%------------------------------------------------------------------------------
% File     : Twee---2.4.2
% Problem  : LCL401-1 : TPTP v8.1.2. Released v2.3.0.
% Transfm  : none
% Format   : tptp:raw
% Command  : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof

% Computer : n024.cluster.edu
% Model    : x86_64 x86_64
% CPU      : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory   : 8042.1875MB
% OS       : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit  : 300s
% DateTime : Thu Aug 31 08:18:47 EDT 2023

% Result   : Unsatisfiable 31.58s 4.39s
% Output   : Proof 32.26s
% Verified : 
% SZS Type : -

% Comments : 
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.06/0.12  % Problem  : LCL401-1 : TPTP v8.1.2. Released v2.3.0.
% 0.06/0.13  % Command  : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.13/0.34  % Computer : n024.cluster.edu
% 0.13/0.34  % Model    : x86_64 x86_64
% 0.13/0.34  % CPU      : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.34  % Memory   : 8042.1875MB
% 0.13/0.34  % OS       : Linux 3.10.0-693.el7.x86_64
% 0.13/0.34  % CPULimit : 300
% 0.13/0.34  % WCLimit  : 300
% 0.13/0.34  % DateTime : Fri Aug 25 05:40:39 EDT 2023
% 0.13/0.34  % CPUTime  : 
% 31.58/4.39  Command-line arguments: --no-flatten-goal
% 31.58/4.39  
% 31.58/4.39  % SZS status Unsatisfiable
% 31.58/4.39  
% 32.26/4.48  % SZS output start Proof
% 32.26/4.48  Take the following subset of the input axioms:
% 32.26/4.48    fof(cn_1, axiom, ![X, Y, Z]: is_a_theorem(implies(implies(X, Y), implies(implies(Y, Z), implies(X, Z))))).
% 32.26/4.48    fof(cn_2, axiom, ![X2]: is_a_theorem(implies(implies(not(X2), X2), X2))).
% 32.26/4.48    fof(cn_3, axiom, ![X2, Y2]: is_a_theorem(implies(X2, implies(not(X2), Y2)))).
% 32.26/4.48    fof(condensed_detachment, axiom, ![X2, Y2]: (~is_a_theorem(implies(X2, Y2)) | (~is_a_theorem(X2) | is_a_theorem(Y2)))).
% 32.26/4.48    fof(prove_cn_67, negated_conjecture, ~is_a_theorem(implies(not(implies(x, y)), not(y)))).
% 32.26/4.48  
% 32.26/4.48  Now clausify the problem and encode Horn clauses using encoding 3 of
% 32.26/4.48  http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 32.26/4.48  We repeatedly replace C & s=t => u=v by the two clauses:
% 32.26/4.48    fresh(y, y, x1...xn) = u
% 32.26/4.48    C => fresh(s, t, x1...xn) = v
% 32.26/4.48  where fresh is a fresh function symbol and x1..xn are the free
% 32.26/4.48  variables of u and v.
% 32.26/4.48  A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 32.26/4.48  input problem has no model of domain size 1).
% 32.26/4.48  
% 32.26/4.48  The encoding turns the above axioms into the following unit equations and goals:
% 32.26/4.48  
% 32.26/4.48  Axiom 1 (condensed_detachment): fresh2(X, X, Y) = true.
% 32.26/4.48  Axiom 2 (condensed_detachment): fresh(X, X, Y, Z) = is_a_theorem(Z).
% 32.26/4.48  Axiom 3 (cn_3): is_a_theorem(implies(X, implies(not(X), Y))) = true.
% 32.26/4.48  Axiom 4 (cn_2): is_a_theorem(implies(implies(not(X), X), X)) = true.
% 32.26/4.48  Axiom 5 (condensed_detachment): fresh(is_a_theorem(implies(X, Y)), true, X, Y) = fresh2(is_a_theorem(X), true, Y).
% 32.26/4.48  Axiom 6 (cn_1): is_a_theorem(implies(implies(X, Y), implies(implies(Y, Z), implies(X, Z)))) = true.
% 32.26/4.48  
% 32.26/4.48  Lemma 7: fresh2(is_a_theorem(X), true, implies(not(X), Y)) = is_a_theorem(implies(not(X), Y)).
% 32.26/4.48  Proof:
% 32.26/4.48    fresh2(is_a_theorem(X), true, implies(not(X), Y))
% 32.26/4.48  = { by axiom 5 (condensed_detachment) R->L }
% 32.26/4.48    fresh(is_a_theorem(implies(X, implies(not(X), Y))), true, X, implies(not(X), Y))
% 32.26/4.48  = { by axiom 3 (cn_3) }
% 32.26/4.48    fresh(true, true, X, implies(not(X), Y))
% 32.26/4.48  = { by axiom 2 (condensed_detachment) }
% 32.26/4.48    is_a_theorem(implies(not(X), Y))
% 32.26/4.48  
% 32.26/4.48  Lemma 8: fresh2(is_a_theorem(implies(not(X), X)), true, X) = is_a_theorem(X).
% 32.26/4.48  Proof:
% 32.26/4.48    fresh2(is_a_theorem(implies(not(X), X)), true, X)
% 32.26/4.48  = { by axiom 5 (condensed_detachment) R->L }
% 32.26/4.48    fresh(is_a_theorem(implies(implies(not(X), X), X)), true, implies(not(X), X), X)
% 32.26/4.48  = { by axiom 4 (cn_2) }
% 32.26/4.48    fresh(true, true, implies(not(X), X), X)
% 32.26/4.48  = { by axiom 2 (condensed_detachment) }
% 32.26/4.48    is_a_theorem(X)
% 32.26/4.48  
% 32.26/4.48  Lemma 9: fresh2(is_a_theorem(implies(X, Y)), true, implies(implies(Y, Z), implies(X, Z))) = is_a_theorem(implies(implies(Y, Z), implies(X, Z))).
% 32.26/4.48  Proof:
% 32.26/4.48    fresh2(is_a_theorem(implies(X, Y)), true, implies(implies(Y, Z), implies(X, Z)))
% 32.26/4.48  = { by axiom 5 (condensed_detachment) R->L }
% 32.26/4.48    fresh(is_a_theorem(implies(implies(X, Y), implies(implies(Y, Z), implies(X, Z)))), true, implies(X, Y), implies(implies(Y, Z), implies(X, Z)))
% 32.26/4.48  = { by axiom 6 (cn_1) }
% 32.26/4.48    fresh(true, true, implies(X, Y), implies(implies(Y, Z), implies(X, Z)))
% 32.26/4.48  = { by axiom 2 (condensed_detachment) }
% 32.26/4.48    is_a_theorem(implies(implies(Y, Z), implies(X, Z)))
% 32.26/4.48  
% 32.26/4.48  Lemma 10: is_a_theorem(implies(implies(X, Y), implies(implies(not(X), X), Y))) = true.
% 32.26/4.48  Proof:
% 32.26/4.48    is_a_theorem(implies(implies(X, Y), implies(implies(not(X), X), Y)))
% 32.26/4.48  = { by lemma 9 R->L }
% 32.26/4.48    fresh2(is_a_theorem(implies(implies(not(X), X), X)), true, implies(implies(X, Y), implies(implies(not(X), X), Y)))
% 32.26/4.48  = { by axiom 4 (cn_2) }
% 32.26/4.48    fresh2(true, true, implies(implies(X, Y), implies(implies(not(X), X), Y)))
% 32.26/4.48  = { by axiom 1 (condensed_detachment) }
% 32.26/4.48    true
% 32.26/4.48  
% 32.26/4.48  Lemma 11: is_a_theorem(implies(implies(implies(not(X), Y), Z), implies(X, Z))) = true.
% 32.26/4.48  Proof:
% 32.26/4.48    is_a_theorem(implies(implies(implies(not(X), Y), Z), implies(X, Z)))
% 32.26/4.48  = { by lemma 9 R->L }
% 32.26/4.48    fresh2(is_a_theorem(implies(X, implies(not(X), Y))), true, implies(implies(implies(not(X), Y), Z), implies(X, Z)))
% 32.26/4.48  = { by axiom 3 (cn_3) }
% 32.26/4.48    fresh2(true, true, implies(implies(implies(not(X), Y), Z), implies(X, Z)))
% 32.26/4.48  = { by axiom 1 (condensed_detachment) }
% 32.26/4.48    true
% 32.26/4.48  
% 32.26/4.48  Lemma 12: fresh2(is_a_theorem(implies(implies(not(X), Y), Z)), true, implies(X, Z)) = is_a_theorem(implies(X, Z)).
% 32.26/4.48  Proof:
% 32.26/4.48    fresh2(is_a_theorem(implies(implies(not(X), Y), Z)), true, implies(X, Z))
% 32.26/4.48  = { by axiom 5 (condensed_detachment) R->L }
% 32.26/4.48    fresh(is_a_theorem(implies(implies(implies(not(X), Y), Z), implies(X, Z))), true, implies(implies(not(X), Y), Z), implies(X, Z))
% 32.26/4.48  = { by lemma 11 }
% 32.26/4.48    fresh(true, true, implies(implies(not(X), Y), Z), implies(X, Z))
% 32.26/4.48  = { by axiom 2 (condensed_detachment) }
% 32.26/4.48    is_a_theorem(implies(X, Z))
% 32.26/4.48  
% 32.26/4.48  Lemma 13: fresh2(is_a_theorem(implies(implies(implies(X, Y), implies(Z, Y)), W)), true, implies(implies(Z, X), W)) = is_a_theorem(implies(implies(Z, X), W)).
% 32.26/4.48  Proof:
% 32.26/4.48    fresh2(is_a_theorem(implies(implies(implies(X, Y), implies(Z, Y)), W)), true, implies(implies(Z, X), W))
% 32.26/4.48  = { by axiom 5 (condensed_detachment) R->L }
% 32.26/4.48    fresh(is_a_theorem(implies(implies(implies(implies(X, Y), implies(Z, Y)), W), implies(implies(Z, X), W))), true, implies(implies(implies(X, Y), implies(Z, Y)), W), implies(implies(Z, X), W))
% 32.26/4.48  = { by lemma 9 R->L }
% 32.26/4.48    fresh(fresh2(is_a_theorem(implies(implies(Z, X), implies(implies(X, Y), implies(Z, Y)))), true, implies(implies(implies(implies(X, Y), implies(Z, Y)), W), implies(implies(Z, X), W))), true, implies(implies(implies(X, Y), implies(Z, Y)), W), implies(implies(Z, X), W))
% 32.26/4.48  = { by axiom 6 (cn_1) }
% 32.26/4.48    fresh(fresh2(true, true, implies(implies(implies(implies(X, Y), implies(Z, Y)), W), implies(implies(Z, X), W))), true, implies(implies(implies(X, Y), implies(Z, Y)), W), implies(implies(Z, X), W))
% 32.26/4.48  = { by axiom 1 (condensed_detachment) }
% 32.26/4.48    fresh(true, true, implies(implies(implies(X, Y), implies(Z, Y)), W), implies(implies(Z, X), W))
% 32.26/4.48  = { by axiom 2 (condensed_detachment) }
% 32.26/4.48    is_a_theorem(implies(implies(Z, X), W))
% 32.26/4.48  
% 32.26/4.48  Lemma 14: fresh2(is_a_theorem(implies(implies(X, implies(not(Y), Z)), W)), true, implies(Y, W)) = is_a_theorem(implies(Y, W)).
