), equivalent(Y, X)))
% 108.98/14.32 = { by axiom 2 (rule_2) }
% 108.98/14.32 fresh(fresh(fresh(fresh(fresh(fresh(true, true, or(or(equivalent(X, Y), equivalent(X, Y)), or(not(equivalent(X, Y)), not(equivalent(X, Y))))), true, or(or(not(equivalent(X, Y)), not(equivalent(X, Y))), or(equivalent(X, Y), equivalent(X, Y)))), true, or(or(not(equivalent(X, Y)), not(equivalent(X, Y))), equivalent(Y, X))), true, or(equivalent(Y, X), or(not(equivalent(X, Y)), not(equivalent(X, Y))))), true, or(equivalent(Y, X), not(equivalent(X, Y)))), true, or(not(equivalent(X, Y)), equivalent(Y, X)))
% 108.98/14.32 = { by axiom 2 (rule_2) }
% 108.98/14.32 fresh(fresh(fresh(fresh(fresh(true, true, or(or(not(equivalent(X, Y)), not(equivalent(X, Y))), or(equivalent(X, Y), equivalent(X, Y)))), true, or(or(not(equivalent(X, Y)), not(equivalent(X, Y))), equivalent(Y, X))), true, or(equivalent(Y, X), or(not(equivalent(X, Y)), not(equivalent(X, Y))))), true, or(equivalent(Y, X), not(equivalent(X, Y)))), true, or(not(equivalent(X, Y)), equivalent(Y, X)))
% 108.98/14.32 = { by axiom 2 (rule_2) }
% 108.98/14.32 fresh(fresh(fresh(fresh(true, true, or(or(not(equivalent(X, Y)), not(equivalent(X, Y))), equivalent(Y, X))), true, or(equivalent(Y, X), or(not(equivalent(X, Y)), not(equivalent(X, Y))))), true, or(equivalent(Y, X), not(equivalent(X, Y)))), true, or(not(equivalent(X, Y)), equivalent(Y, X)))
% 108.98/14.32 = { by axiom 2 (rule_2) }
% 108.98/14.32 fresh(fresh(fresh(true, true, or(equivalent(Y, X), or(not(equivalent(X, Y)), not(equivalent(X, Y))))), true, or(equivalent(Y, X), not(equivalent(X, Y)))), true, or(not(equivalent(X, Y)), equivalent(Y, X)))
% 108.98/14.32 = { by axiom 2 (rule_2) }
% 108.98/14.32 fresh(fresh(true, true, or(equivalent(Y, X), not(equivalent(X, Y)))), true, or(not(equivalent(X, Y)), equivalent(Y, X)))
% 108.98/14.32 = { by axiom 2 (rule_2) }
% 108.98/14.32 fresh(true, true, or(not(equivalent(X, Y)), equivalent(Y, X)))
% 108.98/14.32 = { by axiom 2 (rule_2) }
% 108.98/14.32 true
% 108.98/14.32
% 108.98/14.32 Goal 1 (prove_this): theorem(equivalent(equivalent(p, q), equivalent(q, p))) = true.
