%------------------------------------------------------------------------------
% File     : Twee---2.4.2
% Problem  : LCL095-1 : 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 : n022.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 08:17:24 EDT 2023

% Result   : Unsatisfiable 6.22s 1.17s
% Output   : Proof 7.40s
% Verified : 
% SZS Type : -

% Comments : 
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12  % Problem  : LCL095-1 : TPTP v8.1.2. Released v1.0.0.
% 0.00/0.13  % Command  : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.16/0.34  % Computer : n022.cluster.edu
% 0.16/0.34  % Model    : x86_64 x86_64
% 0.16/0.34  % CPU      : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.16/0.34  % Memory   : 8042.1875MB
% 0.16/0.34  % OS       : Linux 3.10.0-693.el7.x86_64
% 0.16/0.34  % CPULimit : 300
% 0.16/0.34  % WCLimit  : 300
% 0.16/0.34  % DateTime : Thu Aug 24 17:13:10 EDT 2023
% 0.16/0.34  % CPUTime  : 
% 6.22/1.17  Command-line arguments: --no-flatten-goal
% 6.22/1.17  
% 6.22/1.17  % SZS status Unsatisfiable
% 6.22/1.17  
% 6.84/1.28  % SZS output start Proof
% 6.84/1.28  Take the following subset of the input axioms:
% 6.84/1.28    fof(condensed_detachment, axiom, ![X, Y]: (~is_a_theorem(implies(X, Y)) | (~is_a_theorem(X) | is_a_theorem(Y)))).
% 6.84/1.28    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)))))).
% 6.84/1.28    fof(prove_ic_5, negated_conjecture, ~is_a_theorem(implies(a, implies(implies(a, b), b)))).
% 6.84/1.28  
% 6.84/1.28  Now clausify the problem and encode Horn clauses using encoding 3 of
% 6.84/1.28  http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 6.84/1.28  We repeatedly replace C & s=t => u=v by the two clauses:
% 6.84/1.28    fresh(y, y, x1...xn) = u
% 6.84/1.28    C => fresh(s, t, x1...xn) = v
% 6.84/1.28  where fresh is a fresh function symbol and x1..xn are the free
% 6.84/1.28  variables of u and v.
% 6.84/1.28  A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 6.84/1.28  input problem has no model of domain size 1).
% 6.84/1.28  
% 6.84/1.28  The encoding turns the above axioms into the following unit equations and goals:
% 6.84/1.28  
% 6.84/1.28  Axiom 1 (condensed_detachment): fresh2(X, X, Y) = true.
% 6.84/1.28  Axiom 2 (condensed_detachment): fresh(X, X, Y, Z) = is_a_theorem(Z).
% 6.84/1.28  Axiom 3 (condensed_detachment): fresh(is_a_theorem(implies(X, Y)), true, X, Y) = fresh2(is_a_theorem(X), true, Y).
% 6.84/1.28  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.
% 6.84/1.28  
% 6.84/1.28  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)))).
% 6.84/1.28  Proof:
% 6.84/1.28    fresh2(is_a_theorem(implies(implies(X, Y), implies(Z, W))), true, implies(implies(W, X), implies(V, implies(Z, X))))
% 6.84/1.28  = { by axiom 3 (condensed_detachment) R->L }
% 6.84/1.28    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))))
% 6.84/1.28  = { by axiom 4 (ic_JLukasiewicz_5) }
% 6.84/1.28    fresh(true, true, implies(implies(X, Y), implies(Z, W)), implies(implies(W, X), implies(V, implies(Z, X))))
% 6.84/1.28  = { by axiom 2 (condensed_detachment) }
% 6.84/1.28    is_a_theorem(implies(implies(W, X), implies(V, implies(Z, X))))
% 6.84/1.28  
% 6.84/1.28  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.
% 6.84/1.28  Proof:
% 6.84/1.28    is_a_theorem(implies(implies(implies(X, implies(Y, Z)), implies(Z, W)), implies(V, implies(implies(U, Z), implies(Z, W)))))
% 6.84/1.28  = { by lemma 5 R->L }
% 6.84/1.28    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)))))
% 6.84/1.28  = { by axiom 4 (ic_JLukasiewicz_5) }
% 6.84/1.28    fresh2(true, true, implies(implies(implies(X, implies(Y, Z)), implies(Z, W)), implies(V, implies(implies(U, Z), implies(Z, W)))))
% 6.84/1.28  = { by axiom 1 (condensed_detachment) }
% 6.84/1.28    true
% 6.84/1.28  
% 6.84/1.28  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)))).
% 6.84/1.28  Proof:
% 6.84/1.28    fresh2(is_a_theorem(implies(implies(X, implies(Y, Z)), implies(Z, W))), true, implies(V, implies(implies(U, Z), implies(Z, W))))
% 6.84/1.28  = { by axiom 3 (condensed_detachment) R->L }
% 6.84/1.28    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))))
% 6.84/1.28  = { by lemma 6 }
% 6.84/1.28    fresh(true, true, implies(implies(X, implies(Y, Z)), implies(Z, W)), implies(V, implies(implies(U, Z), implies(Z, W))))
% 6.84/1.28  = { by axiom 2 (condensed_detachment) }
% 6.84/1.28    is_a_theorem(implies(V, implies(implies(U, Z), implies(Z, W))))
% 6.84/1.28  
% 6.84/1.28  Lemma 8: is_a_theorem(implies(implies(X, Y), implies(Y, implies(implies(Z, W), implies(W, Y))))) = true.
