TSTP Solution File: LCL092-1 by Twee---2.4.2
View Problem
- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : LCL092-1 : TPTP v8.1.2. Released v1.0.0.
% 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 : n005.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 08:17:23 EDT 2023
% Result : Unsatisfiable 6.06s 1.19s
% Output : Proof 7.77s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.12/0.12 % Problem : LCL092-1 : TPTP v8.1.2. Released v1.0.0.
% 0.12/0.13 % 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 : n005.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 19:00:08 EDT 2023
% 0.13/0.34 % CPUTime :
% 6.06/1.19 Command-line arguments: --no-flatten-goal
% 6.06/1.19
% 6.06/1.19 % SZS status Unsatisfiable
% 6.06/1.19
% 7.07/1.30 % SZS output start Proof
% 7.07/1.30 Take the following subset of the input axioms:
% 7.07/1.30 fof(condensed_detachment, axiom, ![X, Y]: (~is_a_theorem(implies(X, Y)) | (~is_a_theorem(X) | is_a_theorem(Y)))).
% 7.07/1.30 fof(ic_JLukasiewicz_5, axiom, ![P, Q, R, S, T]: is_a_theorem(implies(implies(implies(P, Q), implies(R, S)), implies(implies(S, P), implies(T, implies(R, P)))))).
% 7.07/1.30 fof(prove_ic_3, negated_conjecture, ~is_a_theorem(implies(implies(implies(a, b), a), a))).
% 7.07/1.30
% 7.07/1.30 Now clausify the problem and encode Horn clauses using encoding 3 of
% 7.07/1.30 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 7.07/1.30 We repeatedly replace C & s=t => u=v by the two clauses:
% 7.07/1.30 fresh(y, y, x1...xn) = u
% 7.07/1.30 C => fresh(s, t, x1...xn) = v
% 7.07/1.30 where fresh is a fresh function symbol and x1..xn are the free
% 7.07/1.30 variables of u and v.
% 7.07/1.30 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 7.07/1.30 input problem has no model of domain size 1).
% 7.07/1.30
% 7.07/1.30 The encoding turns the above axioms into the following unit equations and goals:
% 7.07/1.30
% 7.07/1.30 Axiom 1 (condensed_detachment): fresh2(X, X, Y) = true.
% 7.07/1.30 Axiom 2 (condensed_detachment): fresh(X, X, Y, Z) = is_a_theorem(Z).
% 7.07/1.30 Axiom 3 (condensed_detachment): fresh(is_a_theorem(implies(X, Y)), true, X, Y) = fresh2(is_a_theorem(X), true, Y).
% 7.07/1.30 Axiom 4 (ic_JLukasiewicz_5): is_a_theorem(implies(implies(implies(X, Y), implies(Z, W)), implies(implies(W, X), implies(V, implies(Z, X))))) = true.
% 7.07/1.30
% 7.07/1.30 Lemma 5: fresh2(is_a_theorem(implies(implies(X, Y), implies(Z, W))), true, implies(implies(W, X), implies(V, implies(Z, X)))) = is_a_theorem(implies(implies(W, X), implies(V, implies(Z, X)))).
% 7.07/1.30 Proof:
% 7.07/1.30 fresh2(is_a_theorem(implies(implies(X, Y), implies(Z, W))), true, implies(implies(W, X), implies(V, implies(Z, X))))
% 7.07/1.30 = { by axiom 3 (condensed_detachment) R->L }
% 7.07/1.30 fresh(is_a_theorem(implies(implies(implies(X, Y), implies(Z, W)), implies(implies(W, X), implies(V, implies(Z, X))))), true, implies(implies(X, Y), implies(Z, W)), implies(implies(W, X), implies(V, implies(Z, X))))
% 7.07/1.30 = { by axiom 4 (ic_JLukasiewicz_5) }
% 7.07/1.30 fresh(true, true, implies(implies(X, Y), implies(Z, W)), implies(implies(W, X), implies(V, implies(Z, X))))
% 7.07/1.30 = { by axiom 2 (condensed_detachment) }
% 7.07/1.30 is_a_theorem(implies(implies(W, X), implies(V, implies(Z, X))))
% 7.07/1.30
% 7.07/1.30 Lemma 6: is_a_theorem(implies(implies(implies(X, implies(Y, Z)), implies(Z, W)), implies(V, implies(implies(U, Z), implies(Z, W))))) = true.
% 7.07/1.30 Proof:
% 7.07/1.30 is_a_theorem(implies(implies(implies(X, implies(Y, Z)), implies(Z, W)), implies(V, implies(implies(U, Z), implies(Z, W)))))
% 7.07/1.30 = { by lemma 5 R->L }
% 7.07/1.30 fresh2(is_a_theorem(implies(implies(implies(Z, W), implies(Y, U)), implies(implies(U, Z), implies(X, implies(Y, Z))))), true, implies(implies(implies(X, implies(Y, Z)), implies(Z, W)), implies(V, implies(implies(U, Z), implies(Z, W)))))
% 7.07/1.30 = { by axiom 4 (ic_JLukasiewicz_5) }
% 7.07/1.30 fresh2(true, true, implies(implies(implies(X, implies(Y, Z)), implies(Z, W)), implies(V, implies(implies(U, Z), implies(Z, W)))))
% 7.07/1.30 = { by axiom 1 (condensed_detachment) }
% 7.07/1.30 true
% 7.07/1.30
% 7.07/1.30 Lemma 7: fresh2(is_a_theorem(implies(implies(X, implies(Y, Z)), implies(Z, W))), true, implies(V, implies(implies(U, Z), implies(Z, W)))) = is_a_theorem(implies(V, implies(implies(U, Z), implies(Z, W)))).
% 7.07/1.30 Proof:
% 7.07/1.30 fresh2(is_a_theorem(implies(implies(X, implies(Y, Z)), implies(Z, W))), true, implies(V, implies(implies(U, Z), implies(Z, W))))
% 7.07/1.30 = { by axiom 3 (condensed_detachment) R->L }
% 7.07/1.30 fresh(is_a_theorem(implies(implies(implies(X, implies(Y, Z)), implies(Z, W)), implies(V, implies(implies(U, Z), implies(Z, W))))), true, implies(implies(X, implies(Y, Z)), implies(Z, W)), implies(V, implies(implies(U, Z), implies(Z, W))))
% 7.07/1.30 = { by lemma 6 }
% 7.07/1.30 fresh(true, true, implies(implies(X, implies(Y, Z)), implies(Z, W)), implies(V, implies(implies(U, Z), implies(Z, W))))
% 7.07/1.30 = { by axiom 2 (condensed_detachment) }
% 7.07/1.30 is_a_theorem(implies(V, implies(implies(U, Z), implies(Z, W))))
% 7.07/1.30
% 7.07/1.30 Lemma 8: is_a_theorem(implies(implies(X, Y), implies(Y, implies(implies(Z, W), implies(W, Y))))) = true.
% 7.07/1.30 Proof:
% 7.07/1.30 is_a_theorem(implies(implies(X, Y), implies(Y, implies(implies(Z, W), implies(W, Y)))))
% 7.07/1.30 = { by axiom 2 (condensed_detachment) R->L }
% 7.07/1.30 fresh(true, true, implies(implies(implies(V, U), implies(T, S)), implies(implies(S, V), implies(X2, implies(T, V)))), implies(implies(X, Y), implies(Y, implies(implies(Z, W), implies(W, Y)))))
% 7.07/1.31 = { by axiom 1 (condensed_detachment) R->L }
% 7.07/1.31 fresh(fresh2(true, true, implies(implies(implies(implies(V, U), implies(T, S)), implies(implies(S, V), implies(X2, implies(T, V)))), implies(implies(X, Y), implies(Y, implies(implies(Z, W), implies(W, Y)))))), true, implies(implies(implies(V, U), implies(T, S)), implies(implies(S, V), implies(X2, implies(T, V)))), implies(implies(X, Y), implies(Y, implies(implies(Z, W), implies(W, Y)))))
% 7.07/1.31 = { by lemma 6 R->L }
% 7.07/1.31 fresh(fresh2(is_a_theorem(implies(implies(implies(Y2, implies(Z2, W)), implies(W, Y)), implies(Y, implies(implies(Z, W), implies(W, Y))))), true, implies(implies(implies(implies(V, U), implies(T, S)), implies(implies(S, V), implies(X2, implies(T, V)))), implies(implies(X, Y), implies(Y, implies(implies(Z, W), implies(W, Y)))))), true, implies(implies(implies(V, U), implies(T, S)), implies(implies(S, V), implies(X2, implies(T, V)))), implies(implies(X, Y), implies(Y, implies(implies(Z, W), implies(W, Y)))))
% 7.07/1.31 = { by lemma 7 }
% 7.07/1.31 fresh(is_a_theorem(implies(implies(implies(implies(V, U), implies(T, S)), implies(implies(S, V), implies(X2, implies(T, V)))), implies(implies(X, Y), implies(Y, implies(implies(Z, W), implies(W, Y)))))), true, implies(implies(implies(V, U), implies(T, S)), implies(implies(S, V), implies(X2, implies(T, V)))), implies(implies(X, Y), implies(Y, implies(implies(Z, W), implies(W, Y)))))
% 7.07/1.31 = { by axiom 3 (condensed_detachment) }
% 7.07/1.31 fresh2(is_a_theorem(implies(implies(implies(V, U), implies(T, S)), implies(implies(S, V), implies(X2, implies(T, V))))), true, implies(implies(X, Y), implies(Y, implies(implies(Z, W), implies(W, Y)))))
% 7.07/1.31 = { by axiom 4 (ic_JLukasiewicz_5) }
% 7.07/1.31 fresh2(true, true, implies(implies(X, Y), implies(Y, implies(implies(Z, W), implies(W, Y)))))
% 7.07/1.31 = { by axiom 1 (condensed_detachment) }
% 7.07/1.31 true
% 7.07/1.31
% 7.07/1.31 Lemma 9: is_a_theorem(implies(implies(implies(implies(X, Y), implies(Y, Z)), W), implies(V, implies(Z, W)))) = true.
% 7.07/1.31 Proof:
% 7.07/1.31 is_a_theorem(implies(implies(implies(implies(X, Y), implies(Y, Z)), W), implies(V, implies(Z, W))))
% 7.07/1.31 = { by lemma 5 R->L }
% 7.07/1.31 fresh2(is_a_theorem(implies(implies(W, Z), implies(Z, implies(implies(X, Y), implies(Y, Z))))), true, implies(implies(implies(implies(X, Y), implies(Y, Z)), W), implies(V, implies(Z, W))))
% 7.07/1.31 = { by lemma 8 }
% 7.07/1.31 fresh2(true, true, implies(implies(implies(implies(X, Y), implies(Y, Z)), W), implies(V, implies(Z, W))))
% 7.07/1.31 = { by axiom 1 (condensed_detachment) }
% 7.07/1.31 true
% 7.07/1.31
% 7.07/1.31 Lemma 10: fresh2(is_a_theorem(implies(implies(X, Y), implies(Z, implies(W, X)))), true, implies(V, implies(U, implies(Z, implies(W, X))))) = is_a_theorem(implies(V, implies(U, implies(Z, implies(W, X))))).
