TSTP Solution File: LCL054-1 by Twee---2.4.2
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- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : LCL054-1 : TPTP v8.1.2. Released v1.0.0.
% Transfm : none
% Format : tptp:raw
% Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% Computer : n012.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Thu Aug 31 08:17:12 EDT 2023
% Result : Unsatisfiable 108.22s 14.21s
% Output : Proof 109.53s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.14 % Problem : LCL054-1 : TPTP v8.1.2. Released v1.0.0.
% 0.00/0.15 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.15/0.37 % Computer : n012.cluster.edu
% 0.15/0.37 % Model : x86_64 x86_64
% 0.15/0.37 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.15/0.37 % Memory : 8042.1875MB
% 0.15/0.37 % OS : Linux 3.10.0-693.el7.x86_64
% 0.15/0.37 % CPULimit : 300
% 0.15/0.37 % WCLimit : 300
% 0.15/0.37 % DateTime : Fri Aug 25 06:46:26 EDT 2023
% 0.15/0.37 % CPUTime :
% 108.22/14.21 Command-line arguments: --set-join --lhs-weight 1 --no-flatten-goal --complete-subsets --goal-heuristic
% 108.22/14.21
% 108.22/14.21 % SZS status Unsatisfiable
% 108.22/14.21
% 108.54/14.32 % SZS output start Proof
% 108.54/14.32 Take the following subset of the input axioms:
% 108.54/14.32 fof(cn_1, axiom, ![X, Y, Z]: is_a_theorem(implies(implies(X, Y), implies(implies(Y, Z), implies(X, Z))))).
% 108.54/14.32 fof(cn_2, axiom, ![X2]: is_a_theorem(implies(implies(not(X2), X2), X2))).
% 108.54/14.32 fof(cn_3, axiom, ![X2, Y2]: is_a_theorem(implies(X2, implies(not(X2), Y2)))).
% 108.54/14.32 fof(condensed_detachment, axiom, ![X2, Y2]: (~is_a_theorem(implies(X2, Y2)) | (~is_a_theorem(X2) | is_a_theorem(Y2)))).
% 108.54/14.32 fof(prove_cn_35, negated_conjecture, ~is_a_theorem(implies(implies(a, implies(b, c)), implies(implies(a, b), implies(a, c))))).
% 108.54/14.32
% 108.54/14.32 Now clausify the problem and encode Horn clauses using encoding 3 of
% 108.54/14.32 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 108.54/14.32 We repeatedly replace C & s=t => u=v by the two clauses:
% 108.54/14.32 fresh(y, y, x1...xn) = u
% 108.54/14.32 C => fresh(s, t, x1...xn) = v
% 108.54/14.32 where fresh is a fresh function symbol and x1..xn are the free
% 108.54/14.32 variables of u and v.
% 108.54/14.32 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 108.54/14.32 input problem has no model of domain size 1).
% 108.54/14.32
% 108.54/14.32 The encoding turns the above axioms into the following unit equations and goals:
% 108.54/14.32
% 108.54/14.32 Axiom 1 (condensed_detachment): fresh2(X, X, Y) = true.
% 108.54/14.32 Axiom 2 (condensed_detachment): fresh(X, X, Y, Z) = is_a_theorem(Z).
% 108.54/14.32 Axiom 3 (cn_3): is_a_theorem(implies(X, implies(not(X), Y))) = true.
% 108.54/14.32 Axiom 4 (cn_2): is_a_theorem(implies(implies(not(X), X), X)) = true.
% 108.54/14.32 Axiom 5 (condensed_detachment): fresh(is_a_theorem(implies(X, Y)), true, X, Y) = fresh2(is_a_theorem(X), true, Y).
% 108.54/14.32 Axiom 6 (cn_1): is_a_theorem(implies(implies(X, Y), implies(implies(Y, Z), implies(X, Z)))) = true.
% 108.54/14.32
% 108.54/14.32 Lemma 7: 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))).
% 108.54/14.32 Proof:
% 108.54/14.32 fresh2(is_a_theorem(implies(X, Y)), true, implies(implies(Y, Z), implies(X, Z)))
% 108.54/14.32 = { by axiom 5 (condensed_detachment) R->L }
% 108.54/14.32 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)))
% 108.54/14.32 = { by axiom 6 (cn_1) }
% 108.54/14.32 fresh(true, true, implies(X, Y), implies(implies(Y, Z), implies(X, Z)))
% 108.54/14.32 = { by axiom 2 (condensed_detachment) }
% 108.54/14.32 is_a_theorem(implies(implies(Y, Z), implies(X, Z)))
% 108.54/14.32
% 108.54/14.32 Lemma 8: fresh2(is_a_theorem(implies(implies(not(X), Y), Z)), true, implies(X, Z)) = is_a_theorem(implies(X, Z)).
% 108.54/14.32 Proof:
% 108.54/14.32 fresh2(is_a_theorem(implies(implies(not(X), Y), Z)), true, implies(X, Z))
% 108.54/14.32 = { by axiom 5 (condensed_detachment) R->L }
% 108.54/14.32 fresh(is_a_theorem(implies(implies(implies(not(X), Y), Z), implies(X, Z))), true, implies(implies(not(X), Y), Z), implies(X, Z))
% 108.54/14.32 = { by lemma 7 R->L }
% 108.54/14.32 fresh(fresh2(is_a_theorem(implies(X, implies(not(X), Y))), true, implies(implies(implies(not(X), Y), Z), implies(X, Z))), true, implies(implies(not(X), Y), Z), implies(X, Z))
% 108.54/14.32 = { by axiom 3 (cn_3) }
% 108.54/14.32 fresh(fresh2(true, true, implies(implies(implies(not(X), Y), Z), implies(X, Z))), true, implies(implies(not(X), Y), Z), implies(X, Z))
% 108.54/14.32 = { by axiom 1 (condensed_detachment) }
% 108.54/14.32 fresh(true, true, implies(implies(not(X), Y), Z), implies(X, Z))
% 108.54/14.32 = { by axiom 2 (condensed_detachment) }
% 108.54/14.32 is_a_theorem(implies(X, Z))
% 108.54/14.33
% 108.54/14.33 Lemma 9: is_a_theorem(implies(X, X)) = true.
% 108.54/14.33 Proof:
% 108.54/14.33 is_a_theorem(implies(X, X))
% 108.54/14.33 = { by lemma 8 R->L }
% 108.54/14.33 fresh2(is_a_theorem(implies(implies(not(X), X), X)), true, implies(X, X))
% 108.54/14.33 = { by axiom 4 (cn_2) }
% 108.54/14.33 fresh2(true, true, implies(X, X))
% 108.54/14.33 = { by axiom 1 (condensed_detachment) }
% 108.54/14.33 true
% 108.54/14.33
% 108.54/14.33 Lemma 10: 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)).
% 108.54/14.33 Proof:
% 108.54/14.33 fresh2(is_a_theorem(implies(implies(implies(X, Y), implies(Z, Y)), W)), true, implies(implies(Z, X), W))
% 108.54/14.33 = { by axiom 5 (condensed_detachment) R->L }
% 108.54/14.33 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))
% 108.54/14.33 = { by lemma 7 R->L }
% 108.54/14.33 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))
% 108.54/14.33 = { by axiom 6 (cn_1) }
% 108.54/14.33 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))
% 108.54/14.33 = { by axiom 1 (condensed_detachment) }
% 108.54/14.33 fresh(true, true, implies(implies(implies(X, Y), implies(Z, Y)), W), implies(implies(Z, X), W))
% 108.54/14.33 = { by axiom 2 (condensed_detachment) }
% 108.54/14.33 is_a_theorem(implies(implies(Z, X), W))
% 108.54/14.33
% 108.54/14.33 Lemma 11: is_a_theorem(implies(implies(X, Y), implies(implies(not(X), X), Y))) = true.
% 108.54/14.33 Proof:
% 108.54/14.33 is_a_theorem(implies(implies(X, Y), implies(implies(not(X), X), Y)))
% 108.54/14.33 = { by lemma 7 R->L }
% 108.54/14.33 fresh2(is_a_theorem(implies(implies(not(X), X), X)), true, implies(implies(X, Y), implies(implies(not(X), X), Y)))
% 108.54/14.33 = { by axiom 4 (cn_2) }
% 108.54/14.33 fresh2(true, true, implies(implies(X, Y), implies(implies(not(X), X), Y)))
% 108.54/14.33 = { by axiom 1 (condensed_detachment) }
% 108.54/14.33 true
% 108.54/14.33
% 108.54/14.33 Lemma 12: is_a_theorem(implies(implies(X, implies(not(Y), Y)), implies(implies(Y, Z), implies(X, Z)))) = true.
% 108.54/14.33 Proof:
% 108.54/14.33 is_a_theorem(implies(implies(X, implies(not(Y), Y)), implies(implies(Y, Z), implies(X, Z))))
% 108.54/14.33 = { by lemma 10 R->L }
% 108.54/14.33 fresh2(is_a_theorem(implies(implies(implies(implies(not(Y), Y), Z), implies(X, Z)), implies(implies(Y, Z), implies(X, Z)))), true, implies(implies(X, implies(not(Y), Y)), implies(implies(Y, Z), implies(X, Z))))
% 108.54/14.33 = { by lemma 7 R->L }
% 108.54/14.33 fresh2(fresh2(is_a_theorem(implies(implies(Y, Z), implies(implies(not(Y), Y), Z))), true, implies(implies(implies(implies(not(Y), Y), Z), implies(X, Z)), implies(implies(Y, Z), implies(X, Z)))), true, implies(implies(X, implies(not(Y), Y)), implies(implies(Y, Z), implies(X, Z))))
% 108.54/14.33 = { by lemma 11 }
% 108.54/14.33 fresh2(fresh2(true, true, implies(implies(implies(implies(not(Y), Y), Z), implies(X, Z)), implies(implies(Y, Z), implies(X, Z)))), true, implies(implies(X, implies(not(Y), Y)), implies(implies(Y, Z), implies(X, Z))))
% 108.54/14.33 = { by axiom 1 (condensed_detachment) }
% 108.54/14.33 fresh2(true, true, implies(implies(X, implies(not(Y), Y)), implies(implies(Y, Z), implies(X, Z))))
% 108.54/14.33 = { by axiom 1 (condensed_detachment) }
% 108.54/14.33 true
% 108.54/14.33
% 108.54/14.33 Lemma 13: fresh2(is_a_theorem(implies(implies(X, X), Y)), true, implies(implies(Z, implies(X, X)), Y)) = is_a_theorem(implies(implies(Z, implies(X, X)), Y)).