% 32.26/4.48  Proof:
% 32.26/4.48    fresh2(is_a_theorem(implies(implies(X, implies(not(Y), Z)), W)), true, implies(Y, W))
% 32.26/4.48  = { by axiom 5 (condensed_detachment) R->L }
% 32.26/4.48    fresh(is_a_theorem(implies(implies(implies(X, implies(not(Y), Z)), W), implies(Y, W))), true, implies(implies(X, implies(not(Y), Z)), W), implies(Y, W))
% 32.26/4.48  = { by lemma 9 R->L }
% 32.26/4.48    fresh(fresh2(is_a_theorem(implies(Y, implies(X, implies(not(Y), Z)))), true, implies(implies(implies(X, implies(not(Y), Z)), W), implies(Y, W))), true, implies(implies(X, implies(not(Y), Z)), W), implies(Y, W))
% 32.26/4.48  = { by lemma 12 R->L }
% 32.26/4.48    fresh(fresh2(fresh2(is_a_theorem(implies(implies(not(Y), not(X)), implies(X, implies(not(Y), Z)))), true, implies(Y, implies(X, implies(not(Y), Z)))), true, implies(implies(implies(X, implies(not(Y), Z)), W), implies(Y, W))), true, implies(implies(X, implies(not(Y), Z)), W), implies(Y, W))
% 32.26/4.48  = { by lemma 13 R->L }
% 32.26/4.48    fresh(fresh2(fresh2(fresh2(is_a_theorem(implies(implies(implies(not(X), Z), implies(not(Y), Z)), implies(X, implies(not(Y), Z)))), true, implies(implies(not(Y), not(X)), implies(X, implies(not(Y), Z)))), true, implies(Y, implies(X, implies(not(Y), Z)))), true, implies(implies(implies(X, implies(not(Y), Z)), W), implies(Y, W))), true, implies(implies(X, implies(not(Y), Z)), W), implies(Y, W))
% 32.26/4.48  = { by lemma 11 }
% 32.26/4.48    fresh(fresh2(fresh2(fresh2(true, true, implies(implies(not(Y), not(X)), implies(X, implies(not(Y), Z)))), true, implies(Y, implies(X, implies(not(Y), Z)))), true, implies(implies(implies(X, implies(not(Y), Z)), W), implies(Y, W))), true, implies(implies(X, implies(not(Y), Z)), W), implies(Y, W))
% 32.26/4.48  = { by axiom 1 (condensed_detachment) }
% 32.26/4.48    fresh(fresh2(fresh2(true, true, implies(Y, implies(X, implies(not(Y), Z)))), true, implies(implies(implies(X, implies(not(Y), Z)), W), implies(Y, W))), true, implies(implies(X, implies(not(Y), Z)), W), implies(Y, W))
% 32.26/4.48  = { by axiom 1 (condensed_detachment) }
% 32.26/4.48    fresh(fresh2(true, true, implies(implies(implies(X, implies(not(Y), Z)), W), implies(Y, W))), true, implies(implies(X, implies(not(Y), Z)), W), implies(Y, W))
% 32.26/4.48  = { by axiom 1 (condensed_detachment) }
% 32.26/4.48    fresh(true, true, implies(implies(X, implies(not(Y), Z)), W), implies(Y, W))
% 32.26/4.48  = { by axiom 2 (condensed_detachment) }
% 32.26/4.48    is_a_theorem(implies(Y, W))
% 32.26/4.48  
% 32.26/4.48  Lemma 15: fresh2(is_a_theorem(implies(implies(implies(not(X), X), Y), Z)), true, implies(implies(X, Y), Z)) = is_a_theorem(implies(implies(X, Y), Z)).
% 32.26/4.48  Proof:
% 32.26/4.48    fresh2(is_a_theorem(implies(implies(implies(not(X), X), Y), Z)), true, implies(implies(X, Y), Z))
% 32.26/4.48  = { by axiom 5 (condensed_detachment) R->L }
% 32.26/4.48    fresh(is_a_theorem(implies(implies(implies(implies(not(X), X), Y), Z), implies(implies(X, Y), Z))), true, implies(implies(implies(not(X), X), Y), Z), implies(implies(X, Y), Z))
% 32.26/4.48  = { by lemma 9 R->L }
% 32.26/4.48    fresh(fresh2(is_a_theorem(implies(implies(X, Y), implies(implies(not(X), X), Y))), true, implies(implies(implies(implies(not(X), X), Y), Z), implies(implies(X, Y), Z))), true, implies(implies(implies(not(X), X), Y), Z), implies(implies(X, Y), Z))
% 32.26/4.48  = { by lemma 10 }
% 32.26/4.48    fresh(fresh2(true, true, implies(implies(implies(implies(not(X), X), Y), Z), implies(implies(X, Y), Z))), true, implies(implies(implies(not(X), X), Y), Z), implies(implies(X, Y), Z))
% 32.26/4.48  = { by axiom 1 (condensed_detachment) }
% 32.26/4.48    fresh(true, true, implies(implies(implies(not(X), X), Y), Z), implies(implies(X, Y), Z))
% 32.26/4.48  = { by axiom 2 (condensed_detachment) }
% 32.26/4.48    is_a_theorem(implies(implies(X, Y), Z))
% 32.26/4.48  
% 32.26/4.48  Lemma 16: is_a_theorem(implies(implies(implies(implies(not(X), X), X), Y), implies(Z, Y))) = true.
% 32.26/4.48  Proof:
% 32.26/4.48    is_a_theorem(implies(implies(implies(implies(not(X), X), X), Y), implies(Z, Y)))
% 32.26/4.49  = { by lemma 15 R->L }
% 32.26/4.49    fresh2(is_a_theorem(implies(implies(implies(not(implies(implies(not(X), X), X)), implies(implies(not(X), X), X)), Y), implies(Z, Y))), true, implies(implies(implies(implies(not(X), X), X), Y), implies(Z, Y)))
% 32.26/4.49  = { by lemma 9 R->L }
% 32.26/4.49    fresh2(fresh2(is_a_theorem(implies(Z, implies(not(implies(implies(not(X), X), X)), implies(implies(not(X), X), X)))), true, implies(implies(implies(not(implies(implies(not(X), X), X)), implies(implies(not(X), X), X)), Y), implies(Z, Y))), true, implies(implies(implies(implies(not(X), X), X), Y), implies(Z, Y)))
% 32.26/4.49  = { by lemma 12 R->L }
% 32.26/4.49    fresh2(fresh2(fresh2(is_a_theorem(implies(implies(not(Z), implies(implies(not(X), X), X)), implies(not(implies(implies(not(X), X), X)), implies(implies(not(X), X), X)))), true, implies(Z, implies(not(implies(implies(not(X), X), X)), implies(implies(not(X), X), X)))), true, implies(implies(implies(not(implies(implies(not(X), X), X)), implies(implies(not(X), X), X)), Y), implies(Z, Y))), true, implies(implies(implies(implies(not(X), X), X), Y), implies(Z, Y)))
% 32.26/4.49  = { by lemma 9 R->L }
% 32.26/4.49    fresh2(fresh2(fresh2(fresh2(is_a_theorem(implies(not(implies(implies(not(X), X), X)), not(Z))), true, implies(implies(not(Z), implies(implies(not(X), X), X)), implies(not(implies(implies(not(X), X), X)), implies(implies(not(X), X), X)))), true, implies(Z, implies(not(implies(implies(not(X), X), X)), implies(implies(not(X), X), X)))), true, implies(implies(implies(not(implies(implies(not(X), X), X)), implies(implies(not(X), X), X)), Y), implies(Z, Y))), true, implies(implies(implies(implies(not(X), X), X), Y), implies(Z, Y)))
% 32.26/4.49  = { by lemma 7 R->L }
% 32.26/4.49    fresh2(fresh2(fresh2(fresh2(fresh2(is_a_theorem(implies(implies(not(X), X), X)), true, implies(not(implies(implies(not(X), X), X)), not(Z))), true, implies(implies(not(Z), implies(implies(not(X), X), X)), implies(not(implies(implies(not(X), X), X)), implies(implies(not(X), X), X)))), true, implies(Z, implies(not(implies(implies(not(X), X), X)), implies(implies(not(X), X), X)))), true, implies(implies(implies(not(implies(implies(not(X), X), X)), implies(implies(not(X), X), X)), Y), implies(Z, Y))), true, implies(implies(implies(implies(not(X), X), X), Y), implies(Z, Y)))
% 32.26/4.49  = { by axiom 4 (cn_2) }
% 32.26/4.49    fresh2(fresh2(fresh2(fresh2(fresh2(true, true, implies(not(implies(implies(not(X), X), X)), not(Z))), true, implies(implies(not(Z), implies(implies(not(X), X), X)), implies(not(implies(implies(not(X), X), X)), implies(implies(not(X), X), X)))), true, implies(Z, implies(not(implies(implies(not(X), X), X)), implies(implies(not(X), X), X)))), true, implies(implies(implies(not(implies(implies(not(X), X), X)), implies(implies(not(X), X), X)), Y), implies(Z, Y))), true, implies(implies(implies(implies(not(X), X), X), Y), implies(Z, Y)))
% 32.26/4.49  = { by axiom 1 (condensed_detachment) }
% 32.26/4.49    fresh2(fresh2(fresh2(fresh2(true, true, implies(implies(not(Z), implies(implies(not(X), X), X)), implies(not(implies(implies(not(X), X), X)), implies(implies(not(X), X), X)))), true, implies(Z, implies(not(implies(implies(not(X), X), X)), implies(implies(not(X), X), X)))), true, implies(implies(implies(not(implies(implies(not(X), X), X)), implies(implies(not(X), X), X)), Y), implies(Z, Y))), true, implies(implies(implies(implies(not(X), X), X), Y), implies(Z, Y)))
% 32.26/4.49  = { by axiom 1 (condensed_detachment) }
% 32.26/4.49    fresh2(fresh2(fresh2(true, true, implies(Z, implies(not(implies(implies(not(X), X), X)), implies(implies(not(X), X), X)))), true, implies(implies(implies(not(implies(implies(not(X), X), X)), implies(implies(not(X), X), X)), Y), implies(Z, Y))), true, implies(implies(implies(implies(not(X), X), X), Y), implies(Z, Y)))
% 32.26/4.49  = { by axiom 1 (condensed_detachment) }
% 32.26/4.49    fresh2(fresh2(true, true, implies(implies(implies(not(implies(implies(not(X), X), X)), implies(implies(not(X), X), X)), Y), implies(Z, Y))), true, implies(implies(implies(implies(not(X), X), X), Y), implies(Z, Y)))
% 32.26/4.49  = { by axiom 1 (condensed_detachment) }
% 32.26/4.49    fresh2(true, true, implies(implies(implies(implies(not(X), X), X), Y), implies(Z, Y)))
% 32.26/4.49  = { by axiom 1 (condensed_detachment) }
% 32.26/4.49    true
% 32.26/4.49  
% 32.26/4.49  Lemma 17: is_a_theorem(implies(implies(X, implies(not(Y), Y)), implies(Z, implies(X, Y)))) = true.
% 32.26/4.49  Proof:
% 32.26/4.49    is_a_theorem(implies(implies(X, implies(not(Y), Y)), implies(Z, implies(X, Y))))
% 32.26/4.49  = { by lemma 13 R->L }
% 32.26/4.49    fresh2(is_a_theorem(implies(implies(implies(implies(not(Y), Y), Y), implies(X, Y)), implies(Z, implies(X, Y)))), true, implies(implies(X, implies(not(Y), Y)), implies(Z, implies(X, Y))))
% 32.26/4.49  = { by lemma 16 }
% 32.26/4.49    fresh2(true, true, implies(implies(X, implies(not(Y), Y)), implies(Z, implies(X, Y))))
% 32.26/4.49  = { by axiom 1 (condensed_detachment) }
% 32.26/4.49    true
% 32.26/4.49  
% 32.26/4.49  Lemma 18: is_a_theorem(implies(implies(implies(X, implies(Y, Z)), W), implies(Z, W))) = true.
% 32.26/4.49  Proof:
% 32.26/4.49    is_a_theorem(implies(implies(implies(X, implies(Y, Z)), W), implies(Z, W)))
% 32.26/4.49  = { by lemma 9 R->L }
% 32.26/4.49    fresh2(is_a_theorem(implies(Z, implies(X, implies(Y, Z)))), true, implies(implies(implies(X, implies(Y, Z)), W), implies(Z, W)))
% 32.26/4.49  = { by lemma 14 R->L }
% 32.26/4.49    fresh2(fresh2(is_a_theorem(implies(implies(Y, implies(not(Z), Z)), implies(X, implies(Y, Z)))), true, implies(Z, implies(X, implies(Y, Z)))), true, implies(implies(implies(X, implies(Y, Z)), W), implies(Z, W)))
% 32.26/4.49  = { by lemma 17 }
% 32.26/4.49    fresh2(fresh2(true, true, implies(Z, implies(X, implies(Y, Z)))), true, implies(implies(implies(X, implies(Y, Z)), W), implies(Z, W)))
% 32.26/4.49  = { by axiom 1 (condensed_detachment) }
% 32.26/4.49    fresh2(true, true, implies(implies(implies(X, implies(Y, Z)), W), implies(Z, W)))
% 32.26/4.49  = { by axiom 1 (condensed_detachment) }
% 32.26/4.49    true
% 32.26/4.49  
% 32.26/4.49  Lemma 19: fresh2(is_a_theorem(implies(X, implies(not(Y), Y))), true, implies(Z, implies(X, Y))) = is_a_theorem(implies(Z, implies(X, Y))).
% 32.26/4.49  Proof:
% 32.26/4.49    fresh2(is_a_theorem(implies(X, implies(not(Y), Y))), true, implies(Z, implies(X, Y)))
% 32.26/4.49  = { by axiom 5 (condensed_detachment) R->L }
% 32.26/4.49    fresh(is_a_theorem(implies(implies(X, implies(not(Y), Y)), implies(Z, implies(X, Y)))), true, implies(X, implies(not(Y), Y)), implies(Z, implies(X, Y)))
% 32.26/4.49  = { by lemma 17 }
% 32.26/4.49    fresh(true, true, implies(X, implies(not(Y), Y)), implies(Z, implies(X, Y)))
% 32.26/4.49  = { by axiom 2 (condensed_detachment) }
% 32.26/4.49    is_a_theorem(implies(Z, implies(X, Y)))
% 32.26/4.49  
% 32.26/4.49  Lemma 20: is_a_theorem(implies(implies(implies(implies(not(not(X)), not(X)), Y), Z), implies(X, Z))) = true.