% 6.84/1.28  Proof:
% 6.84/1.28    is_a_theorem(implies(implies(X, Y), implies(Y, implies(implies(Z, W), implies(W, Y)))))
% 6.84/1.28  = { by axiom 2 (condensed_detachment) R->L }
% 6.84/1.28    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)))))
% 6.84/1.28  = { by axiom 1 (condensed_detachment) R->L }
% 6.84/1.28    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)))))
% 6.84/1.28  = { by lemma 6 R->L }
% 6.84/1.28    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)))))
% 6.84/1.29  = { by lemma 7 }
% 6.84/1.29    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)))))
% 6.84/1.29  = { by axiom 3 (condensed_detachment) }
% 6.84/1.29    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)))))
% 6.84/1.29  = { by axiom 4 (ic_JLukasiewicz_5) }
% 6.84/1.29    fresh2(true, true, implies(implies(X, Y), implies(Y, implies(implies(Z, W), implies(W, Y)))))
% 6.84/1.29  = { by axiom 1 (condensed_detachment) }
% 6.84/1.29    true
% 6.84/1.29  
% 6.84/1.29  Lemma 9: is_a_theorem(implies(implies(implies(implies(X, Y), implies(Y, Z)), W), implies(V, implies(Z, W)))) = true.
% 6.84/1.29  Proof:
% 6.84/1.29    is_a_theorem(implies(implies(implies(implies(X, Y), implies(Y, Z)), W), implies(V, implies(Z, W))))
% 6.84/1.29  = { by lemma 5 R->L }
% 6.84/1.29    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))))
% 6.84/1.29  = { by lemma 8 }
% 6.84/1.29    fresh2(true, true, implies(implies(implies(implies(X, Y), implies(Y, Z)), W), implies(V, implies(Z, W))))
% 6.84/1.29  = { by axiom 1 (condensed_detachment) }
% 6.84/1.29    true
% 6.84/1.29  
% 6.84/1.29  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))))).
% 6.84/1.29  Proof:
% 6.84/1.29    fresh2(is_a_theorem(implies(implies(X, Y), implies(Z, implies(W, X)))), true, implies(V, implies(U, implies(Z, implies(W, X)))))
% 6.84/1.29  = { by axiom 3 (condensed_detachment) R->L }
% 6.84/1.29    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)))))
% 6.84/1.29  = { by lemma 5 R->L }
% 6.84/1.29    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)))))
% 6.84/1.29  = { by lemma 5 R->L }
% 6.84/1.29    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)))))
% 6.84/1.29  = { by axiom 2 (condensed_detachment) R->L }
% 6.84/1.29    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)))))
% 6.84/1.29  = { by axiom 1 (condensed_detachment) R->L }
% 6.84/1.29    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)))))
% 6.84/1.29  = { by lemma 9 R->L }
% 6.84/1.29    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)))))
% 6.84/1.29  = { by lemma 7 }
% 6.84/1.29    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)))))
% 6.84/1.29  = { by axiom 3 (condensed_detachment) }
% 6.84/1.29    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)))))
% 6.84/1.29  = { by axiom 4 (ic_JLukasiewicz_5) }
% 6.84/1.29    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)))))
% 6.84/1.29  = { by axiom 1 (condensed_detachment) }
% 6.84/1.29    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)))))
% 6.84/1.29  = { by axiom 1 (condensed_detachment) }
% 6.84/1.29    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)))))
% 6.84/1.29  = { by axiom 1 (condensed_detachment) }
% 6.84/1.29    fresh(true, true, implies(implies(X, Y), implies(Z, implies(W, X))), implies(V, implies(U, implies(Z, implies(W, X)))))
% 6.84/1.29  = { by axiom 2 (condensed_detachment) }
% 6.84/1.29    is_a_theorem(implies(V, implies(U, implies(Z, implies(W, X)))))
% 6.84/1.29  
% 6.84/1.29  Lemma 11: is_a_theorem(implies(X, implies(implies(X, Y), implies(Z, implies(W, Y))))) = true.
% 6.84/1.29  Proof:
% 6.84/1.29    is_a_theorem(implies(X, implies(implies(X, Y), implies(Z, implies(W, Y)))))
% 6.84/1.29  = { by axiom 2 (condensed_detachment) R->L }
% 6.84/1.29    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)))))
% 6.84/1.29  = { by axiom 1 (condensed_detachment) R->L }
% 6.84/1.29    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)))))
% 6.84/1.29  = { by axiom 4 (ic_JLukasiewicz_5) R->L }
% 6.84/1.29    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)))))
% 6.84/1.29  = { by axiom 3 (condensed_detachment) R->L }
% 6.84/1.29    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)))))
% 6.84/1.29  = { by lemma 9 }
% 6.84/1.29    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)))))
% 6.84/1.29  = { by axiom 2 (condensed_detachment) }
% 6.84/1.29    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)))))
% 6.84/1.29  = { by axiom 3 (condensed_detachment) }
% 6.84/1.29    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)))))
% 6.84/1.29  = { by axiom 4 (ic_JLukasiewicz_5) }
% 6.84/1.29    fresh2(true, true, implies(X, implies(implies(X, Y), implies(Z, implies(W, Y)))))
% 6.84/1.29  = { by axiom 1 (condensed_detachment) }
% 6.84/1.29    true
% 6.84/1.29  
% 6.84/1.29  Lemma 12: 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))).