% 108.54/14.33 Proof:
% 108.54/14.33 fresh2(is_a_theorem(implies(implies(X, X), Y)), true, implies(implies(Z, implies(X, X)), Y))
% 108.54/14.33 = { by axiom 5 (condensed_detachment) R->L }
% 108.54/14.33 fresh(is_a_theorem(implies(implies(implies(X, X), Y), implies(implies(Z, implies(X, X)), Y))), true, implies(implies(X, X), Y), implies(implies(Z, implies(X, X)), Y))
% 108.54/14.33 = { by axiom 2 (condensed_detachment) R->L }
% 108.54/14.33 fresh(fresh(true, true, implies(X, X), implies(implies(implies(X, X), Y), implies(implies(Z, implies(X, X)), Y))), true, implies(implies(X, X), Y), implies(implies(Z, implies(X, X)), Y))
% 108.54/14.33 = { by axiom 1 (condensed_detachment) R->L }
% 108.54/14.33 fresh(fresh(fresh2(true, true, implies(implies(X, X), implies(implies(implies(X, X), Y), implies(implies(Z, implies(X, X)), Y)))), true, implies(X, X), implies(implies(implies(X, X), Y), implies(implies(Z, implies(X, X)), Y))), true, implies(implies(X, X), Y), implies(implies(Z, implies(X, X)), Y))
% 108.54/14.33 = { by axiom 1 (condensed_detachment) R->L }
% 108.54/14.33 fresh(fresh(fresh2(fresh2(true, true, implies(implies(not(implies(X, X)), Z), implies(implies(implies(X, X), Y), implies(implies(Z, implies(X, X)), Y)))), true, implies(implies(X, X), implies(implies(implies(X, X), Y), implies(implies(Z, implies(X, X)), Y)))), true, implies(X, X), implies(implies(implies(X, X), Y), implies(implies(Z, implies(X, X)), Y))), true, implies(implies(X, X), Y), implies(implies(Z, implies(X, X)), Y))
% 108.54/14.33 = { by lemma 12 R->L }
% 108.54/14.33 fresh(fresh(fresh2(fresh2(is_a_theorem(implies(implies(implies(Z, implies(X, X)), implies(not(implies(X, X)), implies(X, X))), implies(implies(implies(X, X), Y), implies(implies(Z, implies(X, X)), Y)))), true, implies(implies(not(implies(X, X)), Z), implies(implies(implies(X, X), Y), implies(implies(Z, implies(X, X)), Y)))), true, implies(implies(X, X), implies(implies(implies(X, X), Y), implies(implies(Z, implies(X, X)), Y)))), true, implies(X, X), implies(implies(implies(X, X), Y), implies(implies(Z, implies(X, X)), Y))), true, implies(implies(X, X), Y), implies(implies(Z, implies(X, X)), Y))
% 108.54/14.33 = { by lemma 10 }
% 108.54/14.33 fresh(fresh(fresh2(is_a_theorem(implies(implies(not(implies(X, X)), Z), implies(implies(implies(X, X), Y), implies(implies(Z, implies(X, X)), Y)))), true, implies(implies(X, X), implies(implies(implies(X, X), Y), implies(implies(Z, implies(X, X)), Y)))), true, implies(X, X), implies(implies(implies(X, X), Y), implies(implies(Z, implies(X, X)), Y))), true, implies(implies(X, X), Y), implies(implies(Z, implies(X, X)), Y))
% 108.54/14.33 = { by lemma 8 }
% 108.54/14.33 fresh(fresh(is_a_theorem(implies(implies(X, X), implies(implies(implies(X, X), Y), implies(implies(Z, implies(X, X)), Y)))), true, implies(X, X), implies(implies(implies(X, X), Y), implies(implies(Z, implies(X, X)), Y))), true, implies(implies(X, X), Y), implies(implies(Z, implies(X, X)), Y))
% 108.54/14.33 = { by axiom 5 (condensed_detachment) }
% 108.54/14.33 fresh(fresh2(is_a_theorem(implies(X, X)), true, implies(implies(implies(X, X), Y), implies(implies(Z, implies(X, X)), Y))), true, implies(implies(X, X), Y), implies(implies(Z, implies(X, X)), Y))
% 108.54/14.33 = { by lemma 9 }
% 108.54/14.33 fresh(fresh2(true, true, implies(implies(implies(X, X), Y), implies(implies(Z, implies(X, X)), Y))), true, implies(implies(X, X), Y), implies(implies(Z, implies(X, X)), Y))
% 108.54/14.33 = { by axiom 1 (condensed_detachment) }
% 108.54/14.33 fresh(true, true, implies(implies(X, X), Y), implies(implies(Z, implies(X, X)), Y))
% 108.54/14.33 = { by axiom 2 (condensed_detachment) }
% 108.54/14.33 is_a_theorem(implies(implies(Z, implies(X, X)), Y))
% 108.54/14.33
% 108.54/14.33 Lemma 14: is_a_theorem(implies(implies(X, Y), implies(Z, implies(X, Y)))) = true.
% 108.54/14.33 Proof:
% 108.54/14.33 is_a_theorem(implies(implies(X, Y), implies(Z, implies(X, Y))))
% 108.54/14.33 = { by lemma 10 R->L }
% 108.54/14.33 fresh2(is_a_theorem(implies(implies(implies(Y, Y), implies(X, Y)), implies(Z, implies(X, Y)))), true, implies(implies(X, Y), implies(Z, implies(X, Y))))
% 108.54/14.33 = { by lemma 7 R->L }
% 108.54/14.33 fresh2(fresh2(is_a_theorem(implies(Z, implies(Y, Y))), true, implies(implies(implies(Y, Y), implies(X, Y)), implies(Z, implies(X, Y)))), true, implies(implies(X, Y), implies(Z, implies(X, Y))))
% 108.54/14.33 = { by lemma 8 R->L }
% 108.54/14.33 fresh2(fresh2(fresh2(is_a_theorem(implies(implies(not(Z), implies(Y, Y)), implies(Y, Y))), true, implies(Z, implies(Y, Y))), true, implies(implies(implies(Y, Y), implies(X, Y)), implies(Z, implies(X, Y)))), true, implies(implies(X, Y), implies(Z, implies(X, Y))))
% 108.54/14.33 = { by lemma 13 R->L }
% 108.54/14.33 fresh2(fresh2(fresh2(fresh2(is_a_theorem(implies(implies(Y, Y), implies(Y, Y))), true, implies(implies(not(Z), implies(Y, Y)), implies(Y, Y))), true, implies(Z, implies(Y, Y))), true, implies(implies(implies(Y, Y), implies(X, Y)), implies(Z, implies(X, Y)))), true, implies(implies(X, Y), implies(Z, implies(X, Y))))
% 108.54/14.33 = { by lemma 9 }
% 108.54/14.33 fresh2(fresh2(fresh2(fresh2(true, true, implies(implies(not(Z), implies(Y, Y)), implies(Y, Y))), true, implies(Z, implies(Y, Y))), true, implies(implies(implies(Y, Y), implies(X, Y)), implies(Z, implies(X, Y)))), true, implies(implies(X, Y), implies(Z, implies(X, Y))))
% 108.54/14.33 = { by axiom 1 (condensed_detachment) }
% 108.54/14.33 fresh2(fresh2(fresh2(true, true, implies(Z, implies(Y, Y))), true, implies(implies(implies(Y, Y), implies(X, Y)), implies(Z, implies(X, Y)))), true, implies(implies(X, Y), implies(Z, implies(X, Y))))
% 108.54/14.33 = { by axiom 1 (condensed_detachment) }
% 108.54/14.33 fresh2(fresh2(true, true, implies(implies(implies(Y, Y), implies(X, Y)), implies(Z, implies(X, Y)))), true, implies(implies(X, Y), implies(Z, implies(X, Y))))
% 108.54/14.33 = { by axiom 1 (condensed_detachment) }
% 108.54/14.33 fresh2(true, true, implies(implies(X, Y), implies(Z, implies(X, Y))))
% 108.54/14.33 = { by axiom 1 (condensed_detachment) }
% 108.54/14.33 true
% 108.54/14.33
% 108.54/14.33 Lemma 15: fresh2(is_a_theorem(implies(implies(X, implies(Y, Z)), W)), true, implies(implies(Y, Z), W)) = is_a_theorem(implies(implies(Y, Z), W)).
% 108.54/14.33 Proof:
% 108.54/14.33 fresh2(is_a_theorem(implies(implies(X, implies(Y, Z)), W)), true, implies(implies(Y, Z), W))
% 108.54/14.33 = { by axiom 5 (condensed_detachment) R->L }
% 108.54/14.33 fresh(is_a_theorem(implies(implies(implies(X, implies(Y, Z)), W), implies(implies(Y, Z), W))), true, implies(implies(X, implies(Y, Z)), W), implies(implies(Y, Z), W))
% 108.54/14.33 = { by lemma 7 R->L }
% 108.54/14.33 fresh(fresh2(is_a_theorem(implies(implies(Y, Z), implies(X, implies(Y, Z)))), true, implies(implies(implies(X, implies(Y, Z)), W), implies(implies(Y, Z), W))), true, implies(implies(X, implies(Y, Z)), W), implies(implies(Y, Z), W))
% 108.54/14.33 = { by lemma 14 }
% 108.54/14.33 fresh(fresh2(true, true, implies(implies(implies(X, implies(Y, Z)), W), implies(implies(Y, Z), W))), true, implies(implies(X, implies(Y, Z)), W), implies(implies(Y, Z), W))
% 108.54/14.33 = { by axiom 1 (condensed_detachment) }
% 108.54/14.33 fresh(true, true, implies(implies(X, implies(Y, Z)), W), implies(implies(Y, Z), W))
% 108.54/14.33 = { by axiom 2 (condensed_detachment) }
% 108.54/14.33 is_a_theorem(implies(implies(Y, Z), W))
% 108.54/14.33
% 108.54/14.33 Lemma 16: fresh2(is_a_theorem(implies(implies(implies(X, Y), implies(Z, Y)), W)), true, implies(X, W)) = is_a_theorem(implies(X, W)).
% 108.54/14.33 Proof:
% 108.54/14.33 fresh2(is_a_theorem(implies(implies(implies(X, Y), implies(Z, Y)), W)), true, implies(X, W))
% 108.54/14.33 = { by axiom 5 (condensed_detachment) R->L }
% 108.54/14.33 fresh(is_a_theorem(implies(implies(implies(implies(X, Y), implies(Z, Y)), W), implies(X, W))), true, implies(implies(implies(X, Y), implies(Z, Y)), W), implies(X, W))
% 108.54/14.33 = { by lemma 7 R->L }
% 108.54/14.33 fresh(fresh2(is_a_theorem(implies(X, implies(implies(X, Y), implies(Z, Y)))), true, implies(implies(implies(implies(X, Y), implies(Z, Y)), W), implies(X, W))), true, implies(implies(implies(X, Y), implies(Z, Y)), W), implies(X, W))
% 108.54/14.33 = { by lemma 8 R->L }
% 108.54/14.33 fresh(fresh2(fresh2(is_a_theorem(implies(implies(not(X), X), implies(implies(X, Y), implies(Z, Y)))), true, implies(X, implies(implies(X, Y), implies(Z, Y)))), true, implies(implies(implies(implies(X, Y), implies(Z, Y)), W), implies(X, W))), true, implies(implies(implies(X, Y), implies(Z, Y)), W), implies(X, W))
% 108.54/14.33 = { by lemma 15 R->L }
% 108.54/14.33 fresh(fresh2(fresh2(fresh2(is_a_theorem(implies(implies(Z, implies(not(X), X)), implies(implies(X, Y), implies(Z, Y)))), true, implies(implies(not(X), X), implies(implies(X, Y), implies(Z, Y)))), true, implies(X, implies(implies(X, Y), implies(Z, Y)))), true, implies(implies(implies(implies(X, Y), implies(Z, Y)), W), implies(X, W))), true, implies(implies(implies(X, Y), implies(Z, Y)), W), implies(X, W))
% 108.54/14.33 = { by lemma 12 }
% 108.54/14.33 fresh(fresh2(fresh2(fresh2(true, true, implies(implies(not(X), X), implies(implies(X, Y), implies(Z, Y)))), true, implies(X, implies(implies(X, Y), implies(Z, Y)))), true, implies(implies(implies(implies(X, Y), implies(Z, Y)), W), implies(X, W))), true, implies(implies(implies(X, Y), implies(Z, Y)), W), implies(X, W))
% 108.54/14.33 = { by axiom 1 (condensed_detachment) }
% 108.54/14.33 fresh(fresh2(fresh2(true, true, implies(X, implies(implies(X, Y), implies(Z, Y)))), true, implies(implies(implies(implies(X, Y), implies(Z, Y)), W), implies(X, W))), true, implies(implies(implies(X, Y), implies(Z, Y)), W), implies(X, W))
% 108.54/14.33 = { by axiom 1 (condensed_detachment) }
% 108.54/14.33 fresh(fresh2(true, true, implies(implies(implies(implies(X, Y), implies(Z, Y)), W), implies(X, W))), true, implies(implies(implies(X, Y), implies(Z, Y)), W), implies(X, W))
% 108.54/14.33 = { by axiom 1 (condensed_detachment) }
% 108.54/14.33 fresh(true, true, implies(implies(implies(X, Y), implies(Z, Y)), W), implies(X, W))
% 108.54/14.33 = { by axiom 2 (condensed_detachment) }
% 108.54/14.33 is_a_theorem(implies(X, W))
% 108.54/14.33
% 108.54/14.33 Lemma 17: fresh2(is_a_theorem(implies(not(X), X)), true, X) = is_a_theorem(X).