% 108.98/14.32 Proof:
% 108.98/14.32 theorem(equivalent(equivalent(p, q), equivalent(q, p)))
% 108.98/14.32 = { by axiom 4 (rule_2) R->L }
% 108.98/14.32 fresh3(true, true, equivalent(equivalent(p, q), equivalent(q, p)), implies(equivalent(p, q), equivalent(q, p)))
% 108.98/14.32 = { by axiom 2 (rule_2) R->L }
% 108.98/14.32 fresh3(fresh(true, true, implies(implies(equivalent(p, q), equivalent(q, p)), not(or(not(implies(equivalent(p, q), equivalent(q, p))), not(implies(equivalent(q, p), equivalent(p, q))))))), true, equivalent(equivalent(p, q), equivalent(q, p)), implies(equivalent(p, q), equivalent(q, p)))
% 108.98/14.32 = { by axiom 2 (rule_2) R->L }
% 108.98/14.32 fresh3(fresh(fresh(true, true, implies(or(not(implies(equivalent(p, q), equivalent(q, p))), not(implies(equivalent(q, p), equivalent(p, q)))), not(implies(equivalent(p, q), equivalent(q, p))))), true, implies(implies(equivalent(p, q), equivalent(q, p)), not(or(not(implies(equivalent(p, q), equivalent(q, p))), not(implies(equivalent(q, p), equivalent(p, q))))))), true, equivalent(equivalent(p, q), equivalent(q, p)), implies(equivalent(p, q), equivalent(q, p)))
% 108.98/14.32 = { by lemma 20 R->L }
% 108.98/14.32 fresh3(fresh(fresh(theorem(implies(equivalent(q, p), equivalent(p, q))), true, implies(or(not(implies(equivalent(p, q), equivalent(q, p))), not(implies(equivalent(q, p), equivalent(p, q)))), not(implies(equivalent(p, q), equivalent(q, p))))), true, implies(implies(equivalent(p, q), equivalent(q, p)), not(or(not(implies(equivalent(p, q), equivalent(q, p))), not(implies(equivalent(q, p), equivalent(p, q))))))), true, equivalent(equivalent(p, q), equivalent(q, p)), implies(equivalent(p, q), equivalent(q, p)))
% 108.98/14.32 = { by axiom 11 (rule_2) R->L }
% 108.98/14.32 fresh3(fresh(fresh3(theorem(implies(implies(equivalent(q, p), equivalent(p, q)), implies(or(not(implies(equivalent(p, q), equivalent(q, p))), not(implies(equivalent(q, p), equivalent(p, q)))), not(implies(equivalent(p, q), equivalent(q, p)))))), true, implies(or(not(implies(equivalent(p, q), equivalent(q, p))), not(implies(equivalent(q, p), equivalent(p, q)))), not(implies(equivalent(p, q), equivalent(q, p)))), implies(equivalent(q, p), equivalent(p, q))), true, implies(implies(equivalent(p, q), equivalent(q, p)), not(or(not(implies(equivalent(p, q), equivalent(q, p))), not(implies(equivalent(q, p), equivalent(p, q))))))), true, equivalent(equivalent(p, q), equivalent(q, p)), implies(equivalent(p, q), equivalent(q, p)))
% 108.98/14.32 = { by axiom 1 (implies_definition) }
% 108.98/14.32 fresh3(fresh(fresh3(theorem(or(not(implies(equivalent(q, p), equivalent(p, q))), implies(or(not(implies(equivalent(p, q), equivalent(q, p))), not(implies(equivalent(q, p), equivalent(p, q)))), not(implies(equivalent(p, q), equivalent(q, p)))))), true, implies(or(not(implies(equivalent(p, q), equivalent(q, p))), not(implies(equivalent(q, p), equivalent(p, q)))), not(implies(equivalent(p, q), equivalent(q, p)))), implies(equivalent(q, p), equivalent(p, q))), true, implies(implies(equivalent(p, q), equivalent(q, p)), not(or(not(implies(equivalent(p, q), equivalent(q, p))), not(implies(equivalent(q, p), equivalent(p, q))))))), true, equivalent(equivalent(p, q), equivalent(q, p)), implies(equivalent(p, q), equivalent(q, p)))
% 108.98/14.32 = { by axiom 4 (rule_2) R->L }