% 6.84/1.29  Proof:
% 6.84/1.29    fresh2(is_a_theorem(implies(implies(implies(X, Y), Z), Y)), true, implies(W, implies(X, Y)))
% 6.84/1.29  = { by axiom 3 (condensed_detachment) R->L }
% 6.84/1.29    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)))
% 6.84/1.29  = { by axiom 2 (condensed_detachment) R->L }
% 6.84/1.29    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)))
% 6.84/1.29  = { by axiom 1 (condensed_detachment) R->L }
% 6.84/1.29    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)))
% 6.84/1.29  = { by axiom 4 (ic_JLukasiewicz_5) R->L }
% 6.84/1.29    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)))
% 6.84/1.29  = { by axiom 3 (condensed_detachment) R->L }
% 6.84/1.29    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)))
% 6.84/1.29  = { by lemma 10 R->L }
% 6.84/1.29    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)))
% 6.84/1.29  = { by lemma 11 }
% 6.84/1.29    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)))
% 6.84/1.29  = { by axiom 1 (condensed_detachment) }
% 6.84/1.29    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)))
% 6.84/1.29  = { by axiom 2 (condensed_detachment) }
% 6.84/1.29    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)))
% 6.84/1.29  = { by axiom 3 (condensed_detachment) }
% 6.84/1.29    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)))
% 6.84/1.29  = { by axiom 4 (ic_JLukasiewicz_5) }
% 6.84/1.30    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)))
% 6.84/1.30  = { by axiom 1 (condensed_detachment) }
% 6.84/1.30    fresh(true, true, implies(implies(implies(X, Y), Z), Y), implies(W, implies(X, Y)))
% 6.84/1.30  = { by axiom 2 (condensed_detachment) }
% 6.84/1.30    is_a_theorem(implies(W, implies(X, Y)))
% 6.84/1.30  
% 6.84/1.30  Lemma 13: is_a_theorem(implies(X, implies(Y, implies(Z, implies(implies(Y, W), W))))) = true.
% 6.84/1.30  Proof:
% 6.84/1.30    is_a_theorem(implies(X, implies(Y, implies(Z, implies(implies(Y, W), W)))))
% 6.84/1.30  = { by lemma 12 R->L }
% 6.84/1.30    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)))))
% 6.84/1.30  = { by lemma 5 R->L }
% 6.84/1.30    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)))))
% 6.84/1.30  = { by axiom 2 (condensed_detachment) R->L }
% 6.84/1.30    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)))))
% 6.84/1.30  = { by axiom 1 (condensed_detachment) R->L }
% 6.84/1.30    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)))))
% 6.84/1.30  = { by axiom 4 (ic_JLukasiewicz_5) R->L }
% 6.84/1.30    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)))))
% 6.84/1.30  = { by axiom 3 (condensed_detachment) R->L }
% 6.84/1.30    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)))))
% 6.84/1.30  = { by axiom 2 (condensed_detachment) R->L }
% 6.84/1.30    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)))))
% 6.84/1.31  = { by axiom 1 (condensed_detachment) R->L }
% 7.40/1.31    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)))))
% 7.40/1.31  = { by lemma 11 R->L }
% 7.40/1.31    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)))))
% 7.40/1.31  = { by lemma 5 }
% 7.40/1.31    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)))))
% 7.40/1.31  = { by axiom 3 (condensed_detachment) }
% 7.40/1.31    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)))))
% 7.40/1.31  = { by lemma 9 }
% 7.40/1.31    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)))))
% 7.40/1.31  = { by axiom 1 (condensed_detachment) }
% 7.40/1.31    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)))))
% 7.40/1.31  = { by axiom 2 (condensed_detachment) }
% 7.40/1.32    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)))))
% 7.40/1.32  = { by axiom 3 (condensed_detachment) }
% 7.40/1.32    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)))))
% 7.40/1.32  = { by lemma 5 R->L }
% 7.40/1.32    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)))))
% 7.40/1.32  = { by axiom 2 (condensed_detachment) R->L }
% 7.40/1.32    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)))))
% 7.40/1.32  = { by axiom 1 (condensed_detachment) R->L }
% 7.40/1.32    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)))))
% 7.40/1.32  = { by axiom 1 (condensed_detachment) R->L }
% 7.40/1.32    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)))))
% 7.40/1.32  = { by axiom 1 (condensed_detachment) R->L }
% 7.40/1.32    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)))))
% 7.40/1.32  = { by lemma 9 R->L }
% 7.40/1.33    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)))))
% 7.40/1.33  = { by axiom 3 (condensed_detachment) R->L }
% 7.40/1.33    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)))))
% 7.40/1.33  = { by lemma 8 }
% 7.40/1.33    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)))))
% 7.40/1.33  = { by axiom 2 (condensed_detachment) }
% 7.40/1.33    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)))))
% 7.40/1.33  = { by lemma 5 }
% 7.40/1.33    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)))))
% 7.40/1.33  = { by lemma 5 }
% 7.40/1.33    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)))))
% 7.40/1.33  = { by axiom 3 (condensed_detachment) }
% 7.40/1.33    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)))))
% 7.40/1.33  = { by axiom 4 (ic_JLukasiewicz_5) }
% 7.40/1.33    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)))))
% 7.40/1.33  = { by axiom 1 (condensed_detachment) }
% 7.40/1.33    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)))))
% 7.40/1.33  = { by axiom 1 (condensed_detachment) }
% 7.40/1.34    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)))))
% 7.40/1.34  = { by axiom 1 (condensed_detachment) }
% 7.40/1.34    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)))))
% 7.40/1.34  = { by axiom 1 (condensed_detachment) }
% 7.40/1.34    fresh2(true, true, implies(X, implies(Y, implies(Z, implies(implies(Y, W), W)))))
% 7.40/1.34  = { by axiom 1 (condensed_detachment) }
% 7.40/1.34    true
% 7.40/1.34  
% 7.40/1.34  Lemma 14: is_a_theorem(implies(X, implies(Y, implies(implies(X, Z), Z)))) = true.