% 7.07/1.31 Proof:
% 7.07/1.31 fresh2(is_a_theorem(implies(implies(X, Y), implies(Z, implies(W, X)))), true, implies(V, implies(U, implies(Z, implies(W, X)))))
% 7.07/1.31 = { by axiom 3 (condensed_detachment) R->L }
% 7.07/1.31 fresh(is_a_theorem(implies(implies(implies(X, Y), implies(Z, implies(W, X))), implies(V, implies(U, implies(Z, implies(W, X)))))), true, implies(implies(X, Y), implies(Z, implies(W, X))), implies(V, implies(U, implies(Z, implies(W, X)))))
% 7.07/1.31 = { by lemma 5 R->L }
% 7.07/1.31 fresh(fresh2(is_a_theorem(implies(implies(implies(Z, implies(W, X)), Y), implies(U, implies(X, Y)))), true, implies(implies(implies(X, Y), implies(Z, implies(W, X))), implies(V, implies(U, implies(Z, implies(W, X)))))), true, implies(implies(X, Y), implies(Z, implies(W, X))), implies(V, implies(U, implies(Z, implies(W, X)))))
% 7.07/1.31 = { by lemma 5 R->L }
% 7.07/1.31 fresh(fresh2(fresh2(is_a_theorem(implies(implies(Y, X), implies(X, implies(Z, implies(W, X))))), true, implies(implies(implies(Z, implies(W, X)), Y), implies(U, implies(X, Y)))), true, implies(implies(implies(X, Y), implies(Z, implies(W, X))), implies(V, implies(U, implies(Z, implies(W, X)))))), true, implies(implies(X, Y), implies(Z, implies(W, X))), implies(V, implies(U, implies(Z, implies(W, X)))))
% 7.07/1.31 = { by axiom 2 (condensed_detachment) R->L }
% 7.07/1.31 fresh(fresh2(fresh2(fresh(true, true, implies(implies(implies(T, S), implies(X2, Y2)), implies(implies(Y2, T), implies(Z2, implies(X2, T)))), implies(implies(Y, X), implies(X, implies(Z, implies(W, X))))), true, implies(implies(implies(Z, implies(W, X)), Y), implies(U, implies(X, Y)))), true, implies(implies(implies(X, Y), implies(Z, implies(W, X))), implies(V, implies(U, implies(Z, implies(W, X)))))), true, implies(implies(X, Y), implies(Z, implies(W, X))), implies(V, implies(U, implies(Z, implies(W, X)))))
% 7.07/1.31 = { by axiom 1 (condensed_detachment) R->L }
% 7.07/1.31 fresh(fresh2(fresh2(fresh(fresh2(true, true, implies(implies(implies(implies(T, S), implies(X2, Y2)), implies(implies(Y2, T), implies(Z2, implies(X2, T)))), implies(implies(Y, X), implies(X, implies(Z, implies(W, X)))))), true, implies(implies(implies(T, S), implies(X2, Y2)), implies(implies(Y2, T), implies(Z2, implies(X2, T)))), implies(implies(Y, X), implies(X, implies(Z, implies(W, X))))), true, implies(implies(implies(Z, implies(W, X)), Y), implies(U, implies(X, Y)))), true, implies(implies(implies(X, Y), implies(Z, implies(W, X))), implies(V, implies(U, implies(Z, implies(W, X)))))), true, implies(implies(X, Y), implies(Z, implies(W, X))), implies(V, implies(U, implies(Z, implies(W, X)))))
% 7.07/1.31 = { by lemma 9 R->L }
% 7.07/1.31 fresh(fresh2(fresh2(fresh(fresh2(is_a_theorem(implies(implies(implies(implies(W2, V2), implies(V2, Z)), implies(W, X)), implies(X, implies(Z, implies(W, X))))), true, implies(implies(implies(implies(T, S), implies(X2, Y2)), implies(implies(Y2, T), implies(Z2, implies(X2, T)))), implies(implies(Y, X), implies(X, implies(Z, implies(W, X)))))), true, implies(implies(implies(T, S), implies(X2, Y2)), implies(implies(Y2, T), implies(Z2, implies(X2, T)))), implies(implies(Y, X), implies(X, implies(Z, implies(W, X))))), true, implies(implies(implies(Z, implies(W, X)), Y), implies(U, implies(X, Y)))), true, implies(implies(implies(X, Y), implies(Z, implies(W, X))), implies(V, implies(U, implies(Z, implies(W, X)))))), true, implies(implies(X, Y), implies(Z, implies(W, X))), implies(V, implies(U, implies(Z, implies(W, X)))))
% 7.07/1.31 = { by lemma 7 }
% 7.07/1.31 fresh(fresh2(fresh2(fresh(is_a_theorem(implies(implies(implies(implies(T, S), implies(X2, Y2)), implies(implies(Y2, T), implies(Z2, implies(X2, T)))), implies(implies(Y, X), implies(X, implies(Z, implies(W, X)))))), true, implies(implies(implies(T, S), implies(X2, Y2)), implies(implies(Y2, T), implies(Z2, implies(X2, T)))), implies(implies(Y, X), implies(X, implies(Z, implies(W, X))))), true, implies(implies(implies(Z, implies(W, X)), Y), implies(U, implies(X, Y)))), true, implies(implies(implies(X, Y), implies(Z, implies(W, X))), implies(V, implies(U, implies(Z, implies(W, X)))))), true, implies(implies(X, Y), implies(Z, implies(W, X))), implies(V, implies(U, implies(Z, implies(W, X)))))
% 7.07/1.31 = { by axiom 3 (condensed_detachment) }
% 7.07/1.31 fresh(fresh2(fresh2(fresh2(is_a_theorem(implies(implies(implies(T, S), implies(X2, Y2)), implies(implies(Y2, T), implies(Z2, implies(X2, T))))), true, implies(implies(Y, X), implies(X, implies(Z, implies(W, X))))), true, implies(implies(implies(Z, implies(W, X)), Y), implies(U, implies(X, Y)))), true, implies(implies(implies(X, Y), implies(Z, implies(W, X))), implies(V, implies(U, implies(Z, implies(W, X)))))), true, implies(implies(X, Y), implies(Z, implies(W, X))), implies(V, implies(U, implies(Z, implies(W, X)))))
% 7.07/1.31 = { by axiom 4 (ic_JLukasiewicz_5) }
% 7.07/1.31 fresh(fresh2(fresh2(fresh2(true, true, implies(implies(Y, X), implies(X, implies(Z, implies(W, X))))), true, implies(implies(implies(Z, implies(W, X)), Y), implies(U, implies(X, Y)))), true, implies(implies(implies(X, Y), implies(Z, implies(W, X))), implies(V, implies(U, implies(Z, implies(W, X)))))), true, implies(implies(X, Y), implies(Z, implies(W, X))), implies(V, implies(U, implies(Z, implies(W, X)))))
% 7.07/1.31 = { by axiom 1 (condensed_detachment) }
% 7.07/1.31 fresh(fresh2(fresh2(true, true, implies(implies(implies(Z, implies(W, X)), Y), implies(U, implies(X, Y)))), true, implies(implies(implies(X, Y), implies(Z, implies(W, X))), implies(V, implies(U, implies(Z, implies(W, X)))))), true, implies(implies(X, Y), implies(Z, implies(W, X))), implies(V, implies(U, implies(Z, implies(W, X)))))
% 7.07/1.31 = { by axiom 1 (condensed_detachment) }
% 7.07/1.31 fresh(fresh2(true, true, implies(implies(implies(X, Y), implies(Z, implies(W, X))), implies(V, implies(U, implies(Z, implies(W, X)))))), true, implies(implies(X, Y), implies(Z, implies(W, X))), implies(V, implies(U, implies(Z, implies(W, X)))))
% 7.07/1.31 = { by axiom 1 (condensed_detachment) }
% 7.07/1.31 fresh(true, true, implies(implies(X, Y), implies(Z, implies(W, X))), implies(V, implies(U, implies(Z, implies(W, X)))))
% 7.07/1.31 = { by axiom 2 (condensed_detachment) }
% 7.07/1.31 is_a_theorem(implies(V, implies(U, implies(Z, implies(W, X)))))
% 7.07/1.31
% 7.07/1.31 Lemma 11: is_a_theorem(implies(X, implies(Y, Y))) = true.
% 7.07/1.31 Proof:
% 7.07/1.31 is_a_theorem(implies(X, implies(Y, Y)))
% 7.07/1.31 = { by axiom 2 (condensed_detachment) R->L }
% 7.07/1.31 fresh(true, true, implies(implies(implies(Z, W), implies(V, U)), implies(implies(U, Z), implies(T, implies(V, Z)))), implies(X, implies(Y, Y)))
% 7.07/1.31 = { by axiom 1 (condensed_detachment) R->L }
% 7.07/1.31 fresh(fresh2(true, true, implies(implies(implies(implies(Z, W), implies(V, U)), implies(implies(U, Z), implies(T, implies(V, Z)))), implies(X, implies(Y, Y)))), true, implies(implies(implies(Z, W), implies(V, U)), implies(implies(U, Z), implies(T, implies(V, Z)))), implies(X, implies(Y, Y)))
% 7.07/1.31 = { by axiom 4 (ic_JLukasiewicz_5) R->L }
% 7.07/1.31 fresh(fresh2(is_a_theorem(implies(implies(implies(S, X2), implies(Y2, Z2)), implies(implies(Z2, S), implies(W2, implies(Y2, S))))), true, implies(implies(implies(implies(Z, W), implies(V, U)), implies(implies(U, Z), implies(T, implies(V, Z)))), implies(X, implies(Y, Y)))), true, implies(implies(implies(Z, W), implies(V, U)), implies(implies(U, Z), implies(T, implies(V, Z)))), implies(X, implies(Y, Y)))
% 7.07/1.31 = { by axiom 3 (condensed_detachment) R->L }
% 7.07/1.31 fresh(fresh(is_a_theorem(implies(implies(implies(implies(S, X2), implies(Y2, Z2)), implies(implies(Z2, S), implies(W2, implies(Y2, S)))), implies(implies(implies(implies(Z, W), implies(V, U)), implies(implies(U, Z), implies(T, implies(V, Z)))), implies(X, implies(Y, Y))))), true, implies(implies(implies(S, X2), implies(Y2, Z2)), implies(implies(Z2, S), implies(W2, implies(Y2, S)))), implies(implies(implies(implies(Z, W), implies(V, U)), implies(implies(U, Z), implies(T, implies(V, Z)))), implies(X, implies(Y, Y)))), true, implies(implies(implies(Z, W), implies(V, U)), implies(implies(U, Z), implies(T, implies(V, Z)))), implies(X, implies(Y, Y)))
% 7.07/1.31 = { by lemma 10 R->L }
% 7.07/1.31 fresh(fresh(fresh2(is_a_theorem(implies(implies(Y, Y), implies(X, implies(Y, Y)))), true, implies(implies(implies(implies(S, X2), implies(Y2, Z2)), implies(implies(Z2, S), implies(W2, implies(Y2, S)))), implies(implies(implies(implies(Z, W), implies(V, U)), implies(implies(U, Z), implies(T, implies(V, Z)))), implies(X, implies(Y, Y))))), true, implies(implies(implies(S, X2), implies(Y2, Z2)), implies(implies(Z2, S), implies(W2, implies(Y2, S)))), implies(implies(implies(implies(Z, W), implies(V, U)), implies(implies(U, Z), implies(T, implies(V, Z)))), implies(X, implies(Y, Y)))), true, implies(implies(implies(Z, W), implies(V, U)), implies(implies(U, Z), implies(T, implies(V, Z)))), implies(X, implies(Y, Y)))
% 7.07/1.31 = { by axiom 2 (condensed_detachment) R->L }
% 7.07/1.31 fresh(fresh(fresh2(fresh(true, true, implies(implies(implies(V2, U2), implies(T2, S2)), implies(implies(S2, V2), implies(X3, implies(T2, V2)))), implies(implies(Y, Y), implies(X, implies(Y, Y)))), true, implies(implies(implies(implies(S, X2), implies(Y2, Z2)), implies(implies(Z2, S), implies(W2, implies(Y2, S)))), implies(implies(implies(implies(Z, W), implies(V, U)), implies(implies(U, Z), implies(T, implies(V, Z)))), implies(X, implies(Y, Y))))), true, implies(implies(implies(S, X2), implies(Y2, Z2)), implies(implies(Z2, S), implies(W2, implies(Y2, S)))), implies(implies(implies(implies(Z, W), implies(V, U)), implies(implies(U, Z), implies(T, implies(V, Z)))), implies(X, implies(Y, Y)))), true, implies(implies(implies(Z, W), implies(V, U)), implies(implies(U, Z), implies(T, implies(V, Z)))), implies(X, implies(Y, Y)))
% 7.07/1.31 = { by axiom 1 (condensed_detachment) R->L }
% 7.07/1.31 fresh(fresh(fresh2(fresh(fresh2(true, true, implies(implies(implies(implies(V2, U2), implies(T2, S2)), implies(implies(S2, V2), implies(X3, implies(T2, V2)))), implies(implies(Y, Y), implies(X, implies(Y, Y))))), true, implies(implies(implies(V2, U2), implies(T2, S2)), implies(implies(S2, V2), implies(X3, implies(T2, V2)))), implies(implies(Y, Y), implies(X, implies(Y, Y)))), true, implies(implies(implies(implies(S, X2), implies(Y2, Z2)), implies(implies(Z2, S), implies(W2, implies(Y2, S)))), implies(implies(implies(implies(Z, W), implies(V, U)), implies(implies(U, Z), implies(T, implies(V, Z)))), implies(X, implies(Y, Y))))), true, implies(implies(implies(S, X2), implies(Y2, Z2)), implies(implies(Z2, S), implies(W2, implies(Y2, S)))), implies(implies(implies(implies(Z, W), implies(V, U)), implies(implies(U, Z), implies(T, implies(V, Z)))), implies(X, implies(Y, Y)))), true, implies(implies(implies(Z, W), implies(V, U)), implies(implies(U, Z), implies(T, implies(V, Z)))), implies(X, implies(Y, Y)))
% 7.07/1.31 = { by axiom 4 (ic_JLukasiewicz_5) R->L }
% 7.07/1.31 fresh(fresh(fresh2(fresh(fresh2(is_a_theorem(implies(implies(implies(Y3, Z3), implies(W3, V3)), implies(implies(V3, Y3), implies(U3, implies(W3, Y3))))), true, implies(implies(implies(implies(V2, U2), implies(T2, S2)), implies(implies(S2, V2), implies(X3, implies(T2, V2)))), implies(implies(Y, Y), implies(X, implies(Y, Y))))), true, implies(implies(implies(V2, U2), implies(T2, S2)), implies(implies(S2, V2), implies(X3, implies(T2, V2)))), implies(implies(Y, Y), implies(X, implies(Y, Y)))), true, implies(implies(implies(implies(S, X2), implies(Y2, Z2)), implies(implies(Z2, S), implies(W2, implies(Y2, S)))), implies(implies(implies(implies(Z, W), implies(V, U)), implies(implies(U, Z), implies(T, implies(V, Z)))), implies(X, implies(Y, Y))))), true, implies(implies(implies(S, X2), implies(Y2, Z2)), implies(implies(Z2, S), implies(W2, implies(Y2, S)))), implies(implies(implies(implies(Z, W), implies(V, U)), implies(implies(U, Z), implies(T, implies(V, Z)))), implies(X, implies(Y, Y)))), true, implies(implies(implies(Z, W), implies(V, U)), implies(implies(U, Z), implies(T, implies(V, Z)))), implies(X, implies(Y, Y)))
% 7.07/1.32 = { by axiom 3 (condensed_detachment) R->L }
% 7.07/1.32 fresh(fresh(fresh2(fresh(fresh(is_a_theorem(implies(implies(implies(implies(Y3, Z3), implies(W3, V3)), implies(implies(V3, Y3), implies(U3, implies(W3, Y3)))), implies(implies(implies(implies(V2, U2), implies(T2, S2)), implies(implies(S2, V2), implies(X3, implies(T2, V2)))), implies(implies(Y, Y), implies(X, implies(Y, Y)))))), true, implies(implies(implies(Y3, Z3), implies(W3, V3)), implies(implies(V3, Y3), implies(U3, implies(W3, Y3)))), implies(implies(implies(implies(V2, U2), implies(T2, S2)), implies(implies(S2, V2), implies(X3, implies(T2, V2)))), implies(implies(Y, Y), implies(X, implies(Y, Y))))), true, implies(implies(implies(V2, U2), implies(T2, S2)), implies(implies(S2, V2), implies(X3, implies(T2, V2)))), implies(implies(Y, Y), implies(X, implies(Y, Y)))), true, implies(implies(implies(implies(S, X2), implies(Y2, Z2)), implies(implies(Z2, S), implies(W2, implies(Y2, S)))), implies(implies(implies(implies(Z, W), implies(V, U)), implies(implies(U, Z), implies(T, implies(V, Z)))), implies(X, implies(Y, Y))))), true, implies(implies(implies(S, X2), implies(Y2, Z2)), implies(implies(Z2, S), implies(W2, implies(Y2, S)))), implies(implies(implies(implies(Z, W), implies(V, U)), implies(implies(U, Z), implies(T, implies(V, Z)))), implies(X, implies(Y, Y)))), true, implies(implies(implies(Z, W), implies(V, U)), implies(implies(U, Z), implies(T, implies(V, Z)))), implies(X, implies(Y, Y)))