% 32.26/4.49  Proof:
% 32.26/4.49    is_a_theorem(implies(implies(implies(implies(not(not(X)), not(X)), Y), Z), implies(X, Z)))
% 32.26/4.49  = { by lemma 9 R->L }
% 32.26/4.49    fresh2(is_a_theorem(implies(X, implies(implies(not(not(X)), not(X)), Y))), true, implies(implies(implies(implies(not(not(X)), not(X)), Y), Z), implies(X, Z)))
% 32.26/4.49  = { by lemma 12 R->L }
% 32.26/4.49    fresh2(fresh2(is_a_theorem(implies(implies(not(X), Y), implies(implies(not(not(X)), not(X)), Y))), true, implies(X, implies(implies(not(not(X)), not(X)), Y))), true, implies(implies(implies(implies(not(not(X)), not(X)), Y), Z), implies(X, Z)))
% 32.26/4.49  = { by lemma 10 }
% 32.26/4.49    fresh2(fresh2(true, true, implies(X, implies(implies(not(not(X)), not(X)), Y))), true, implies(implies(implies(implies(not(not(X)), not(X)), Y), Z), implies(X, Z)))
% 32.26/4.49  = { by axiom 1 (condensed_detachment) }
% 32.26/4.49    fresh2(true, true, implies(implies(implies(implies(not(not(X)), not(X)), Y), Z), implies(X, Z)))
% 32.26/4.49  = { by axiom 1 (condensed_detachment) }
% 32.26/4.50    true
% 32.26/4.50  
% 32.26/4.50  Goal 1 (prove_cn_67): is_a_theorem(implies(not(implies(x, y)), not(y))) = true.
% 32.26/4.50  Proof:
% 32.26/4.50    is_a_theorem(implies(not(implies(x, y)), not(y)))
% 32.26/4.50  = { by axiom 2 (condensed_detachment) R->L }
% 32.26/4.50    fresh(true, true, implies(not(implies(not(implies(x, y)), not(y))), implies(x, y)), implies(not(implies(x, y)), not(y)))
% 32.26/4.50  = { by axiom 1 (condensed_detachment) R->L }
% 32.26/4.50    fresh(fresh2(true, true, implies(implies(not(implies(not(implies(x, y)), not(y))), implies(x, y)), implies(not(implies(x, y)), not(y)))), true, implies(not(implies(not(implies(x, y)), not(y))), implies(x, y)), implies(not(implies(x, y)), not(y)))
% 32.26/4.50  = { by axiom 1 (condensed_detachment) R->L }
% 32.26/4.50    fresh(fresh2(fresh2(true, true, implies(not(implies(implies(not(implies(not(implies(x, y)), not(y))), implies(x, y)), implies(not(implies(x, y)), not(y)))), implies(implies(not(implies(not(implies(x, y)), not(y))), implies(x, y)), implies(not(implies(x, y)), not(y))))), true, implies(implies(not(implies(not(implies(x, y)), not(y))), implies(x, y)), implies(not(implies(x, y)), not(y)))), true, implies(not(implies(not(implies(x, y)), not(y))), implies(x, y)), implies(not(implies(x, y)), not(y)))
% 32.26/4.50  = { by axiom 1 (condensed_detachment) R->L }
% 32.26/4.50    fresh(fresh2(fresh2(fresh2(true, true, implies(implies(not(implies(not(implies(x, y)), not(y))), implies(x, y)), implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y))))), true, implies(not(implies(implies(not(implies(not(implies(x, y)), not(y))), implies(x, y)), implies(not(implies(x, y)), not(y)))), implies(implies(not(implies(not(implies(x, y)), not(y))), implies(x, y)), implies(not(implies(x, y)), not(y))))), true, implies(implies(not(implies(not(implies(x, y)), not(y))), implies(x, y)), implies(not(implies(x, y)), not(y)))), true, implies(not(implies(not(implies(x, y)), not(y))), implies(x, y)), implies(not(implies(x, y)), not(y)))
% 32.26/4.50  = { by axiom 1 (condensed_detachment) R->L }
% 32.26/4.50    fresh(fresh2(fresh2(fresh2(fresh2(true, true, implies(implies(implies(implies(x, y), implies(not(implies(x, y)), not(y))), implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y)))), implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y))))), true, implies(implies(not(implies(not(implies(x, y)), not(y))), implies(x, y)), implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y))))), true, implies(not(implies(implies(not(implies(not(implies(x, y)), not(y))), implies(x, y)), implies(not(implies(x, y)), not(y)))), implies(implies(not(implies(not(implies(x, y)), not(y))), implies(x, y)), implies(not(implies(x, y)), not(y))))), true, implies(implies(not(implies(not(implies(x, y)), not(y))), implies(x, y)), implies(not(implies(x, y)), not(y)))), true, implies(not(implies(not(implies(x, y)), not(y))), implies(x, y)), implies(not(implies(x, y)), not(y)))
% 32.26/4.50  = { by axiom 1 (condensed_detachment) R->L }
% 32.26/4.50    fresh(fresh2(fresh2(fresh2(fresh2(fresh2(true, true, implies(not(implies(implies(implies(implies(x, y), implies(not(implies(x, y)), not(y))), implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y)))), implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y))))), implies(implies(implies(implies(x, y), implies(not(implies(x, y)), not(y))), implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y)))), implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y)))))), true, implies(implies(implies(implies(x, y), implies(not(implies(x, y)), not(y))), implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y)))), implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y))))), true, implies(implies(not(implies(not(implies(x, y)), not(y))), implies(x, y)), implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y))))), true, implies(not(implies(implies(not(implies(not(implies(x, y)), not(y))), implies(x, y)), implies(not(implies(x, y)), not(y)))), implies(implies(not(implies(not(implies(x, y)), not(y))), implies(x, y)), implies(not(implies(x, y)), not(y))))), true, implies(implies(not(implies(not(implies(x, y)), not(y))), implies(x, y)), implies(not(implies(x, y)), not(y)))), true, implies(not(implies(not(implies(x, y)), not(y))), implies(x, y)), implies(not(implies(x, y)), not(y)))
% 32.26/4.51  = { by axiom 1 (condensed_detachment) R->L }
% 32.26/4.51    fresh(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(true, true, implies(implies(implies(implies(x, y), implies(not(implies(x, y)), not(y))), implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y)))), implies(not(implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y)))), implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y)))))), true, implies(not(implies(implies(implies(implies(x, y), implies(not(implies(x, y)), not(y))), implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y)))), implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y))))), implies(implies(implies(implies(x, y), implies(not(implies(x, y)), not(y))), implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y)))), implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y)))))), true, implies(implies(implies(implies(x, y), implies(not(implies(x, y)), not(y))), implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y)))), implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y))))), true, implies(implies(not(implies(not(implies(x, y)), not(y))), implies(x, y)), implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y))))), true, implies(not(implies(implies(not(implies(not(implies(x, y)), not(y))), implies(x, y)), implies(not(implies(x, y)), not(y)))), implies(implies(not(implies(not(implies(x, y)), not(y))), implies(x, y)), implies(not(implies(x, y)), not(y))))), true, implies(implies(not(implies(not(implies(x, y)), not(y))), implies(x, y)), implies(not(implies(x, y)), not(y)))), true, implies(not(implies(not(implies(x, y)), not(y))), implies(x, y)), implies(not(implies(x, y)), not(y)))
% 32.26/4.51  = { by axiom 1 (condensed_detachment) R->L }
% 32.26/4.51    fresh(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(true, true, implies(implies(implies(not(implies(implies(x, y), implies(not(implies(x, y)), not(y)))), implies(implies(x, y), implies(not(implies(x, y)), not(y)))), implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y)))), implies(not(implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y)))), implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y)))))), true, implies(implies(implies(implies(x, y), implies(not(implies(x, y)), not(y))), implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y)))), implies(not(implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y)))), implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y)))))), true, implies(not(implies(implies(implies(implies(x, y), implies(not(implies(x, y)), not(y))), implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y)))), implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y))))), implies(implies(implies(implies(x, y), implies(not(implies(x, y)), not(y))), implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y)))), implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y)))))), true, implies(implies(implies(implies(x, y), implies(not(implies(x, y)), not(y))), implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y)))), implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y))))), true, implies(implies(not(implies(not(implies(x, y)), not(y))), implies(x, y)), implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y))))), true, implies(not(implies(implies(not(implies(not(implies(x, y)), not(y))), implies(x, y)), implies(not(implies(x, y)), not(y)))), implies(implies(not(implies(not(implies(x, y)), not(y))), implies(x, y)), implies(not(implies(x, y)), not(y))))), true, implies(implies(not(implies(not(implies(x, y)), not(y))), implies(x, y)), implies(not(implies(x, y)), not(y)))), true, implies(not(implies(not(implies(x, y)), not(y))), implies(x, y)), implies(not(implies(x, y)), not(y)))
% 32.26/4.51  = { by axiom 1 (condensed_detachment) R->L }
% 32.26/4.51    fresh(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(true, true, implies(not(implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y)))), implies(not(implies(implies(x, y), implies(not(implies(x, y)), not(y)))), implies(implies(x, y), implies(not(implies(x, y)), not(y)))))), true, implies(implies(implies(not(implies(implies(x, y), implies(not(implies(x, y)), not(y)))), implies(implies(x, y), implies(not(implies(x, y)), not(y)))), implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y)))), implies(not(implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y)))), implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y)))))), true, implies(implies(implies(implies(x, y), implies(not(implies(x, y)), not(y))), implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y)))), implies(not(implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y)))), implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y)))))), true, implies(not(implies(implies(implies(implies(x, y), implies(not(implies(x, y)), not(y))), implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y)))), implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y))))), implies(implies(implies(implies(x, y), implies(not(implies(x, y)), not(y))), implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y)))), implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y)))))), true, implies(implies(implies(implies(x, y), implies(not(implies(x, y)), not(y))), implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y)))), implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y))))), true, implies(implies(not(implies(not(implies(x, y)), not(y))), implies(x, y)), implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y))))), true, implies(not(implies(implies(not(implies(not(implies(x, y)), not(y))), implies(x, y)), implies(not(implies(x, y)), not(y)))), implies(implies(not(implies(not(implies(x, y)), not(y))), implies(x, y)), implies(not(implies(x, y)), not(y))))), true, implies(implies(not(implies(not(implies(x, y)), not(y))), implies(x, y)), implies(not(implies(x, y)), not(y)))), true, implies(not(implies(not(implies(x, y)), not(y))), implies(x, y)), implies(not(implies(x, y)), not(y)))
% 32.26/4.52  = { by axiom 1 (condensed_detachment) R->L }
% 32.26/4.52    fresh(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(true, true, implies(implies(not(not(implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y))))), implies(implies(x, y), implies(not(implies(x, y)), not(y)))), implies(not(implies(implies(x, y), implies(not(implies(x, y)), not(y)))), implies(implies(x, y), implies(not(implies(x, y)), not(y)))))), true, implies(not(implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y)))), implies(not(implies(implies(x, y), implies(not(implies(x, y)), not(y)))), implies(implies(x, y), implies(not(implies(x, y)), not(y)))))), true, implies(implies(implies(not(implies(implies(x, y), implies(not(implies(x, y)), not(y)))), implies(implies(x, y), implies(not(implies(x, y)), not(y)))), implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y)))), implies(not(implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y)))), implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y)))))), true, implies(implies(implies(implies(x, y), implies(not(implies(x, y)), not(y))), implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y)))), implies(not(implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y)))), implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y)))))), true, implies(not(implies(implies(implies(implies(x, y), implies(not(implies(x, y)), not(y))), implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y)))), implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y))))), implies(implies(implies(implies(x, y), implies(not(implies(x, y)), not(y))), implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y)))), implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y)))))), true, implies(implies(implies(implies(x, y), implies(not(implies(x, y)), not(y))), implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y)))), implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y))))), true, implies(implies(not(implies(not(implies(x, y)), not(y))), implies(x, y)), implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y))))), true, implies(not(implies(implies(not(implies(not(implies(x, y)), not(y))), implies(x, y)), implies(not(implies(x, y)), not(y)))), implies(implies(not(implies(not(implies(x, y)), not(y))), implies(x, y)), implies(not(implies(x, y)), not(y))))), true, implies(implies(not(implies(not(implies(x, y)), not(y))), implies(x, y)), implies(not(implies(x, y)), not(y)))), true, implies(not(implies(not(implies(x, y)), not(y))), implies(x, y)), implies(not(implies(x, y)), not(y)))
% 32.26/4.52  = { by axiom 1 (condensed_detachment) R->L }