% 108.54/14.33 Proof:
% 108.54/14.33 fresh2(is_a_theorem(implies(not(X), X)), true, X)
% 108.54/14.33 = { by axiom 5 (condensed_detachment) R->L }
% 108.54/14.33 fresh(is_a_theorem(implies(implies(not(X), X), X)), true, implies(not(X), X), X)
% 108.54/14.33 = { by axiom 4 (cn_2) }
% 108.54/14.33 fresh(true, true, implies(not(X), X), X)
% 108.54/14.33 = { by axiom 2 (condensed_detachment) }
% 108.54/14.33 is_a_theorem(X)
% 108.54/14.33
% 108.54/14.33 Lemma 18: is_a_theorem(implies(implies(X, implies(not(Y), Y)), implies(Z, implies(X, Y)))) = true.
% 108.54/14.33 Proof:
% 108.54/14.33 is_a_theorem(implies(implies(X, implies(not(Y), Y)), implies(Z, implies(X, Y))))
% 108.54/14.33 = { by lemma 10 R->L }
% 108.54/14.33 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))))
% 108.54/14.33 = { by lemma 7 R->L }
% 108.54/14.33 fresh2(fresh2(is_a_theorem(implies(Z, implies(implies(not(Y), Y), Y))), true, 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))))
% 108.54/14.33 = { by lemma 8 R->L }
% 108.54/14.33 fresh2(fresh2(fresh2(is_a_theorem(implies(implies(not(Z), implies(Y, Y)), implies(implies(not(Y), Y), Y))), true, implies(Z, implies(implies(not(Y), Y), Y))), true, 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))))
% 108.54/14.33 = { by lemma 13 R->L }
% 108.54/14.33 fresh2(fresh2(fresh2(fresh2(is_a_theorem(implies(implies(Y, Y), implies(implies(not(Y), Y), Y))), true, implies(implies(not(Z), implies(Y, Y)), implies(implies(not(Y), Y), Y))), true, implies(Z, implies(implies(not(Y), Y), Y))), true, 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))))
% 108.54/14.33 = { by lemma 11 }
% 108.54/14.33 fresh2(fresh2(fresh2(fresh2(true, true, implies(implies(not(Z), implies(Y, Y)), implies(implies(not(Y), Y), Y))), true, implies(Z, implies(implies(not(Y), Y), Y))), true, 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))))
% 108.54/14.33 = { by axiom 1 (condensed_detachment) }
% 108.54/14.33 fresh2(fresh2(fresh2(true, true, implies(Z, implies(implies(not(Y), Y), Y))), true, 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))))
% 108.54/14.33 = { by axiom 1 (condensed_detachment) }
% 108.54/14.33 fresh2(fresh2(true, true, 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))))
% 108.54/14.33 = { by axiom 1 (condensed_detachment) }
% 108.54/14.33 fresh2(true, true, implies(implies(X, implies(not(Y), Y)), implies(Z, implies(X, Y))))
% 108.54/14.33 = { by axiom 1 (condensed_detachment) }
% 108.54/14.33 true
% 108.54/14.33
% 108.54/14.33 Lemma 19: fresh2(is_a_theorem(X), true, implies(Y, implies(implies(X, Z), Z))) = is_a_theorem(implies(Y, implies(implies(X, Z), Z))).
% 108.54/14.33 Proof:
% 108.54/14.33 fresh2(is_a_theorem(X), true, implies(Y, implies(implies(X, Z), Z)))
% 108.54/14.33 = { by axiom 5 (condensed_detachment) R->L }
% 108.54/14.33 fresh(is_a_theorem(implies(X, implies(Y, implies(implies(X, Z), Z)))), true, X, implies(Y, implies(implies(X, Z), Z)))
% 108.54/14.33 = { by lemma 16 R->L }
% 108.54/14.33 fresh(fresh2(is_a_theorem(implies(implies(implies(X, Z), implies(not(Z), Z)), implies(Y, implies(implies(X, Z), Z)))), true, implies(X, implies(Y, implies(implies(X, Z), Z)))), true, X, implies(Y, implies(implies(X, Z), Z)))
% 108.54/14.34 = { by lemma 18 }
% 108.54/14.34 fresh(fresh2(true, true, implies(X, implies(Y, implies(implies(X, Z), Z)))), true, X, implies(Y, implies(implies(X, Z), Z)))
% 108.54/14.34 = { by axiom 1 (condensed_detachment) }
% 108.54/14.34 fresh(true, true, X, implies(Y, implies(implies(X, Z), Z)))
% 108.54/14.34 = { by axiom 2 (condensed_detachment) }
% 109.19/14.34 is_a_theorem(implies(Y, implies(implies(X, Z), Z)))
% 109.19/14.34
% 109.19/14.34 Lemma 20: is_a_theorem(implies(X, implies(implies(X, Y), Y))) = true.
% 109.19/14.34 Proof:
% 109.19/14.34 is_a_theorem(implies(X, implies(implies(X, Y), Y)))
% 109.19/14.34 = { by lemma 16 R->L }
% 109.19/14.34 fresh2(is_a_theorem(implies(implies(implies(X, Y), implies(implies(Z, Z), Y)), implies(implies(X, Y), Y))), true, implies(X, implies(implies(X, Y), Y)))
% 109.19/14.34 = { by lemma 10 R->L }
% 109.19/14.34 fresh2(fresh2(is_a_theorem(implies(implies(implies(implies(implies(Z, Z), Y), Y), implies(implies(X, Y), Y)), implies(implies(X, Y), Y))), true, implies(implies(implies(X, Y), implies(implies(Z, Z), Y)), implies(implies(X, Y), Y))), true, implies(X, implies(implies(X, Y), Y)))
% 109.19/14.34 = { by lemma 17 R->L }
% 109.19/14.34 fresh2(fresh2(fresh2(is_a_theorem(implies(not(implies(implies(implies(implies(implies(Z, Z), Y), Y), implies(implies(X, Y), Y)), implies(implies(X, Y), Y))), implies(implies(implies(implies(implies(Z, Z), Y), Y), implies(implies(X, Y), Y)), implies(implies(X, Y), Y)))), true, implies(implies(implies(implies(implies(Z, Z), Y), Y), implies(implies(X, Y), Y)), implies(implies(X, Y), Y))), true, implies(implies(implies(X, Y), implies(implies(Z, Z), Y)), implies(implies(X, Y), Y))), true, implies(X, implies(implies(X, Y), Y)))
% 109.19/14.34 = { by lemma 19 R->L }
% 109.19/14.34 fresh2(fresh2(fresh2(fresh2(is_a_theorem(implies(implies(implies(Z, Z), Y), Y)), true, implies(not(implies(implies(implies(implies(implies(Z, Z), Y), Y), implies(implies(X, Y), Y)), implies(implies(X, Y), Y))), implies(implies(implies(implies(implies(Z, Z), Y), Y), implies(implies(X, Y), Y)), implies(implies(X, Y), Y)))), true, implies(implies(implies(implies(implies(Z, Z), Y), Y), implies(implies(X, Y), Y)), implies(implies(X, Y), Y))), true, implies(implies(implies(X, Y), implies(implies(Z, Z), Y)), implies(implies(X, Y), Y))), true, implies(X, implies(implies(X, Y), Y)))
% 109.19/14.34 = { by lemma 17 R->L }
% 109.19/14.34 fresh2(fresh2(fresh2(fresh2(fresh2(is_a_theorem(implies(not(implies(implies(implies(Z, Z), Y), Y)), implies(implies(implies(Z, Z), Y), Y))), true, implies(implies(implies(Z, Z), Y), Y)), true, implies(not(implies(implies(implies(implies(implies(Z, Z), Y), Y), implies(implies(X, Y), Y)), implies(implies(X, Y), Y))), implies(implies(implies(implies(implies(Z, Z), Y), Y), implies(implies(X, Y), Y)), implies(implies(X, Y), Y)))), true, implies(implies(implies(implies(implies(Z, Z), Y), Y), implies(implies(X, Y), Y)), implies(implies(X, Y), Y))), true, implies(implies(implies(X, Y), implies(implies(Z, Z), Y)), implies(implies(X, Y), Y))), true, implies(X, implies(implies(X, Y), Y)))
% 109.19/14.34 = { by lemma 17 R->L }
% 109.19/14.34 fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(is_a_theorem(implies(not(implies(not(implies(implies(implies(Z, Z), Y), Y)), implies(implies(implies(Z, Z), Y), Y))), implies(not(implies(implies(implies(Z, Z), Y), Y)), implies(implies(implies(Z, Z), Y), Y)))), true, implies(not(implies(implies(implies(Z, Z), Y), Y)), implies(implies(implies(Z, Z), Y), Y))), true, implies(implies(implies(Z, Z), Y), Y)), true, implies(not(implies(implies(implies(implies(implies(Z, Z), Y), Y), implies(implies(X, Y), Y)), implies(implies(X, Y), Y))), implies(implies(implies(implies(implies(Z, Z), Y), Y), implies(implies(X, Y), Y)), implies(implies(X, Y), Y)))), true, implies(implies(implies(implies(implies(Z, Z), Y), Y), implies(implies(X, Y), Y)), implies(implies(X, Y), Y))), true, implies(implies(implies(X, Y), implies(implies(Z, Z), Y)), implies(implies(X, Y), Y))), true, implies(X, implies(implies(X, Y), Y)))
% 109.19/14.34 = { by axiom 2 (condensed_detachment) R->L }
% 109.19/14.34 fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh(true, true, implies(implies(not(Y), implies(Z, Z)), implies(not(implies(implies(implies(Z, Z), Y), Y)), implies(implies(implies(Z, Z), Y), Y))), implies(not(implies(not(implies(implies(implies(Z, Z), Y), Y)), implies(implies(implies(Z, Z), Y), Y))), implies(not(implies(implies(implies(Z, Z), Y), Y)), implies(implies(implies(Z, Z), Y), Y)))), true, implies(not(implies(implies(implies(Z, Z), Y), Y)), implies(implies(implies(Z, Z), Y), Y))), true, implies(implies(implies(Z, Z), Y), Y)), true, implies(not(implies(implies(implies(implies(implies(Z, Z), Y), Y), implies(implies(X, Y), Y)), implies(implies(X, Y), Y))), implies(implies(implies(implies(implies(Z, Z), Y), Y), implies(implies(X, Y), Y)), implies(implies(X, Y), Y)))), true, implies(implies(implies(implies(implies(Z, Z), Y), Y), implies(implies(X, Y), Y)), implies(implies(X, Y), Y))), true, implies(implies(implies(X, Y), implies(implies(Z, Z), Y)), implies(implies(X, Y), Y))), true, implies(X, implies(implies(X, Y), Y)))
% 109.19/14.34 = { by axiom 1 (condensed_detachment) R->L }
% 109.19/14.34 fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh(fresh2(true, true, implies(implies(implies(not(Y), implies(Z, Z)), implies(not(implies(implies(implies(Z, Z), Y), Y)), implies(implies(implies(Z, Z), Y), Y))), implies(not(implies(not(implies(implies(implies(Z, Z), Y), Y)), implies(implies(implies(Z, Z), Y), Y))), implies(not(implies(implies(implies(Z, Z), Y), Y)), implies(implies(implies(Z, Z), Y), Y))))), true, implies(implies(not(Y), implies(Z, Z)), implies(not(implies(implies(implies(Z, Z), Y), Y)), implies(implies(implies(Z, Z), Y), Y))), implies(not(implies(not(implies(implies(implies(Z, Z), Y), Y)), implies(implies(implies(Z, Z), Y), Y))), implies(not(implies(implies(implies(Z, Z), Y), Y)), implies(implies(implies(Z, Z), Y), Y)))), true, implies(not(implies(implies(implies(Z, Z), Y), Y)), implies(implies(implies(Z, Z), Y), Y))), true, implies(implies(implies(Z, Z), Y), Y)), true, implies(not(implies(implies(implies(implies(implies(Z, Z), Y), Y), implies(implies(X, Y), Y)), implies(implies(X, Y), Y))), implies(implies(implies(implies(implies(Z, Z), Y), Y), implies(implies(X, Y), Y)), implies(implies(X, Y), Y)))), true, implies(implies(implies(implies(implies(Z, Z), Y), Y), implies(implies(X, Y), Y)), implies(implies(X, Y), Y))), true, implies(implies(implies(X, Y), implies(implies(Z, Z), Y)), implies(implies(X, Y), Y))), true, implies(X, implies(implies(X, Y), Y)))