% 7.07/1.32 = { by lemma 10 R->L }
% 7.07/1.32 fresh(fresh(fresh2(fresh(fresh(fresh2(is_a_theorem(implies(implies(implies(Y, Y), implies(Y, Y)), implies(implies(Y, Y), implies(X, implies(Y, Y))))), true, implies(implies(implies(implies(Y3, Z3), implies(W3, V3)), implies(implies(V3, Y3), implies(U3, implies(W3, Y3)))), implies(implies(implies(implies(V2, U2), implies(T2, S2)), implies(implies(S2, V2), implies(X3, implies(T2, V2)))), implies(implies(Y, Y), implies(X, implies(Y, Y)))))), true, implies(implies(implies(Y3, Z3), implies(W3, V3)), implies(implies(V3, Y3), implies(U3, implies(W3, Y3)))), implies(implies(implies(implies(V2, U2), implies(T2, S2)), implies(implies(S2, V2), implies(X3, implies(T2, V2)))), implies(implies(Y, Y), implies(X, implies(Y, Y))))), true, implies(implies(implies(V2, U2), implies(T2, S2)), implies(implies(S2, V2), implies(X3, implies(T2, V2)))), implies(implies(Y, Y), implies(X, implies(Y, Y)))), true, implies(implies(implies(implies(S, X2), implies(Y2, Z2)), implies(implies(Z2, S), implies(W2, implies(Y2, S)))), implies(implies(implies(implies(Z, W), implies(V, U)), implies(implies(U, Z), implies(T, implies(V, Z)))), implies(X, implies(Y, Y))))), true, implies(implies(implies(S, X2), implies(Y2, Z2)), implies(implies(Z2, S), implies(W2, implies(Y2, S)))), implies(implies(implies(implies(Z, W), implies(V, U)), implies(implies(U, Z), implies(T, implies(V, Z)))), implies(X, implies(Y, Y)))), true, implies(implies(implies(Z, W), implies(V, U)), implies(implies(U, Z), implies(T, implies(V, Z)))), implies(X, implies(Y, Y)))
% 7.07/1.32 = { by axiom 4 (ic_JLukasiewicz_5) }
% 7.07/1.32 fresh(fresh(fresh2(fresh(fresh(fresh2(true, true, implies(implies(implies(implies(Y3, Z3), implies(W3, V3)), implies(implies(V3, Y3), implies(U3, implies(W3, Y3)))), implies(implies(implies(implies(V2, U2), implies(T2, S2)), implies(implies(S2, V2), implies(X3, implies(T2, V2)))), implies(implies(Y, Y), implies(X, implies(Y, Y)))))), true, implies(implies(implies(Y3, Z3), implies(W3, V3)), implies(implies(V3, Y3), implies(U3, implies(W3, Y3)))), implies(implies(implies(implies(V2, U2), implies(T2, S2)), implies(implies(S2, V2), implies(X3, implies(T2, V2)))), implies(implies(Y, Y), implies(X, implies(Y, Y))))), true, implies(implies(implies(V2, U2), implies(T2, S2)), implies(implies(S2, V2), implies(X3, implies(T2, V2)))), implies(implies(Y, Y), implies(X, implies(Y, Y)))), true, implies(implies(implies(implies(S, X2), implies(Y2, Z2)), implies(implies(Z2, S), implies(W2, implies(Y2, S)))), implies(implies(implies(implies(Z, W), implies(V, U)), implies(implies(U, Z), implies(T, implies(V, Z)))), implies(X, implies(Y, Y))))), true, implies(implies(implies(S, X2), implies(Y2, Z2)), implies(implies(Z2, S), implies(W2, implies(Y2, S)))), implies(implies(implies(implies(Z, W), implies(V, U)), implies(implies(U, Z), implies(T, implies(V, Z)))), implies(X, implies(Y, Y)))), true, implies(implies(implies(Z, W), implies(V, U)), implies(implies(U, Z), implies(T, implies(V, Z)))), implies(X, implies(Y, Y)))
% 7.07/1.32 = { by axiom 1 (condensed_detachment) }
% 7.07/1.32 fresh(fresh(fresh2(fresh(fresh(true, true, implies(implies(implies(Y3, Z3), implies(W3, V3)), implies(implies(V3, Y3), implies(U3, implies(W3, Y3)))), implies(implies(implies(implies(V2, U2), implies(T2, S2)), implies(implies(S2, V2), implies(X3, implies(T2, V2)))), implies(implies(Y, Y), implies(X, implies(Y, Y))))), true, implies(implies(implies(V2, U2), implies(T2, S2)), implies(implies(S2, V2), implies(X3, implies(T2, V2)))), implies(implies(Y, Y), implies(X, implies(Y, Y)))), true, implies(implies(implies(implies(S, X2), implies(Y2, Z2)), implies(implies(Z2, S), implies(W2, implies(Y2, S)))), implies(implies(implies(implies(Z, W), implies(V, U)), implies(implies(U, Z), implies(T, implies(V, Z)))), implies(X, implies(Y, Y))))), true, implies(implies(implies(S, X2), implies(Y2, Z2)), implies(implies(Z2, S), implies(W2, implies(Y2, S)))), implies(implies(implies(implies(Z, W), implies(V, U)), implies(implies(U, Z), implies(T, implies(V, Z)))), implies(X, implies(Y, Y)))), true, implies(implies(implies(Z, W), implies(V, U)), implies(implies(U, Z), implies(T, implies(V, Z)))), implies(X, implies(Y, Y)))
% 7.07/1.32 = { by axiom 2 (condensed_detachment) }
% 7.07/1.32 fresh(fresh(fresh2(fresh(is_a_theorem(implies(implies(implies(implies(V2, U2), implies(T2, S2)), implies(implies(S2, V2), implies(X3, implies(T2, V2)))), implies(implies(Y, Y), implies(X, implies(Y, Y))))), true, implies(implies(implies(V2, U2), implies(T2, S2)), implies(implies(S2, V2), implies(X3, implies(T2, V2)))), implies(implies(Y, Y), implies(X, implies(Y, Y)))), true, implies(implies(implies(implies(S, X2), implies(Y2, Z2)), implies(implies(Z2, S), implies(W2, implies(Y2, S)))), implies(implies(implies(implies(Z, W), implies(V, U)), implies(implies(U, Z), implies(T, implies(V, Z)))), implies(X, implies(Y, Y))))), true, implies(implies(implies(S, X2), implies(Y2, Z2)), implies(implies(Z2, S), implies(W2, implies(Y2, S)))), implies(implies(implies(implies(Z, W), implies(V, U)), implies(implies(U, Z), implies(T, implies(V, Z)))), implies(X, implies(Y, Y)))), true, implies(implies(implies(Z, W), implies(V, U)), implies(implies(U, Z), implies(T, implies(V, Z)))), implies(X, implies(Y, Y)))
% 7.07/1.32 = { by axiom 3 (condensed_detachment) }
% 7.07/1.32 fresh(fresh(fresh2(fresh2(is_a_theorem(implies(implies(implies(V2, U2), implies(T2, S2)), implies(implies(S2, V2), implies(X3, implies(T2, V2))))), true, implies(implies(Y, Y), implies(X, implies(Y, Y)))), true, implies(implies(implies(implies(S, X2), implies(Y2, Z2)), implies(implies(Z2, S), implies(W2, implies(Y2, S)))), implies(implies(implies(implies(Z, W), implies(V, U)), implies(implies(U, Z), implies(T, implies(V, Z)))), implies(X, implies(Y, Y))))), true, implies(implies(implies(S, X2), implies(Y2, Z2)), implies(implies(Z2, S), implies(W2, implies(Y2, S)))), implies(implies(implies(implies(Z, W), implies(V, U)), implies(implies(U, Z), implies(T, implies(V, Z)))), implies(X, implies(Y, Y)))), true, implies(implies(implies(Z, W), implies(V, U)), implies(implies(U, Z), implies(T, implies(V, Z)))), implies(X, implies(Y, Y)))
% 7.07/1.32 = { by axiom 4 (ic_JLukasiewicz_5) }
% 7.07/1.32 fresh(fresh(fresh2(fresh2(true, true, implies(implies(Y, Y), implies(X, implies(Y, Y)))), true, implies(implies(implies(implies(S, X2), implies(Y2, Z2)), implies(implies(Z2, S), implies(W2, implies(Y2, S)))), implies(implies(implies(implies(Z, W), implies(V, U)), implies(implies(U, Z), implies(T, implies(V, Z)))), implies(X, implies(Y, Y))))), true, implies(implies(implies(S, X2), implies(Y2, Z2)), implies(implies(Z2, S), implies(W2, implies(Y2, S)))), implies(implies(implies(implies(Z, W), implies(V, U)), implies(implies(U, Z), implies(T, implies(V, Z)))), implies(X, implies(Y, Y)))), true, implies(implies(implies(Z, W), implies(V, U)), implies(implies(U, Z), implies(T, implies(V, Z)))), implies(X, implies(Y, Y)))
% 7.07/1.32 = { by axiom 1 (condensed_detachment) }
% 7.07/1.32 fresh(fresh(fresh2(true, true, implies(implies(implies(implies(S, X2), implies(Y2, Z2)), implies(implies(Z2, S), implies(W2, implies(Y2, S)))), implies(implies(implies(implies(Z, W), implies(V, U)), implies(implies(U, Z), implies(T, implies(V, Z)))), implies(X, implies(Y, Y))))), true, implies(implies(implies(S, X2), implies(Y2, Z2)), implies(implies(Z2, S), implies(W2, implies(Y2, S)))), implies(implies(implies(implies(Z, W), implies(V, U)), implies(implies(U, Z), implies(T, implies(V, Z)))), implies(X, implies(Y, Y)))), true, implies(implies(implies(Z, W), implies(V, U)), implies(implies(U, Z), implies(T, implies(V, Z)))), implies(X, implies(Y, Y)))
% 7.07/1.32 = { by axiom 1 (condensed_detachment) }
% 7.07/1.32 fresh(fresh(true, true, implies(implies(implies(S, X2), implies(Y2, Z2)), implies(implies(Z2, S), implies(W2, implies(Y2, S)))), implies(implies(implies(implies(Z, W), implies(V, U)), implies(implies(U, Z), implies(T, implies(V, Z)))), implies(X, implies(Y, Y)))), true, implies(implies(implies(Z, W), implies(V, U)), implies(implies(U, Z), implies(T, implies(V, Z)))), implies(X, implies(Y, Y)))
% 7.07/1.32 = { by axiom 2 (condensed_detachment) }
% 7.07/1.32 fresh(is_a_theorem(implies(implies(implies(implies(Z, W), implies(V, U)), implies(implies(U, Z), implies(T, implies(V, Z)))), implies(X, implies(Y, Y)))), true, implies(implies(implies(Z, W), implies(V, U)), implies(implies(U, Z), implies(T, implies(V, Z)))), implies(X, implies(Y, Y)))
% 7.07/1.32 = { by axiom 3 (condensed_detachment) }
% 7.07/1.32 fresh2(is_a_theorem(implies(implies(implies(Z, W), implies(V, U)), implies(implies(U, Z), implies(T, implies(V, Z))))), true, implies(X, implies(Y, Y)))
% 7.07/1.32 = { by axiom 4 (ic_JLukasiewicz_5) }
% 7.07/1.32 fresh2(true, true, implies(X, implies(Y, Y)))
% 7.07/1.32 = { by axiom 1 (condensed_detachment) }
% 7.07/1.32 true
% 7.07/1.32
% 7.07/1.32 Lemma 12: is_a_theorem(implies(X, implies(implies(X, Y), implies(Z, implies(W, Y))))) = true.
% 7.07/1.32 Proof:
% 7.07/1.32 is_a_theorem(implies(X, implies(implies(X, Y), implies(Z, implies(W, Y)))))
% 7.07/1.32 = { by axiom 2 (condensed_detachment) R->L }
% 7.07/1.32 fresh(true, true, implies(implies(implies(V, U), implies(T, S)), implies(implies(S, V), implies(X2, implies(T, V)))), implies(X, implies(implies(X, Y), implies(Z, implies(W, Y)))))
% 7.07/1.32 = { by axiom 1 (condensed_detachment) R->L }
% 7.07/1.32 fresh(fresh2(true, true, implies(implies(implies(implies(V, U), implies(T, S)), implies(implies(S, V), implies(X2, implies(T, V)))), implies(X, implies(implies(X, Y), implies(Z, implies(W, Y)))))), true, implies(implies(implies(V, U), implies(T, S)), implies(implies(S, V), implies(X2, implies(T, V)))), implies(X, implies(implies(X, Y), implies(Z, implies(W, Y)))))
% 7.07/1.32 = { by axiom 4 (ic_JLukasiewicz_5) R->L }
% 7.07/1.32 fresh(fresh2(is_a_theorem(implies(implies(implies(Y, W), implies(W, X)), implies(implies(X, Y), implies(Z, implies(W, Y))))), true, implies(implies(implies(implies(V, U), implies(T, S)), implies(implies(S, V), implies(X2, implies(T, V)))), implies(X, implies(implies(X, Y), implies(Z, implies(W, Y)))))), true, implies(implies(implies(V, U), implies(T, S)), implies(implies(S, V), implies(X2, implies(T, V)))), implies(X, implies(implies(X, Y), implies(Z, implies(W, Y)))))
% 7.07/1.32 = { by axiom 3 (condensed_detachment) R->L }
% 7.07/1.32 fresh(fresh(is_a_theorem(implies(implies(implies(implies(Y, W), implies(W, X)), implies(implies(X, Y), implies(Z, implies(W, Y)))), implies(implies(implies(implies(V, U), implies(T, S)), implies(implies(S, V), implies(X2, implies(T, V)))), implies(X, implies(implies(X, Y), implies(Z, implies(W, Y))))))), true, implies(implies(implies(Y, W), implies(W, X)), implies(implies(X, Y), implies(Z, implies(W, Y)))), implies(implies(implies(implies(V, U), implies(T, S)), implies(implies(S, V), implies(X2, implies(T, V)))), implies(X, implies(implies(X, Y), implies(Z, implies(W, Y)))))), true, implies(implies(implies(V, U), implies(T, S)), implies(implies(S, V), implies(X2, implies(T, V)))), implies(X, implies(implies(X, Y), implies(Z, implies(W, Y)))))
% 7.07/1.32 = { by lemma 9 }
% 7.07/1.32 fresh(fresh(true, true, implies(implies(implies(Y, W), implies(W, X)), implies(implies(X, Y), implies(Z, implies(W, Y)))), implies(implies(implies(implies(V, U), implies(T, S)), implies(implies(S, V), implies(X2, implies(T, V)))), implies(X, implies(implies(X, Y), implies(Z, implies(W, Y)))))), true, implies(implies(implies(V, U), implies(T, S)), implies(implies(S, V), implies(X2, implies(T, V)))), implies(X, implies(implies(X, Y), implies(Z, implies(W, Y)))))
% 7.07/1.32 = { by axiom 2 (condensed_detachment) }
% 7.07/1.32 fresh(is_a_theorem(implies(implies(implies(implies(V, U), implies(T, S)), implies(implies(S, V), implies(X2, implies(T, V)))), implies(X, implies(implies(X, Y), implies(Z, implies(W, Y)))))), true, implies(implies(implies(V, U), implies(T, S)), implies(implies(S, V), implies(X2, implies(T, V)))), implies(X, implies(implies(X, Y), implies(Z, implies(W, Y)))))
% 7.07/1.32 = { by axiom 3 (condensed_detachment) }
% 7.07/1.32 fresh2(is_a_theorem(implies(implies(implies(V, U), implies(T, S)), implies(implies(S, V), implies(X2, implies(T, V))))), true, implies(X, implies(implies(X, Y), implies(Z, implies(W, Y)))))
% 7.07/1.32 = { by axiom 4 (ic_JLukasiewicz_5) }
% 7.07/1.32 fresh2(true, true, implies(X, implies(implies(X, Y), implies(Z, implies(W, Y)))))
% 7.07/1.32 = { by axiom 1 (condensed_detachment) }
% 7.07/1.32 true
% 7.07/1.32
% 7.07/1.32 Lemma 13: fresh2(is_a_theorem(implies(implies(implies(X, Y), Z), Y)), true, implies(W, implies(X, Y))) = is_a_theorem(implies(W, implies(X, Y))).