% 7.40/1.34  Proof:
% 7.40/1.34    is_a_theorem(implies(X, implies(Y, implies(implies(X, Z), Z))))
% 7.40/1.34  = { by axiom 2 (condensed_detachment) R->L }
% 7.40/1.34    fresh(true, true, implies(implies(implies(W, V), implies(U, T)), implies(implies(T, W), implies(S, implies(U, W)))), implies(X, implies(Y, implies(implies(X, Z), Z))))
% 7.40/1.34  = { by lemma 13 R->L }
% 7.40/1.34    fresh(is_a_theorem(implies(implies(implies(implies(W, V), implies(U, T)), implies(implies(T, W), implies(S, implies(U, W)))), implies(X, implies(Y, implies(implies(X, Z), Z))))), true, implies(implies(implies(W, V), implies(U, T)), implies(implies(T, W), implies(S, implies(U, W)))), implies(X, implies(Y, implies(implies(X, Z), Z))))
% 7.40/1.34  = { by axiom 3 (condensed_detachment) }
% 7.40/1.34    fresh2(is_a_theorem(implies(implies(implies(W, V), implies(U, T)), implies(implies(T, W), implies(S, implies(U, W))))), true, implies(X, implies(Y, implies(implies(X, Z), Z))))
% 7.40/1.34  = { by axiom 4 (ic_JLukasiewicz_5) }
% 7.40/1.34    fresh2(true, true, implies(X, implies(Y, implies(implies(X, Z), Z))))
% 7.40/1.34  = { by axiom 1 (condensed_detachment) }
% 7.40/1.34    true
% 7.40/1.34  
% 7.40/1.34  Lemma 15: fresh2(is_a_theorem(implies(implies(X, X), Y)), true, Y) = is_a_theorem(Y).
% 7.40/1.34  Proof:
% 7.40/1.34    fresh2(is_a_theorem(implies(implies(X, X), Y)), true, Y)
% 7.40/1.34  = { by axiom 3 (condensed_detachment) R->L }
% 7.40/1.34    fresh(is_a_theorem(implies(implies(implies(X, X), Y), Y)), true, implies(implies(X, X), Y), Y)
% 7.40/1.34  = { by axiom 2 (condensed_detachment) R->L }
% 7.40/1.34    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)
% 7.40/1.34  = { by axiom 1 (condensed_detachment) R->L }
% 7.40/1.34    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)
% 7.40/1.34  = { by axiom 1 (condensed_detachment) R->L }
% 7.40/1.34    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)
% 7.40/1.34  = { by axiom 4 (ic_JLukasiewicz_5) R->L }
% 7.40/1.34    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)
% 7.40/1.34  = { by axiom 3 (condensed_detachment) R->L }
% 7.40/1.34    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)
% 7.40/1.34  = { by axiom 2 (condensed_detachment) R->L }
% 7.40/1.34    fresh(fresh(fresh2(fresh(fresh(true, true, implies(implies(implies(V2, U2), implies(T2, S2)), implies(implies(S2, V2), implies(X3, implies(T2, V2)))), 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)
% 7.40/1.34  = { by axiom 1 (condensed_detachment) R->L }
% 7.40/1.34    fresh(fresh(fresh2(fresh(fresh(fresh2(true, true, implies(implies(implies(implies(V2, U2), implies(T2, S2)), implies(implies(S2, V2), implies(X3, implies(T2, V2)))), 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(V2, U2), implies(T2, S2)), implies(implies(S2, V2), implies(X3, implies(T2, V2)))), 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)
% 7.40/1.34  = { by axiom 4 (ic_JLukasiewicz_5) R->L }
% 7.40/1.34    fresh(fresh(fresh2(fresh(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(implies(implies(S, X2), implies(Y2, Z2)), implies(implies(Z2, S), implies(W2, implies(Y2, S)))), implies(X, X)))), true, implies(implies(implies(V2, U2), implies(T2, S2)), implies(implies(S2, V2), implies(X3, implies(T2, V2)))), 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)
% 7.40/1.34  = { by axiom 3 (condensed_detachment) R->L }
% 7.40/1.34    fresh(fresh(fresh2(fresh(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(implies(implies(S, X2), implies(Y2, Z2)), implies(implies(Z2, S), implies(W2, implies(Y2, S)))), implies(X, X))))), 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(implies(implies(S, X2), implies(Y2, Z2)), implies(implies(Z2, S), implies(W2, implies(Y2, S)))), implies(X, X)))), true, implies(implies(implies(V2, U2), implies(T2, S2)), implies(implies(S2, V2), implies(X3, implies(T2, V2)))), 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)
% 7.40/1.35  = { by lemma 10 R->L }
% 7.40/1.35    fresh(fresh(fresh2(fresh(fresh(fresh(fresh2(is_a_theorem(implies(implies(X, X), 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(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(implies(implies(S, X2), implies(Y2, Z2)), implies(implies(Z2, S), implies(W2, implies(Y2, S)))), implies(X, X))))), 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(implies(implies(S, X2), implies(Y2, Z2)), implies(implies(Z2, S), implies(W2, implies(Y2, S)))), implies(X, X)))), true, implies(implies(implies(V2, U2), implies(T2, S2)), implies(implies(S2, V2), implies(X3, implies(T2, V2)))), 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)
% 7.40/1.35  = { by axiom 2 (condensed_detachment) R->L }
% 7.40/1.35    fresh(fresh(fresh2(fresh(fresh(fresh(fresh2(fresh(true, true, implies(implies(implies(T3, S3), implies(X4, Y4)), implies(implies(Y4, T3), implies(Z4, implies(X4, T3)))), implies(implies(X, X), 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(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(implies(implies(S, X2), implies(Y2, Z2)), implies(implies(Z2, S), implies(W2, implies(Y2, S)))), implies(X, X))))), 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(implies(implies(S, X2), implies(Y2, Z2)), implies(implies(Z2, S), implies(W2, implies(Y2, S)))), implies(X, X)))), true, implies(implies(implies(V2, U2), implies(T2, S2)), implies(implies(S2, V2), implies(X3, implies(T2, V2)))), 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)
% 7.40/1.35  = { by axiom 1 (condensed_detachment) R->L }
% 7.40/1.35    fresh(fresh(fresh2(fresh(fresh(fresh(fresh2(fresh(fresh2(true, true, implies(implies(implies(implies(T3, S3), implies(X4, Y4)), implies(implies(Y4, T3), implies(Z4, implies(X4, T3)))), implies(implies(X, X), 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(T3, S3), implies(X4, Y4)), implies(implies(Y4, T3), implies(Z4, implies(X4, T3)))), implies(implies(X, X), 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(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(implies(implies(S, X2), implies(Y2, Z2)), implies(implies(Z2, S), implies(W2, implies(Y2, S)))), implies(X, X))))), 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(implies(implies(S, X2), implies(Y2, Z2)), implies(implies(Z2, S), implies(W2, implies(Y2, S)))), implies(X, X)))), true, implies(implies(implies(V2, U2), implies(T2, S2)), implies(implies(S2, V2), implies(X3, implies(T2, V2)))), 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)
% 7.40/1.35  = { by axiom 4 (ic_JLukasiewicz_5) R->L }
% 7.40/1.35    fresh(fresh(fresh2(fresh(fresh(fresh(fresh2(fresh(fresh2(is_a_theorem(implies(implies(implies(W4, V4), implies(U4, T4)), implies(implies(T4, W4), implies(S4, implies(U4, W4))))), true, implies(implies(implies(implies(T3, S3), implies(X4, Y4)), implies(implies(Y4, T3), implies(Z4, implies(X4, T3)))), implies(implies(X, X), 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(T3, S3), implies(X4, Y4)), implies(implies(Y4, T3), implies(Z4, implies(X4, T3)))), implies(implies(X, X), 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(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(implies(implies(S, X2), implies(Y2, Z2)), implies(implies(Z2, S), implies(W2, implies(Y2, S)))), implies(X, X))))), 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(implies(implies(S, X2), implies(Y2, Z2)), implies(implies(Z2, S), implies(W2, implies(Y2, S)))), implies(X, X)))), true, implies(implies(implies(V2, U2), implies(T2, S2)), implies(implies(S2, V2), implies(X3, implies(T2, V2)))), 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)
% 7.40/1.35  = { by axiom 3 (condensed_detachment) R->L }
% 7.40/1.35    fresh(fresh(fresh2(fresh(fresh(fresh(fresh2(fresh(fresh(is_a_theorem(implies(implies(implies(implies(W4, V4), implies(U4, T4)), implies(implies(T4, W4), implies(S4, implies(U4, W4)))), implies(implies(implies(implies(T3, S3), implies(X4, Y4)), implies(implies(Y4, T3), implies(Z4, implies(X4, T3)))), implies(implies(X, X), 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(W4, V4), implies(U4, T4)), implies(implies(T4, W4), implies(S4, implies(U4, W4)))), implies(implies(implies(implies(T3, S3), implies(X4, Y4)), implies(implies(Y4, T3), implies(Z4, implies(X4, T3)))), implies(implies(X, X), 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(T3, S3), implies(X4, Y4)), implies(implies(Y4, T3), implies(Z4, implies(X4, T3)))), implies(implies(X, X), 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(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(implies(implies(S, X2), implies(Y2, Z2)), implies(implies(Z2, S), implies(W2, implies(Y2, S)))), implies(X, X))))), 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(implies(implies(S, X2), implies(Y2, Z2)), implies(implies(Z2, S), implies(W2, implies(Y2, S)))), implies(X, X)))), true, implies(implies(implies(V2, U2), implies(T2, S2)), implies(implies(S2, V2), implies(X3, implies(T2, V2)))), 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)
% 7.40/1.36  = { by lemma 10 R->L }
% 7.40/1.36    fresh(fresh(fresh2(fresh(fresh(fresh(fresh2(fresh(fresh(fresh2(is_a_theorem(implies(implies(implies(X, X), implies(X, X)), implies(implies(X, X), 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(implies(W4, V4), implies(U4, T4)), implies(implies(T4, W4), implies(S4, implies(U4, W4)))), implies(implies(implies(implies(T3, S3), implies(X4, Y4)), implies(implies(Y4, T3), implies(Z4, implies(X4, T3)))), implies(implies(X, X), 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(W4, V4), implies(U4, T4)), implies(implies(T4, W4), implies(S4, implies(U4, W4)))), implies(implies(implies(implies(T3, S3), implies(X4, Y4)), implies(implies(Y4, T3), implies(Z4, implies(X4, T3)))), implies(implies(X, X), 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(T3, S3), implies(X4, Y4)), implies(implies(Y4, T3), implies(Z4, implies(X4, T3)))), implies(implies(X, X), 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(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(implies(implies(S, X2), implies(Y2, Z2)), implies(implies(Z2, S), implies(W2, implies(Y2, S)))), implies(X, X))))), 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(implies(implies(S, X2), implies(Y2, Z2)), implies(implies(Z2, S), implies(W2, implies(Y2, S)))), implies(X, X)))), true, implies(implies(implies(V2, U2), implies(T2, S2)), implies(implies(S2, V2), implies(X3, implies(T2, V2)))), 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)