% 32.26/4.52    fresh(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(true, true, implies(not(implies(implies(x, y), implies(not(implies(x, y)), not(y)))), not(not(implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y))))))), true, implies(implies(not(not(implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y))))), implies(implies(x, y), implies(not(implies(x, y)), not(y)))), implies(not(implies(implies(x, y), implies(not(implies(x, y)), not(y)))), implies(implies(x, y), implies(not(implies(x, y)), not(y)))))), true, implies(not(implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y)))), implies(not(implies(implies(x, y), implies(not(implies(x, y)), not(y)))), implies(implies(x, y), implies(not(implies(x, y)), not(y)))))), true, implies(implies(implies(not(implies(implies(x, y), implies(not(implies(x, y)), not(y)))), implies(implies(x, y), implies(not(implies(x, y)), not(y)))), implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y)))), implies(not(implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y)))), implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y)))))), true, implies(implies(implies(implies(x, y), implies(not(implies(x, y)), not(y))), implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y)))), implies(not(implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y)))), implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y)))))), true, implies(not(implies(implies(implies(implies(x, y), implies(not(implies(x, y)), not(y))), implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y)))), implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y))))), implies(implies(implies(implies(x, y), implies(not(implies(x, y)), not(y))), implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y)))), implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y)))))), true, implies(implies(implies(implies(x, y), implies(not(implies(x, y)), not(y))), implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y)))), implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y))))), true, implies(implies(not(implies(not(implies(x, y)), not(y))), implies(x, y)), implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y))))), true, implies(not(implies(implies(not(implies(not(implies(x, y)), not(y))), implies(x, y)), implies(not(implies(x, y)), not(y)))), implies(implies(not(implies(not(implies(x, y)), not(y))), implies(x, y)), implies(not(implies(x, y)), not(y))))), true, implies(implies(not(implies(not(implies(x, y)), not(y))), implies(x, y)), implies(not(implies(x, y)), not(y)))), true, implies(not(implies(not(implies(x, y)), not(y))), implies(x, y)), implies(not(implies(x, y)), not(y)))
% 32.26/4.52  = { by axiom 3 (cn_3) R->L }
% 32.26/4.53    fresh(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(is_a_theorem(implies(implies(x, y), implies(not(implies(x, y)), not(y)))), true, implies(not(implies(implies(x, y), implies(not(implies(x, y)), not(y)))), not(not(implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y))))))), true, implies(implies(not(not(implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y))))), implies(implies(x, y), implies(not(implies(x, y)), not(y)))), implies(not(implies(implies(x, y), implies(not(implies(x, y)), not(y)))), implies(implies(x, y), implies(not(implies(x, y)), not(y)))))), true, implies(not(implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y)))), implies(not(implies(implies(x, y), implies(not(implies(x, y)), not(y)))), implies(implies(x, y), implies(not(implies(x, y)), not(y)))))), true, implies(implies(implies(not(implies(implies(x, y), implies(not(implies(x, y)), not(y)))), implies(implies(x, y), implies(not(implies(x, y)), not(y)))), implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y)))), implies(not(implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y)))), implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y)))))), true, implies(implies(implies(implies(x, y), implies(not(implies(x, y)), not(y))), implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y)))), implies(not(implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y)))), implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y)))))), true, implies(not(implies(implies(implies(implies(x, y), implies(not(implies(x, y)), not(y))), implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y)))), implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y))))), implies(implies(implies(implies(x, y), implies(not(implies(x, y)), not(y))), implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y)))), implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y)))))), true, implies(implies(implies(implies(x, y), implies(not(implies(x, y)), not(y))), implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y)))), implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y))))), true, implies(implies(not(implies(not(implies(x, y)), not(y))), implies(x, y)), implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y))))), true, implies(not(implies(implies(not(implies(not(implies(x, y)), not(y))), implies(x, y)), implies(not(implies(x, y)), not(y)))), implies(implies(not(implies(not(implies(x, y)), not(y))), implies(x, y)), implies(not(implies(x, y)), not(y))))), true, implies(implies(not(implies(not(implies(x, y)), not(y))), implies(x, y)), implies(not(implies(x, y)), not(y)))), true, implies(not(implies(not(implies(x, y)), not(y))), implies(x, y)), implies(not(implies(x, y)), not(y)))
% 32.26/4.53  = { by lemma 7 }
% 32.26/4.53    fresh(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(is_a_theorem(implies(not(implies(implies(x, y), implies(not(implies(x, y)), not(y)))), not(not(implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y))))))), true, implies(implies(not(not(implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y))))), implies(implies(x, y), implies(not(implies(x, y)), not(y)))), implies(not(implies(implies(x, y), implies(not(implies(x, y)), not(y)))), implies(implies(x, y), implies(not(implies(x, y)), not(y)))))), true, implies(not(implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y)))), implies(not(implies(implies(x, y), implies(not(implies(x, y)), not(y)))), implies(implies(x, y), implies(not(implies(x, y)), not(y)))))), true, implies(implies(implies(not(implies(implies(x, y), implies(not(implies(x, y)), not(y)))), implies(implies(x, y), implies(not(implies(x, y)), not(y)))), implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y)))), implies(not(implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y)))), implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y)))))), true, implies(implies(implies(implies(x, y), implies(not(implies(x, y)), not(y))), implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y)))), implies(not(implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y)))), implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y)))))), true, implies(not(implies(implies(implies(implies(x, y), implies(not(implies(x, y)), not(y))), implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y)))), implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y))))), implies(implies(implies(implies(x, y), implies(not(implies(x, y)), not(y))), implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y)))), implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y)))))), true, implies(implies(implies(implies(x, y), implies(not(implies(x, y)), not(y))), implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y)))), implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y))))), true, implies(implies(not(implies(not(implies(x, y)), not(y))), implies(x, y)), implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y))))), true, implies(not(implies(implies(not(implies(not(implies(x, y)), not(y))), implies(x, y)), implies(not(implies(x, y)), not(y)))), implies(implies(not(implies(not(implies(x, y)), not(y))), implies(x, y)), implies(not(implies(x, y)), not(y))))), true, implies(implies(not(implies(not(implies(x, y)), not(y))), implies(x, y)), implies(not(implies(x, y)), not(y)))), true, implies(not(implies(not(implies(x, y)), not(y))), implies(x, y)), implies(not(implies(x, y)), not(y)))
% 32.26/4.53  = { by lemma 9 }
% 32.26/4.53    fresh(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(is_a_theorem(implies(implies(not(not(implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y))))), implies(implies(x, y), implies(not(implies(x, y)), not(y)))), implies(not(implies(implies(x, y), implies(not(implies(x, y)), not(y)))), implies(implies(x, y), implies(not(implies(x, y)), not(y)))))), true, implies(not(implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y)))), implies(not(implies(implies(x, y), implies(not(implies(x, y)), not(y)))), implies(implies(x, y), implies(not(implies(x, y)), not(y)))))), true, implies(implies(implies(not(implies(implies(x, y), implies(not(implies(x, y)), not(y)))), implies(implies(x, y), implies(not(implies(x, y)), not(y)))), implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y)))), implies(not(implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y)))), implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y)))))), true, implies(implies(implies(implies(x, y), implies(not(implies(x, y)), not(y))), implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y)))), implies(not(implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y)))), implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y)))))), true, implies(not(implies(implies(implies(implies(x, y), implies(not(implies(x, y)), not(y))), implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y)))), implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y))))), implies(implies(implies(implies(x, y), implies(not(implies(x, y)), not(y))), implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y)))), implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y)))))), true, implies(implies(implies(implies(x, y), implies(not(implies(x, y)), not(y))), implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y)))), implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y))))), true, implies(implies(not(implies(not(implies(x, y)), not(y))), implies(x, y)), implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y))))), true, implies(not(implies(implies(not(implies(not(implies(x, y)), not(y))), implies(x, y)), implies(not(implies(x, y)), not(y)))), implies(implies(not(implies(not(implies(x, y)), not(y))), implies(x, y)), implies(not(implies(x, y)), not(y))))), true, implies(implies(not(implies(not(implies(x, y)), not(y))), implies(x, y)), implies(not(implies(x, y)), not(y)))), true, implies(not(implies(not(implies(x, y)), not(y))), implies(x, y)), implies(not(implies(x, y)), not(y)))
% 32.26/4.53  = { by lemma 12 }
% 32.26/4.53    fresh(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(is_a_theorem(implies(not(implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y)))), implies(not(implies(implies(x, y), implies(not(implies(x, y)), not(y)))), implies(implies(x, y), implies(not(implies(x, y)), not(y)))))), true, implies(implies(implies(not(implies(implies(x, y), implies(not(implies(x, y)), not(y)))), implies(implies(x, y), implies(not(implies(x, y)), not(y)))), implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y)))), implies(not(implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y)))), implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y)))))), true, implies(implies(implies(implies(x, y), implies(not(implies(x, y)), not(y))), implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y)))), implies(not(implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y)))), implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y)))))), true, implies(not(implies(implies(implies(implies(x, y), implies(not(implies(x, y)), not(y))), implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y)))), implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y))))), implies(implies(implies(implies(x, y), implies(not(implies(x, y)), not(y))), implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y)))), implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y)))))), true, implies(implies(implies(implies(x, y), implies(not(implies(x, y)), not(y))), implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y)))), implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y))))), true, implies(implies(not(implies(not(implies(x, y)), not(y))), implies(x, y)), implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y))))), true, implies(not(implies(implies(not(implies(not(implies(x, y)), not(y))), implies(x, y)), implies(not(implies(x, y)), not(y)))), implies(implies(not(implies(not(implies(x, y)), not(y))), implies(x, y)), implies(not(implies(x, y)), not(y))))), true, implies(implies(not(implies(not(implies(x, y)), not(y))), implies(x, y)), implies(not(implies(x, y)), not(y)))), true, implies(not(implies(not(implies(x, y)), not(y))), implies(x, y)), implies(not(implies(x, y)), not(y)))
% 32.26/4.53  = { by lemma 9 }
% 32.26/4.54    fresh(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(is_a_theorem(implies(implies(implies(not(implies(implies(x, y), implies(not(implies(x, y)), not(y)))), implies(implies(x, y), implies(not(implies(x, y)), not(y)))), implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y)))), implies(not(implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y)))), implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y)))))), true, implies(implies(implies(implies(x, y), implies(not(implies(x, y)), not(y))), implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y)))), implies(not(implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y)))), implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y)))))), true, implies(not(implies(implies(implies(implies(x, y), implies(not(implies(x, y)), not(y))), implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y)))), implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y))))), implies(implies(implies(implies(x, y), implies(not(implies(x, y)), not(y))), implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y)))), implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y)))))), true, implies(implies(implies(implies(x, y), implies(not(implies(x, y)), not(y))), implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y)))), implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y))))), true, implies(implies(not(implies(not(implies(x, y)), not(y))), implies(x, y)), implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y))))), true, implies(not(implies(implies(not(implies(not(implies(x, y)), not(y))), implies(x, y)), implies(not(implies(x, y)), not(y)))), implies(implies(not(implies(not(implies(x, y)), not(y))), implies(x, y)), implies(not(implies(x, y)), not(y))))), true, implies(implies(not(implies(not(implies(x, y)), not(y))), implies(x, y)), implies(not(implies(x, y)), not(y)))), true, implies(not(implies(not(implies(x, y)), not(y))), implies(x, y)), implies(not(implies(x, y)), not(y)))
% 32.26/4.54  = { by lemma 15 }
% 32.26/4.54    fresh(fresh2(fresh2(fresh2(fresh2(fresh2(is_a_theorem(implies(implies(implies(implies(x, y), implies(not(implies(x, y)), not(y))), implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y)))), implies(not(implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y)))), implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y)))))), true, implies(not(implies(implies(implies(implies(x, y), implies(not(implies(x, y)), not(y))), implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y)))), implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y))))), implies(implies(implies(implies(x, y), implies(not(implies(x, y)), not(y))), implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y)))), implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y)))))), true, implies(implies(implies(implies(x, y), implies(not(implies(x, y)), not(y))), implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y)))), implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y))))), true, implies(implies(not(implies(not(implies(x, y)), not(y))), implies(x, y)), implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y))))), true, implies(not(implies(implies(not(implies(not(implies(x, y)), not(y))), implies(x, y)), implies(not(implies(x, y)), not(y)))), implies(implies(not(implies(not(implies(x, y)), not(y))), implies(x, y)), implies(not(implies(x, y)), not(y))))), true, implies(implies(not(implies(not(implies(x, y)), not(y))), implies(x, y)), implies(not(implies(x, y)), not(y)))), true, implies(not(implies(not(implies(x, y)), not(y))), implies(x, y)), implies(not(implies(x, y)), not(y)))