% 7.07/1.32 Proof:
% 7.07/1.32 fresh2(is_a_theorem(implies(implies(implies(X, Y), Z), Y)), true, implies(W, implies(X, Y)))
% 7.07/1.32 = { by axiom 3 (condensed_detachment) R->L }
% 7.07/1.32 fresh(is_a_theorem(implies(implies(implies(implies(X, Y), Z), Y), implies(W, implies(X, Y)))), true, implies(implies(implies(X, Y), Z), Y), implies(W, implies(X, Y)))
% 7.07/1.32 = { by axiom 2 (condensed_detachment) R->L }
% 7.07/1.32 fresh(fresh(true, true, implies(implies(implies(V, U), implies(T, S)), implies(implies(S, V), implies(X2, implies(T, V)))), implies(implies(implies(implies(X, Y), Z), Y), implies(W, implies(X, Y)))), true, implies(implies(implies(X, Y), Z), Y), implies(W, implies(X, Y)))
% 7.07/1.32 = { by axiom 1 (condensed_detachment) R->L }
% 7.07/1.32 fresh(fresh(fresh2(true, true, implies(implies(implies(implies(V, U), implies(T, S)), implies(implies(S, V), implies(X2, implies(T, V)))), implies(implies(implies(implies(X, Y), Z), Y), implies(W, implies(X, Y))))), true, implies(implies(implies(V, U), implies(T, S)), implies(implies(S, V), implies(X2, implies(T, V)))), implies(implies(implies(implies(X, Y), Z), Y), implies(W, implies(X, Y)))), true, implies(implies(implies(X, Y), Z), Y), implies(W, implies(X, Y)))
% 7.07/1.32 = { by axiom 4 (ic_JLukasiewicz_5) R->L }
% 7.07/1.32 fresh(fresh(fresh2(is_a_theorem(implies(implies(implies(Y2, Z2), implies(W2, V2)), implies(implies(V2, Y2), implies(U2, implies(W2, Y2))))), true, implies(implies(implies(implies(V, U), implies(T, S)), implies(implies(S, V), implies(X2, implies(T, V)))), implies(implies(implies(implies(X, Y), Z), Y), implies(W, implies(X, Y))))), true, implies(implies(implies(V, U), implies(T, S)), implies(implies(S, V), implies(X2, implies(T, V)))), implies(implies(implies(implies(X, Y), Z), Y), implies(W, implies(X, Y)))), true, implies(implies(implies(X, Y), Z), Y), implies(W, implies(X, Y)))
% 7.07/1.32 = { by axiom 3 (condensed_detachment) R->L }
% 7.07/1.32 fresh(fresh(fresh(is_a_theorem(implies(implies(implies(implies(Y2, Z2), implies(W2, V2)), implies(implies(V2, Y2), implies(U2, implies(W2, Y2)))), implies(implies(implies(implies(V, U), implies(T, S)), implies(implies(S, V), implies(X2, implies(T, V)))), implies(implies(implies(implies(X, Y), Z), Y), implies(W, implies(X, Y)))))), true, implies(implies(implies(Y2, Z2), implies(W2, V2)), implies(implies(V2, Y2), implies(U2, implies(W2, Y2)))), implies(implies(implies(implies(V, U), implies(T, S)), implies(implies(S, V), implies(X2, implies(T, V)))), implies(implies(implies(implies(X, Y), Z), Y), implies(W, implies(X, Y))))), true, implies(implies(implies(V, U), implies(T, S)), implies(implies(S, V), implies(X2, implies(T, V)))), implies(implies(implies(implies(X, Y), Z), Y), implies(W, implies(X, Y)))), true, implies(implies(implies(X, Y), Z), Y), implies(W, implies(X, Y)))
% 7.07/1.32 = { by lemma 10 R->L }
% 7.07/1.32 fresh(fresh(fresh(fresh2(is_a_theorem(implies(implies(implies(X, Y), Z), implies(implies(implies(implies(X, Y), Z), Y), implies(W, implies(X, Y))))), true, implies(implies(implies(implies(Y2, Z2), implies(W2, V2)), implies(implies(V2, Y2), implies(U2, implies(W2, Y2)))), implies(implies(implies(implies(V, U), implies(T, S)), implies(implies(S, V), implies(X2, implies(T, V)))), implies(implies(implies(implies(X, Y), Z), Y), implies(W, implies(X, Y)))))), true, implies(implies(implies(Y2, Z2), implies(W2, V2)), implies(implies(V2, Y2), implies(U2, implies(W2, Y2)))), implies(implies(implies(implies(V, U), implies(T, S)), implies(implies(S, V), implies(X2, implies(T, V)))), implies(implies(implies(implies(X, Y), Z), Y), implies(W, implies(X, Y))))), true, implies(implies(implies(V, U), implies(T, S)), implies(implies(S, V), implies(X2, implies(T, V)))), implies(implies(implies(implies(X, Y), Z), Y), implies(W, implies(X, Y)))), true, implies(implies(implies(X, Y), Z), Y), implies(W, implies(X, Y)))
% 7.07/1.32 = { by lemma 12 }
% 7.07/1.32 fresh(fresh(fresh(fresh2(true, true, implies(implies(implies(implies(Y2, Z2), implies(W2, V2)), implies(implies(V2, Y2), implies(U2, implies(W2, Y2)))), implies(implies(implies(implies(V, U), implies(T, S)), implies(implies(S, V), implies(X2, implies(T, V)))), implies(implies(implies(implies(X, Y), Z), Y), implies(W, implies(X, Y)))))), true, implies(implies(implies(Y2, Z2), implies(W2, V2)), implies(implies(V2, Y2), implies(U2, implies(W2, Y2)))), implies(implies(implies(implies(V, U), implies(T, S)), implies(implies(S, V), implies(X2, implies(T, V)))), implies(implies(implies(implies(X, Y), Z), Y), implies(W, implies(X, Y))))), true, implies(implies(implies(V, U), implies(T, S)), implies(implies(S, V), implies(X2, implies(T, V)))), implies(implies(implies(implies(X, Y), Z), Y), implies(W, implies(X, Y)))), true, implies(implies(implies(X, Y), Z), Y), implies(W, implies(X, Y)))
% 7.07/1.32 = { by axiom 1 (condensed_detachment) }
% 7.07/1.32 fresh(fresh(fresh(true, true, implies(implies(implies(Y2, Z2), implies(W2, V2)), implies(implies(V2, Y2), implies(U2, implies(W2, Y2)))), implies(implies(implies(implies(V, U), implies(T, S)), implies(implies(S, V), implies(X2, implies(T, V)))), implies(implies(implies(implies(X, Y), Z), Y), implies(W, implies(X, Y))))), true, implies(implies(implies(V, U), implies(T, S)), implies(implies(S, V), implies(X2, implies(T, V)))), implies(implies(implies(implies(X, Y), Z), Y), implies(W, implies(X, Y)))), true, implies(implies(implies(X, Y), Z), Y), implies(W, implies(X, Y)))
% 7.07/1.32 = { by axiom 2 (condensed_detachment) }
% 7.07/1.32 fresh(fresh(is_a_theorem(implies(implies(implies(implies(V, U), implies(T, S)), implies(implies(S, V), implies(X2, implies(T, V)))), implies(implies(implies(implies(X, Y), Z), Y), implies(W, implies(X, Y))))), true, implies(implies(implies(V, U), implies(T, S)), implies(implies(S, V), implies(X2, implies(T, V)))), implies(implies(implies(implies(X, Y), Z), Y), implies(W, implies(X, Y)))), true, implies(implies(implies(X, Y), Z), Y), implies(W, implies(X, Y)))
% 7.07/1.32 = { by axiom 3 (condensed_detachment) }
% 7.07/1.32 fresh(fresh2(is_a_theorem(implies(implies(implies(V, U), implies(T, S)), implies(implies(S, V), implies(X2, implies(T, V))))), true, implies(implies(implies(implies(X, Y), Z), Y), implies(W, implies(X, Y)))), true, implies(implies(implies(X, Y), Z), Y), implies(W, implies(X, Y)))
% 7.07/1.32 = { by axiom 4 (ic_JLukasiewicz_5) }
% 7.07/1.32 fresh(fresh2(true, true, implies(implies(implies(implies(X, Y), Z), Y), implies(W, implies(X, Y)))), true, implies(implies(implies(X, Y), Z), Y), implies(W, implies(X, Y)))
% 7.07/1.32 = { by axiom 1 (condensed_detachment) }
% 7.07/1.32 fresh(true, true, implies(implies(implies(X, Y), Z), Y), implies(W, implies(X, Y)))
% 7.07/1.32 = { by axiom 2 (condensed_detachment) }
% 7.07/1.33 is_a_theorem(implies(W, implies(X, Y)))
% 7.07/1.33
% 7.07/1.33 Lemma 14: is_a_theorem(implies(X, implies(Y, implies(Z, implies(implies(Y, W), W))))) = true.