% 7.40/1.36  = { by axiom 4 (ic_JLukasiewicz_5) }
% 7.40/1.36    fresh(fresh(fresh2(fresh(fresh(fresh(fresh2(fresh(fresh(fresh2(true, true, implies(implies(implies(implies(W4, V4), implies(U4, T4)), implies(implies(T4, W4), implies(S4, implies(U4, W4)))), implies(implies(implies(implies(T3, S3), implies(X4, Y4)), implies(implies(Y4, T3), implies(Z4, implies(X4, T3)))), implies(implies(X, X), 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(W4, V4), implies(U4, T4)), implies(implies(T4, W4), implies(S4, implies(U4, W4)))), implies(implies(implies(implies(T3, S3), implies(X4, Y4)), implies(implies(Y4, T3), implies(Z4, implies(X4, T3)))), implies(implies(X, X), 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(T3, S3), implies(X4, Y4)), implies(implies(Y4, T3), implies(Z4, implies(X4, T3)))), implies(implies(X, X), 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(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(implies(implies(S, X2), implies(Y2, Z2)), implies(implies(Z2, S), implies(W2, implies(Y2, S)))), implies(X, X))))), 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(implies(implies(S, X2), implies(Y2, Z2)), implies(implies(Z2, S), implies(W2, implies(Y2, S)))), implies(X, X)))), true, implies(implies(implies(V2, U2), implies(T2, S2)), implies(implies(S2, V2), implies(X3, implies(T2, V2)))), 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)
% 7.40/1.36  = { by axiom 1 (condensed_detachment) }
% 7.40/1.36    fresh(fresh(fresh2(fresh(fresh(fresh(fresh2(fresh(fresh(true, true, implies(implies(implies(W4, V4), implies(U4, T4)), implies(implies(T4, W4), implies(S4, implies(U4, W4)))), implies(implies(implies(implies(T3, S3), implies(X4, Y4)), implies(implies(Y4, T3), implies(Z4, implies(X4, T3)))), implies(implies(X, X), 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(T3, S3), implies(X4, Y4)), implies(implies(Y4, T3), implies(Z4, implies(X4, T3)))), implies(implies(X, X), 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(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(implies(implies(S, X2), implies(Y2, Z2)), implies(implies(Z2, S), implies(W2, implies(Y2, S)))), implies(X, X))))), 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(implies(implies(S, X2), implies(Y2, Z2)), implies(implies(Z2, S), implies(W2, implies(Y2, S)))), implies(X, X)))), true, implies(implies(implies(V2, U2), implies(T2, S2)), implies(implies(S2, V2), implies(X3, implies(T2, V2)))), 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)
% 7.40/1.36  = { by axiom 2 (condensed_detachment) }
% 7.40/1.36    fresh(fresh(fresh2(fresh(fresh(fresh(fresh2(fresh(is_a_theorem(implies(implies(implies(implies(T3, S3), implies(X4, Y4)), implies(implies(Y4, T3), implies(Z4, implies(X4, T3)))), implies(implies(X, X), 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(T3, S3), implies(X4, Y4)), implies(implies(Y4, T3), implies(Z4, implies(X4, T3)))), implies(implies(X, X), 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(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(implies(implies(S, X2), implies(Y2, Z2)), implies(implies(Z2, S), implies(W2, implies(Y2, S)))), implies(X, X))))), 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(implies(implies(S, X2), implies(Y2, Z2)), implies(implies(Z2, S), implies(W2, implies(Y2, S)))), implies(X, X)))), true, implies(implies(implies(V2, U2), implies(T2, S2)), implies(implies(S2, V2), implies(X3, implies(T2, V2)))), 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)
% 7.40/1.36  = { by axiom 3 (condensed_detachment) }
% 7.40/1.36    fresh(fresh(fresh2(fresh(fresh(fresh(fresh2(fresh2(is_a_theorem(implies(implies(implies(T3, S3), implies(X4, Y4)), implies(implies(Y4, T3), implies(Z4, implies(X4, T3))))), true, implies(implies(X, X), 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(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(implies(implies(S, X2), implies(Y2, Z2)), implies(implies(Z2, S), implies(W2, implies(Y2, S)))), implies(X, X))))), 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(implies(implies(S, X2), implies(Y2, Z2)), implies(implies(Z2, S), implies(W2, implies(Y2, S)))), implies(X, X)))), true, implies(implies(implies(V2, U2), implies(T2, S2)), implies(implies(S2, V2), implies(X3, implies(T2, V2)))), 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)