% 32.26/4.54  = { by lemma 19 }
% 32.26/4.54    fresh(fresh2(fresh2(fresh2(fresh2(is_a_theorem(implies(not(implies(implies(implies(implies(x, y), implies(not(implies(x, y)), not(y))), implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y)))), implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y))))), implies(implies(implies(implies(x, y), implies(not(implies(x, y)), not(y))), implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y)))), implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y)))))), true, implies(implies(implies(implies(x, y), implies(not(implies(x, y)), not(y))), implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y)))), implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y))))), true, implies(implies(not(implies(not(implies(x, y)), not(y))), implies(x, y)), implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y))))), true, implies(not(implies(implies(not(implies(not(implies(x, y)), not(y))), implies(x, y)), implies(not(implies(x, y)), not(y)))), implies(implies(not(implies(not(implies(x, y)), not(y))), implies(x, y)), implies(not(implies(x, y)), not(y))))), true, implies(implies(not(implies(not(implies(x, y)), not(y))), implies(x, y)), implies(not(implies(x, y)), not(y)))), true, implies(not(implies(not(implies(x, y)), not(y))), implies(x, y)), implies(not(implies(x, y)), not(y)))
% 32.26/4.54  = { by lemma 8 }
% 32.26/4.54    fresh(fresh2(fresh2(fresh2(is_a_theorem(implies(implies(implies(implies(x, y), implies(not(implies(x, y)), not(y))), implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y)))), implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y))))), true, implies(implies(not(implies(not(implies(x, y)), not(y))), implies(x, y)), implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y))))), true, implies(not(implies(implies(not(implies(not(implies(x, y)), not(y))), implies(x, y)), implies(not(implies(x, y)), not(y)))), implies(implies(not(implies(not(implies(x, y)), not(y))), implies(x, y)), implies(not(implies(x, y)), not(y))))), true, implies(implies(not(implies(not(implies(x, y)), not(y))), implies(x, y)), implies(not(implies(x, y)), not(y)))), true, implies(not(implies(not(implies(x, y)), not(y))), implies(x, y)), implies(not(implies(x, y)), not(y)))
% 32.26/4.54  = { by lemma 13 }
% 32.26/4.54    fresh(fresh2(fresh2(is_a_theorem(implies(implies(not(implies(not(implies(x, y)), not(y))), implies(x, y)), implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y))))), true, implies(not(implies(implies(not(implies(not(implies(x, y)), not(y))), implies(x, y)), implies(not(implies(x, y)), not(y)))), implies(implies(not(implies(not(implies(x, y)), not(y))), implies(x, y)), implies(not(implies(x, y)), not(y))))), true, implies(implies(not(implies(not(implies(x, y)), not(y))), implies(x, y)), implies(not(implies(x, y)), not(y)))), true, implies(not(implies(not(implies(x, y)), not(y))), implies(x, y)), implies(not(implies(x, y)), not(y)))
% 32.26/4.54  = { by lemma 19 }
% 32.26/4.54    fresh(fresh2(is_a_theorem(implies(not(implies(implies(not(implies(not(implies(x, y)), not(y))), implies(x, y)), implies(not(implies(x, y)), not(y)))), implies(implies(not(implies(not(implies(x, y)), not(y))), implies(x, y)), implies(not(implies(x, y)), not(y))))), true, implies(implies(not(implies(not(implies(x, y)), not(y))), implies(x, y)), implies(not(implies(x, y)), not(y)))), true, implies(not(implies(not(implies(x, y)), not(y))), implies(x, y)), implies(not(implies(x, y)), not(y)))
% 32.26/4.54  = { by lemma 8 }
% 32.26/4.54    fresh(is_a_theorem(implies(implies(not(implies(not(implies(x, y)), not(y))), implies(x, y)), implies(not(implies(x, y)), not(y)))), true, implies(not(implies(not(implies(x, y)), not(y))), implies(x, y)), implies(not(implies(x, y)), not(y)))
% 32.26/4.54  = { by axiom 5 (condensed_detachment) }
% 32.26/4.54    fresh2(is_a_theorem(implies(not(implies(not(implies(x, y)), not(y))), implies(x, y))), true, implies(not(implies(x, y)), not(y)))
% 32.26/4.54  = { by axiom 2 (condensed_detachment) R->L }
% 32.26/4.54    fresh2(fresh(true, true, implies(not(implies(not(implies(x, y)), not(y))), y), implies(not(implies(not(implies(x, y)), not(y))), implies(x, y))), true, implies(not(implies(x, y)), not(y)))
% 32.26/4.54  = { by axiom 1 (condensed_detachment) R->L }
% 32.26/4.54    fresh2(fresh(fresh2(true, true, implies(implies(not(implies(not(implies(x, y)), not(y))), y), implies(not(implies(not(implies(x, y)), not(y))), implies(x, y)))), true, implies(not(implies(not(implies(x, y)), not(y))), y), implies(not(implies(not(implies(x, y)), not(y))), implies(x, y))), true, implies(not(implies(x, y)), not(y)))
% 32.26/4.54  = { by axiom 1 (condensed_detachment) R->L }
% 32.26/4.54    fresh2(fresh(fresh2(fresh2(true, true, implies(implies(implies(y, implies(x, y)), implies(not(implies(not(implies(x, y)), not(y))), implies(x, y))), implies(not(implies(not(implies(x, y)), not(y))), implies(x, y)))), true, implies(implies(not(implies(not(implies(x, y)), not(y))), y), implies(not(implies(not(implies(x, y)), not(y))), implies(x, y)))), true, implies(not(implies(not(implies(x, y)), not(y))), y), implies(not(implies(not(implies(x, y)), not(y))), implies(x, y))), true, implies(not(implies(x, y)), not(y)))
% 32.26/4.54  = { by axiom 1 (condensed_detachment) R->L }
% 32.26/4.54    fresh2(fresh(fresh2(fresh2(fresh2(true, true, implies(not(implies(implies(implies(y, implies(x, y)), implies(not(implies(not(implies(x, y)), not(y))), implies(x, y))), implies(not(implies(not(implies(x, y)), not(y))), implies(x, y)))), implies(implies(implies(y, implies(x, y)), implies(not(implies(not(implies(x, y)), not(y))), implies(x, y))), implies(not(implies(not(implies(x, y)), not(y))), implies(x, y))))), true, implies(implies(implies(y, implies(x, y)), implies(not(implies(not(implies(x, y)), not(y))), implies(x, y))), implies(not(implies(not(implies(x, y)), not(y))), implies(x, y)))), true, implies(implies(not(implies(not(implies(x, y)), not(y))), y), implies(not(implies(not(implies(x, y)), not(y))), implies(x, y)))), true, implies(not(implies(not(implies(x, y)), not(y))), y), implies(not(implies(not(implies(x, y)), not(y))), implies(x, y))), true, implies(not(implies(x, y)), not(y)))
% 32.26/4.54  = { by axiom 1 (condensed_detachment) R->L }
% 32.26/4.54    fresh2(fresh(fresh2(fresh2(fresh2(fresh2(true, true, implies(implies(implies(y, implies(x, y)), implies(not(implies(not(implies(x, y)), not(y))), implies(x, y))), implies(not(implies(not(implies(not(implies(x, y)), not(y))), implies(x, y))), implies(not(implies(not(implies(x, y)), not(y))), implies(x, y))))), true, implies(not(implies(implies(implies(y, implies(x, y)), implies(not(implies(not(implies(x, y)), not(y))), implies(x, y))), implies(not(implies(not(implies(x, y)), not(y))), implies(x, y)))), implies(implies(implies(y, implies(x, y)), implies(not(implies(not(implies(x, y)), not(y))), implies(x, y))), implies(not(implies(not(implies(x, y)), not(y))), implies(x, y))))), true, implies(implies(implies(y, implies(x, y)), implies(not(implies(not(implies(x, y)), not(y))), implies(x, y))), implies(not(implies(not(implies(x, y)), not(y))), implies(x, y)))), true, implies(implies(not(implies(not(implies(x, y)), not(y))), y), implies(not(implies(not(implies(x, y)), not(y))), implies(x, y)))), true, implies(not(implies(not(implies(x, y)), not(y))), y), implies(not(implies(not(implies(x, y)), not(y))), implies(x, y))), true, implies(not(implies(x, y)), not(y)))
% 32.26/4.54  = { by axiom 1 (condensed_detachment) R->L }
% 32.26/4.54    fresh2(fresh(fresh2(fresh2(fresh2(fresh2(fresh2(true, true, implies(not(implies(not(implies(not(implies(x, y)), not(y))), implies(x, y))), implies(y, implies(x, y)))), true, implies(implies(implies(y, implies(x, y)), implies(not(implies(not(implies(x, y)), not(y))), implies(x, y))), implies(not(implies(not(implies(not(implies(x, y)), not(y))), implies(x, y))), implies(not(implies(not(implies(x, y)), not(y))), implies(x, y))))), true, implies(not(implies(implies(implies(y, implies(x, y)), implies(not(implies(not(implies(x, y)), not(y))), implies(x, y))), implies(not(implies(not(implies(x, y)), not(y))), implies(x, y)))), implies(implies(implies(y, implies(x, y)), implies(not(implies(not(implies(x, y)), not(y))), implies(x, y))), implies(not(implies(not(implies(x, y)), not(y))), implies(x, y))))), true, implies(implies(implies(y, implies(x, y)), implies(not(implies(not(implies(x, y)), not(y))), implies(x, y))), implies(not(implies(not(implies(x, y)), not(y))), implies(x, y)))), true, implies(implies(not(implies(not(implies(x, y)), not(y))), y), implies(not(implies(not(implies(x, y)), not(y))), implies(x, y)))), true, implies(not(implies(not(implies(x, y)), not(y))), y), implies(not(implies(not(implies(x, y)), not(y))), implies(x, y))), true, implies(not(implies(x, y)), not(y)))
% 32.26/4.55  = { by lemma 18 R->L }
% 32.26/4.55    fresh2(fresh(fresh2(fresh2(fresh2(fresh2(fresh2(is_a_theorem(implies(implies(implies(not(implies(x, y)), implies(x, y)), implies(x, y)), implies(y, implies(x, y)))), true, implies(not(implies(not(implies(not(implies(x, y)), not(y))), implies(x, y))), implies(y, implies(x, y)))), true, implies(implies(implies(y, implies(x, y)), implies(not(implies(not(implies(x, y)), not(y))), implies(x, y))), implies(not(implies(not(implies(not(implies(x, y)), not(y))), implies(x, y))), implies(not(implies(not(implies(x, y)), not(y))), implies(x, y))))), true, implies(not(implies(implies(implies(y, implies(x, y)), implies(not(implies(not(implies(x, y)), not(y))), implies(x, y))), implies(not(implies(not(implies(x, y)), not(y))), implies(x, y)))), implies(implies(implies(y, implies(x, y)), implies(not(implies(not(implies(x, y)), not(y))), implies(x, y))), implies(not(implies(not(implies(x, y)), not(y))), implies(x, y))))), true, implies(implies(implies(y, implies(x, y)), implies(not(implies(not(implies(x, y)), not(y))), implies(x, y))), implies(not(implies(not(implies(x, y)), not(y))), implies(x, y)))), true, implies(implies(not(implies(not(implies(x, y)), not(y))), y), implies(not(implies(not(implies(x, y)), not(y))), implies(x, y)))), true, implies(not(implies(not(implies(x, y)), not(y))), y), implies(not(implies(not(implies(x, y)), not(y))), implies(x, y))), true, implies(not(implies(x, y)), not(y)))
% 32.26/4.55  = { by axiom 5 (condensed_detachment) R->L }
% 32.26/4.55    fresh2(fresh(fresh2(fresh2(fresh2(fresh2(fresh(is_a_theorem(implies(implies(implies(implies(not(implies(x, y)), implies(x, y)), implies(x, y)), implies(y, implies(x, y))), implies(not(implies(not(implies(not(implies(x, y)), not(y))), implies(x, y))), implies(y, implies(x, y))))), true, implies(implies(implies(not(implies(x, y)), implies(x, y)), implies(x, y)), implies(y, implies(x, y))), implies(not(implies(not(implies(not(implies(x, y)), not(y))), implies(x, y))), implies(y, implies(x, y)))), true, implies(implies(implies(y, implies(x, y)), implies(not(implies(not(implies(x, y)), not(y))), implies(x, y))), implies(not(implies(not(implies(not(implies(x, y)), not(y))), implies(x, y))), implies(not(implies(not(implies(x, y)), not(y))), implies(x, y))))), true, implies(not(implies(implies(implies(y, implies(x, y)), implies(not(implies(not(implies(x, y)), not(y))), implies(x, y))), implies(not(implies(not(implies(x, y)), not(y))), implies(x, y)))), implies(implies(implies(y, implies(x, y)), implies(not(implies(not(implies(x, y)), not(y))), implies(x, y))), implies(not(implies(not(implies(x, y)), not(y))), implies(x, y))))), true, implies(implies(implies(y, implies(x, y)), implies(not(implies(not(implies(x, y)), not(y))), implies(x, y))), implies(not(implies(not(implies(x, y)), not(y))), implies(x, y)))), true, implies(implies(not(implies(not(implies(x, y)), not(y))), y), implies(not(implies(not(implies(x, y)), not(y))), implies(x, y)))), true, implies(not(implies(not(implies(x, y)), not(y))), y), implies(not(implies(not(implies(x, y)), not(y))), implies(x, y))), true, implies(not(implies(x, y)), not(y)))
% 32.26/4.55  = { by lemma 16 }
% 32.26/4.55    fresh2(fresh(fresh2(fresh2(fresh2(fresh2(fresh(true, true, implies(implies(implies(not(implies(x, y)), implies(x, y)), implies(x, y)), implies(y, implies(x, y))), implies(not(implies(not(implies(not(implies(x, y)), not(y))), implies(x, y))), implies(y, implies(x, y)))), true, implies(implies(implies(y, implies(x, y)), implies(not(implies(not(implies(x, y)), not(y))), implies(x, y))), implies(not(implies(not(implies(not(implies(x, y)), not(y))), implies(x, y))), implies(not(implies(not(implies(x, y)), not(y))), implies(x, y))))), true, implies(not(implies(implies(implies(y, implies(x, y)), implies(not(implies(not(implies(x, y)), not(y))), implies(x, y))), implies(not(implies(not(implies(x, y)), not(y))), implies(x, y)))), implies(implies(implies(y, implies(x, y)), implies(not(implies(not(implies(x, y)), not(y))), implies(x, y))), implies(not(implies(not(implies(x, y)), not(y))), implies(x, y))))), true, implies(implies(implies(y, implies(x, y)), implies(not(implies(not(implies(x, y)), not(y))), implies(x, y))), implies(not(implies(not(implies(x, y)), not(y))), implies(x, y)))), true, implies(implies(not(implies(not(implies(x, y)), not(y))), y), implies(not(implies(not(implies(x, y)), not(y))), implies(x, y)))), true, implies(not(implies(not(implies(x, y)), not(y))), y), implies(not(implies(not(implies(x, y)), not(y))), implies(x, y))), true, implies(not(implies(x, y)), not(y)))
% 32.26/4.55  = { by axiom 2 (condensed_detachment) }
% 32.26/4.55    fresh2(fresh(fresh2(fresh2(fresh2(fresh2(is_a_theorem(implies(not(implies(not(implies(not(implies(x, y)), not(y))), implies(x, y))), implies(y, implies(x, y)))), true, implies(implies(implies(y, implies(x, y)), implies(not(implies(not(implies(x, y)), not(y))), implies(x, y))), implies(not(implies(not(implies(not(implies(x, y)), not(y))), implies(x, y))), implies(not(implies(not(implies(x, y)), not(y))), implies(x, y))))), true, implies(not(implies(implies(implies(y, implies(x, y)), implies(not(implies(not(implies(x, y)), not(y))), implies(x, y))), implies(not(implies(not(implies(x, y)), not(y))), implies(x, y)))), implies(implies(implies(y, implies(x, y)), implies(not(implies(not(implies(x, y)), not(y))), implies(x, y))), implies(not(implies(not(implies(x, y)), not(y))), implies(x, y))))), true, implies(implies(implies(y, implies(x, y)), implies(not(implies(not(implies(x, y)), not(y))), implies(x, y))), implies(not(implies(not(implies(x, y)), not(y))), implies(x, y)))), true, implies(implies(not(implies(not(implies(x, y)), not(y))), y), implies(not(implies(not(implies(x, y)), not(y))), implies(x, y)))), true, implies(not(implies(not(implies(x, y)), not(y))), y), implies(not(implies(not(implies(x, y)), not(y))), implies(x, y))), true, implies(not(implies(x, y)), not(y)))