% 108.98/14.32 fresh3(fresh(fresh3(fresh3(true, true, or(not(implies(equivalent(q, p), equivalent(p, q))), implies(or(not(implies(equivalent(p, q), equivalent(q, p))), not(implies(equivalent(q, p), equivalent(p, q)))), not(implies(equivalent(p, q), equivalent(q, p))))), implies(or(not(implies(equivalent(p, q), equivalent(q, p))), not(implies(equivalent(q, p), equivalent(p, q)))), or(not(implies(equivalent(q, p), equivalent(p, q))), not(implies(equivalent(p, q), equivalent(q, p)))))), true, implies(or(not(implies(equivalent(p, q), equivalent(q, p))), not(implies(equivalent(q, p), equivalent(p, q)))), not(implies(equivalent(p, q), equivalent(q, p)))), implies(equivalent(q, p), equivalent(p, q))), true, implies(implies(equivalent(p, q), equivalent(q, p)), not(or(not(implies(equivalent(p, q), equivalent(q, p))), not(implies(equivalent(q, p), equivalent(p, q))))))), true, equivalent(equivalent(p, q), equivalent(q, p)), implies(equivalent(p, q), equivalent(q, p)))
% 108.98/14.32 = { by axiom 3 (rule_1) R->L }
% 108.98/14.32 fresh3(fresh(fresh3(fresh3(fresh2(true, true, implies(implies(or(not(implies(equivalent(p, q), equivalent(q, p))), not(implies(equivalent(q, p), equivalent(p, q)))), or(not(implies(equivalent(q, p), equivalent(p, q))), not(implies(equivalent(p, q), equivalent(q, p))))), or(not(implies(equivalent(q, p), equivalent(p, q))), implies(or(not(implies(equivalent(p, q), equivalent(q, p))), not(implies(equivalent(q, p), equivalent(p, q)))), not(implies(equivalent(p, q), equivalent(q, p))))))), true, or(not(implies(equivalent(q, p), equivalent(p, q))), implies(or(not(implies(equivalent(p, q), equivalent(q, p))), not(implies(equivalent(q, p), equivalent(p, q)))), not(implies(equivalent(p, q), equivalent(q, p))))), implies(or(not(implies(equivalent(p, q), equivalent(q, p))), not(implies(equivalent(q, p), equivalent(p, q)))), or(not(implies(equivalent(q, p), equivalent(p, q))), not(implies(equivalent(p, q), equivalent(q, p)))))), true, implies(or(not(implies(equivalent(p, q), equivalent(q, p))), not(implies(equivalent(q, p), equivalent(p, q)))), not(implies(equivalent(p, q), equivalent(q, p)))), implies(equivalent(q, p), equivalent(p, q))), true, implies(implies(equivalent(p, q), equivalent(q, p)), not(or(not(implies(equivalent(p, q), equivalent(q, p))), not(implies(equivalent(q, p), equivalent(p, q))))))), true, equivalent(equivalent(p, q), equivalent(q, p)), implies(equivalent(p, q), equivalent(q, p)))
% 108.98/14.33 = { by axiom 13 (axiom_1_5) R->L }
% 108.98/14.33 fresh3(fresh(fresh3(fresh3(fresh2(axiom(implies(or(not(or(not(implies(equivalent(p, q), equivalent(q, p))), not(implies(equivalent(q, p), equivalent(p, q))))), or(not(implies(equivalent(q, p), equivalent(p, q))), not(implies(equivalent(p, q), equivalent(q, p))))), or(not(implies(equivalent(q, p), equivalent(p, q))), or(not(or(not(implies(equivalent(p, q), equivalent(q, p))), not(implies(equivalent(q, p), equivalent(p, q))))), not(implies(equivalent(p, q), equivalent(q, p))))))), true, implies(implies(or(not(implies(equivalent(p, q), equivalent(q, p))), not(implies(equivalent(q, p), equivalent(p, q)))), or(not(implies(equivalent(q, p), equivalent(p, q))), not(implies(equivalent(p, q), equivalent(q, p))))), or(not(implies(equivalent(q, p), equivalent(p, q))), implies(or(not(implies(equivalent(p, q), equivalent(q, p))), not(implies(equivalent(q, p), equivalent(p, q)))), not(implies(equivalent(p, q), equivalent(q, p))))))), true, or(not(implies(equivalent(q, p), equivalent(p, q))), implies(or(not(implies(equivalent(p, q), equivalent(q, p))), not(implies(equivalent(q, p), equivalent(p, q)))), not(implies(equivalent(p, q), equivalent(q, p))))), implies(or(not(implies(equivalent(p, q), equivalent(q, p))), not(implies(equivalent(q, p), equivalent(p, q)))), or(not(implies(equivalent(q, p), equivalent(p, q))), not(implies(equivalent(p, q), equivalent(q, p)))))), true, implies(or(not(implies(equivalent(p, q), equivalent(q, p))), not(implies(equivalent(q, p), equivalent(p, q)))), not(implies(equivalent(p, q), equivalent(q, p)))), implies(equivalent(q, p), equivalent(p, q))), true, implies(implies(equivalent(p, q), equivalent(q, p)), not(or(not(implies(equivalent(p, q), equivalent(q, p))), not(implies(equivalent(q, p), equivalent(p, q))))))), true, equivalent(equivalent(p, q), equivalent(q, p)), implies(equivalent(p, q), equivalent(q, p)))
% 108.98/14.33 = { by axiom 1 (implies_definition) R->L }
% 108.98/14.33 fresh3(fresh(fresh3(fresh3(fresh2(axiom(implies(implies(or(not(implies(equivalent(p, q), equivalent(q, p))), not(implies(equivalent(q, p), equivalent(p, q)))), or(not(implies(equivalent(q, p), equivalent(p, q))), not(implies(equivalent(p, q), equivalent(q, p))))), or(not(implies(equivalent(q, p), equivalent(p, q))), or(not(or(not(implies(equivalent(p, q), equivalent(q, p))), not(implies(equivalent(q, p), equivalent(p, q))))), not(implies(equivalent(p, q), equivalent(q, p))))))), true, implies(implies(or(not(implies(equivalent(p, q), equivalent(q, p))), not(implies(equivalent(q, p), equivalent(p, q)))), or(not(implies(equivalent(q, p), equivalent(p, q))), not(implies(equivalent(p, q), equivalent(q, p))))), or(not(implies(equivalent(q, p), equivalent(p, q))), implies(or(not(implies(equivalent(p, q), equivalent(q, p))), not(implies(equivalent(q, p), equivalent(p, q)))), not(implies(equivalent(p, q), equivalent(q, p))))))), true, or(not(implies(equivalent(q, p), equivalent(p, q))), implies(or(not(implies(equivalent(p, q), equivalent(q, p))), not(implies(equivalent(q, p), equivalent(p, q)))), not(implies(equivalent(p, q), equivalent(q, p))))), implies(or(not(implies(equivalent(p, q), equivalent(q, p))), not(implies(equivalent(q, p), equivalent(p, q)))), or(not(implies(equivalent(q, p), equivalent(p, q))), not(implies(equivalent(p, q), equivalent(q, p)))))), true, implies(or(not(implies(equivalent(p, q), equivalent(q, p))), not(implies(equivalent(q, p), equivalent(p, q)))), not(implies(equivalent(p, q), equivalent(q, p)))), implies(equivalent(q, p), equivalent(p, q))), true, implies(implies(equivalent(p, q), equivalent(q, p)), not(or(not(implies(equivalent(p, q), equivalent(q, p))), not(implies(equivalent(q, p), equivalent(p, q))))))), true, equivalent(equivalent(p, q), equivalent(q, p)), implies(equivalent(p, q), equivalent(q, p)))
% 108.98/14.33 = { by axiom 1 (implies_definition) R->L }
% 108.98/14.33 fresh3(fresh(fresh3(fresh3(fresh2(axiom(implies(implies(or(not(implies(equivalent(p, q), equivalent(q, p))), not(implies(equivalent(q, p), equivalent(p, q)))), or(not(implies(equivalent(q, p), equivalent(p, q))), not(implies(equivalent(p, q), equivalent(q, p))))), or(not(implies(equivalent(q, p), equivalent(p, q))), implies(or(not(implies(equivalent(p, q), equivalent(q, p))), not(implies(equivalent(q, p), equivalent(p, q)))), not(implies(equivalent(p, q), equivalent(q, p))))))), true, implies(implies(or(not(implies(equivalent(p, q), equivalent(q, p))), not(implies(equivalent(q, p), equivalent(p, q)))), or(not(implies(equivalent(q, p), equivalent(p, q))), not(implies(equivalent(p, q), equivalent(q, p))))), or(not(implies(equivalent(q, p), equivalent(p, q))), implies(or(not(implies(equivalent(p, q), equivalent(q, p))), not(implies(equivalent(q, p), equivalent(p, q)))), not(implies(equivalent(p, q), equivalent(q, p))))))), true, or(not(implies(equivalent(q, p), equivalent(p, q))), implies(or(not(implies(equivalent(p, q), equivalent(q, p))), not(implies(equivalent(q, p), equivalent(p, q)))), not(implies(equivalent(p, q), equivalent(q, p))))), implies(or(not(implies(equivalent(p, q), equivalent(q, p))), not(implies(equivalent(q, p), equivalent(p, q)))), or(not(implies(equivalent(q, p), equivalent(p, q))), not(implies(equivalent(p, q), equivalent(q, p)))))), true, implies(or(not(implies(equivalent(p, q), equivalent(q, p))), not(implies(equivalent(q, p), equivalent(p, q)))), not(implies(equivalent(p, q), equivalent(q, p)))), implies(equivalent(q, p), equivalent(p, q))), true, implies(implies(equivalent(p, q), equivalent(q, p)), not(or(not(implies(equivalent(p, q), equivalent(q, p))), not(implies(equivalent(q, p), equivalent(p, q))))))), true, equivalent(equivalent(p, q), equivalent(q, p)), implies(equivalent(p, q), equivalent(q, p)))
% 108.98/14.33 = { by axiom 5 (rule_1) }
% 108.98/14.33 fresh3(fresh(fresh3(fresh3(theorem(implies(implies(or(not(implies(equivalent(p, q), equivalent(q, p))), not(implies(equivalent(q, p), equivalent(p, q)))), or(not(implies(equivalent(q, p), equivalent(p, q))), not(implies(equivalent(p, q), equivalent(q, p))))), or(not(implies(equivalent(q, p), equivalent(p, q))), implies(or(not(implies(equivalent(p, q), equivalent(q, p))), not(implies(equivalent(q, p), equivalent(p, q)))), not(implies(equivalent(p, q), equivalent(q, p))))))), true, or(not(implies(equivalent(q, p), equivalent(p, q))), implies(or(not(implies(equivalent(p, q), equivalent(q, p))), not(implies(equivalent(q, p), equivalent(p, q)))), not(implies(equivalent(p, q), equivalent(q, p))))), implies(or(not(implies(equivalent(p, q), equivalent(q, p))), not(implies(equivalent(q, p), equivalent(p, q)))), or(not(implies(equivalent(q, p), equivalent(p, q))), not(implies(equivalent(p, q), equivalent(q, p)))))), true, implies(or(not(implies(equivalent(p, q), equivalent(q, p))), not(implies(equivalent(q, p), equivalent(p, q)))), not(implies(equivalent(p, q), equivalent(q, p)))), implies(equivalent(q, p), equivalent(p, q))), true, implies(implies(equivalent(p, q), equivalent(q, p)), not(or(not(implies(equivalent(p, q), equivalent(q, p))), not(implies(equivalent(q, p), equivalent(p, q))))))), true, equivalent(equivalent(p, q), equivalent(q, p)), implies(equivalent(p, q), equivalent(q, p)))
% 108.98/14.33 = { by axiom 11 (rule_2) }
% 108.98/14.33 fresh3(fresh(fresh3(fresh(theorem(implies(or(not(implies(equivalent(p, q), equivalent(q, p))), not(implies(equivalent(q, p), equivalent(p, q)))), or(not(implies(equivalent(q, p), equivalent(p, q))), not(implies(equivalent(p, q), equivalent(q, p)))))), true, or(not(implies(equivalent(q, p), equivalent(p, q))), implies(or(not(implies(equivalent(p, q), equivalent(q, p))), not(implies(equivalent(q, p), equivalent(p, q)))), not(implies(equivalent(p, q), equivalent(q, p)))))), true, implies(or(not(implies(equivalent(p, q), equivalent(q, p))), not(implies(equivalent(q, p), equivalent(p, q)))), not(implies(equivalent(p, q), equivalent(q, p)))), implies(equivalent(q, p), equivalent(p, q))), true, implies(implies(equivalent(p, q), equivalent(q, p)), not(or(not(implies(equivalent(p, q), equivalent(q, p))), not(implies(equivalent(q, p), equivalent(p, q))))))), true, equivalent(equivalent(p, q), equivalent(q, p)), implies(equivalent(p, q), equivalent(q, p)))