% 7.07/1.33 Proof:
% 7.07/1.33 is_a_theorem(implies(X, implies(Y, implies(Z, implies(implies(Y, W), W)))))
% 7.07/1.33 = { by lemma 13 R->L }
% 7.07/1.33 fresh2(is_a_theorem(implies(implies(implies(Y, implies(Z, implies(implies(Y, W), W))), W), implies(Z, implies(implies(Y, W), W)))), true, implies(X, implies(Y, implies(Z, implies(implies(Y, W), W)))))
% 7.07/1.33 = { by lemma 5 R->L }
% 7.07/1.33 fresh2(fresh2(is_a_theorem(implies(implies(W, implies(Z, implies(implies(Y, W), W))), implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W)))))), true, implies(implies(implies(Y, implies(Z, implies(implies(Y, W), W))), W), implies(Z, implies(implies(Y, W), W)))), true, implies(X, implies(Y, implies(Z, implies(implies(Y, W), W)))))
% 7.07/1.33 = { by axiom 2 (condensed_detachment) R->L }
% 7.07/1.33 fresh2(fresh2(fresh(true, true, implies(implies(implies(implies(W, implies(Z, implies(implies(Y, W), W))), implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W))))), W), implies(Z, implies(implies(Y, W), W))), implies(implies(W, implies(Z, implies(implies(Y, W), W))), implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W)))))), true, implies(implies(implies(Y, implies(Z, implies(implies(Y, W), W))), W), implies(Z, implies(implies(Y, W), W)))), true, implies(X, implies(Y, implies(Z, implies(implies(Y, W), W)))))
% 7.07/1.33 = { by axiom 1 (condensed_detachment) R->L }
% 7.07/1.33 fresh2(fresh2(fresh(fresh2(true, true, implies(implies(implies(implies(implies(W, implies(Z, implies(implies(Y, W), W))), implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W))))), W), implies(Z, implies(implies(Y, W), W))), implies(implies(W, implies(Z, implies(implies(Y, W), W))), implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W))))))), true, implies(implies(implies(implies(W, implies(Z, implies(implies(Y, W), W))), implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W))))), W), implies(Z, implies(implies(Y, W), W))), implies(implies(W, implies(Z, implies(implies(Y, W), W))), implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W)))))), true, implies(implies(implies(Y, implies(Z, implies(implies(Y, W), W))), W), implies(Z, implies(implies(Y, W), W)))), true, implies(X, implies(Y, implies(Z, implies(implies(Y, W), W)))))
% 7.07/1.33 = { by axiom 4 (ic_JLukasiewicz_5) R->L }
% 7.07/1.33 fresh2(fresh2(fresh(fresh2(is_a_theorem(implies(implies(implies(V, U), implies(T, S)), implies(implies(S, V), implies(X2, implies(T, V))))), true, implies(implies(implies(implies(implies(W, implies(Z, implies(implies(Y, W), W))), implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W))))), W), implies(Z, implies(implies(Y, W), W))), implies(implies(W, implies(Z, implies(implies(Y, W), W))), implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W))))))), true, implies(implies(implies(implies(W, implies(Z, implies(implies(Y, W), W))), implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W))))), W), implies(Z, implies(implies(Y, W), W))), implies(implies(W, implies(Z, implies(implies(Y, W), W))), implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W)))))), true, implies(implies(implies(Y, implies(Z, implies(implies(Y, W), W))), W), implies(Z, implies(implies(Y, W), W)))), true, implies(X, implies(Y, implies(Z, implies(implies(Y, W), W)))))
% 7.07/1.33 = { by axiom 3 (condensed_detachment) R->L }
% 7.07/1.34 fresh2(fresh2(fresh(fresh(is_a_theorem(implies(implies(implies(implies(V, U), implies(T, S)), implies(implies(S, V), implies(X2, implies(T, V)))), implies(implies(implies(implies(implies(W, implies(Z, implies(implies(Y, W), W))), implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W))))), W), implies(Z, implies(implies(Y, W), W))), implies(implies(W, implies(Z, implies(implies(Y, W), W))), implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W)))))))), true, implies(implies(implies(V, U), implies(T, S)), implies(implies(S, V), implies(X2, implies(T, V)))), implies(implies(implies(implies(implies(W, implies(Z, implies(implies(Y, W), W))), implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W))))), W), implies(Z, implies(implies(Y, W), W))), implies(implies(W, implies(Z, implies(implies(Y, W), W))), implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W))))))), true, implies(implies(implies(implies(W, implies(Z, implies(implies(Y, W), W))), implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W))))), W), implies(Z, implies(implies(Y, W), W))), implies(implies(W, implies(Z, implies(implies(Y, W), W))), implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W)))))), true, implies(implies(implies(Y, implies(Z, implies(implies(Y, W), W))), W), implies(Z, implies(implies(Y, W), W)))), true, implies(X, implies(Y, implies(Z, implies(implies(Y, W), W)))))
% 7.07/1.34 = { by axiom 2 (condensed_detachment) R->L }
% 7.07/1.34 fresh2(fresh2(fresh(fresh(fresh(true, true, implies(implies(implies(implies(Y2, Z2), implies(Z2, implies(Y, W))), implies(Y, implies(Z, implies(implies(Y, W), W)))), implies(implies(W, implies(Z, implies(implies(Y, W), W))), implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W)))))), implies(implies(implies(implies(V, U), implies(T, S)), implies(implies(S, V), implies(X2, implies(T, V)))), implies(implies(implies(implies(implies(W, implies(Z, implies(implies(Y, W), W))), implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W))))), W), implies(Z, implies(implies(Y, W), W))), implies(implies(W, implies(Z, implies(implies(Y, W), W))), implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W)))))))), true, implies(implies(implies(V, U), implies(T, S)), implies(implies(S, V), implies(X2, implies(T, V)))), implies(implies(implies(implies(implies(W, implies(Z, implies(implies(Y, W), W))), implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W))))), W), implies(Z, implies(implies(Y, W), W))), implies(implies(W, implies(Z, implies(implies(Y, W), W))), implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W))))))), true, implies(implies(implies(implies(W, implies(Z, implies(implies(Y, W), W))), implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W))))), W), implies(Z, implies(implies(Y, W), W))), implies(implies(W, implies(Z, implies(implies(Y, W), W))), implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W)))))), true, implies(implies(implies(Y, implies(Z, implies(implies(Y, W), W))), W), implies(Z, implies(implies(Y, W), W)))), true, implies(X, implies(Y, implies(Z, implies(implies(Y, W), W)))))
% 7.07/1.34 = { by axiom 1 (condensed_detachment) R->L }
% 7.07/1.34 fresh2(fresh2(fresh(fresh(fresh(fresh2(true, true, implies(implies(implies(implies(implies(Y2, Z2), implies(Z2, implies(Y, W))), implies(Y, implies(Z, implies(implies(Y, W), W)))), implies(implies(W, implies(Z, implies(implies(Y, W), W))), implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W)))))), implies(implies(implies(implies(V, U), implies(T, S)), implies(implies(S, V), implies(X2, implies(T, V)))), implies(implies(implies(implies(implies(W, implies(Z, implies(implies(Y, W), W))), implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W))))), W), implies(Z, implies(implies(Y, W), W))), implies(implies(W, implies(Z, implies(implies(Y, W), W))), implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W))))))))), true, implies(implies(implies(implies(Y2, Z2), implies(Z2, implies(Y, W))), implies(Y, implies(Z, implies(implies(Y, W), W)))), implies(implies(W, implies(Z, implies(implies(Y, W), W))), implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W)))))), implies(implies(implies(implies(V, U), implies(T, S)), implies(implies(S, V), implies(X2, implies(T, V)))), implies(implies(implies(implies(implies(W, implies(Z, implies(implies(Y, W), W))), implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W))))), W), implies(Z, implies(implies(Y, W), W))), implies(implies(W, implies(Z, implies(implies(Y, W), W))), implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W)))))))), true, implies(implies(implies(V, U), implies(T, S)), implies(implies(S, V), implies(X2, implies(T, V)))), implies(implies(implies(implies(implies(W, implies(Z, implies(implies(Y, W), W))), implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W))))), W), implies(Z, implies(implies(Y, W), W))), implies(implies(W, implies(Z, implies(implies(Y, W), W))), implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W))))))), true, implies(implies(implies(implies(W, implies(Z, implies(implies(Y, W), W))), implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W))))), W), implies(Z, implies(implies(Y, W), W))), implies(implies(W, implies(Z, implies(implies(Y, W), W))), implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W)))))), true, implies(implies(implies(Y, implies(Z, implies(implies(Y, W), W))), W), implies(Z, implies(implies(Y, W), W)))), true, implies(X, implies(Y, implies(Z, implies(implies(Y, W), W)))))
% 7.07/1.34 = { by lemma 12 R->L }
% 7.07/1.35 fresh2(fresh2(fresh(fresh(fresh(fresh2(is_a_theorem(implies(implies(implies(implies(W, implies(Z, implies(implies(Y, W), W))), implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W))))), W), implies(implies(implies(implies(implies(W, implies(Z, implies(implies(Y, W), W))), implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W))))), W), implies(Z, implies(implies(Y, W), W))), implies(implies(implies(Y2, Z2), implies(Z2, implies(Y, W))), implies(Y, implies(Z, implies(implies(Y, W), W))))))), true, implies(implies(implies(implies(implies(Y2, Z2), implies(Z2, implies(Y, W))), implies(Y, implies(Z, implies(implies(Y, W), W)))), implies(implies(W, implies(Z, implies(implies(Y, W), W))), implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W)))))), implies(implies(implies(implies(V, U), implies(T, S)), implies(implies(S, V), implies(X2, implies(T, V)))), implies(implies(implies(implies(implies(W, implies(Z, implies(implies(Y, W), W))), implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W))))), W), implies(Z, implies(implies(Y, W), W))), implies(implies(W, implies(Z, implies(implies(Y, W), W))), implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W))))))))), true, implies(implies(implies(implies(Y2, Z2), implies(Z2, implies(Y, W))), implies(Y, implies(Z, implies(implies(Y, W), W)))), implies(implies(W, implies(Z, implies(implies(Y, W), W))), implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W)))))), implies(implies(implies(implies(V, U), implies(T, S)), implies(implies(S, V), implies(X2, implies(T, V)))), implies(implies(implies(implies(implies(W, implies(Z, implies(implies(Y, W), W))), implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W))))), W), implies(Z, implies(implies(Y, W), W))), implies(implies(W, implies(Z, implies(implies(Y, W), W))), implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W)))))))), true, implies(implies(implies(V, U), implies(T, S)), implies(implies(S, V), implies(X2, implies(T, V)))), implies(implies(implies(implies(implies(W, implies(Z, implies(implies(Y, W), W))), implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W))))), W), implies(Z, implies(implies(Y, W), W))), implies(implies(W, implies(Z, implies(implies(Y, W), W))), implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W))))))), true, implies(implies(implies(implies(W, implies(Z, implies(implies(Y, W), W))), implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W))))), W), implies(Z, implies(implies(Y, W), W))), implies(implies(W, implies(Z, implies(implies(Y, W), W))), implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W)))))), true, implies(implies(implies(Y, implies(Z, implies(implies(Y, W), W))), W), implies(Z, implies(implies(Y, W), W)))), true, implies(X, implies(Y, implies(Z, implies(implies(Y, W), W)))))
% 7.07/1.35 = { by lemma 5 }
% 7.07/1.35 fresh2(fresh2(fresh(fresh(fresh(is_a_theorem(implies(implies(implies(implies(implies(Y2, Z2), implies(Z2, implies(Y, W))), implies(Y, implies(Z, implies(implies(Y, W), W)))), implies(implies(W, implies(Z, implies(implies(Y, W), W))), implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W)))))), implies(implies(implies(implies(V, U), implies(T, S)), implies(implies(S, V), implies(X2, implies(T, V)))), implies(implies(implies(implies(implies(W, implies(Z, implies(implies(Y, W), W))), implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W))))), W), implies(Z, implies(implies(Y, W), W))), implies(implies(W, implies(Z, implies(implies(Y, W), W))), implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W))))))))), true, implies(implies(implies(implies(Y2, Z2), implies(Z2, implies(Y, W))), implies(Y, implies(Z, implies(implies(Y, W), W)))), implies(implies(W, implies(Z, implies(implies(Y, W), W))), implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W)))))), implies(implies(implies(implies(V, U), implies(T, S)), implies(implies(S, V), implies(X2, implies(T, V)))), implies(implies(implies(implies(implies(W, implies(Z, implies(implies(Y, W), W))), implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W))))), W), implies(Z, implies(implies(Y, W), W))), implies(implies(W, implies(Z, implies(implies(Y, W), W))), implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W)))))))), true, implies(implies(implies(V, U), implies(T, S)), implies(implies(S, V), implies(X2, implies(T, V)))), implies(implies(implies(implies(implies(W, implies(Z, implies(implies(Y, W), W))), implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W))))), W), implies(Z, implies(implies(Y, W), W))), implies(implies(W, implies(Z, implies(implies(Y, W), W))), implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W))))))), true, implies(implies(implies(implies(W, implies(Z, implies(implies(Y, W), W))), implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W))))), W), implies(Z, implies(implies(Y, W), W))), implies(implies(W, implies(Z, implies(implies(Y, W), W))), implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W)))))), true, implies(implies(implies(Y, implies(Z, implies(implies(Y, W), W))), W), implies(Z, implies(implies(Y, W), W)))), true, implies(X, implies(Y, implies(Z, implies(implies(Y, W), W)))))
% 7.07/1.35 = { by axiom 3 (condensed_detachment) }