% 32.26/4.55  = { by lemma 9 }
% 32.26/4.55    fresh2(fresh(fresh2(fresh2(fresh2(is_a_theorem(implies(implies(implies(y, implies(x, y)), implies(not(implies(not(implies(x, y)), not(y))), implies(x, y))), implies(not(implies(not(implies(not(implies(x, y)), not(y))), implies(x, y))), implies(not(implies(not(implies(x, y)), not(y))), implies(x, y))))), true, implies(not(implies(implies(implies(y, implies(x, y)), implies(not(implies(not(implies(x, y)), not(y))), implies(x, y))), implies(not(implies(not(implies(x, y)), not(y))), implies(x, y)))), implies(implies(implies(y, implies(x, y)), implies(not(implies(not(implies(x, y)), not(y))), implies(x, y))), implies(not(implies(not(implies(x, y)), not(y))), implies(x, y))))), true, implies(implies(implies(y, implies(x, y)), implies(not(implies(not(implies(x, y)), not(y))), implies(x, y))), implies(not(implies(not(implies(x, y)), not(y))), implies(x, y)))), true, implies(implies(not(implies(not(implies(x, y)), not(y))), y), implies(not(implies(not(implies(x, y)), not(y))), implies(x, y)))), true, implies(not(implies(not(implies(x, y)), not(y))), y), implies(not(implies(not(implies(x, y)), not(y))), implies(x, y))), true, implies(not(implies(x, y)), not(y)))
% 32.26/4.55  = { by lemma 19 }
% 32.26/4.55    fresh2(fresh(fresh2(fresh2(is_a_theorem(implies(not(implies(implies(implies(y, implies(x, y)), implies(not(implies(not(implies(x, y)), not(y))), implies(x, y))), implies(not(implies(not(implies(x, y)), not(y))), implies(x, y)))), implies(implies(implies(y, implies(x, y)), implies(not(implies(not(implies(x, y)), not(y))), implies(x, y))), implies(not(implies(not(implies(x, y)), not(y))), implies(x, y))))), true, implies(implies(implies(y, implies(x, y)), implies(not(implies(not(implies(x, y)), not(y))), implies(x, y))), implies(not(implies(not(implies(x, y)), not(y))), implies(x, y)))), true, implies(implies(not(implies(not(implies(x, y)), not(y))), y), implies(not(implies(not(implies(x, y)), not(y))), implies(x, y)))), true, implies(not(implies(not(implies(x, y)), not(y))), y), implies(not(implies(not(implies(x, y)), not(y))), implies(x, y))), true, implies(not(implies(x, y)), not(y)))
% 32.26/4.55  = { by lemma 8 }
% 32.26/4.55    fresh2(fresh(fresh2(is_a_theorem(implies(implies(implies(y, implies(x, y)), implies(not(implies(not(implies(x, y)), not(y))), implies(x, y))), implies(not(implies(not(implies(x, y)), not(y))), implies(x, y)))), true, implies(implies(not(implies(not(implies(x, y)), not(y))), y), implies(not(implies(not(implies(x, y)), not(y))), implies(x, y)))), true, implies(not(implies(not(implies(x, y)), not(y))), y), implies(not(implies(not(implies(x, y)), not(y))), implies(x, y))), true, implies(not(implies(x, y)), not(y)))
% 32.26/4.55  = { by lemma 13 }
% 32.26/4.55    fresh2(fresh(is_a_theorem(implies(implies(not(implies(not(implies(x, y)), not(y))), y), implies(not(implies(not(implies(x, y)), not(y))), implies(x, y)))), true, implies(not(implies(not(implies(x, y)), not(y))), y), implies(not(implies(not(implies(x, y)), not(y))), implies(x, y))), true, implies(not(implies(x, y)), not(y)))
% 32.26/4.55  = { by axiom 5 (condensed_detachment) }
% 32.26/4.55    fresh2(fresh2(is_a_theorem(implies(not(implies(not(implies(x, y)), not(y))), y)), true, implies(not(implies(not(implies(x, y)), not(y))), implies(x, y))), true, implies(not(implies(x, y)), not(y)))
% 32.26/4.55  = { by axiom 2 (condensed_detachment) R->L }
% 32.26/4.55    fresh2(fresh2(fresh(true, true, implies(not(implies(not(implies(x, y)), not(y))), implies(not(y), y)), implies(not(implies(not(implies(x, y)), not(y))), y)), true, implies(not(implies(not(implies(x, y)), not(y))), implies(x, y))), true, implies(not(implies(x, y)), not(y)))
% 32.26/4.55  = { by axiom 1 (condensed_detachment) R->L }
% 32.26/4.55    fresh2(fresh2(fresh(fresh2(true, true, implies(implies(not(implies(not(implies(x, y)), not(y))), implies(not(y), y)), implies(not(implies(not(implies(x, y)), not(y))), y))), true, implies(not(implies(not(implies(x, y)), not(y))), implies(not(y), y)), implies(not(implies(not(implies(x, y)), not(y))), y)), true, implies(not(implies(not(implies(x, y)), not(y))), implies(x, y))), true, implies(not(implies(x, y)), not(y)))
% 32.26/4.55  = { by axiom 1 (condensed_detachment) R->L }
% 32.26/4.55    fresh2(fresh2(fresh(fresh2(fresh2(true, true, implies(implies(implies(implies(not(y), y), y), implies(not(implies(not(implies(x, y)), not(y))), y)), implies(not(implies(not(implies(x, y)), not(y))), y))), true, implies(implies(not(implies(not(implies(x, y)), not(y))), implies(not(y), y)), implies(not(implies(not(implies(x, y)), not(y))), y))), true, implies(not(implies(not(implies(x, y)), not(y))), implies(not(y), y)), implies(not(implies(not(implies(x, y)), not(y))), y)), true, implies(not(implies(not(implies(x, y)), not(y))), implies(x, y))), true, implies(not(implies(x, y)), not(y)))
% 32.26/4.55  = { by axiom 1 (condensed_detachment) R->L }
% 32.26/4.55    fresh2(fresh2(fresh(fresh2(fresh2(fresh2(true, true, implies(not(implies(implies(implies(implies(not(y), y), y), implies(not(implies(not(implies(x, y)), not(y))), y)), implies(not(implies(not(implies(x, y)), not(y))), y))), implies(implies(implies(implies(not(y), y), y), implies(not(implies(not(implies(x, y)), not(y))), y)), implies(not(implies(not(implies(x, y)), not(y))), y)))), true, implies(implies(implies(implies(not(y), y), y), implies(not(implies(not(implies(x, y)), not(y))), y)), implies(not(implies(not(implies(x, y)), not(y))), y))), true, implies(implies(not(implies(not(implies(x, y)), not(y))), implies(not(y), y)), implies(not(implies(not(implies(x, y)), not(y))), y))), true, implies(not(implies(not(implies(x, y)), not(y))), implies(not(y), y)), implies(not(implies(not(implies(x, y)), not(y))), y)), true, implies(not(implies(not(implies(x, y)), not(y))), implies(x, y))), true, implies(not(implies(x, y)), not(y)))
% 32.26/4.55  = { by lemma 16 R->L }
% 32.26/4.55    fresh2(fresh2(fresh(fresh2(fresh2(fresh2(is_a_theorem(implies(implies(implies(implies(not(y), y), y), implies(not(implies(not(implies(x, y)), not(y))), y)), implies(not(implies(not(implies(not(implies(x, y)), not(y))), y)), implies(not(implies(not(implies(x, y)), not(y))), y)))), true, implies(not(implies(implies(implies(implies(not(y), y), y), implies(not(implies(not(implies(x, y)), not(y))), y)), implies(not(implies(not(implies(x, y)), not(y))), y))), implies(implies(implies(implies(not(y), y), y), implies(not(implies(not(implies(x, y)), not(y))), y)), implies(not(implies(not(implies(x, y)), not(y))), y)))), true, implies(implies(implies(implies(not(y), y), y), implies(not(implies(not(implies(x, y)), not(y))), y)), implies(not(implies(not(implies(x, y)), not(y))), y))), true, implies(implies(not(implies(not(implies(x, y)), not(y))), implies(not(y), y)), implies(not(implies(not(implies(x, y)), not(y))), y))), true, implies(not(implies(not(implies(x, y)), not(y))), implies(not(y), y)), implies(not(implies(not(implies(x, y)), not(y))), y)), true, implies(not(implies(not(implies(x, y)), not(y))), implies(x, y))), true, implies(not(implies(x, y)), not(y)))
% 32.26/4.55  = { by lemma 19 }
% 32.26/4.55    fresh2(fresh2(fresh(fresh2(fresh2(is_a_theorem(implies(not(implies(implies(implies(implies(not(y), y), y), implies(not(implies(not(implies(x, y)), not(y))), y)), implies(not(implies(not(implies(x, y)), not(y))), y))), implies(implies(implies(implies(not(y), y), y), implies(not(implies(not(implies(x, y)), not(y))), y)), implies(not(implies(not(implies(x, y)), not(y))), y)))), true, implies(implies(implies(implies(not(y), y), y), implies(not(implies(not(implies(x, y)), not(y))), y)), implies(not(implies(not(implies(x, y)), not(y))), y))), true, implies(implies(not(implies(not(implies(x, y)), not(y))), implies(not(y), y)), implies(not(implies(not(implies(x, y)), not(y))), y))), true, implies(not(implies(not(implies(x, y)), not(y))), implies(not(y), y)), implies(not(implies(not(implies(x, y)), not(y))), y)), true, implies(not(implies(not(implies(x, y)), not(y))), implies(x, y))), true, implies(not(implies(x, y)), not(y)))
% 32.26/4.56  = { by lemma 8 }
% 32.26/4.56    fresh2(fresh2(fresh(fresh2(is_a_theorem(implies(implies(implies(implies(not(y), y), y), implies(not(implies(not(implies(x, y)), not(y))), y)), implies(not(implies(not(implies(x, y)), not(y))), y))), true, implies(implies(not(implies(not(implies(x, y)), not(y))), implies(not(y), y)), implies(not(implies(not(implies(x, y)), not(y))), y))), true, implies(not(implies(not(implies(x, y)), not(y))), implies(not(y), y)), implies(not(implies(not(implies(x, y)), not(y))), y)), true, implies(not(implies(not(implies(x, y)), not(y))), implies(x, y))), true, implies(not(implies(x, y)), not(y)))
% 32.26/4.56  = { by lemma 13 }
% 32.26/4.56    fresh2(fresh2(fresh(is_a_theorem(implies(implies(not(implies(not(implies(x, y)), not(y))), implies(not(y), y)), implies(not(implies(not(implies(x, y)), not(y))), y))), true, implies(not(implies(not(implies(x, y)), not(y))), implies(not(y), y)), implies(not(implies(not(implies(x, y)), not(y))), y)), true, implies(not(implies(not(implies(x, y)), not(y))), implies(x, y))), true, implies(not(implies(x, y)), not(y)))
% 32.26/4.56  = { by axiom 5 (condensed_detachment) }
% 32.26/4.56    fresh2(fresh2(fresh2(is_a_theorem(implies(not(implies(not(implies(x, y)), not(y))), implies(not(y), y))), true, implies(not(implies(not(implies(x, y)), not(y))), y)), true, implies(not(implies(not(implies(x, y)), not(y))), implies(x, y))), true, implies(not(implies(x, y)), not(y)))
% 32.26/4.56  = { by axiom 2 (condensed_detachment) R->L }
% 32.26/4.56    fresh2(fresh2(fresh2(fresh(true, true, implies(not(y), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y)))))), implies(not(implies(not(implies(x, y)), not(y))), implies(not(y), y))), true, implies(not(implies(not(implies(x, y)), not(y))), y)), true, implies(not(implies(not(implies(x, y)), not(y))), implies(x, y))), true, implies(not(implies(x, y)), not(y)))
% 32.26/4.56  = { by axiom 1 (condensed_detachment) R->L }
% 32.26/4.56    fresh2(fresh2(fresh2(fresh(fresh2(true, true, implies(implies(not(y), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y)))))), implies(not(implies(not(implies(x, y)), not(y))), implies(not(y), y)))), true, implies(not(y), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y)))))), implies(not(implies(not(implies(x, y)), not(y))), implies(not(y), y))), true, implies(not(implies(not(implies(x, y)), not(y))), y)), true, implies(not(implies(not(implies(x, y)), not(y))), implies(x, y))), true, implies(not(implies(x, y)), not(y)))
% 32.26/4.56  = { by axiom 1 (condensed_detachment) R->L }
% 32.26/4.56    fresh2(fresh2(fresh2(fresh(fresh2(fresh2(true, true, implies(implies(implies(not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))), y), implies(not(y), y)), implies(not(implies(not(implies(x, y)), not(y))), implies(not(y), y)))), true, implies(implies(not(y), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y)))))), implies(not(implies(not(implies(x, y)), not(y))), implies(not(y), y)))), true, implies(not(y), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y)))))), implies(not(implies(not(implies(x, y)), not(y))), implies(not(y), y))), true, implies(not(implies(not(implies(x, y)), not(y))), y)), true, implies(not(implies(not(implies(x, y)), not(y))), implies(x, y))), true, implies(not(implies(x, y)), not(y)))
% 32.26/4.56  = { by axiom 1 (condensed_detachment) R->L }
% 32.26/4.56    fresh2(fresh2(fresh2(fresh(fresh2(fresh2(fresh2(true, true, implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))), y))), true, implies(implies(implies(not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))), y), implies(not(y), y)), implies(not(implies(not(implies(x, y)), not(y))), implies(not(y), y)))), true, implies(implies(not(y), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y)))))), implies(not(implies(not(implies(x, y)), not(y))), implies(not(y), y)))), true, implies(not(y), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y)))))), implies(not(implies(not(implies(x, y)), not(y))), implies(not(y), y))), true, implies(not(implies(not(implies(x, y)), not(y))), y)), true, implies(not(implies(not(implies(x, y)), not(y))), implies(x, y))), true, implies(not(implies(x, y)), not(y)))
% 32.26/4.56  = { by axiom 3 (cn_3) R->L }
% 32.26/4.56    fresh2(fresh2(fresh2(fresh(fresh2(fresh2(fresh2(is_a_theorem(implies(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y)))), implies(not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))), y))), true, implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))), y))), true, implies(implies(implies(not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))), y), implies(not(y), y)), implies(not(implies(not(implies(x, y)), not(y))), implies(not(y), y)))), true, implies(implies(not(y), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y)))))), implies(not(implies(not(implies(x, y)), not(y))), implies(not(y), y)))), true, implies(not(y), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y)))))), implies(not(implies(not(implies(x, y)), not(y))), implies(not(y), y))), true, implies(not(implies(not(implies(x, y)), not(y))), y)), true, implies(not(implies(not(implies(x, y)), not(y))), implies(x, y))), true, implies(not(implies(x, y)), not(y)))
% 32.26/4.56  = { by lemma 12 }
% 32.26/4.56    fresh2(fresh2(fresh2(fresh(fresh2(fresh2(is_a_theorem(implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))), y))), true, implies(implies(implies(not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))), y), implies(not(y), y)), implies(not(implies(not(implies(x, y)), not(y))), implies(not(y), y)))), true, implies(implies(not(y), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y)))))), implies(not(implies(not(implies(x, y)), not(y))), implies(not(y), y)))), true, implies(not(y), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y)))))), implies(not(implies(not(implies(x, y)), not(y))), implies(not(y), y))), true, implies(not(implies(not(implies(x, y)), not(y))), y)), true, implies(not(implies(not(implies(x, y)), not(y))), implies(x, y))), true, implies(not(implies(x, y)), not(y)))
% 32.26/4.56  = { by lemma 9 }
% 32.26/4.56    fresh2(fresh2(fresh2(fresh(fresh2(is_a_theorem(implies(implies(implies(not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))), y), implies(not(y), y)), implies(not(implies(not(implies(x, y)), not(y))), implies(not(y), y)))), true, implies(implies(not(y), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y)))))), implies(not(implies(not(implies(x, y)), not(y))), implies(not(y), y)))), true, implies(not(y), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y)))))), implies(not(implies(not(implies(x, y)), not(y))), implies(not(y), y))), true, implies(not(implies(not(implies(x, y)), not(y))), y)), true, implies(not(implies(not(implies(x, y)), not(y))), implies(x, y))), true, implies(not(implies(x, y)), not(y)))