% 7.40/1.37  = { by axiom 4 (ic_JLukasiewicz_5) }
% 7.40/1.37    fresh(fresh(fresh2(fresh(fresh(fresh(fresh2(fresh2(true, true, implies(implies(X, X), 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(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(implies(implies(S, X2), implies(Y2, Z2)), implies(implies(Z2, S), implies(W2, implies(Y2, S)))), implies(X, X))))), 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(implies(implies(S, X2), implies(Y2, Z2)), implies(implies(Z2, S), implies(W2, implies(Y2, S)))), implies(X, X)))), true, implies(implies(implies(V2, U2), implies(T2, S2)), implies(implies(S2, V2), implies(X3, implies(T2, V2)))), 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)
% 7.40/1.37  = { by axiom 1 (condensed_detachment) }
% 7.40/1.37    fresh(fresh(fresh2(fresh(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(implies(implies(S, X2), implies(Y2, Z2)), implies(implies(Z2, S), implies(W2, implies(Y2, S)))), implies(X, X))))), 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(implies(implies(S, X2), implies(Y2, Z2)), implies(implies(Z2, S), implies(W2, implies(Y2, S)))), implies(X, X)))), true, implies(implies(implies(V2, U2), implies(T2, S2)), implies(implies(S2, V2), implies(X3, implies(T2, V2)))), 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)
% 7.40/1.37  = { by axiom 1 (condensed_detachment) }
% 7.40/1.37    fresh(fresh(fresh2(fresh(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(implies(implies(S, X2), implies(Y2, Z2)), implies(implies(Z2, S), implies(W2, implies(Y2, S)))), implies(X, X)))), true, implies(implies(implies(V2, U2), implies(T2, S2)), implies(implies(S2, V2), implies(X3, implies(T2, V2)))), 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)
% 7.40/1.37  = { by axiom 2 (condensed_detachment) }
% 7.40/1.37    fresh(fresh(fresh2(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(implies(implies(S, X2), implies(Y2, Z2)), implies(implies(Z2, S), implies(W2, implies(Y2, S)))), implies(X, X)))), true, implies(implies(implies(V2, U2), implies(T2, S2)), implies(implies(S2, V2), implies(X3, implies(T2, V2)))), 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)
% 7.40/1.37  = { by axiom 3 (condensed_detachment) }
% 7.40/1.37    fresh(fresh(fresh2(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(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)
% 7.40/1.37  = { by axiom 4 (ic_JLukasiewicz_5) }
% 7.40/1.37    fresh(fresh(fresh2(fresh(fresh2(true, true, 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)
% 7.40/1.37  = { by axiom 1 (condensed_detachment) }
% 7.40/1.37    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)
% 7.40/1.37  = { by axiom 2 (condensed_detachment) }
% 7.40/1.37    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)
% 7.40/1.37  = { by axiom 3 (condensed_detachment) R->L }
% 7.40/1.37    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)
% 7.40/1.37  = { by lemma 14 }
% 7.40/1.37    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)
% 7.40/1.37  = { by axiom 2 (condensed_detachment) }
% 7.40/1.37    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)
% 7.40/1.37  = { by axiom 3 (condensed_detachment) }
% 7.40/1.37    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)
% 7.40/1.37  = { by axiom 4 (ic_JLukasiewicz_5) }
% 7.40/1.37    fresh(fresh2(true, true, implies(implies(implies(X, X), Y), Y)), true, implies(implies(X, X), Y), Y)
% 7.40/1.37  = { by axiom 1 (condensed_detachment) }
% 7.40/1.37    fresh(true, true, implies(implies(X, X), Y), Y)
% 7.40/1.37  = { by axiom 2 (condensed_detachment) }
% 7.40/1.37    is_a_theorem(Y)
% 7.40/1.37  
% 7.40/1.37  Goal 1 (prove_ic_5): is_a_theorem(implies(a, implies(implies(a, b), b))) = true.
% 7.40/1.37  Proof:
% 7.40/1.37    is_a_theorem(implies(a, implies(implies(a, b), b)))
% 7.40/1.37  = { by axiom 2 (condensed_detachment) R->L }
% 7.40/1.37    fresh(true, true, implies(a, implies(a, implies(implies(a, b), b))), implies(a, implies(implies(a, b), b)))
% 7.40/1.37  = { by axiom 1 (condensed_detachment) R->L }
% 7.40/1.37    fresh(fresh2(true, true, implies(implies(a, implies(a, implies(implies(a, b), b))), implies(a, implies(implies(a, b), b)))), true, implies(a, implies(a, implies(implies(a, b), b))), implies(a, implies(implies(a, b), b)))
% 7.40/1.37  = { by axiom 1 (condensed_detachment) R->L }
% 7.40/1.37    fresh(fresh2(fresh2(true, true, implies(implies(X, X), implies(implies(a, implies(a, implies(implies(a, b), b))), implies(a, implies(implies(a, b), b))))), true, implies(implies(a, implies(a, implies(implies(a, b), b))), implies(a, implies(implies(a, b), b)))), true, implies(a, implies(a, implies(implies(a, b), b))), implies(a, implies(implies(a, b), b)))
% 7.40/1.37  = { by axiom 1 (condensed_detachment) R->L }