% 108.98/14.33 = { by lemma 17 }
% 108.98/14.33 fresh3(fresh(fresh3(fresh(true, true, or(not(implies(equivalent(q, p), equivalent(p, q))), implies(or(not(implies(equivalent(p, q), equivalent(q, p))), not(implies(equivalent(q, p), equivalent(p, q)))), not(implies(equivalent(p, q), equivalent(q, p)))))), true, implies(or(not(implies(equivalent(p, q), equivalent(q, p))), not(implies(equivalent(q, p), equivalent(p, q)))), not(implies(equivalent(p, q), equivalent(q, p)))), implies(equivalent(q, p), equivalent(p, q))), true, implies(implies(equivalent(p, q), equivalent(q, p)), not(or(not(implies(equivalent(p, q), equivalent(q, p))), not(implies(equivalent(q, p), equivalent(p, q))))))), true, equivalent(equivalent(p, q), equivalent(q, p)), implies(equivalent(p, q), equivalent(q, p)))
% 108.98/14.33 = { by axiom 2 (rule_2) }
% 108.98/14.33 fresh3(fresh(fresh3(true, true, implies(or(not(implies(equivalent(p, q), equivalent(q, p))), not(implies(equivalent(q, p), equivalent(p, q)))), not(implies(equivalent(p, q), equivalent(q, p)))), implies(equivalent(q, p), equivalent(p, q))), true, implies(implies(equivalent(p, q), equivalent(q, p)), not(or(not(implies(equivalent(p, q), equivalent(q, p))), not(implies(equivalent(q, p), equivalent(p, q))))))), true, equivalent(equivalent(p, q), equivalent(q, p)), implies(equivalent(p, q), equivalent(q, p)))
% 108.98/14.33 = { by axiom 4 (rule_2) }
% 108.98/14.33 fresh3(fresh(theorem(implies(or(not(implies(equivalent(p, q), equivalent(q, p))), not(implies(equivalent(q, p), equivalent(p, q)))), not(implies(equivalent(p, q), equivalent(q, p))))), true, implies(implies(equivalent(p, q), equivalent(q, p)), not(or(not(implies(equivalent(p, q), equivalent(q, p))), not(implies(equivalent(q, p), equivalent(p, q))))))), true, equivalent(equivalent(p, q), equivalent(q, p)), implies(equivalent(p, q), equivalent(q, p)))
% 108.98/14.33 = { by axiom 1 (implies_definition) }
% 108.98/14.33 fresh3(fresh(theorem(or(not(or(not(implies(equivalent(p, q), equivalent(q, p))), not(implies(equivalent(q, p), equivalent(p, q))))), not(implies(equivalent(p, q), equivalent(q, p))))), true, implies(implies(equivalent(p, q), equivalent(q, p)), not(or(not(implies(equivalent(p, q), equivalent(q, p))), not(implies(equivalent(q, p), equivalent(p, q))))))), true, equivalent(equivalent(p, q), equivalent(q, p)), implies(equivalent(p, q), equivalent(q, p)))
% 108.98/14.33 = { by axiom 11 (rule_2) R->L }
% 108.98/14.33 fresh3(fresh3(theorem(implies(or(not(or(not(implies(equivalent(p, q), equivalent(q, p))), not(implies(equivalent(q, p), equivalent(p, q))))), not(implies(equivalent(p, q), equivalent(q, p)))), implies(implies(equivalent(p, q), equivalent(q, p)), not(or(not(implies(equivalent(p, q), equivalent(q, p))), not(implies(equivalent(q, p), equivalent(p, q)))))))), true, implies(implies(equivalent(p, q), equivalent(q, p)), not(or(not(implies(equivalent(p, q), equivalent(q, p))), not(implies(equivalent(q, p), equivalent(p, q)))))), or(not(or(not(implies(equivalent(p, q), equivalent(q, p))), not(implies(equivalent(q, p), equivalent(p, q))))), not(implies(equivalent(p, q), equivalent(q, p))))), true, equivalent(equivalent(p, q), equivalent(q, p)), implies(equivalent(p, q), equivalent(q, p)))