% 7.07/1.35 fresh2(fresh2(fresh(fresh(fresh2(is_a_theorem(implies(implies(implies(implies(Y2, Z2), implies(Z2, implies(Y, W))), implies(Y, implies(Z, implies(implies(Y, W), W)))), implies(implies(W, implies(Z, implies(implies(Y, W), W))), implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W))))))), true, implies(implies(implies(implies(V, U), implies(T, S)), implies(implies(S, V), implies(X2, implies(T, V)))), implies(implies(implies(implies(implies(W, implies(Z, implies(implies(Y, W), W))), implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W))))), W), implies(Z, implies(implies(Y, W), W))), implies(implies(W, implies(Z, implies(implies(Y, W), W))), implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W)))))))), true, implies(implies(implies(V, U), implies(T, S)), implies(implies(S, V), implies(X2, implies(T, V)))), implies(implies(implies(implies(implies(W, implies(Z, implies(implies(Y, W), W))), implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W))))), W), implies(Z, implies(implies(Y, W), W))), implies(implies(W, implies(Z, implies(implies(Y, W), W))), implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W))))))), true, implies(implies(implies(implies(W, implies(Z, implies(implies(Y, W), W))), implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W))))), W), implies(Z, implies(implies(Y, W), W))), implies(implies(W, implies(Z, implies(implies(Y, W), W))), implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W)))))), true, implies(implies(implies(Y, implies(Z, implies(implies(Y, W), W))), W), implies(Z, implies(implies(Y, W), W)))), true, implies(X, implies(Y, implies(Z, implies(implies(Y, W), W)))))
% 7.07/1.35 = { by lemma 9 }
% 7.07/1.35 fresh2(fresh2(fresh(fresh(fresh2(true, true, implies(implies(implies(implies(V, U), implies(T, S)), implies(implies(S, V), implies(X2, implies(T, V)))), implies(implies(implies(implies(implies(W, implies(Z, implies(implies(Y, W), W))), implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W))))), W), implies(Z, implies(implies(Y, W), W))), implies(implies(W, implies(Z, implies(implies(Y, W), W))), implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W)))))))), true, implies(implies(implies(V, U), implies(T, S)), implies(implies(S, V), implies(X2, implies(T, V)))), implies(implies(implies(implies(implies(W, implies(Z, implies(implies(Y, W), W))), implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W))))), W), implies(Z, implies(implies(Y, W), W))), implies(implies(W, implies(Z, implies(implies(Y, W), W))), implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W))))))), true, implies(implies(implies(implies(W, implies(Z, implies(implies(Y, W), W))), implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W))))), W), implies(Z, implies(implies(Y, W), W))), implies(implies(W, implies(Z, implies(implies(Y, W), W))), implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W)))))), true, implies(implies(implies(Y, implies(Z, implies(implies(Y, W), W))), W), implies(Z, implies(implies(Y, W), W)))), true, implies(X, implies(Y, implies(Z, implies(implies(Y, W), W)))))
% 7.07/1.35 = { by axiom 1 (condensed_detachment) }
% 7.07/1.35 fresh2(fresh2(fresh(fresh(true, true, implies(implies(implies(V, U), implies(T, S)), implies(implies(S, V), implies(X2, implies(T, V)))), implies(implies(implies(implies(implies(W, implies(Z, implies(implies(Y, W), W))), implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W))))), W), implies(Z, implies(implies(Y, W), W))), implies(implies(W, implies(Z, implies(implies(Y, W), W))), implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W))))))), true, implies(implies(implies(implies(W, implies(Z, implies(implies(Y, W), W))), implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W))))), W), implies(Z, implies(implies(Y, W), W))), implies(implies(W, implies(Z, implies(implies(Y, W), W))), implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W)))))), true, implies(implies(implies(Y, implies(Z, implies(implies(Y, W), W))), W), implies(Z, implies(implies(Y, W), W)))), true, implies(X, implies(Y, implies(Z, implies(implies(Y, W), W)))))
% 7.07/1.35 = { by axiom 2 (condensed_detachment) }
% 7.07/1.35 fresh2(fresh2(fresh(is_a_theorem(implies(implies(implies(implies(implies(W, implies(Z, implies(implies(Y, W), W))), implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W))))), W), implies(Z, implies(implies(Y, W), W))), implies(implies(W, implies(Z, implies(implies(Y, W), W))), implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W))))))), true, implies(implies(implies(implies(W, implies(Z, implies(implies(Y, W), W))), implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W))))), W), implies(Z, implies(implies(Y, W), W))), implies(implies(W, implies(Z, implies(implies(Y, W), W))), implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W)))))), true, implies(implies(implies(Y, implies(Z, implies(implies(Y, W), W))), W), implies(Z, implies(implies(Y, W), W)))), true, implies(X, implies(Y, implies(Z, implies(implies(Y, W), W)))))
% 7.07/1.35 = { by axiom 3 (condensed_detachment) }
% 7.77/1.35 fresh2(fresh2(fresh2(is_a_theorem(implies(implies(implies(implies(W, implies(Z, implies(implies(Y, W), W))), implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W))))), W), implies(Z, implies(implies(Y, W), W)))), true, implies(implies(W, implies(Z, implies(implies(Y, W), W))), implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W)))))), true, implies(implies(implies(Y, implies(Z, implies(implies(Y, W), W))), W), implies(Z, implies(implies(Y, W), W)))), true, implies(X, implies(Y, implies(Z, implies(implies(Y, W), W)))))
% 7.77/1.35 = { by lemma 5 R->L }
% 7.77/1.35 fresh2(fresh2(fresh2(fresh2(is_a_theorem(implies(implies(W, W2), implies(implies(Y, W), implies(implies(W, implies(Z, implies(implies(Y, W), W))), implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W)))))))), true, implies(implies(implies(implies(W, implies(Z, implies(implies(Y, W), W))), implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W))))), W), implies(Z, implies(implies(Y, W), W)))), true, implies(implies(W, implies(Z, implies(implies(Y, W), W))), implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W)))))), true, implies(implies(implies(Y, implies(Z, implies(implies(Y, W), W))), W), implies(Z, implies(implies(Y, W), W)))), true, implies(X, implies(Y, implies(Z, implies(implies(Y, W), W)))))
% 7.77/1.35 = { by axiom 2 (condensed_detachment) R->L }
% 7.77/1.35 fresh2(fresh2(fresh2(fresh2(fresh(true, true, implies(implies(implies(implies(Z, implies(implies(Y, W), W)), implies(W, implies(Z, implies(implies(Y, W), W)))), implies(Y, W)), implies(implies(W, implies(Z, implies(implies(Y, W), W))), implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W)))))), implies(implies(W, W2), implies(implies(Y, W), implies(implies(W, implies(Z, implies(implies(Y, W), W))), implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W)))))))), true, implies(implies(implies(implies(W, implies(Z, implies(implies(Y, W), W))), implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W))))), W), implies(Z, implies(implies(Y, W), W)))), true, implies(implies(W, implies(Z, implies(implies(Y, W), W))), implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W)))))), true, implies(implies(implies(Y, implies(Z, implies(implies(Y, W), W))), W), implies(Z, implies(implies(Y, W), W)))), true, implies(X, implies(Y, implies(Z, implies(implies(Y, W), W)))))
% 7.77/1.36 = { by axiom 1 (condensed_detachment) R->L }
% 7.77/1.36 fresh2(fresh2(fresh2(fresh2(fresh(fresh2(true, true, implies(implies(implies(implies(implies(Z, implies(implies(Y, W), W)), implies(W, implies(Z, implies(implies(Y, W), W)))), implies(Y, W)), implies(implies(W, implies(Z, implies(implies(Y, W), W))), implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W)))))), implies(implies(W, W2), implies(implies(Y, W), implies(implies(W, implies(Z, implies(implies(Y, W), W))), implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W))))))))), true, implies(implies(implies(implies(Z, implies(implies(Y, W), W)), implies(W, implies(Z, implies(implies(Y, W), W)))), implies(Y, W)), implies(implies(W, implies(Z, implies(implies(Y, W), W))), implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W)))))), implies(implies(W, W2), implies(implies(Y, W), implies(implies(W, implies(Z, implies(implies(Y, W), W))), implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W)))))))), true, implies(implies(implies(implies(W, implies(Z, implies(implies(Y, W), W))), implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W))))), W), implies(Z, implies(implies(Y, W), W)))), true, implies(implies(W, implies(Z, implies(implies(Y, W), W))), implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W)))))), true, implies(implies(implies(Y, implies(Z, implies(implies(Y, W), W))), W), implies(Z, implies(implies(Y, W), W)))), true, implies(X, implies(Y, implies(Z, implies(implies(Y, W), W)))))
% 7.77/1.36 = { by axiom 1 (condensed_detachment) R->L }
% 7.77/1.36 fresh2(fresh2(fresh2(fresh2(fresh(fresh2(fresh2(true, true, implies(implies(implies(implies(W, implies(Z, implies(implies(Y, W), W))), implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W))))), implies(Y, W)), implies(implies(Y, W), implies(implies(implies(Z, implies(implies(Y, W), W)), implies(W, implies(Z, implies(implies(Y, W), W)))), implies(Y, W))))), true, implies(implies(implies(implies(implies(Z, implies(implies(Y, W), W)), implies(W, implies(Z, implies(implies(Y, W), W)))), implies(Y, W)), implies(implies(W, implies(Z, implies(implies(Y, W), W))), implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W)))))), implies(implies(W, W2), implies(implies(Y, W), implies(implies(W, implies(Z, implies(implies(Y, W), W))), implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W))))))))), true, implies(implies(implies(implies(Z, implies(implies(Y, W), W)), implies(W, implies(Z, implies(implies(Y, W), W)))), implies(Y, W)), implies(implies(W, implies(Z, implies(implies(Y, W), W))), implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W)))))), implies(implies(W, W2), implies(implies(Y, W), implies(implies(W, implies(Z, implies(implies(Y, W), W))), implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W)))))))), true, implies(implies(implies(implies(W, implies(Z, implies(implies(Y, W), W))), implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W))))), W), implies(Z, implies(implies(Y, W), W)))), true, implies(implies(W, implies(Z, implies(implies(Y, W), W))), implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W)))))), true, implies(implies(implies(Y, implies(Z, implies(implies(Y, W), W))), W), implies(Z, implies(implies(Y, W), W)))), true, implies(X, implies(Y, implies(Z, implies(implies(Y, W), W)))))
% 7.77/1.36 = { by axiom 1 (condensed_detachment) R->L }
% 7.77/1.36 fresh2(fresh2(fresh2(fresh2(fresh(fresh2(fresh2(fresh2(true, true, implies(implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W)))), implies(implies(implies(Z, implies(implies(Y, W), W)), implies(W, implies(Z, implies(implies(Y, W), W)))), implies(implies(W, implies(Z, implies(implies(Y, W), W))), implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W)))))))), true, implies(implies(implies(implies(W, implies(Z, implies(implies(Y, W), W))), implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W))))), implies(Y, W)), implies(implies(Y, W), implies(implies(implies(Z, implies(implies(Y, W), W)), implies(W, implies(Z, implies(implies(Y, W), W)))), implies(Y, W))))), true, implies(implies(implies(implies(implies(Z, implies(implies(Y, W), W)), implies(W, implies(Z, implies(implies(Y, W), W)))), implies(Y, W)), implies(implies(W, implies(Z, implies(implies(Y, W), W))), implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W)))))), implies(implies(W, W2), implies(implies(Y, W), implies(implies(W, implies(Z, implies(implies(Y, W), W))), implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W))))))))), true, implies(implies(implies(implies(Z, implies(implies(Y, W), W)), implies(W, implies(Z, implies(implies(Y, W), W)))), implies(Y, W)), implies(implies(W, implies(Z, implies(implies(Y, W), W))), implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W)))))), implies(implies(W, W2), implies(implies(Y, W), implies(implies(W, implies(Z, implies(implies(Y, W), W))), implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W)))))))), true, implies(implies(implies(implies(W, implies(Z, implies(implies(Y, W), W))), implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W))))), W), implies(Z, implies(implies(Y, W), W)))), true, implies(implies(W, implies(Z, implies(implies(Y, W), W))), implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W)))))), true, implies(implies(implies(Y, implies(Z, implies(implies(Y, W), W))), W), implies(Z, implies(implies(Y, W), W)))), true, implies(X, implies(Y, implies(Z, implies(implies(Y, W), W)))))
% 7.77/1.36 = { by lemma 9 R->L }
% 7.77/1.36 fresh2(fresh2(fresh2(fresh2(fresh(fresh2(fresh2(fresh2(is_a_theorem(implies(implies(implies(implies(V2, U2), implies(U2, Y)), implies(Z, implies(implies(Y, W), W))), implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W)))))), true, implies(implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W)))), implies(implies(implies(Z, implies(implies(Y, W), W)), implies(W, implies(Z, implies(implies(Y, W), W)))), implies(implies(W, implies(Z, implies(implies(Y, W), W))), implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W)))))))), true, implies(implies(implies(implies(W, implies(Z, implies(implies(Y, W), W))), implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W))))), implies(Y, W)), implies(implies(Y, W), implies(implies(implies(Z, implies(implies(Y, W), W)), implies(W, implies(Z, implies(implies(Y, W), W)))), implies(Y, W))))), true, implies(implies(implies(implies(implies(Z, implies(implies(Y, W), W)), implies(W, implies(Z, implies(implies(Y, W), W)))), implies(Y, W)), implies(implies(W, implies(Z, implies(implies(Y, W), W))), implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W)))))), implies(implies(W, W2), implies(implies(Y, W), implies(implies(W, implies(Z, implies(implies(Y, W), W))), implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W))))))))), true, implies(implies(implies(implies(Z, implies(implies(Y, W), W)), implies(W, implies(Z, implies(implies(Y, W), W)))), implies(Y, W)), implies(implies(W, implies(Z, implies(implies(Y, W), W))), implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W)))))), implies(implies(W, W2), implies(implies(Y, W), implies(implies(W, implies(Z, implies(implies(Y, W), W))), implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W)))))))), true, implies(implies(implies(implies(W, implies(Z, implies(implies(Y, W), W))), implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W))))), W), implies(Z, implies(implies(Y, W), W)))), true, implies(implies(W, implies(Z, implies(implies(Y, W), W))), implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W)))))), true, implies(implies(implies(Y, implies(Z, implies(implies(Y, W), W))), W), implies(Z, implies(implies(Y, W), W)))), true, implies(X, implies(Y, implies(Z, implies(implies(Y, W), W)))))
% 7.77/1.36 = { by axiom 3 (condensed_detachment) R->L }