% 109.19/14.34 = { by axiom 1 (condensed_detachment) R->L }
% 109.19/14.34 fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh(fresh2(fresh2(true, true, implies(not(implies(not(implies(implies(implies(Z, Z), Y), Y)), implies(implies(implies(Z, Z), Y), Y))), implies(not(Y), implies(Z, Z)))), true, implies(implies(implies(not(Y), implies(Z, Z)), implies(not(implies(implies(implies(Z, Z), Y), Y)), implies(implies(implies(Z, Z), Y), Y))), implies(not(implies(not(implies(implies(implies(Z, Z), Y), Y)), implies(implies(implies(Z, Z), Y), Y))), implies(not(implies(implies(implies(Z, Z), Y), Y)), implies(implies(implies(Z, Z), Y), Y))))), true, implies(implies(not(Y), implies(Z, Z)), implies(not(implies(implies(implies(Z, Z), Y), Y)), implies(implies(implies(Z, Z), Y), Y))), implies(not(implies(not(implies(implies(implies(Z, Z), Y), Y)), implies(implies(implies(Z, Z), Y), Y))), implies(not(implies(implies(implies(Z, Z), Y), Y)), implies(implies(implies(Z, Z), Y), Y)))), true, implies(not(implies(implies(implies(Z, Z), Y), Y)), implies(implies(implies(Z, Z), Y), Y))), true, implies(implies(implies(Z, Z), Y), Y)), true, implies(not(implies(implies(implies(implies(implies(Z, Z), Y), Y), implies(implies(X, Y), Y)), implies(implies(X, Y), Y))), implies(implies(implies(implies(implies(Z, Z), Y), Y), implies(implies(X, Y), Y)), implies(implies(X, Y), Y)))), true, implies(implies(implies(implies(implies(Z, Z), Y), Y), implies(implies(X, Y), Y)), implies(implies(X, Y), Y))), true, implies(implies(implies(X, Y), implies(implies(Z, Z), Y)), implies(implies(X, Y), Y))), true, implies(X, implies(implies(X, Y), Y)))
% 109.19/14.34 = { by axiom 1 (condensed_detachment) R->L }
% 109.19/14.35 fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh(fresh2(fresh2(fresh2(true, true, implies(implies(not(not(implies(not(implies(implies(implies(Z, Z), Y), Y)), implies(implies(implies(Z, Z), Y), Y)))), implies(Z, Z)), implies(not(Y), implies(Z, Z)))), true, implies(not(implies(not(implies(implies(implies(Z, Z), Y), Y)), implies(implies(implies(Z, Z), Y), Y))), implies(not(Y), implies(Z, Z)))), true, implies(implies(implies(not(Y), implies(Z, Z)), implies(not(implies(implies(implies(Z, Z), Y), Y)), implies(implies(implies(Z, Z), Y), Y))), implies(not(implies(not(implies(implies(implies(Z, Z), Y), Y)), implies(implies(implies(Z, Z), Y), Y))), implies(not(implies(implies(implies(Z, Z), Y), Y)), implies(implies(implies(Z, Z), Y), Y))))), true, implies(implies(not(Y), implies(Z, Z)), implies(not(implies(implies(implies(Z, Z), Y), Y)), implies(implies(implies(Z, Z), Y), Y))), implies(not(implies(not(implies(implies(implies(Z, Z), Y), Y)), implies(implies(implies(Z, Z), Y), Y))), implies(not(implies(implies(implies(Z, Z), Y), Y)), implies(implies(implies(Z, Z), Y), Y)))), true, implies(not(implies(implies(implies(Z, Z), Y), Y)), implies(implies(implies(Z, Z), Y), Y))), true, implies(implies(implies(Z, Z), Y), Y)), true, implies(not(implies(implies(implies(implies(implies(Z, Z), Y), Y), implies(implies(X, Y), Y)), implies(implies(X, Y), Y))), implies(implies(implies(implies(implies(Z, Z), Y), Y), implies(implies(X, Y), Y)), implies(implies(X, Y), Y)))), true, implies(implies(implies(implies(implies(Z, Z), Y), Y), implies(implies(X, Y), Y)), implies(implies(X, Y), Y))), true, implies(implies(implies(X, Y), implies(implies(Z, Z), Y)), implies(implies(X, Y), Y))), true, implies(X, implies(implies(X, Y), Y)))
% 109.19/14.35 = { by lemma 14 R->L }
% 109.19/14.35 fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh(fresh2(fresh2(fresh2(is_a_theorem(implies(implies(Z, Z), implies(not(Y), implies(Z, Z)))), true, implies(implies(not(not(implies(not(implies(implies(implies(Z, Z), Y), Y)), implies(implies(implies(Z, Z), Y), Y)))), implies(Z, Z)), implies(not(Y), implies(Z, Z)))), true, implies(not(implies(not(implies(implies(implies(Z, Z), Y), Y)), implies(implies(implies(Z, Z), Y), Y))), implies(not(Y), implies(Z, Z)))), true, implies(implies(implies(not(Y), implies(Z, Z)), implies(not(implies(implies(implies(Z, Z), Y), Y)), implies(implies(implies(Z, Z), Y), Y))), implies(not(implies(not(implies(implies(implies(Z, Z), Y), Y)), implies(implies(implies(Z, Z), Y), Y))), implies(not(implies(implies(implies(Z, Z), Y), Y)), implies(implies(implies(Z, Z), Y), Y))))), true, implies(implies(not(Y), implies(Z, Z)), implies(not(implies(implies(implies(Z, Z), Y), Y)), implies(implies(implies(Z, Z), Y), Y))), implies(not(implies(not(implies(implies(implies(Z, Z), Y), Y)), implies(implies(implies(Z, Z), Y), Y))), implies(not(implies(implies(implies(Z, Z), Y), Y)), implies(implies(implies(Z, Z), Y), Y)))), true, implies(not(implies(implies(implies(Z, Z), Y), Y)), implies(implies(implies(Z, Z), Y), Y))), true, implies(implies(implies(Z, Z), Y), Y)), true, implies(not(implies(implies(implies(implies(implies(Z, Z), Y), Y), implies(implies(X, Y), Y)), implies(implies(X, Y), Y))), implies(implies(implies(implies(implies(Z, Z), Y), Y), implies(implies(X, Y), Y)), implies(implies(X, Y), Y)))), true, implies(implies(implies(implies(implies(Z, Z), Y), Y), implies(implies(X, Y), Y)), implies(implies(X, Y), Y))), true, implies(implies(implies(X, Y), implies(implies(Z, Z), Y)), implies(implies(X, Y), Y))), true, implies(X, implies(implies(X, Y), Y)))
% 109.19/14.35 = { by lemma 13 }
% 109.19/14.35 fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh(fresh2(fresh2(is_a_theorem(implies(implies(not(not(implies(not(implies(implies(implies(Z, Z), Y), Y)), implies(implies(implies(Z, Z), Y), Y)))), implies(Z, Z)), implies(not(Y), implies(Z, Z)))), true, implies(not(implies(not(implies(implies(implies(Z, Z), Y), Y)), implies(implies(implies(Z, Z), Y), Y))), implies(not(Y), implies(Z, Z)))), true, implies(implies(implies(not(Y), implies(Z, Z)), implies(not(implies(implies(implies(Z, Z), Y), Y)), implies(implies(implies(Z, Z), Y), Y))), implies(not(implies(not(implies(implies(implies(Z, Z), Y), Y)), implies(implies(implies(Z, Z), Y), Y))), implies(not(implies(implies(implies(Z, Z), Y), Y)), implies(implies(implies(Z, Z), Y), Y))))), true, implies(implies(not(Y), implies(Z, Z)), implies(not(implies(implies(implies(Z, Z), Y), Y)), implies(implies(implies(Z, Z), Y), Y))), implies(not(implies(not(implies(implies(implies(Z, Z), Y), Y)), implies(implies(implies(Z, Z), Y), Y))), implies(not(implies(implies(implies(Z, Z), Y), Y)), implies(implies(implies(Z, Z), Y), Y)))), true, implies(not(implies(implies(implies(Z, Z), Y), Y)), implies(implies(implies(Z, Z), Y), Y))), true, implies(implies(implies(Z, Z), Y), Y)), true, implies(not(implies(implies(implies(implies(implies(Z, Z), Y), Y), implies(implies(X, Y), Y)), implies(implies(X, Y), Y))), implies(implies(implies(implies(implies(Z, Z), Y), Y), implies(implies(X, Y), Y)), implies(implies(X, Y), Y)))), true, implies(implies(implies(implies(implies(Z, Z), Y), Y), implies(implies(X, Y), Y)), implies(implies(X, Y), Y))), true, implies(implies(implies(X, Y), implies(implies(Z, Z), Y)), implies(implies(X, Y), Y))), true, implies(X, implies(implies(X, Y), Y)))
% 109.19/14.35 = { by lemma 8 }
% 109.19/14.35 fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh(fresh2(is_a_theorem(implies(not(implies(not(implies(implies(implies(Z, Z), Y), Y)), implies(implies(implies(Z, Z), Y), Y))), implies(not(Y), implies(Z, Z)))), true, implies(implies(implies(not(Y), implies(Z, Z)), implies(not(implies(implies(implies(Z, Z), Y), Y)), implies(implies(implies(Z, Z), Y), Y))), implies(not(implies(not(implies(implies(implies(Z, Z), Y), Y)), implies(implies(implies(Z, Z), Y), Y))), implies(not(implies(implies(implies(Z, Z), Y), Y)), implies(implies(implies(Z, Z), Y), Y))))), true, implies(implies(not(Y), implies(Z, Z)), implies(not(implies(implies(implies(Z, Z), Y), Y)), implies(implies(implies(Z, Z), Y), Y))), implies(not(implies(not(implies(implies(implies(Z, Z), Y), Y)), implies(implies(implies(Z, Z), Y), Y))), implies(not(implies(implies(implies(Z, Z), Y), Y)), implies(implies(implies(Z, Z), Y), Y)))), true, implies(not(implies(implies(implies(Z, Z), Y), Y)), implies(implies(implies(Z, Z), Y), Y))), true, implies(implies(implies(Z, Z), Y), Y)), true, implies(not(implies(implies(implies(implies(implies(Z, Z), Y), Y), implies(implies(X, Y), Y)), implies(implies(X, Y), Y))), implies(implies(implies(implies(implies(Z, Z), Y), Y), implies(implies(X, Y), Y)), implies(implies(X, Y), Y)))), true, implies(implies(implies(implies(implies(Z, Z), Y), Y), implies(implies(X, Y), Y)), implies(implies(X, Y), Y))), true, implies(implies(implies(X, Y), implies(implies(Z, Z), Y)), implies(implies(X, Y), Y))), true, implies(X, implies(implies(X, Y), Y)))
% 109.19/14.35 = { by lemma 7 }