% 108.98/14.33 = { by axiom 1 (implies_definition) }
% 108.98/14.33 fresh3(fresh3(theorem(implies(or(not(or(not(implies(equivalent(p, q), equivalent(q, p))), not(implies(equivalent(q, p), equivalent(p, q))))), not(implies(equivalent(p, q), equivalent(q, p)))), or(not(implies(equivalent(p, q), equivalent(q, p))), not(or(not(implies(equivalent(p, q), equivalent(q, p))), not(implies(equivalent(q, p), equivalent(p, q)))))))), true, implies(implies(equivalent(p, q), equivalent(q, p)), not(or(not(implies(equivalent(p, q), equivalent(q, p))), not(implies(equivalent(q, p), equivalent(p, q)))))), or(not(or(not(implies(equivalent(p, q), equivalent(q, p))), not(implies(equivalent(q, p), equivalent(p, q))))), not(implies(equivalent(p, q), equivalent(q, p))))), true, equivalent(equivalent(p, q), equivalent(q, p)), implies(equivalent(p, q), equivalent(q, p)))
% 108.98/14.33 = { by lemma 17 }
% 108.98/14.33 fresh3(fresh3(true, true, implies(implies(equivalent(p, q), equivalent(q, p)), not(or(not(implies(equivalent(p, q), equivalent(q, p))), not(implies(equivalent(q, p), equivalent(p, q)))))), or(not(or(not(implies(equivalent(p, q), equivalent(q, p))), not(implies(equivalent(q, p), equivalent(p, q))))), not(implies(equivalent(p, q), equivalent(q, p))))), true, equivalent(equivalent(p, q), equivalent(q, p)), implies(equivalent(p, q), equivalent(q, p)))
% 108.98/14.33 = { by axiom 4 (rule_2) }
% 108.98/14.33 fresh3(theorem(implies(implies(equivalent(p, q), equivalent(q, p)), not(or(not(implies(equivalent(p, q), equivalent(q, p))), not(implies(equivalent(q, p), equivalent(p, q))))))), true, equivalent(equivalent(p, q), equivalent(q, p)), implies(equivalent(p, q), equivalent(q, p)))
% 108.98/14.33 = { by axiom 1 (implies_definition) R->L }
% 108.98/14.33 fresh3(theorem(implies(implies(equivalent(p, q), equivalent(q, p)), not(implies(implies(equivalent(p, q), equivalent(q, p)), not(implies(equivalent(q, p), equivalent(p, q))))))), true, equivalent(equivalent(p, q), equivalent(q, p)), implies(equivalent(p, q), equivalent(q, p)))
% 108.98/14.33 = { by lemma 14 }
% 108.98/14.33 fresh3(theorem(implies(implies(equivalent(p, q), equivalent(q, p)), and(implies(equivalent(p, q), equivalent(q, p)), implies(equivalent(q, p), equivalent(p, q))))), true, equivalent(equivalent(p, q), equivalent(q, p)), implies(equivalent(p, q), equivalent(q, p)))
% 108.98/14.33 = { by axiom 9 (equivalent_defn) R->L }
% 108.98/14.33 fresh3(theorem(implies(implies(equivalent(p, q), equivalent(q, p)), equivalent(equivalent(p, q), equivalent(q, p)))), true, equivalent(equivalent(p, q), equivalent(q, p)), implies(equivalent(p, q), equivalent(q, p)))
% 108.98/14.33 = { by axiom 11 (rule_2) }
% 108.98/14.33 fresh(theorem(implies(equivalent(p, q), equivalent(q, p))), true, equivalent(equivalent(p, q), equivalent(q, p)))
% 108.98/14.33 = { by lemma 20 }
% 108.98/14.33 fresh(true, true, equivalent(equivalent(p, q), equivalent(q, p)))
% 108.98/14.33 = { by axiom 2 (rule_2) }
% 108.98/14.33 true
% 108.98/14.33 % SZS output end Proof
% 108.98/14.33
% 108.98/14.33 RESULT: Unsatisfiable (the axioms are contradictory).
%------------------------------------------------------------------------------