% 32.26/4.56  = { by lemma 13 }
% 32.26/4.56    fresh2(fresh2(fresh2(fresh(is_a_theorem(implies(implies(not(y), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y)))))), implies(not(implies(not(implies(x, y)), not(y))), implies(not(y), y)))), true, implies(not(y), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y)))))), implies(not(implies(not(implies(x, y)), not(y))), implies(not(y), y))), true, implies(not(implies(not(implies(x, y)), not(y))), y)), true, implies(not(implies(not(implies(x, y)), not(y))), implies(x, y))), true, implies(not(implies(x, y)), not(y)))
% 32.26/4.57  = { by axiom 5 (condensed_detachment) }
% 32.26/4.57    fresh2(fresh2(fresh2(fresh2(is_a_theorem(implies(not(y), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))))), true, implies(not(implies(not(implies(x, y)), not(y))), implies(not(y), y))), true, implies(not(implies(not(implies(x, y)), not(y))), y)), true, implies(not(implies(not(implies(x, y)), not(y))), implies(x, y))), true, implies(not(implies(x, y)), not(y)))
% 32.26/4.57  = { by axiom 2 (condensed_detachment) R->L }
% 32.26/4.57    fresh2(fresh2(fresh2(fresh2(fresh(true, true, implies(implies(not(implies(x, y)), not(y)), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y)))))), implies(not(y), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))))), true, implies(not(implies(not(implies(x, y)), not(y))), implies(not(y), y))), true, implies(not(implies(not(implies(x, y)), not(y))), y)), true, implies(not(implies(not(implies(x, y)), not(y))), implies(x, y))), true, implies(not(implies(x, y)), not(y)))
% 32.26/4.57  = { by axiom 1 (condensed_detachment) R->L }
% 32.26/4.57    fresh2(fresh2(fresh2(fresh2(fresh(fresh2(true, true, implies(implies(implies(not(implies(x, y)), not(y)), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y)))))), implies(not(y), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y)))))))), true, implies(implies(not(implies(x, y)), not(y)), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y)))))), implies(not(y), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))))), true, implies(not(implies(not(implies(x, y)), not(y))), implies(not(y), y))), true, implies(not(implies(not(implies(x, y)), not(y))), y)), true, implies(not(implies(not(implies(x, y)), not(y))), implies(x, y))), true, implies(not(implies(x, y)), not(y)))
% 32.26/4.57  = { by lemma 18 R->L }
% 32.26/4.57    fresh2(fresh2(fresh2(fresh2(fresh(fresh2(is_a_theorem(implies(implies(implies(not(implies(not(implies(x, y)), not(y))), implies(not(implies(x, y)), not(y))), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y)))))), implies(not(y), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y)))))))), true, implies(implies(implies(not(implies(x, y)), not(y)), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y)))))), implies(not(y), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y)))))))), true, implies(implies(not(implies(x, y)), not(y)), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y)))))), implies(not(y), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))))), true, implies(not(implies(not(implies(x, y)), not(y))), implies(not(y), y))), true, implies(not(implies(not(implies(x, y)), not(y))), y)), true, implies(not(implies(not(implies(x, y)), not(y))), implies(x, y))), true, implies(not(implies(x, y)), not(y)))
% 32.26/4.57  = { by lemma 15 }
% 32.26/4.57    fresh2(fresh2(fresh2(fresh2(fresh(is_a_theorem(implies(implies(implies(not(implies(x, y)), not(y)), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y)))))), implies(not(y), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y)))))))), true, implies(implies(not(implies(x, y)), not(y)), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y)))))), implies(not(y), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))))), true, implies(not(implies(not(implies(x, y)), not(y))), implies(not(y), y))), true, implies(not(implies(not(implies(x, y)), not(y))), y)), true, implies(not(implies(not(implies(x, y)), not(y))), implies(x, y))), true, implies(not(implies(x, y)), not(y)))
% 32.26/4.57  = { by axiom 5 (condensed_detachment) }
% 32.26/4.57    fresh2(fresh2(fresh2(fresh2(fresh2(is_a_theorem(implies(implies(not(implies(x, y)), not(y)), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))))), true, implies(not(y), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))))), true, implies(not(implies(not(implies(x, y)), not(y))), implies(not(y), y))), true, implies(not(implies(not(implies(x, y)), not(y))), y)), true, implies(not(implies(not(implies(x, y)), not(y))), implies(x, y))), true, implies(not(implies(x, y)), not(y)))
% 32.26/4.57  = { by axiom 2 (condensed_detachment) R->L }
% 32.26/4.57    fresh2(fresh2(fresh2(fresh2(fresh2(fresh(true, true, implies(implies(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y)))), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y)))))), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y)))))), implies(implies(not(implies(x, y)), not(y)), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))))), true, implies(not(y), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))))), true, implies(not(implies(not(implies(x, y)), not(y))), implies(not(y), y))), true, implies(not(implies(not(implies(x, y)), not(y))), y)), true, implies(not(implies(not(implies(x, y)), not(y))), implies(x, y))), true, implies(not(implies(x, y)), not(y)))
% 32.26/4.57  = { by lemma 20 R->L }
% 32.26/4.57    fresh2(fresh2(fresh2(fresh2(fresh2(fresh(is_a_theorem(implies(implies(implies(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y)))), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y)))))), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y)))))), implies(implies(not(implies(x, y)), not(y)), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y)))))))), true, implies(implies(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y)))), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y)))))), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y)))))), implies(implies(not(implies(x, y)), not(y)), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))))), true, implies(not(y), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))))), true, implies(not(implies(not(implies(x, y)), not(y))), implies(not(y), y))), true, implies(not(implies(not(implies(x, y)), not(y))), y)), true, implies(not(implies(not(implies(x, y)), not(y))), implies(x, y))), true, implies(not(implies(x, y)), not(y)))
% 32.26/4.57  = { by axiom 5 (condensed_detachment) }
% 32.26/4.57    fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(is_a_theorem(implies(implies(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y)))), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y)))))), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))))), true, implies(implies(not(implies(x, y)), not(y)), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))))), true, implies(not(y), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))))), true, implies(not(implies(not(implies(x, y)), not(y))), implies(not(y), y))), true, implies(not(implies(not(implies(x, y)), not(y))), y)), true, implies(not(implies(not(implies(x, y)), not(y))), implies(x, y))), true, implies(not(implies(x, y)), not(y)))
% 32.26/4.58  = { by lemma 15 R->L }
% 32.26/4.58    fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(is_a_theorem(implies(implies(implies(not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))), implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y)))))), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))))), true, implies(implies(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y)))), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y)))))), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))))), true, implies(implies(not(implies(x, y)), not(y)), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))))), true, implies(not(y), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))))), true, implies(not(implies(not(implies(x, y)), not(y))), implies(not(y), y))), true, implies(not(implies(not(implies(x, y)), not(y))), y)), true, implies(not(implies(not(implies(x, y)), not(y))), implies(x, y))), true, implies(not(implies(x, y)), not(y)))
% 32.26/4.58  = { by lemma 8 R->L }
% 32.26/4.58    fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(is_a_theorem(implies(not(implies(implies(implies(not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))), implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y)))))), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))))), implies(implies(implies(not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))), implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y)))))), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y)))))))), true, implies(implies(implies(not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))), implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y)))))), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))))), true, implies(implies(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y)))), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y)))))), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))))), true, implies(implies(not(implies(x, y)), not(y)), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))))), true, implies(not(y), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))))), true, implies(not(implies(not(implies(x, y)), not(y))), implies(not(y), y))), true, implies(not(implies(not(implies(x, y)), not(y))), y)), true, implies(not(implies(not(implies(x, y)), not(y))), implies(x, y))), true, implies(not(implies(x, y)), not(y)))
% 32.26/4.58  = { by lemma 19 R->L }
% 32.26/4.59    fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(is_a_theorem(implies(implies(implies(not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))), implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y)))))), implies(not(not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y)))))), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y)))))))), true, implies(not(implies(implies(implies(not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))), implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y)))))), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))))), implies(implies(implies(not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))), implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y)))))), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y)))))))), true, implies(implies(implies(not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))), implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y)))))), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))))), true, implies(implies(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y)))), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y)))))), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))))), true, implies(implies(not(implies(x, y)), not(y)), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))))), true, implies(not(y), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))))), true, implies(not(implies(not(implies(x, y)), not(y))), implies(not(y), y))), true, implies(not(implies(not(implies(x, y)), not(y))), y)), true, implies(not(implies(not(implies(x, y)), not(y))), implies(x, y))), true, implies(not(implies(x, y)), not(y)))
% 32.26/4.59  = { by lemma 9 R->L }
% 32.26/4.59    fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(is_a_theorem(implies(not(not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y)))))), implies(not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))), implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))))), true, implies(implies(implies(not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))), implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y)))))), implies(not(not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y)))))), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y)))))))), true, implies(not(implies(implies(implies(not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))), implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y)))))), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))))), implies(implies(implies(not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))), implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y)))))), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y)))))))), true, implies(implies(implies(not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))), implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y)))))), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))))), true, implies(implies(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y)))), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y)))))), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))))), true, implies(implies(not(implies(x, y)), not(y)), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))))), true, implies(not(y), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))))), true, implies(not(implies(not(implies(x, y)), not(y))), implies(not(y), y))), true, implies(not(implies(not(implies(x, y)), not(y))), y)), true, implies(not(implies(not(implies(x, y)), not(y))), implies(x, y))), true, implies(not(implies(x, y)), not(y)))
% 32.26/4.60  = { by lemma 14 R->L }
% 32.26/4.60    fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(is_a_theorem(implies(implies(not(not(implies(not(implies(x, y)), not(y)))), implies(not(not(not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))))), not(not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y)))))))), implies(not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))), implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))))), true, implies(not(not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y)))))), implies(not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))), implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))))), true, implies(implies(implies(not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))), implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y)))))), implies(not(not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y)))))), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y)))))))), true, implies(not(implies(implies(implies(not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))), implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y)))))), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))))), implies(implies(implies(not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))), implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y)))))), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y)))))))), true, implies(implies(implies(not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))), implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y)))))), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))))), true, implies(implies(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y)))), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y)))))), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))))), true, implies(implies(not(implies(x, y)), not(y)), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))))), true, implies(not(y), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))))), true, implies(not(implies(not(implies(x, y)), not(y))), implies(not(y), y))), true, implies(not(implies(not(implies(x, y)), not(y))), y)), true, implies(not(implies(not(implies(x, y)), not(y))), implies(x, y))), true, implies(not(implies(x, y)), not(y)))