% 109.19/14.35 fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh(is_a_theorem(implies(implies(implies(not(Y), implies(Z, Z)), implies(not(implies(implies(implies(Z, Z), Y), Y)), implies(implies(implies(Z, Z), Y), Y))), implies(not(implies(not(implies(implies(implies(Z, Z), Y), Y)), implies(implies(implies(Z, Z), Y), Y))), implies(not(implies(implies(implies(Z, Z), Y), Y)), implies(implies(implies(Z, Z), Y), Y))))), true, implies(implies(not(Y), implies(Z, Z)), implies(not(implies(implies(implies(Z, Z), Y), Y)), implies(implies(implies(Z, Z), Y), Y))), implies(not(implies(not(implies(implies(implies(Z, Z), Y), Y)), implies(implies(implies(Z, Z), Y), Y))), implies(not(implies(implies(implies(Z, Z), Y), Y)), implies(implies(implies(Z, Z), Y), Y)))), true, implies(not(implies(implies(implies(Z, Z), Y), Y)), implies(implies(implies(Z, Z), Y), Y))), true, implies(implies(implies(Z, Z), Y), Y)), true, implies(not(implies(implies(implies(implies(implies(Z, Z), Y), Y), implies(implies(X, Y), Y)), implies(implies(X, Y), Y))), implies(implies(implies(implies(implies(Z, Z), Y), Y), implies(implies(X, Y), Y)), implies(implies(X, Y), Y)))), true, implies(implies(implies(implies(implies(Z, Z), Y), Y), implies(implies(X, Y), Y)), implies(implies(X, Y), Y))), true, implies(implies(implies(X, Y), implies(implies(Z, Z), Y)), implies(implies(X, Y), Y))), true, implies(X, implies(implies(X, Y), Y)))
% 109.19/14.35 = { by axiom 5 (condensed_detachment) }
% 109.19/14.35 fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(is_a_theorem(implies(implies(not(Y), implies(Z, Z)), implies(not(implies(implies(implies(Z, Z), Y), Y)), implies(implies(implies(Z, Z), Y), Y)))), true, implies(not(implies(not(implies(implies(implies(Z, Z), Y), Y)), implies(implies(implies(Z, Z), Y), Y))), implies(not(implies(implies(implies(Z, Z), Y), Y)), implies(implies(implies(Z, Z), Y), Y)))), true, implies(not(implies(implies(implies(Z, Z), Y), Y)), implies(implies(implies(Z, Z), Y), Y))), true, implies(implies(implies(Z, Z), Y), Y)), true, implies(not(implies(implies(implies(implies(implies(Z, Z), Y), Y), implies(implies(X, Y), Y)), implies(implies(X, Y), Y))), implies(implies(implies(implies(implies(Z, Z), Y), Y), implies(implies(X, Y), Y)), implies(implies(X, Y), Y)))), true, implies(implies(implies(implies(implies(Z, Z), Y), Y), implies(implies(X, Y), Y)), implies(implies(X, Y), Y))), true, implies(implies(implies(X, Y), implies(implies(Z, Z), Y)), implies(implies(X, Y), Y))), true, implies(X, implies(implies(X, Y), Y)))
% 109.19/14.35 = { by lemma 10 R->L }
% 109.19/14.35 fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(is_a_theorem(implies(implies(implies(implies(Z, Z), Y), implies(not(Y), Y)), implies(not(implies(implies(implies(Z, Z), Y), Y)), implies(implies(implies(Z, Z), Y), Y)))), true, implies(implies(not(Y), implies(Z, Z)), implies(not(implies(implies(implies(Z, Z), Y), Y)), implies(implies(implies(Z, Z), Y), Y)))), true, implies(not(implies(not(implies(implies(implies(Z, Z), Y), Y)), implies(implies(implies(Z, Z), Y), Y))), implies(not(implies(implies(implies(Z, Z), Y), Y)), implies(implies(implies(Z, Z), Y), Y)))), true, implies(not(implies(implies(implies(Z, Z), Y), Y)), implies(implies(implies(Z, Z), Y), Y))), true, implies(implies(implies(Z, Z), Y), Y)), true, implies(not(implies(implies(implies(implies(implies(Z, Z), Y), Y), implies(implies(X, Y), Y)), implies(implies(X, Y), Y))), implies(implies(implies(implies(implies(Z, Z), Y), Y), implies(implies(X, Y), Y)), implies(implies(X, Y), Y)))), true, implies(implies(implies(implies(implies(Z, Z), Y), Y), implies(implies(X, Y), Y)), implies(implies(X, Y), Y))), true, implies(implies(implies(X, Y), implies(implies(Z, Z), Y)), implies(implies(X, Y), Y))), true, implies(X, implies(implies(X, Y), Y)))
% 109.19/14.35 = { by lemma 18 }
% 109.19/14.36 fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(true, true, implies(implies(not(Y), implies(Z, Z)), implies(not(implies(implies(implies(Z, Z), Y), Y)), implies(implies(implies(Z, Z), Y), Y)))), true, implies(not(implies(not(implies(implies(implies(Z, Z), Y), Y)), implies(implies(implies(Z, Z), Y), Y))), implies(not(implies(implies(implies(Z, Z), Y), Y)), implies(implies(implies(Z, Z), Y), Y)))), true, implies(not(implies(implies(implies(Z, Z), Y), Y)), implies(implies(implies(Z, Z), Y), Y))), true, implies(implies(implies(Z, Z), Y), Y)), true, implies(not(implies(implies(implies(implies(implies(Z, Z), Y), Y), implies(implies(X, Y), Y)), implies(implies(X, Y), Y))), implies(implies(implies(implies(implies(Z, Z), Y), Y), implies(implies(X, Y), Y)), implies(implies(X, Y), Y)))), true, implies(implies(implies(implies(implies(Z, Z), Y), Y), implies(implies(X, Y), Y)), implies(implies(X, Y), Y))), true, implies(implies(implies(X, Y), implies(implies(Z, Z), Y)), implies(implies(X, Y), Y))), true, implies(X, implies(implies(X, Y), Y)))
% 109.19/14.36 = { by axiom 1 (condensed_detachment) }
% 109.19/14.36 fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(true, true, implies(not(implies(not(implies(implies(implies(Z, Z), Y), Y)), implies(implies(implies(Z, Z), Y), Y))), implies(not(implies(implies(implies(Z, Z), Y), Y)), implies(implies(implies(Z, Z), Y), Y)))), true, implies(not(implies(implies(implies(Z, Z), Y), Y)), implies(implies(implies(Z, Z), Y), Y))), true, implies(implies(implies(Z, Z), Y), Y)), true, implies(not(implies(implies(implies(implies(implies(Z, Z), Y), Y), implies(implies(X, Y), Y)), implies(implies(X, Y), Y))), implies(implies(implies(implies(implies(Z, Z), Y), Y), implies(implies(X, Y), Y)), implies(implies(X, Y), Y)))), true, implies(implies(implies(implies(implies(Z, Z), Y), Y), implies(implies(X, Y), Y)), implies(implies(X, Y), Y))), true, implies(implies(implies(X, Y), implies(implies(Z, Z), Y)), implies(implies(X, Y), Y))), true, implies(X, implies(implies(X, Y), Y)))
% 109.19/14.36 = { by axiom 1 (condensed_detachment) }
% 109.19/14.36 fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(true, true, implies(not(implies(implies(implies(Z, Z), Y), Y)), implies(implies(implies(Z, Z), Y), Y))), true, implies(implies(implies(Z, Z), Y), Y)), true, implies(not(implies(implies(implies(implies(implies(Z, Z), Y), Y), implies(implies(X, Y), Y)), implies(implies(X, Y), Y))), implies(implies(implies(implies(implies(Z, Z), Y), Y), implies(implies(X, Y), Y)), implies(implies(X, Y), Y)))), true, implies(implies(implies(implies(implies(Z, Z), Y), Y), implies(implies(X, Y), Y)), implies(implies(X, Y), Y))), true, implies(implies(implies(X, Y), implies(implies(Z, Z), Y)), implies(implies(X, Y), Y))), true, implies(X, implies(implies(X, Y), Y)))
% 109.19/14.36 = { by axiom 1 (condensed_detachment) }
% 109.19/14.36 fresh2(fresh2(fresh2(fresh2(fresh2(true, true, implies(implies(implies(Z, Z), Y), Y)), true, implies(not(implies(implies(implies(implies(implies(Z, Z), Y), Y), implies(implies(X, Y), Y)), implies(implies(X, Y), Y))), implies(implies(implies(implies(implies(Z, Z), Y), Y), implies(implies(X, Y), Y)), implies(implies(X, Y), Y)))), true, implies(implies(implies(implies(implies(Z, Z), Y), Y), implies(implies(X, Y), Y)), implies(implies(X, Y), Y))), true, implies(implies(implies(X, Y), implies(implies(Z, Z), Y)), implies(implies(X, Y), Y))), true, implies(X, implies(implies(X, Y), Y)))
% 109.19/14.36 = { by axiom 1 (condensed_detachment) }
% 109.19/14.36 fresh2(fresh2(fresh2(fresh2(true, true, implies(not(implies(implies(implies(implies(implies(Z, Z), Y), Y), implies(implies(X, Y), Y)), implies(implies(X, Y), Y))), implies(implies(implies(implies(implies(Z, Z), Y), Y), implies(implies(X, Y), Y)), implies(implies(X, Y), Y)))), true, implies(implies(implies(implies(implies(Z, Z), Y), Y), implies(implies(X, Y), Y)), implies(implies(X, Y), Y))), true, implies(implies(implies(X, Y), implies(implies(Z, Z), Y)), implies(implies(X, Y), Y))), true, implies(X, implies(implies(X, Y), Y)))
% 109.19/14.36 = { by axiom 1 (condensed_detachment) }
% 109.19/14.36 fresh2(fresh2(fresh2(true, true, implies(implies(implies(implies(implies(Z, Z), Y), Y), implies(implies(X, Y), Y)), implies(implies(X, Y), Y))), true, implies(implies(implies(X, Y), implies(implies(Z, Z), Y)), implies(implies(X, Y), Y))), true, implies(X, implies(implies(X, Y), Y)))
% 109.19/14.36 = { by axiom 1 (condensed_detachment) }
% 109.19/14.36 fresh2(fresh2(true, true, implies(implies(implies(X, Y), implies(implies(Z, Z), Y)), implies(implies(X, Y), Y))), true, implies(X, implies(implies(X, Y), Y)))
% 109.19/14.36 = { by axiom 1 (condensed_detachment) }
% 109.19/14.36 fresh2(true, true, implies(X, implies(implies(X, Y), Y)))
% 109.19/14.36 = { by axiom 1 (condensed_detachment) }
% 109.19/14.36 true
% 109.19/14.36
% 109.19/14.36 Lemma 21: is_a_theorem(implies(implies(implies(implies(X, Y), Y), Z), implies(X, Z))) = true.
% 109.19/14.36 Proof:
% 109.19/14.36 is_a_theorem(implies(implies(implies(implies(X, Y), Y), Z), implies(X, Z)))
% 109.19/14.36 = { by lemma 7 R->L }
% 109.19/14.36 fresh2(is_a_theorem(implies(X, implies(implies(X, Y), Y))), true, implies(implies(implies(implies(X, Y), Y), Z), implies(X, Z)))
% 109.19/14.36 = { by lemma 20 }
% 109.19/14.36 fresh2(true, true, implies(implies(implies(implies(X, Y), Y), Z), implies(X, Z)))
% 109.19/14.36 = { by axiom 1 (condensed_detachment) }
% 109.19/14.36 true
% 109.19/14.36
% 109.19/14.36 Lemma 22: fresh2(is_a_theorem(implies(X, implies(Y, Z))), true, implies(Y, implies(X, Z))) = is_a_theorem(implies(Y, implies(X, Z))).