% 7.77/1.37 fresh2(fresh2(fresh2(fresh2(fresh(fresh2(fresh2(fresh(is_a_theorem(implies(implies(implies(implies(implies(V2, U2), implies(U2, Y)), implies(Z, implies(implies(Y, W), W))), implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W))))), implies(implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W)))), implies(implies(implies(Z, implies(implies(Y, W), W)), implies(W, implies(Z, implies(implies(Y, W), W)))), implies(implies(W, implies(Z, implies(implies(Y, W), W))), implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W))))))))), true, implies(implies(implies(implies(V2, U2), implies(U2, Y)), implies(Z, implies(implies(Y, W), W))), implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W))))), implies(implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W)))), implies(implies(implies(Z, implies(implies(Y, W), W)), implies(W, implies(Z, implies(implies(Y, W), W)))), implies(implies(W, implies(Z, implies(implies(Y, W), W))), implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W)))))))), true, implies(implies(implies(implies(W, implies(Z, implies(implies(Y, W), W))), implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W))))), implies(Y, W)), implies(implies(Y, W), implies(implies(implies(Z, implies(implies(Y, W), W)), implies(W, implies(Z, implies(implies(Y, W), W)))), implies(Y, W))))), true, implies(implies(implies(implies(implies(Z, implies(implies(Y, W), W)), implies(W, implies(Z, implies(implies(Y, W), W)))), implies(Y, W)), implies(implies(W, implies(Z, implies(implies(Y, W), W))), implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W)))))), implies(implies(W, W2), implies(implies(Y, W), implies(implies(W, implies(Z, implies(implies(Y, W), W))), implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W))))))))), true, implies(implies(implies(implies(Z, implies(implies(Y, W), W)), implies(W, implies(Z, implies(implies(Y, W), W)))), implies(Y, W)), implies(implies(W, implies(Z, implies(implies(Y, W), W))), implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W)))))), implies(implies(W, W2), implies(implies(Y, W), implies(implies(W, implies(Z, implies(implies(Y, W), W))), implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W)))))))), true, implies(implies(implies(implies(W, implies(Z, implies(implies(Y, W), W))), implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W))))), W), implies(Z, implies(implies(Y, W), W)))), true, implies(implies(W, implies(Z, implies(implies(Y, W), W))), implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W)))))), true, implies(implies(implies(Y, implies(Z, implies(implies(Y, W), W))), W), implies(Z, implies(implies(Y, W), W)))), true, implies(X, implies(Y, implies(Z, implies(implies(Y, W), W)))))
% 7.77/1.37 = { by lemma 8 }
% 7.77/1.37 fresh2(fresh2(fresh2(fresh2(fresh(fresh2(fresh2(fresh(true, true, implies(implies(implies(implies(V2, U2), implies(U2, Y)), implies(Z, implies(implies(Y, W), W))), implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W))))), implies(implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W)))), implies(implies(implies(Z, implies(implies(Y, W), W)), implies(W, implies(Z, implies(implies(Y, W), W)))), implies(implies(W, implies(Z, implies(implies(Y, W), W))), implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W)))))))), true, implies(implies(implies(implies(W, implies(Z, implies(implies(Y, W), W))), implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W))))), implies(Y, W)), implies(implies(Y, W), implies(implies(implies(Z, implies(implies(Y, W), W)), implies(W, implies(Z, implies(implies(Y, W), W)))), implies(Y, W))))), true, implies(implies(implies(implies(implies(Z, implies(implies(Y, W), W)), implies(W, implies(Z, implies(implies(Y, W), W)))), implies(Y, W)), implies(implies(W, implies(Z, implies(implies(Y, W), W))), implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W)))))), implies(implies(W, W2), implies(implies(Y, W), implies(implies(W, implies(Z, implies(implies(Y, W), W))), implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W))))))))), true, implies(implies(implies(implies(Z, implies(implies(Y, W), W)), implies(W, implies(Z, implies(implies(Y, W), W)))), implies(Y, W)), implies(implies(W, implies(Z, implies(implies(Y, W), W))), implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W)))))), implies(implies(W, W2), implies(implies(Y, W), implies(implies(W, implies(Z, implies(implies(Y, W), W))), implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W)))))))), true, implies(implies(implies(implies(W, implies(Z, implies(implies(Y, W), W))), implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W))))), W), implies(Z, implies(implies(Y, W), W)))), true, implies(implies(W, implies(Z, implies(implies(Y, W), W))), implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W)))))), true, implies(implies(implies(Y, implies(Z, implies(implies(Y, W), W))), W), implies(Z, implies(implies(Y, W), W)))), true, implies(X, implies(Y, implies(Z, implies(implies(Y, W), W)))))
% 7.77/1.37 = { by axiom 2 (condensed_detachment) }
% 7.77/1.37 fresh2(fresh2(fresh2(fresh2(fresh(fresh2(fresh2(is_a_theorem(implies(implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W)))), implies(implies(implies(Z, implies(implies(Y, W), W)), implies(W, implies(Z, implies(implies(Y, W), W)))), implies(implies(W, implies(Z, implies(implies(Y, W), W))), implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W)))))))), true, implies(implies(implies(implies(W, implies(Z, implies(implies(Y, W), W))), implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W))))), implies(Y, W)), implies(implies(Y, W), implies(implies(implies(Z, implies(implies(Y, W), W)), implies(W, implies(Z, implies(implies(Y, W), W)))), implies(Y, W))))), true, implies(implies(implies(implies(implies(Z, implies(implies(Y, W), W)), implies(W, implies(Z, implies(implies(Y, W), W)))), implies(Y, W)), implies(implies(W, implies(Z, implies(implies(Y, W), W))), implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W)))))), implies(implies(W, W2), implies(implies(Y, W), implies(implies(W, implies(Z, implies(implies(Y, W), W))), implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W))))))))), true, implies(implies(implies(implies(Z, implies(implies(Y, W), W)), implies(W, implies(Z, implies(implies(Y, W), W)))), implies(Y, W)), implies(implies(W, implies(Z, implies(implies(Y, W), W))), implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W)))))), implies(implies(W, W2), implies(implies(Y, W), implies(implies(W, implies(Z, implies(implies(Y, W), W))), implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W)))))))), true, implies(implies(implies(implies(W, implies(Z, implies(implies(Y, W), W))), implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W))))), W), implies(Z, implies(implies(Y, W), W)))), true, implies(implies(W, implies(Z, implies(implies(Y, W), W))), implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W)))))), true, implies(implies(implies(Y, implies(Z, implies(implies(Y, W), W))), W), implies(Z, implies(implies(Y, W), W)))), true, implies(X, implies(Y, implies(Z, implies(implies(Y, W), W)))))
% 7.77/1.37 = { by lemma 5 }
% 7.77/1.37 fresh2(fresh2(fresh2(fresh2(fresh(fresh2(is_a_theorem(implies(implies(implies(implies(W, implies(Z, implies(implies(Y, W), W))), implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W))))), implies(Y, W)), implies(implies(Y, W), implies(implies(implies(Z, implies(implies(Y, W), W)), implies(W, implies(Z, implies(implies(Y, W), W)))), implies(Y, W))))), true, implies(implies(implies(implies(implies(Z, implies(implies(Y, W), W)), implies(W, implies(Z, implies(implies(Y, W), W)))), implies(Y, W)), implies(implies(W, implies(Z, implies(implies(Y, W), W))), implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W)))))), implies(implies(W, W2), implies(implies(Y, W), implies(implies(W, implies(Z, implies(implies(Y, W), W))), implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W))))))))), true, implies(implies(implies(implies(Z, implies(implies(Y, W), W)), implies(W, implies(Z, implies(implies(Y, W), W)))), implies(Y, W)), implies(implies(W, implies(Z, implies(implies(Y, W), W))), implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W)))))), implies(implies(W, W2), implies(implies(Y, W), implies(implies(W, implies(Z, implies(implies(Y, W), W))), implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W)))))))), true, implies(implies(implies(implies(W, implies(Z, implies(implies(Y, W), W))), implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W))))), W), implies(Z, implies(implies(Y, W), W)))), true, implies(implies(W, implies(Z, implies(implies(Y, W), W))), implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W)))))), true, implies(implies(implies(Y, implies(Z, implies(implies(Y, W), W))), W), implies(Z, implies(implies(Y, W), W)))), true, implies(X, implies(Y, implies(Z, implies(implies(Y, W), W)))))
% 7.77/1.37 = { by lemma 5 }
% 7.77/1.37 fresh2(fresh2(fresh2(fresh2(fresh(is_a_theorem(implies(implies(implies(implies(implies(Z, implies(implies(Y, W), W)), implies(W, implies(Z, implies(implies(Y, W), W)))), implies(Y, W)), implies(implies(W, implies(Z, implies(implies(Y, W), W))), implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W)))))), implies(implies(W, W2), implies(implies(Y, W), implies(implies(W, implies(Z, implies(implies(Y, W), W))), implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W))))))))), true, implies(implies(implies(implies(Z, implies(implies(Y, W), W)), implies(W, implies(Z, implies(implies(Y, W), W)))), implies(Y, W)), implies(implies(W, implies(Z, implies(implies(Y, W), W))), implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W)))))), implies(implies(W, W2), implies(implies(Y, W), implies(implies(W, implies(Z, implies(implies(Y, W), W))), implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W)))))))), true, implies(implies(implies(implies(W, implies(Z, implies(implies(Y, W), W))), implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W))))), W), implies(Z, implies(implies(Y, W), W)))), true, implies(implies(W, implies(Z, implies(implies(Y, W), W))), implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W)))))), true, implies(implies(implies(Y, implies(Z, implies(implies(Y, W), W))), W), implies(Z, implies(implies(Y, W), W)))), true, implies(X, implies(Y, implies(Z, implies(implies(Y, W), W)))))
% 7.77/1.37 = { by axiom 3 (condensed_detachment) }
% 7.77/1.37 fresh2(fresh2(fresh2(fresh2(fresh2(is_a_theorem(implies(implies(implies(implies(Z, implies(implies(Y, W), W)), implies(W, implies(Z, implies(implies(Y, W), W)))), implies(Y, W)), implies(implies(W, implies(Z, implies(implies(Y, W), W))), implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W))))))), true, implies(implies(W, W2), implies(implies(Y, W), implies(implies(W, implies(Z, implies(implies(Y, W), W))), implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W)))))))), true, implies(implies(implies(implies(W, implies(Z, implies(implies(Y, W), W))), implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W))))), W), implies(Z, implies(implies(Y, W), W)))), true, implies(implies(W, implies(Z, implies(implies(Y, W), W))), implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W)))))), true, implies(implies(implies(Y, implies(Z, implies(implies(Y, W), W))), W), implies(Z, implies(implies(Y, W), W)))), true, implies(X, implies(Y, implies(Z, implies(implies(Y, W), W)))))
% 7.77/1.37 = { by axiom 4 (ic_JLukasiewicz_5) }
% 7.77/1.37 fresh2(fresh2(fresh2(fresh2(fresh2(true, true, implies(implies(W, W2), implies(implies(Y, W), implies(implies(W, implies(Z, implies(implies(Y, W), W))), implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W)))))))), true, implies(implies(implies(implies(W, implies(Z, implies(implies(Y, W), W))), implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W))))), W), implies(Z, implies(implies(Y, W), W)))), true, implies(implies(W, implies(Z, implies(implies(Y, W), W))), implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W)))))), true, implies(implies(implies(Y, implies(Z, implies(implies(Y, W), W))), W), implies(Z, implies(implies(Y, W), W)))), true, implies(X, implies(Y, implies(Z, implies(implies(Y, W), W)))))
% 7.77/1.37 = { by axiom 1 (condensed_detachment) }
% 7.77/1.37 fresh2(fresh2(fresh2(fresh2(true, true, implies(implies(implies(implies(W, implies(Z, implies(implies(Y, W), W))), implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W))))), W), implies(Z, implies(implies(Y, W), W)))), true, implies(implies(W, implies(Z, implies(implies(Y, W), W))), implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W)))))), true, implies(implies(implies(Y, implies(Z, implies(implies(Y, W), W))), W), implies(Z, implies(implies(Y, W), W)))), true, implies(X, implies(Y, implies(Z, implies(implies(Y, W), W)))))
% 7.77/1.37 = { by axiom 1 (condensed_detachment) }
% 7.77/1.37 fresh2(fresh2(fresh2(true, true, implies(implies(W, implies(Z, implies(implies(Y, W), W))), implies(implies(Y, W), implies(Y, implies(Z, implies(implies(Y, W), W)))))), true, implies(implies(implies(Y, implies(Z, implies(implies(Y, W), W))), W), implies(Z, implies(implies(Y, W), W)))), true, implies(X, implies(Y, implies(Z, implies(implies(Y, W), W)))))
% 7.77/1.37 = { by axiom 1 (condensed_detachment) }
% 7.77/1.37 fresh2(fresh2(true, true, implies(implies(implies(Y, implies(Z, implies(implies(Y, W), W))), W), implies(Z, implies(implies(Y, W), W)))), true, implies(X, implies(Y, implies(Z, implies(implies(Y, W), W)))))
% 7.77/1.37 = { by axiom 1 (condensed_detachment) }
% 7.77/1.37 fresh2(true, true, implies(X, implies(Y, implies(Z, implies(implies(Y, W), W)))))
% 7.77/1.37 = { by axiom 1 (condensed_detachment) }
% 7.77/1.37 true
% 7.77/1.37
% 7.77/1.37 Lemma 15: fresh2(is_a_theorem(implies(implies(X, X), Y)), true, Y) = is_a_theorem(Y).
% 7.77/1.37 Proof:
% 7.77/1.37 fresh2(is_a_theorem(implies(implies(X, X), Y)), true, Y)
% 7.77/1.37 = { by axiom 3 (condensed_detachment) R->L }
% 7.77/1.37 fresh(is_a_theorem(implies(implies(implies(X, X), Y), Y)), true, implies(implies(X, X), Y), Y)
% 7.77/1.37 = { by axiom 2 (condensed_detachment) R->L }
% 7.77/1.37 fresh(fresh(true, true, implies(implies(implies(Z, W), implies(V, U)), implies(implies(U, Z), implies(T, implies(V, Z)))), implies(implies(implies(X, X), Y), Y)), true, implies(implies(X, X), Y), Y)
% 7.77/1.37 = { by axiom 1 (condensed_detachment) R->L }
% 7.77/1.37 fresh(fresh(fresh2(true, true, implies(implies(implies(implies(Z, W), implies(V, U)), implies(implies(U, Z), implies(T, implies(V, Z)))), implies(implies(implies(X, X), Y), Y))), true, implies(implies(implies(Z, W), implies(V, U)), implies(implies(U, Z), implies(T, implies(V, Z)))), implies(implies(implies(X, X), Y), Y)), true, implies(implies(X, X), Y), Y)