% 32.26/4.60  = { by lemma 13 R->L }
% 32.26/4.60    fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(is_a_theorem(implies(implies(implies(implies(not(not(not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))))), not(not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))))), not(implies(not(implies(x, y)), not(y)))), implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))), implies(not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))), implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))))), true, implies(implies(not(not(implies(not(implies(x, y)), not(y)))), implies(not(not(not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))))), not(not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y)))))))), implies(not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))), implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))))), true, implies(not(not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y)))))), implies(not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))), implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))))), true, implies(implies(implies(not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))), implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y)))))), implies(not(not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y)))))), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y)))))))), true, implies(not(implies(implies(implies(not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))), implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y)))))), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))))), implies(implies(implies(not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))), implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y)))))), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y)))))))), true, implies(implies(implies(not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))), implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y)))))), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))))), true, implies(implies(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y)))), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y)))))), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))))), true, implies(implies(not(implies(x, y)), not(y)), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))))), true, implies(not(y), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))))), true, implies(not(implies(not(implies(x, y)), not(y))), implies(not(y), y))), true, implies(not(implies(not(implies(x, y)), not(y))), y)), true, implies(not(implies(not(implies(x, y)), not(y))), implies(x, y))), true, implies(not(implies(x, y)), not(y)))
% 32.26/4.61  = { by lemma 20 }
% 32.26/4.61    fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(true, true, implies(implies(not(not(implies(not(implies(x, y)), not(y)))), implies(not(not(not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))))), not(not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y)))))))), implies(not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))), implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))))), true, implies(not(not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y)))))), implies(not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))), implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))))), true, implies(implies(implies(not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))), implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y)))))), implies(not(not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y)))))), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y)))))))), true, implies(not(implies(implies(implies(not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))), implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y)))))), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))))), implies(implies(implies(not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))), implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y)))))), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y)))))))), true, implies(implies(implies(not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))), implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y)))))), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))))), true, implies(implies(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y)))), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y)))))), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))))), true, implies(implies(not(implies(x, y)), not(y)), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))))), true, implies(not(y), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))))), true, implies(not(implies(not(implies(x, y)), not(y))), implies(not(y), y))), true, implies(not(implies(not(implies(x, y)), not(y))), y)), true, implies(not(implies(not(implies(x, y)), not(y))), implies(x, y))), true, implies(not(implies(x, y)), not(y)))
% 32.26/4.61  = { by axiom 1 (condensed_detachment) }
% 32.26/4.61    fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(true, true, implies(not(not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y)))))), implies(not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))), implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))))), true, implies(implies(implies(not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))), implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y)))))), implies(not(not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y)))))), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y)))))))), true, implies(not(implies(implies(implies(not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))), implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y)))))), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))))), implies(implies(implies(not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))), implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y)))))), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y)))))))), true, implies(implies(implies(not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))), implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y)))))), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))))), true, implies(implies(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y)))), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y)))))), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))))), true, implies(implies(not(implies(x, y)), not(y)), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))))), true, implies(not(y), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))))), true, implies(not(implies(not(implies(x, y)), not(y))), implies(not(y), y))), true, implies(not(implies(not(implies(x, y)), not(y))), y)), true, implies(not(implies(not(implies(x, y)), not(y))), implies(x, y))), true, implies(not(implies(x, y)), not(y)))
% 32.26/4.61  = { by axiom 1 (condensed_detachment) }
% 32.26/4.62    fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(true, true, implies(implies(implies(not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))), implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y)))))), implies(not(not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y)))))), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y)))))))), true, implies(not(implies(implies(implies(not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))), implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y)))))), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))))), implies(implies(implies(not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))), implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y)))))), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y)))))))), true, implies(implies(implies(not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))), implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y)))))), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))))), true, implies(implies(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y)))), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y)))))), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))))), true, implies(implies(not(implies(x, y)), not(y)), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))))), true, implies(not(y), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))))), true, implies(not(implies(not(implies(x, y)), not(y))), implies(not(y), y))), true, implies(not(implies(not(implies(x, y)), not(y))), y)), true, implies(not(implies(not(implies(x, y)), not(y))), implies(x, y))), true, implies(not(implies(x, y)), not(y)))
% 32.26/4.62  = { by axiom 1 (condensed_detachment) }
% 32.26/4.62    fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(true, true, implies(not(implies(implies(implies(not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))), implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y)))))), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))))), implies(implies(implies(not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))), implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y)))))), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y)))))))), true, implies(implies(implies(not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))), implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y)))))), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))))), true, implies(implies(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y)))), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y)))))), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))))), true, implies(implies(not(implies(x, y)), not(y)), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))))), true, implies(not(y), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))))), true, implies(not(implies(not(implies(x, y)), not(y))), implies(not(y), y))), true, implies(not(implies(not(implies(x, y)), not(y))), y)), true, implies(not(implies(not(implies(x, y)), not(y))), implies(x, y))), true, implies(not(implies(x, y)), not(y)))
% 32.26/4.62  = { by axiom 1 (condensed_detachment) }
% 32.26/4.62    fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(true, true, implies(implies(implies(not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))), implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y)))))), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))))), true, implies(implies(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y)))), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y)))))), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))))), true, implies(implies(not(implies(x, y)), not(y)), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))))), true, implies(not(y), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))))), true, implies(not(implies(not(implies(x, y)), not(y))), implies(not(y), y))), true, implies(not(implies(not(implies(x, y)), not(y))), y)), true, implies(not(implies(not(implies(x, y)), not(y))), implies(x, y))), true, implies(not(implies(x, y)), not(y)))
% 32.26/4.62  = { by axiom 1 (condensed_detachment) }
% 32.26/4.62    fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(true, true, implies(implies(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y)))), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y)))))), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))))), true, implies(implies(not(implies(x, y)), not(y)), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))))), true, implies(not(y), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))))), true, implies(not(implies(not(implies(x, y)), not(y))), implies(not(y), y))), true, implies(not(implies(not(implies(x, y)), not(y))), y)), true, implies(not(implies(not(implies(x, y)), not(y))), implies(x, y))), true, implies(not(implies(x, y)), not(y)))
% 32.26/4.62  = { by axiom 1 (condensed_detachment) }
% 32.26/4.62    fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(true, true, implies(implies(not(implies(x, y)), not(y)), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))))), true, implies(not(y), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))))), true, implies(not(implies(not(implies(x, y)), not(y))), implies(not(y), y))), true, implies(not(implies(not(implies(x, y)), not(y))), y)), true, implies(not(implies(not(implies(x, y)), not(y))), implies(x, y))), true, implies(not(implies(x, y)), not(y)))
% 32.26/4.62  = { by axiom 1 (condensed_detachment) }
% 32.26/4.62    fresh2(fresh2(fresh2(fresh2(fresh2(true, true, implies(not(y), not(implies(not(not(implies(not(implies(x, y)), not(y)))), not(implies(not(implies(x, y)), not(y))))))), true, implies(not(implies(not(implies(x, y)), not(y))), implies(not(y), y))), true, implies(not(implies(not(implies(x, y)), not(y))), y)), true, implies(not(implies(not(implies(x, y)), not(y))), implies(x, y))), true, implies(not(implies(x, y)), not(y)))
% 32.26/4.62  = { by axiom 1 (condensed_detachment) }
% 32.26/4.62    fresh2(fresh2(fresh2(fresh2(true, true, implies(not(implies(not(implies(x, y)), not(y))), implies(not(y), y))), true, implies(not(implies(not(implies(x, y)), not(y))), y)), true, implies(not(implies(not(implies(x, y)), not(y))), implies(x, y))), true, implies(not(implies(x, y)), not(y)))
% 32.26/4.62  = { by axiom 1 (condensed_detachment) }
% 32.26/4.62    fresh2(fresh2(fresh2(true, true, implies(not(implies(not(implies(x, y)), not(y))), y)), true, implies(not(implies(not(implies(x, y)), not(y))), implies(x, y))), true, implies(not(implies(x, y)), not(y)))
% 32.26/4.62  = { by axiom 1 (condensed_detachment) }
% 32.26/4.62    fresh2(fresh2(true, true, implies(not(implies(not(implies(x, y)), not(y))), implies(x, y))), true, implies(not(implies(x, y)), not(y)))
% 32.26/4.62  = { by axiom 1 (condensed_detachment) }
% 32.26/4.62    fresh2(true, true, implies(not(implies(x, y)), not(y)))
% 32.26/4.62  = { by axiom 1 (condensed_detachment) }
% 32.26/4.62    true
% 32.26/4.62  % SZS output end Proof
% 32.26/4.62  
% 32.26/4.62  RESULT: Unsatisfiable (the axioms are contradictory).
%------------------------------------------------------------------------------