% 109.19/14.36 Proof:
% 109.19/14.36 fresh2(is_a_theorem(implies(X, implies(Y, Z))), true, implies(Y, implies(X, Z)))
% 109.19/14.36 = { by axiom 5 (condensed_detachment) R->L }
% 109.19/14.36 fresh(is_a_theorem(implies(implies(X, implies(Y, Z)), implies(Y, implies(X, Z)))), true, implies(X, implies(Y, Z)), implies(Y, implies(X, Z)))
% 109.19/14.36 = { by lemma 10 R->L }
% 109.19/14.36 fresh(fresh2(is_a_theorem(implies(implies(implies(implies(Y, Z), Z), implies(X, Z)), implies(Y, implies(X, Z)))), true, implies(implies(X, implies(Y, Z)), implies(Y, implies(X, Z)))), true, implies(X, implies(Y, Z)), implies(Y, implies(X, Z)))
% 109.19/14.36 = { by lemma 21 }
% 109.19/14.36 fresh(fresh2(true, true, implies(implies(X, implies(Y, Z)), implies(Y, implies(X, Z)))), true, implies(X, implies(Y, Z)), implies(Y, implies(X, Z)))
% 109.19/14.36 = { by axiom 1 (condensed_detachment) }
% 109.19/14.36 fresh(true, true, implies(X, implies(Y, Z)), implies(Y, implies(X, Z)))
% 109.19/14.36 = { by axiom 2 (condensed_detachment) }
% 109.19/14.36 is_a_theorem(implies(Y, implies(X, Z)))
% 109.19/14.36
% 109.19/14.36 Lemma 23: fresh2(is_a_theorem(implies(implies(implies(X, Y), Y), Z)), true, implies(X, Z)) = is_a_theorem(implies(X, Z)).
% 109.19/14.36 Proof:
% 109.19/14.36 fresh2(is_a_theorem(implies(implies(implies(X, Y), Y), Z)), true, implies(X, Z))
% 109.19/14.36 = { by axiom 5 (condensed_detachment) R->L }
% 109.19/14.36 fresh(is_a_theorem(implies(implies(implies(implies(X, Y), Y), Z), implies(X, Z))), true, implies(implies(implies(X, Y), Y), Z), implies(X, Z))
% 109.19/14.36 = { by lemma 21 }
% 109.19/14.36 fresh(true, true, implies(implies(implies(X, Y), Y), Z), implies(X, Z))
% 109.19/14.36 = { by axiom 2 (condensed_detachment) }
% 109.19/14.36 is_a_theorem(implies(X, Z))
% 109.19/14.36
% 109.19/14.36 Goal 1 (prove_cn_35): is_a_theorem(implies(implies(a, implies(b, c)), implies(implies(a, b), implies(a, c)))) = true.
% 109.19/14.36 Proof:
% 109.19/14.36 is_a_theorem(implies(implies(a, implies(b, c)), implies(implies(a, b), implies(a, c))))
% 109.19/14.36 = { by lemma 22 R->L }
% 109.19/14.36 fresh2(is_a_theorem(implies(implies(a, b), implies(implies(a, implies(b, c)), implies(a, c)))), true, implies(implies(a, implies(b, c)), implies(implies(a, b), implies(a, c))))
% 109.19/14.36 = { by lemma 10 R->L }
% 109.19/14.36 fresh2(fresh2(is_a_theorem(implies(implies(implies(b, c), implies(a, c)), implies(implies(a, implies(b, c)), implies(a, c)))), true, implies(implies(a, b), implies(implies(a, implies(b, c)), implies(a, c)))), true, implies(implies(a, implies(b, c)), implies(implies(a, b), implies(a, c))))
% 109.19/14.37 = { by lemma 22 R->L }
% 109.19/14.37 fresh2(fresh2(fresh2(is_a_theorem(implies(implies(a, implies(b, c)), implies(implies(implies(b, c), implies(a, c)), implies(a, c)))), true, implies(implies(implies(b, c), implies(a, c)), implies(implies(a, implies(b, c)), implies(a, c)))), true, implies(implies(a, b), implies(implies(a, implies(b, c)), implies(a, c)))), true, implies(implies(a, implies(b, c)), implies(implies(a, b), implies(a, c))))
% 109.19/14.37 = { by lemma 10 R->L }
% 109.19/14.37 fresh2(fresh2(fresh2(fresh2(is_a_theorem(implies(implies(implies(implies(b, c), implies(a, c)), implies(a, implies(a, c))), implies(implies(implies(b, c), implies(a, c)), implies(a, c)))), true, implies(implies(a, implies(b, c)), implies(implies(implies(b, c), implies(a, c)), implies(a, c)))), true, implies(implies(implies(b, c), implies(a, c)), implies(implies(a, implies(b, c)), implies(a, c)))), true, implies(implies(a, b), implies(implies(a, implies(b, c)), implies(a, c)))), true, implies(implies(a, implies(b, c)), implies(implies(a, b), implies(a, c))))
% 109.19/14.37 = { by lemma 10 R->L }
% 109.19/14.37 fresh2(fresh2(fresh2(fresh2(fresh2(is_a_theorem(implies(implies(implies(implies(a, implies(a, c)), implies(a, c)), implies(implies(implies(b, c), implies(a, c)), implies(a, c))), implies(implies(implies(b, c), implies(a, c)), implies(a, c)))), true, implies(implies(implies(implies(b, c), implies(a, c)), implies(a, implies(a, c))), implies(implies(implies(b, c), implies(a, c)), implies(a, c)))), true, implies(implies(a, implies(b, c)), implies(implies(implies(b, c), implies(a, c)), implies(a, c)))), true, implies(implies(implies(b, c), implies(a, c)), implies(implies(a, implies(b, c)), implies(a, c)))), true, implies(implies(a, b), implies(implies(a, implies(b, c)), implies(a, c)))), true, implies(implies(a, implies(b, c)), implies(implies(a, b), implies(a, c))))
% 109.19/14.37 = { by axiom 2 (condensed_detachment) R->L }
% 109.19/14.37 fresh2(fresh2(fresh2(fresh2(fresh2(fresh(true, true, implies(implies(a, implies(a, c)), implies(a, c)), implies(implies(implies(implies(a, implies(a, c)), implies(a, c)), implies(implies(implies(b, c), implies(a, c)), implies(a, c))), implies(implies(implies(b, c), implies(a, c)), implies(a, c)))), true, implies(implies(implies(implies(b, c), implies(a, c)), implies(a, implies(a, c))), implies(implies(implies(b, c), implies(a, c)), implies(a, c)))), true, implies(implies(a, implies(b, c)), implies(implies(implies(b, c), implies(a, c)), implies(a, c)))), true, implies(implies(implies(b, c), implies(a, c)), implies(implies(a, implies(b, c)), implies(a, c)))), true, implies(implies(a, b), implies(implies(a, implies(b, c)), implies(a, c)))), true, implies(implies(a, implies(b, c)), implies(implies(a, b), implies(a, c))))
% 109.19/14.37 = { by lemma 20 R->L }
% 109.19/14.37 fresh2(fresh2(fresh2(fresh2(fresh2(fresh(is_a_theorem(implies(implies(implies(a, implies(a, c)), implies(a, c)), implies(implies(implies(implies(a, implies(a, c)), implies(a, c)), implies(implies(implies(b, c), implies(a, c)), implies(a, c))), implies(implies(implies(b, c), implies(a, c)), implies(a, c))))), true, implies(implies(a, implies(a, c)), implies(a, c)), implies(implies(implies(implies(a, implies(a, c)), implies(a, c)), implies(implies(implies(b, c), implies(a, c)), implies(a, c))), implies(implies(implies(b, c), implies(a, c)), implies(a, c)))), true, implies(implies(implies(implies(b, c), implies(a, c)), implies(a, implies(a, c))), implies(implies(implies(b, c), implies(a, c)), implies(a, c)))), true, implies(implies(a, implies(b, c)), implies(implies(implies(b, c), implies(a, c)), implies(a, c)))), true, implies(implies(implies(b, c), implies(a, c)), implies(implies(a, implies(b, c)), implies(a, c)))), true, implies(implies(a, b), implies(implies(a, implies(b, c)), implies(a, c)))), true, implies(implies(a, implies(b, c)), implies(implies(a, b), implies(a, c))))
% 109.19/14.37 = { by axiom 5 (condensed_detachment) }
% 109.19/14.37 fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(is_a_theorem(implies(implies(a, implies(a, c)), implies(a, c))), true, implies(implies(implies(implies(a, implies(a, c)), implies(a, c)), implies(implies(implies(b, c), implies(a, c)), implies(a, c))), implies(implies(implies(b, c), implies(a, c)), implies(a, c)))), true, implies(implies(implies(implies(b, c), implies(a, c)), implies(a, implies(a, c))), implies(implies(implies(b, c), implies(a, c)), implies(a, c)))), true, implies(implies(a, implies(b, c)), implies(implies(implies(b, c), implies(a, c)), implies(a, c)))), true, implies(implies(implies(b, c), implies(a, c)), implies(implies(a, implies(b, c)), implies(a, c)))), true, implies(implies(a, b), implies(implies(a, implies(b, c)), implies(a, c)))), true, implies(implies(a, implies(b, c)), implies(implies(a, b), implies(a, c))))
% 109.19/14.37 = { by lemma 17 R->L }
% 109.19/14.37 fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(is_a_theorem(implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, implies(a, c)), implies(a, c)))), true, implies(implies(a, implies(a, c)), implies(a, c))), true, implies(implies(implies(implies(a, implies(a, c)), implies(a, c)), implies(implies(implies(b, c), implies(a, c)), implies(a, c))), implies(implies(implies(b, c), implies(a, c)), implies(a, c)))), true, implies(implies(implies(implies(b, c), implies(a, c)), implies(a, implies(a, c))), implies(implies(implies(b, c), implies(a, c)), implies(a, c)))), true, implies(implies(a, implies(b, c)), implies(implies(implies(b, c), implies(a, c)), implies(a, c)))), true, implies(implies(implies(b, c), implies(a, c)), implies(implies(a, implies(b, c)), implies(a, c)))), true, implies(implies(a, b), implies(implies(a, implies(b, c)), implies(a, c)))), true, implies(implies(a, implies(b, c)), implies(implies(a, b), implies(a, c))))
% 109.19/14.37 = { by axiom 2 (condensed_detachment) R->L }
% 109.19/14.37 fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh(true, true, implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(a, c)), implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, implies(a, c)), implies(a, c)))), true, implies(implies(a, implies(a, c)), implies(a, c))), true, implies(implies(implies(implies(a, implies(a, c)), implies(a, c)), implies(implies(implies(b, c), implies(a, c)), implies(a, c))), implies(implies(implies(b, c), implies(a, c)), implies(a, c)))), true, implies(implies(implies(implies(b, c), implies(a, c)), implies(a, implies(a, c))), implies(implies(implies(b, c), implies(a, c)), implies(a, c)))), true, implies(implies(a, implies(b, c)), implies(implies(implies(b, c), implies(a, c)), implies(a, c)))), true, implies(implies(implies(b, c), implies(a, c)), implies(implies(a, implies(b, c)), implies(a, c)))), true, implies(implies(a, b), implies(implies(a, implies(b, c)), implies(a, c)))), true, implies(implies(a, implies(b, c)), implies(implies(a, b), implies(a, c))))
% 109.19/14.37 = { by axiom 1 (condensed_detachment) R->L }
% 109.19/14.37 fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh(fresh2(true, true, implies(implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(a, c)), implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, implies(a, c)), implies(a, c))))), true, implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(a, c)), implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, implies(a, c)), implies(a, c)))), true, implies(implies(a, implies(a, c)), implies(a, c))), true, implies(implies(implies(implies(a, implies(a, c)), implies(a, c)), implies(implies(implies(b, c), implies(a, c)), implies(a, c))), implies(implies(implies(b, c), implies(a, c)), implies(a, c)))), true, implies(implies(implies(implies(b, c), implies(a, c)), implies(a, implies(a, c))), implies(implies(implies(b, c), implies(a, c)), implies(a, c)))), true, implies(implies(a, implies(b, c)), implies(implies(implies(b, c), implies(a, c)), implies(a, c)))), true, implies(implies(implies(b, c), implies(a, c)), implies(implies(a, implies(b, c)), implies(a, c)))), true, implies(implies(a, b), implies(implies(a, implies(b, c)), implies(a, c)))), true, implies(implies(a, implies(b, c)), implies(implies(a, b), implies(a, c))))
% 109.19/14.37 = { by axiom 1 (condensed_detachment) R->L }
% 109.19/14.37 fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh(fresh2(fresh2(true, true, implies(implies(implies(implies(a, c), implies(implies(a, implies(a, c)), implies(a, c))), implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, implies(a, c)), implies(a, c)))), implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, implies(a, c)), implies(a, c))))), true, implies(implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(a, c)), implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, implies(a, c)), implies(a, c))))), true, implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(a, c)), implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, implies(a, c)), implies(a, c)))), true, implies(implies(a, implies(a, c)), implies(a, c))), true, implies(implies(implies(implies(a, implies(a, c)), implies(a, c)), implies(implies(implies(b, c), implies(a, c)), implies(a, c))), implies(implies(implies(b, c), implies(a, c)), implies(a, c)))), true, implies(implies(implies(implies(b, c), implies(a, c)), implies(a, implies(a, c))), implies(implies(implies(b, c), implies(a, c)), implies(a, c)))), true, implies(implies(a, implies(b, c)), implies(implies(implies(b, c), implies(a, c)), implies(a, c)))), true, implies(implies(implies(b, c), implies(a, c)), implies(implies(a, implies(b, c)), implies(a, c)))), true, implies(implies(a, b), implies(implies(a, implies(b, c)), implies(a, c)))), true, implies(implies(a, implies(b, c)), implies(implies(a, b), implies(a, c))))
% 109.19/14.38 = { by axiom 1 (condensed_detachment) R->L }