% 7.40/1.37    fresh(fresh2(fresh2(fresh2(true, true, implies(implies(Y, Y), implies(implies(X, X), implies(implies(a, implies(a, implies(implies(a, b), b))), implies(a, implies(implies(a, b), b)))))), true, implies(implies(X, X), implies(implies(a, implies(a, implies(implies(a, b), b))), implies(a, implies(implies(a, b), b))))), true, implies(implies(a, implies(a, implies(implies(a, b), b))), implies(a, implies(implies(a, b), b)))), true, implies(a, implies(a, implies(implies(a, b), b))), implies(a, implies(implies(a, b), b)))
% 7.40/1.37  = { by axiom 1 (condensed_detachment) R->L }
% 7.40/1.37    fresh(fresh2(fresh2(fresh2(fresh2(true, true, implies(implies(implies(implies(X, X), implies(implies(a, implies(a, implies(implies(a, b), b))), implies(a, implies(implies(a, b), b)))), implies(implies(a, b), b)), implies(implies(a, implies(a, implies(implies(a, b), b))), implies(a, implies(implies(a, b), b))))), true, implies(implies(Y, Y), implies(implies(X, X), implies(implies(a, implies(a, implies(implies(a, b), b))), implies(a, implies(implies(a, b), b)))))), true, implies(implies(X, X), implies(implies(a, implies(a, implies(implies(a, b), b))), implies(a, implies(implies(a, b), b))))), true, implies(implies(a, implies(a, implies(implies(a, b), b))), implies(a, implies(implies(a, b), b)))), true, implies(a, implies(a, implies(implies(a, b), b))), implies(a, implies(implies(a, b), b)))
% 7.40/1.37  = { by lemma 13 R->L }
% 7.40/1.37    fresh(fresh2(fresh2(fresh2(fresh2(is_a_theorem(implies(implies(implies(implies(a, b), b), Z), implies(a, implies(implies(X, X), implies(implies(a, implies(a, implies(implies(a, b), b))), implies(a, implies(implies(a, b), b))))))), true, implies(implies(implies(implies(X, X), implies(implies(a, implies(a, implies(implies(a, b), b))), implies(a, implies(implies(a, b), b)))), implies(implies(a, b), b)), implies(implies(a, implies(a, implies(implies(a, b), b))), implies(a, implies(implies(a, b), b))))), true, implies(implies(Y, Y), implies(implies(X, X), implies(implies(a, implies(a, implies(implies(a, b), b))), implies(a, implies(implies(a, b), b)))))), true, implies(implies(X, X), implies(implies(a, implies(a, implies(implies(a, b), b))), implies(a, implies(implies(a, b), b))))), true, implies(implies(a, implies(a, implies(implies(a, b), b))), implies(a, implies(implies(a, b), b)))), true, implies(a, implies(a, implies(implies(a, b), b))), implies(a, implies(implies(a, b), b)))
% 7.40/1.37  = { by lemma 5 }
% 7.40/1.37    fresh(fresh2(fresh2(fresh2(is_a_theorem(implies(implies(implies(implies(X, X), implies(implies(a, implies(a, implies(implies(a, b), b))), implies(a, implies(implies(a, b), b)))), implies(implies(a, b), b)), implies(implies(a, implies(a, implies(implies(a, b), b))), implies(a, implies(implies(a, b), b))))), true, implies(implies(Y, Y), implies(implies(X, X), implies(implies(a, implies(a, implies(implies(a, b), b))), implies(a, implies(implies(a, b), b)))))), true, implies(implies(X, X), implies(implies(a, implies(a, implies(implies(a, b), b))), implies(a, implies(implies(a, b), b))))), true, implies(implies(a, implies(a, implies(implies(a, b), b))), implies(a, implies(implies(a, b), b)))), true, implies(a, implies(a, implies(implies(a, b), b))), implies(a, implies(implies(a, b), b)))
% 7.40/1.37  = { by lemma 12 }
% 7.40/1.37    fresh(fresh2(fresh2(is_a_theorem(implies(implies(Y, Y), implies(implies(X, X), implies(implies(a, implies(a, implies(implies(a, b), b))), implies(a, implies(implies(a, b), b)))))), true, implies(implies(X, X), implies(implies(a, implies(a, implies(implies(a, b), b))), implies(a, implies(implies(a, b), b))))), true, implies(implies(a, implies(a, implies(implies(a, b), b))), implies(a, implies(implies(a, b), b)))), true, implies(a, implies(a, implies(implies(a, b), b))), implies(a, implies(implies(a, b), b)))
% 7.40/1.37  = { by lemma 15 }
% 7.40/1.37    fresh(fresh2(is_a_theorem(implies(implies(X, X), implies(implies(a, implies(a, implies(implies(a, b), b))), implies(a, implies(implies(a, b), b))))), true, implies(implies(a, implies(a, implies(implies(a, b), b))), implies(a, implies(implies(a, b), b)))), true, implies(a, implies(a, implies(implies(a, b), b))), implies(a, implies(implies(a, b), b)))
% 7.40/1.37  = { by lemma 15 }
% 7.40/1.37    fresh(is_a_theorem(implies(implies(a, implies(a, implies(implies(a, b), b))), implies(a, implies(implies(a, b), b)))), true, implies(a, implies(a, implies(implies(a, b), b))), implies(a, implies(implies(a, b), b)))
% 7.40/1.37  = { by axiom 3 (condensed_detachment) }
% 7.40/1.37    fresh2(is_a_theorem(implies(a, implies(a, implies(implies(a, b), b)))), true, implies(a, implies(implies(a, b), b)))
% 7.40/1.37  = { by lemma 14 }
% 7.40/1.37    fresh2(true, true, implies(a, implies(implies(a, b), b)))
% 7.40/1.37  = { by axiom 1 (condensed_detachment) }
% 7.40/1.37    true
% 7.40/1.37  % SZS output end Proof
% 7.40/1.37  
% 7.40/1.37  RESULT: Unsatisfiable (the axioms are contradictory).
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