% 7.77/1.37 = { by axiom 1 (condensed_detachment) R->L }
% 7.77/1.37 fresh(fresh(fresh2(fresh2(true, true, implies(X, X)), true, implies(implies(implies(implies(Z, W), implies(V, U)), implies(implies(U, Z), implies(T, implies(V, Z)))), implies(implies(implies(X, X), Y), Y))), true, implies(implies(implies(Z, W), implies(V, U)), implies(implies(U, Z), implies(T, implies(V, Z)))), implies(implies(implies(X, X), Y), Y)), true, implies(implies(X, X), Y), Y)
% 7.77/1.37 = { by axiom 4 (ic_JLukasiewicz_5) R->L }
% 7.77/1.37 fresh(fresh(fresh2(fresh2(is_a_theorem(implies(implies(implies(S, X2), implies(Y2, Z2)), implies(implies(Z2, S), implies(W2, implies(Y2, S))))), true, implies(X, X)), true, implies(implies(implies(implies(Z, W), implies(V, U)), implies(implies(U, Z), implies(T, implies(V, Z)))), implies(implies(implies(X, X), Y), Y))), true, implies(implies(implies(Z, W), implies(V, U)), implies(implies(U, Z), implies(T, implies(V, Z)))), implies(implies(implies(X, X), Y), Y)), true, implies(implies(X, X), Y), Y)
% 7.77/1.37 = { by axiom 3 (condensed_detachment) R->L }
% 7.77/1.37 fresh(fresh(fresh2(fresh(is_a_theorem(implies(implies(implies(implies(S, X2), implies(Y2, Z2)), implies(implies(Z2, S), implies(W2, implies(Y2, S)))), implies(X, X))), true, implies(implies(implies(S, X2), implies(Y2, Z2)), implies(implies(Z2, S), implies(W2, implies(Y2, S)))), implies(X, X)), true, implies(implies(implies(implies(Z, W), implies(V, U)), implies(implies(U, Z), implies(T, implies(V, Z)))), implies(implies(implies(X, X), Y), Y))), true, implies(implies(implies(Z, W), implies(V, U)), implies(implies(U, Z), implies(T, implies(V, Z)))), implies(implies(implies(X, X), Y), Y)), true, implies(implies(X, X), Y), Y)
% 7.77/1.37 = { by lemma 11 }
% 7.77/1.38 fresh(fresh(fresh2(fresh(true, true, implies(implies(implies(S, X2), implies(Y2, Z2)), implies(implies(Z2, S), implies(W2, implies(Y2, S)))), implies(X, X)), true, implies(implies(implies(implies(Z, W), implies(V, U)), implies(implies(U, Z), implies(T, implies(V, Z)))), implies(implies(implies(X, X), Y), Y))), true, implies(implies(implies(Z, W), implies(V, U)), implies(implies(U, Z), implies(T, implies(V, Z)))), implies(implies(implies(X, X), Y), Y)), true, implies(implies(X, X), Y), Y)
% 7.77/1.38 = { by axiom 2 (condensed_detachment) }
% 7.77/1.38 fresh(fresh(fresh2(is_a_theorem(implies(X, X)), true, implies(implies(implies(implies(Z, W), implies(V, U)), implies(implies(U, Z), implies(T, implies(V, Z)))), implies(implies(implies(X, X), Y), Y))), true, implies(implies(implies(Z, W), implies(V, U)), implies(implies(U, Z), implies(T, implies(V, Z)))), implies(implies(implies(X, X), Y), Y)), true, implies(implies(X, X), Y), Y)
% 7.77/1.38 = { by axiom 3 (condensed_detachment) R->L }
% 7.77/1.38 fresh(fresh(fresh(is_a_theorem(implies(implies(X, X), implies(implies(implies(implies(Z, W), implies(V, U)), implies(implies(U, Z), implies(T, implies(V, Z)))), implies(implies(implies(X, X), Y), Y)))), true, implies(X, X), implies(implies(implies(implies(Z, W), implies(V, U)), implies(implies(U, Z), implies(T, implies(V, Z)))), implies(implies(implies(X, X), Y), Y))), true, implies(implies(implies(Z, W), implies(V, U)), implies(implies(U, Z), implies(T, implies(V, Z)))), implies(implies(implies(X, X), Y), Y)), true, implies(implies(X, X), Y), Y)
% 7.77/1.38 = { by axiom 2 (condensed_detachment) R->L }
% 7.77/1.38 fresh(fresh(fresh(fresh(true, true, implies(implies(implies(V2, U2), implies(T2, S2)), implies(implies(S2, V2), implies(X3, implies(T2, V2)))), implies(implies(X, X), implies(implies(implies(implies(Z, W), implies(V, U)), implies(implies(U, Z), implies(T, implies(V, Z)))), implies(implies(implies(X, X), Y), Y)))), true, implies(X, X), implies(implies(implies(implies(Z, W), implies(V, U)), implies(implies(U, Z), implies(T, implies(V, Z)))), implies(implies(implies(X, X), Y), Y))), true, implies(implies(implies(Z, W), implies(V, U)), implies(implies(U, Z), implies(T, implies(V, Z)))), implies(implies(implies(X, X), Y), Y)), true, implies(implies(X, X), Y), Y)
% 7.77/1.38 = { by lemma 14 R->L }
% 7.77/1.38 fresh(fresh(fresh(fresh(is_a_theorem(implies(implies(implies(implies(V2, U2), implies(T2, S2)), implies(implies(S2, V2), implies(X3, implies(T2, V2)))), implies(implies(X, X), implies(implies(implies(implies(Z, W), implies(V, U)), implies(implies(U, Z), implies(T, implies(V, Z)))), implies(implies(implies(X, X), Y), Y))))), true, implies(implies(implies(V2, U2), implies(T2, S2)), implies(implies(S2, V2), implies(X3, implies(T2, V2)))), implies(implies(X, X), implies(implies(implies(implies(Z, W), implies(V, U)), implies(implies(U, Z), implies(T, implies(V, Z)))), implies(implies(implies(X, X), Y), Y)))), true, implies(X, X), implies(implies(implies(implies(Z, W), implies(V, U)), implies(implies(U, Z), implies(T, implies(V, Z)))), implies(implies(implies(X, X), Y), Y))), true, implies(implies(implies(Z, W), implies(V, U)), implies(implies(U, Z), implies(T, implies(V, Z)))), implies(implies(implies(X, X), Y), Y)), true, implies(implies(X, X), Y), Y)
% 7.77/1.38 = { by axiom 3 (condensed_detachment) }
% 7.77/1.38 fresh(fresh(fresh(fresh2(is_a_theorem(implies(implies(implies(V2, U2), implies(T2, S2)), implies(implies(S2, V2), implies(X3, implies(T2, V2))))), true, implies(implies(X, X), implies(implies(implies(implies(Z, W), implies(V, U)), implies(implies(U, Z), implies(T, implies(V, Z)))), implies(implies(implies(X, X), Y), Y)))), true, implies(X, X), implies(implies(implies(implies(Z, W), implies(V, U)), implies(implies(U, Z), implies(T, implies(V, Z)))), implies(implies(implies(X, X), Y), Y))), true, implies(implies(implies(Z, W), implies(V, U)), implies(implies(U, Z), implies(T, implies(V, Z)))), implies(implies(implies(X, X), Y), Y)), true, implies(implies(X, X), Y), Y)
% 7.77/1.38 = { by axiom 4 (ic_JLukasiewicz_5) }
% 7.77/1.38 fresh(fresh(fresh(fresh2(true, true, implies(implies(X, X), implies(implies(implies(implies(Z, W), implies(V, U)), implies(implies(U, Z), implies(T, implies(V, Z)))), implies(implies(implies(X, X), Y), Y)))), true, implies(X, X), implies(implies(implies(implies(Z, W), implies(V, U)), implies(implies(U, Z), implies(T, implies(V, Z)))), implies(implies(implies(X, X), Y), Y))), true, implies(implies(implies(Z, W), implies(V, U)), implies(implies(U, Z), implies(T, implies(V, Z)))), implies(implies(implies(X, X), Y), Y)), true, implies(implies(X, X), Y), Y)
% 7.77/1.38 = { by axiom 1 (condensed_detachment) }
% 7.77/1.38 fresh(fresh(fresh(true, true, implies(X, X), implies(implies(implies(implies(Z, W), implies(V, U)), implies(implies(U, Z), implies(T, implies(V, Z)))), implies(implies(implies(X, X), Y), Y))), true, implies(implies(implies(Z, W), implies(V, U)), implies(implies(U, Z), implies(T, implies(V, Z)))), implies(implies(implies(X, X), Y), Y)), true, implies(implies(X, X), Y), Y)
% 7.77/1.38 = { by axiom 2 (condensed_detachment) }
% 7.77/1.38 fresh(fresh(is_a_theorem(implies(implies(implies(implies(Z, W), implies(V, U)), implies(implies(U, Z), implies(T, implies(V, Z)))), implies(implies(implies(X, X), Y), Y))), true, implies(implies(implies(Z, W), implies(V, U)), implies(implies(U, Z), implies(T, implies(V, Z)))), implies(implies(implies(X, X), Y), Y)), true, implies(implies(X, X), Y), Y)
% 7.77/1.38 = { by axiom 3 (condensed_detachment) }
% 7.77/1.38 fresh(fresh2(is_a_theorem(implies(implies(implies(Z, W), implies(V, U)), implies(implies(U, Z), implies(T, implies(V, Z))))), true, implies(implies(implies(X, X), Y), Y)), true, implies(implies(X, X), Y), Y)
% 7.77/1.38 = { by axiom 4 (ic_JLukasiewicz_5) }
% 7.77/1.38 fresh(fresh2(true, true, implies(implies(implies(X, X), Y), Y)), true, implies(implies(X, X), Y), Y)
% 7.77/1.38 = { by axiom 1 (condensed_detachment) }
% 7.77/1.38 fresh(true, true, implies(implies(X, X), Y), Y)
% 7.77/1.38 = { by axiom 2 (condensed_detachment) }
% 7.77/1.38 is_a_theorem(Y)
% 7.77/1.38
% 7.77/1.38 Lemma 16: fresh2(is_a_theorem(implies(X, implies(X, Y))), true, implies(X, Y)) = is_a_theorem(implies(X, Y)).
% 7.77/1.38 Proof:
% 7.77/1.38 fresh2(is_a_theorem(implies(X, implies(X, Y))), true, implies(X, Y))
% 7.77/1.38 = { by axiom 3 (condensed_detachment) R->L }
% 7.77/1.38 fresh(is_a_theorem(implies(implies(X, implies(X, Y)), implies(X, Y))), true, implies(X, implies(X, Y)), implies(X, Y))
% 7.77/1.38 = { by lemma 15 R->L }
% 7.77/1.38 fresh(fresh2(is_a_theorem(implies(implies(Z, Z), implies(implies(X, implies(X, Y)), implies(X, Y)))), true, implies(implies(X, implies(X, Y)), implies(X, Y))), true, implies(X, implies(X, Y)), implies(X, Y))
% 7.77/1.38 = { by lemma 15 R->L }
% 7.77/1.38 fresh(fresh2(fresh2(is_a_theorem(implies(implies(W, W), implies(implies(Z, Z), implies(implies(X, implies(X, Y)), implies(X, Y))))), true, implies(implies(Z, Z), implies(implies(X, implies(X, Y)), implies(X, Y)))), true, implies(implies(X, implies(X, Y)), implies(X, Y))), true, implies(X, implies(X, Y)), implies(X, Y))
% 7.77/1.38 = { by lemma 13 R->L }
% 7.77/1.38 fresh(fresh2(fresh2(fresh2(is_a_theorem(implies(implies(implies(implies(Z, Z), implies(implies(X, implies(X, Y)), implies(X, Y))), Y), implies(implies(X, implies(X, Y)), implies(X, Y)))), true, implies(implies(W, W), implies(implies(Z, Z), implies(implies(X, implies(X, Y)), implies(X, Y))))), true, implies(implies(Z, Z), implies(implies(X, implies(X, Y)), implies(X, Y)))), true, implies(implies(X, implies(X, Y)), implies(X, Y))), true, implies(X, implies(X, Y)), implies(X, Y))
% 7.77/1.38 = { by lemma 5 R->L }
% 7.77/1.38 fresh(fresh2(fresh2(fresh2(fresh2(is_a_theorem(implies(implies(Y, V), implies(X, implies(implies(Z, Z), implies(implies(X, implies(X, Y)), implies(X, Y)))))), true, implies(implies(implies(implies(Z, Z), implies(implies(X, implies(X, Y)), implies(X, Y))), Y), implies(implies(X, implies(X, Y)), implies(X, Y)))), true, implies(implies(W, W), implies(implies(Z, Z), implies(implies(X, implies(X, Y)), implies(X, Y))))), true, implies(implies(Z, Z), implies(implies(X, implies(X, Y)), implies(X, Y)))), true, implies(implies(X, implies(X, Y)), implies(X, Y))), true, implies(X, implies(X, Y)), implies(X, Y))
% 7.77/1.38 = { by lemma 14 }
% 7.77/1.38 fresh(fresh2(fresh2(fresh2(fresh2(true, true, implies(implies(implies(implies(Z, Z), implies(implies(X, implies(X, Y)), implies(X, Y))), Y), implies(implies(X, implies(X, Y)), implies(X, Y)))), true, implies(implies(W, W), implies(implies(Z, Z), implies(implies(X, implies(X, Y)), implies(X, Y))))), true, implies(implies(Z, Z), implies(implies(X, implies(X, Y)), implies(X, Y)))), true, implies(implies(X, implies(X, Y)), implies(X, Y))), true, implies(X, implies(X, Y)), implies(X, Y))
% 7.77/1.38 = { by axiom 1 (condensed_detachment) }
% 7.77/1.38 fresh(fresh2(fresh2(fresh2(true, true, implies(implies(W, W), implies(implies(Z, Z), implies(implies(X, implies(X, Y)), implies(X, Y))))), true, implies(implies(Z, Z), implies(implies(X, implies(X, Y)), implies(X, Y)))), true, implies(implies(X, implies(X, Y)), implies(X, Y))), true, implies(X, implies(X, Y)), implies(X, Y))
% 7.77/1.38 = { by axiom 1 (condensed_detachment) }
% 7.77/1.38 fresh(fresh2(fresh2(true, true, implies(implies(Z, Z), implies(implies(X, implies(X, Y)), implies(X, Y)))), true, implies(implies(X, implies(X, Y)), implies(X, Y))), true, implies(X, implies(X, Y)), implies(X, Y))
% 7.77/1.38 = { by axiom 1 (condensed_detachment) }
% 7.77/1.38 fresh(fresh2(true, true, implies(implies(X, implies(X, Y)), implies(X, Y))), true, implies(X, implies(X, Y)), implies(X, Y))
% 7.77/1.38 = { by axiom 1 (condensed_detachment) }
% 7.77/1.38 fresh(true, true, implies(X, implies(X, Y)), implies(X, Y))
% 7.77/1.38 = { by axiom 2 (condensed_detachment) }
% 7.77/1.38 is_a_theorem(implies(X, Y))
% 7.77/1.38
% 7.77/1.38 Goal 1 (prove_ic_3): is_a_theorem(implies(implies(implies(a, b), a), a)) = true.
% 7.77/1.38 Proof:
% 7.77/1.38 is_a_theorem(implies(implies(implies(a, b), a), a))
% 7.77/1.38 = { by lemma 16 R->L }
% 7.77/1.38 fresh2(is_a_theorem(implies(implies(implies(a, b), a), implies(implies(implies(a, b), a), a))), true, implies(implies(implies(a, b), a), a))
% 7.77/1.38 = { by lemma 16 R->L }
% 7.77/1.38 fresh2(fresh2(is_a_theorem(implies(implies(implies(a, b), a), implies(implies(implies(a, b), a), implies(implies(implies(a, b), a), a)))), true, implies(implies(implies(a, b), a), implies(implies(implies(a, b), a), a))), true, implies(implies(implies(a, b), a), a))
% 7.77/1.38 = { by lemma 5 R->L }
% 7.77/1.38 fresh2(fresh2(fresh2(is_a_theorem(implies(implies(a, b), implies(implies(implies(a, b), a), implies(a, b)))), true, implies(implies(implies(a, b), a), implies(implies(implies(a, b), a), implies(implies(implies(a, b), a), a)))), true, implies(implies(implies(a, b), a), implies(implies(implies(a, b), a), a))), true, implies(implies(implies(a, b), a), a))
% 7.77/1.38 = { by lemma 5 R->L }
% 7.77/1.38 fresh2(fresh2(fresh2(fresh2(is_a_theorem(implies(implies(b, X), implies(a, a))), true, implies(implies(a, b), implies(implies(implies(a, b), a), implies(a, b)))), true, implies(implies(implies(a, b), a), implies(implies(implies(a, b), a), implies(implies(implies(a, b), a), a)))), true, implies(implies(implies(a, b), a), implies(implies(implies(a, b), a), a))), true, implies(implies(implies(a, b), a), a))
% 7.77/1.38 = { by lemma 11 }
% 7.77/1.38 fresh2(fresh2(fresh2(fresh2(true, true, implies(implies(a, b), implies(implies(implies(a, b), a), implies(a, b)))), true, implies(implies(implies(a, b), a), implies(implies(implies(a, b), a), implies(implies(implies(a, b), a), a)))), true, implies(implies(implies(a, b), a), implies(implies(implies(a, b), a), a))), true, implies(implies(implies(a, b), a), a))
% 7.77/1.38 = { by axiom 1 (condensed_detachment) }
% 7.77/1.38 fresh2(fresh2(fresh2(true, true, implies(implies(implies(a, b), a), implies(implies(implies(a, b), a), implies(implies(implies(a, b), a), a)))), true, implies(implies(implies(a, b), a), implies(implies(implies(a, b), a), a))), true, implies(implies(implies(a, b), a), a))
% 7.77/1.38 = { by axiom 1 (condensed_detachment) }
% 7.77/1.38 fresh2(fresh2(true, true, implies(implies(implies(a, b), a), implies(implies(implies(a, b), a), a))), true, implies(implies(implies(a, b), a), a))
% 7.77/1.38 = { by axiom 1 (condensed_detachment) }
% 7.77/1.38 fresh2(true, true, implies(implies(implies(a, b), a), a))
% 7.77/1.38 = { by axiom 1 (condensed_detachment) }
% 7.77/1.38 true
% 7.77/1.38 % SZS output end Proof
% 7.77/1.38
% 7.77/1.38 RESULT: Unsatisfiable (the axioms are contradictory).
%------------------------------------------------------------------------------