% 109.19/14.38 fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh(fresh2(fresh2(fresh2(true, true, implies(not(implies(implies(implies(implies(a, c), implies(implies(a, implies(a, c)), implies(a, c))), implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, implies(a, c)), implies(a, c)))), implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, implies(a, c)), implies(a, c))))), implies(implies(implies(implies(a, c), implies(implies(a, implies(a, c)), implies(a, c))), implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, implies(a, c)), implies(a, c)))), implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, implies(a, c)), implies(a, c)))))), true, implies(implies(implies(implies(a, c), implies(implies(a, implies(a, c)), implies(a, c))), implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, implies(a, c)), implies(a, c)))), implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, implies(a, c)), implies(a, c))))), true, implies(implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(a, c)), implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, implies(a, c)), implies(a, c))))), true, implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(a, c)), implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, implies(a, c)), implies(a, c)))), true, implies(implies(a, implies(a, c)), implies(a, c))), true, implies(implies(implies(implies(a, implies(a, c)), implies(a, c)), implies(implies(implies(b, c), implies(a, c)), implies(a, c))), implies(implies(implies(b, c), implies(a, c)), implies(a, c)))), true, implies(implies(implies(implies(b, c), implies(a, c)), implies(a, implies(a, c))), implies(implies(implies(b, c), implies(a, c)), implies(a, c)))), true, implies(implies(a, implies(b, c)), implies(implies(implies(b, c), implies(a, c)), implies(a, c)))), true, implies(implies(implies(b, c), implies(a, c)), implies(implies(a, implies(b, c)), implies(a, c)))), true, implies(implies(a, b), implies(implies(a, implies(b, c)), implies(a, c)))), true, implies(implies(a, implies(b, c)), implies(implies(a, b), implies(a, c))))
% 109.19/14.38 = { by axiom 1 (condensed_detachment) R->L }
% 109.53/14.38 fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh(fresh2(fresh2(fresh2(fresh2(true, true, implies(implies(a, c), implies(implies(a, implies(a, c)), implies(a, c)))), true, implies(not(implies(implies(implies(implies(a, c), implies(implies(a, implies(a, c)), implies(a, c))), implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, implies(a, c)), implies(a, c)))), implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, implies(a, c)), implies(a, c))))), implies(implies(implies(implies(a, c), implies(implies(a, implies(a, c)), implies(a, c))), implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, implies(a, c)), implies(a, c)))), implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, implies(a, c)), implies(a, c)))))), true, implies(implies(implies(implies(a, c), implies(implies(a, implies(a, c)), implies(a, c))), implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, implies(a, c)), implies(a, c)))), implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, implies(a, c)), implies(a, c))))), true, implies(implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(a, c)), implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, implies(a, c)), implies(a, c))))), true, implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(a, c)), implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, implies(a, c)), implies(a, c)))), true, implies(implies(a, implies(a, c)), implies(a, c))), true, implies(implies(implies(implies(a, implies(a, c)), implies(a, c)), implies(implies(implies(b, c), implies(a, c)), implies(a, c))), implies(implies(implies(b, c), implies(a, c)), implies(a, c)))), true, implies(implies(implies(implies(b, c), implies(a, c)), implies(a, implies(a, c))), implies(implies(implies(b, c), implies(a, c)), implies(a, c)))), true, implies(implies(a, implies(b, c)), implies(implies(implies(b, c), implies(a, c)), implies(a, c)))), true, implies(implies(implies(b, c), implies(a, c)), implies(implies(a, implies(b, c)), implies(a, c)))), true, implies(implies(a, b), implies(implies(a, implies(b, c)), implies(a, c)))), true, implies(implies(a, implies(b, c)), implies(implies(a, b), implies(a, c))))
% 109.53/14.38 = { by axiom 4 (cn_2) R->L }
% 109.53/14.38 fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh(fresh2(fresh2(fresh2(fresh2(is_a_theorem(implies(implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, implies(a, c)), implies(a, c)))), true, implies(implies(a, c), implies(implies(a, implies(a, c)), implies(a, c)))), true, implies(not(implies(implies(implies(implies(a, c), implies(implies(a, implies(a, c)), implies(a, c))), implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, implies(a, c)), implies(a, c)))), implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, implies(a, c)), implies(a, c))))), implies(implies(implies(implies(a, c), implies(implies(a, implies(a, c)), implies(a, c))), implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, implies(a, c)), implies(a, c)))), implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, implies(a, c)), implies(a, c)))))), true, implies(implies(implies(implies(a, c), implies(implies(a, implies(a, c)), implies(a, c))), implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, implies(a, c)), implies(a, c)))), implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, implies(a, c)), implies(a, c))))), true, implies(implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(a, c)), implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, implies(a, c)), implies(a, c))))), true, implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(a, c)), implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, implies(a, c)), implies(a, c)))), true, implies(implies(a, implies(a, c)), implies(a, c))), true, implies(implies(implies(implies(a, implies(a, c)), implies(a, c)), implies(implies(implies(b, c), implies(a, c)), implies(a, c))), implies(implies(implies(b, c), implies(a, c)), implies(a, c)))), true, implies(implies(implies(implies(b, c), implies(a, c)), implies(a, implies(a, c))), implies(implies(implies(b, c), implies(a, c)), implies(a, c)))), true, implies(implies(a, implies(b, c)), implies(implies(implies(b, c), implies(a, c)), implies(a, c)))), true, implies(implies(implies(b, c), implies(a, c)), implies(implies(a, implies(b, c)), implies(a, c)))), true, implies(implies(a, b), implies(implies(a, implies(b, c)), implies(a, c)))), true, implies(implies(a, implies(b, c)), implies(implies(a, b), implies(a, c))))
% 109.53/14.39 = { by axiom 5 (condensed_detachment) R->L }
% 109.53/14.39 fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh(fresh2(fresh2(fresh2(fresh(is_a_theorem(implies(implies(implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, c), implies(implies(a, implies(a, c)), implies(a, c))))), true, implies(implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, c), implies(implies(a, implies(a, c)), implies(a, c)))), true, implies(not(implies(implies(implies(implies(a, c), implies(implies(a, implies(a, c)), implies(a, c))), implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, implies(a, c)), implies(a, c)))), implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, implies(a, c)), implies(a, c))))), implies(implies(implies(implies(a, c), implies(implies(a, implies(a, c)), implies(a, c))), implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, implies(a, c)), implies(a, c)))), implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, implies(a, c)), implies(a, c)))))), true, implies(implies(implies(implies(a, c), implies(implies(a, implies(a, c)), implies(a, c))), implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, implies(a, c)), implies(a, c)))), implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, implies(a, c)), implies(a, c))))), true, implies(implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(a, c)), implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, implies(a, c)), implies(a, c))))), true, implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(a, c)), implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, implies(a, c)), implies(a, c)))), true, implies(implies(a, implies(a, c)), implies(a, c))), true, implies(implies(implies(implies(a, implies(a, c)), implies(a, c)), implies(implies(implies(b, c), implies(a, c)), implies(a, c))), implies(implies(implies(b, c), implies(a, c)), implies(a, c)))), true, implies(implies(implies(implies(b, c), implies(a, c)), implies(a, implies(a, c))), implies(implies(implies(b, c), implies(a, c)), implies(a, c)))), true, implies(implies(a, implies(b, c)), implies(implies(implies(b, c), implies(a, c)), implies(a, c)))), true, implies(implies(implies(b, c), implies(a, c)), implies(implies(a, implies(b, c)), implies(a, c)))), true, implies(implies(a, b), implies(implies(a, implies(b, c)), implies(a, c)))), true, implies(implies(a, implies(b, c)), implies(implies(a, b), implies(a, c))))
% 109.53/14.39 = { by lemma 7 R->L }
% 109.53/14.39 fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh(fresh2(fresh2(fresh2(fresh(fresh2(is_a_theorem(implies(implies(a, c), implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, implies(a, c)), implies(a, c))))), true, implies(implies(implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, c), implies(implies(a, implies(a, c)), implies(a, c))))), true, implies(implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, c), implies(implies(a, implies(a, c)), implies(a, c)))), true, implies(not(implies(implies(implies(implies(a, c), implies(implies(a, implies(a, c)), implies(a, c))), implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, implies(a, c)), implies(a, c)))), implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, implies(a, c)), implies(a, c))))), implies(implies(implies(implies(a, c), implies(implies(a, implies(a, c)), implies(a, c))), implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, implies(a, c)), implies(a, c)))), implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, implies(a, c)), implies(a, c)))))), true, implies(implies(implies(implies(a, c), implies(implies(a, implies(a, c)), implies(a, c))), implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, implies(a, c)), implies(a, c)))), implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, implies(a, c)), implies(a, c))))), true, implies(implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(a, c)), implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, implies(a, c)), implies(a, c))))), true, implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(a, c)), implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, implies(a, c)), implies(a, c)))), true, implies(implies(a, implies(a, c)), implies(a, c))), true, implies(implies(implies(implies(a, implies(a, c)), implies(a, c)), implies(implies(implies(b, c), implies(a, c)), implies(a, c))), implies(implies(implies(b, c), implies(a, c)), implies(a, c)))), true, implies(implies(implies(implies(b, c), implies(a, c)), implies(a, implies(a, c))), implies(implies(implies(b, c), implies(a, c)), implies(a, c)))), true, implies(implies(a, implies(b, c)), implies(implies(implies(b, c), implies(a, c)), implies(a, c)))), true, implies(implies(implies(b, c), implies(a, c)), implies(implies(a, implies(b, c)), implies(a, c)))), true, implies(implies(a, b), implies(implies(a, implies(b, c)), implies(a, c)))), true, implies(implies(a, implies(b, c)), implies(implies(a, b), implies(a, c))))
% 109.53/14.39 = { by lemma 8 R->L }
% 109.53/14.39 fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh(fresh2(fresh2(fresh2(fresh(fresh2(fresh2(is_a_theorem(implies(implies(not(implies(a, c)), implies(a, c)), implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, implies(a, c)), implies(a, c))))), true, implies(implies(a, c), implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, implies(a, c)), implies(a, c))))), true, implies(implies(implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, c), implies(implies(a, implies(a, c)), implies(a, c))))), true, implies(implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, c), implies(implies(a, implies(a, c)), implies(a, c)))), true, implies(not(implies(implies(implies(implies(a, c), implies(implies(a, implies(a, c)), implies(a, c))), implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, implies(a, c)), implies(a, c)))), implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, implies(a, c)), implies(a, c))))), implies(implies(implies(implies(a, c), implies(implies(a, implies(a, c)), implies(a, c))), implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, implies(a, c)), implies(a, c)))), implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, implies(a, c)), implies(a, c)))))), true, implies(implies(implies(implies(a, c), implies(implies(a, implies(a, c)), implies(a, c))), implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, implies(a, c)), implies(a, c)))), implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, implies(a, c)), implies(a, c))))), true, implies(implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(a, c)), implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, implies(a, c)), implies(a, c))))), true, implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(a, c)), implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, implies(a, c)), implies(a, c)))), true, implies(implies(a, implies(a, c)), implies(a, c))), true, implies(implies(implies(implies(a, implies(a, c)), implies(a, c)), implies(implies(implies(b, c), implies(a, c)), implies(a, c))), implies(implies(implies(b, c), implies(a, c)), implies(a, c)))), true, implies(implies(implies(implies(b, c), implies(a, c)), implies(a, implies(a, c))), implies(implies(implies(b, c), implies(a, c)), implies(a, c)))), true, implies(implies(a, implies(b, c)), implies(implies(implies(b, c), implies(a, c)), implies(a, c)))), true, implies(implies(implies(b, c), implies(a, c)), implies(implies(a, implies(b, c)), implies(a, c)))), true, implies(implies(a, b), implies(implies(a, implies(b, c)), implies(a, c)))), true, implies(implies(a, implies(b, c)), implies(implies(a, b), implies(a, c))))
% 109.53/14.40 = { by lemma 15 R->L }
% 109.53/14.40 fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh(fresh2(fresh2(fresh2(fresh(fresh2(fresh2(fresh2(is_a_theorem(implies(implies(implies(a, implies(a, c)), implies(not(implies(a, c)), implies(a, c))), implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, implies(a, c)), implies(a, c))))), true, implies(implies(not(implies(a, c)), implies(a, c)), implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, implies(a, c)), implies(a, c))))), true, implies(implies(a, c), implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, implies(a, c)), implies(a, c))))), true, implies(implies(implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, c), implies(implies(a, implies(a, c)), implies(a, c))))), true, implies(implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, c), implies(implies(a, implies(a, c)), implies(a, c)))), true, implies(not(implies(implies(implies(implies(a, c), implies(implies(a, implies(a, c)), implies(a, c))), implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, implies(a, c)), implies(a, c)))), implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, implies(a, c)), implies(a, c))))), implies(implies(implies(implies(a, c), implies(implies(a, implies(a, c)), implies(a, c))), implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, implies(a, c)), implies(a, c)))), implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, implies(a, c)), implies(a, c)))))), true, implies(implies(implies(implies(a, c), implies(implies(a, implies(a, c)), implies(a, c))), implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, implies(a, c)), implies(a, c)))), implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, implies(a, c)), implies(a, c))))), true, implies(implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(a, c)), implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, implies(a, c)), implies(a, c))))), true, implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(a, c)), implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, implies(a, c)), implies(a, c)))), true, implies(implies(a, implies(a, c)), implies(a, c))), true, implies(implies(implies(implies(a, implies(a, c)), implies(a, c)), implies(implies(implies(b, c), implies(a, c)), implies(a, c))), implies(implies(implies(b, c), implies(a, c)), implies(a, c)))), true, implies(implies(implies(implies(b, c), implies(a, c)), implies(a, implies(a, c))), implies(implies(implies(b, c), implies(a, c)), implies(a, c)))), true, implies(implies(a, implies(b, c)), implies(implies(implies(b, c), implies(a, c)), implies(a, c)))), true, implies(implies(implies(b, c), implies(a, c)), implies(implies(a, implies(b, c)), implies(a, c)))), true, implies(implies(a, b), implies(implies(a, implies(b, c)), implies(a, c)))), true, implies(implies(a, implies(b, c)), implies(implies(a, b), implies(a, c))))
% 109.53/14.40 = { by lemma 18 }
% 109.53/14.40 fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh(fresh2(fresh2(fresh2(fresh(fresh2(fresh2(fresh2(true, true, implies(implies(not(implies(a, c)), implies(a, c)), implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, implies(a, c)), implies(a, c))))), true, implies(implies(a, c), implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, implies(a, c)), implies(a, c))))), true, implies(implies(implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, c), implies(implies(a, implies(a, c)), implies(a, c))))), true, implies(implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, c), implies(implies(a, implies(a, c)), implies(a, c)))), true, implies(not(implies(implies(implies(implies(a, c), implies(implies(a, implies(a, c)), implies(a, c))), implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, implies(a, c)), implies(a, c)))), implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, implies(a, c)), implies(a, c))))), implies(implies(implies(implies(a, c), implies(implies(a, implies(a, c)), implies(a, c))), implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, implies(a, c)), implies(a, c)))), implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, implies(a, c)), implies(a, c)))))), true, implies(implies(implies(implies(a, c), implies(implies(a, implies(a, c)), implies(a, c))), implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, implies(a, c)), implies(a, c)))), implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, implies(a, c)), implies(a, c))))), true, implies(implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(a, c)), implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, implies(a, c)), implies(a, c))))), true, implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(a, c)), implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, implies(a, c)), implies(a, c)))), true, implies(implies(a, implies(a, c)), implies(a, c))), true, implies(implies(implies(implies(a, implies(a, c)), implies(a, c)), implies(implies(implies(b, c), implies(a, c)), implies(a, c))), implies(implies(implies(b, c), implies(a, c)), implies(a, c)))), true, implies(implies(implies(implies(b, c), implies(a, c)), implies(a, implies(a, c))), implies(implies(implies(b, c), implies(a, c)), implies(a, c)))), true, implies(implies(a, implies(b, c)), implies(implies(implies(b, c), implies(a, c)), implies(a, c)))), true, implies(implies(implies(b, c), implies(a, c)), implies(implies(a, implies(b, c)), implies(a, c)))), true, implies(implies(a, b), implies(implies(a, implies(b, c)), implies(a, c)))), true, implies(implies(a, implies(b, c)), implies(implies(a, b), implies(a, c))))
% 109.53/14.40 = { by axiom 1 (condensed_detachment) }
% 109.53/14.41 fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh(fresh2(fresh2(fresh2(fresh(fresh2(fresh2(true, true, implies(implies(a, c), implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, implies(a, c)), implies(a, c))))), true, implies(implies(implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, c), implies(implies(a, implies(a, c)), implies(a, c))))), true, implies(implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, c), implies(implies(a, implies(a, c)), implies(a, c)))), true, implies(not(implies(implies(implies(implies(a, c), implies(implies(a, implies(a, c)), implies(a, c))), implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, implies(a, c)), implies(a, c)))), implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, implies(a, c)), implies(a, c))))), implies(implies(implies(implies(a, c), implies(implies(a, implies(a, c)), implies(a, c))), implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, implies(a, c)), implies(a, c)))), implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, implies(a, c)), implies(a, c)))))), true, implies(implies(implies(implies(a, c), implies(implies(a, implies(a, c)), implies(a, c))), implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, implies(a, c)), implies(a, c)))), implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, implies(a, c)), implies(a, c))))), true, implies(implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(a, c)), implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, implies(a, c)), implies(a, c))))), true, implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(a, c)), implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, implies(a, c)), implies(a, c)))), true, implies(implies(a, implies(a, c)), implies(a, c))), true, implies(implies(implies(implies(a, implies(a, c)), implies(a, c)), implies(implies(implies(b, c), implies(a, c)), implies(a, c))), implies(implies(implies(b, c), implies(a, c)), implies(a, c)))), true, implies(implies(implies(implies(b, c), implies(a, c)), implies(a, implies(a, c))), implies(implies(implies(b, c), implies(a, c)), implies(a, c)))), true, implies(implies(a, implies(b, c)), implies(implies(implies(b, c), implies(a, c)), implies(a, c)))), true, implies(implies(implies(b, c), implies(a, c)), implies(implies(a, implies(b, c)), implies(a, c)))), true, implies(implies(a, b), implies(implies(a, implies(b, c)), implies(a, c)))), true, implies(implies(a, implies(b, c)), implies(implies(a, b), implies(a, c))))
% 109.53/14.41 = { by axiom 1 (condensed_detachment) }
% 109.53/14.41 fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh(fresh2(fresh2(fresh2(fresh(fresh2(true, true, implies(implies(implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, c), implies(implies(a, implies(a, c)), implies(a, c))))), true, implies(implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, c), implies(implies(a, implies(a, c)), implies(a, c)))), true, implies(not(implies(implies(implies(implies(a, c), implies(implies(a, implies(a, c)), implies(a, c))), implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, implies(a, c)), implies(a, c)))), implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, implies(a, c)), implies(a, c))))), implies(implies(implies(implies(a, c), implies(implies(a, implies(a, c)), implies(a, c))), implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, implies(a, c)), implies(a, c)))), implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, implies(a, c)), implies(a, c)))))), true, implies(implies(implies(implies(a, c), implies(implies(a, implies(a, c)), implies(a, c))), implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, implies(a, c)), implies(a, c)))), implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, implies(a, c)), implies(a, c))))), true, implies(implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(a, c)), implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, implies(a, c)), implies(a, c))))), true, implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(a, c)), implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, implies(a, c)), implies(a, c)))), true, implies(implies(a, implies(a, c)), implies(a, c))), true, implies(implies(implies(implies(a, implies(a, c)), implies(a, c)), implies(implies(implies(b, c), implies(a, c)), implies(a, c))), implies(implies(implies(b, c), implies(a, c)), implies(a, c)))), true, implies(implies(implies(implies(b, c), implies(a, c)), implies(a, implies(a, c))), implies(implies(implies(b, c), implies(a, c)), implies(a, c)))), true, implies(implies(a, implies(b, c)), implies(implies(implies(b, c), implies(a, c)), implies(a, c)))), true, implies(implies(implies(b, c), implies(a, c)), implies(implies(a, implies(b, c)), implies(a, c)))), true, implies(implies(a, b), implies(implies(a, implies(b, c)), implies(a, c)))), true, implies(implies(a, implies(b, c)), implies(implies(a, b), implies(a, c))))
% 109.53/14.41 = { by axiom 1 (condensed_detachment) }
% 109.53/14.41 fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh(fresh2(fresh2(fresh2(fresh(true, true, implies(implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, c), implies(implies(a, implies(a, c)), implies(a, c)))), true, implies(not(implies(implies(implies(implies(a, c), implies(implies(a, implies(a, c)), implies(a, c))), implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, implies(a, c)), implies(a, c)))), implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, implies(a, c)), implies(a, c))))), implies(implies(implies(implies(a, c), implies(implies(a, implies(a, c)), implies(a, c))), implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, implies(a, c)), implies(a, c)))), implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, implies(a, c)), implies(a, c)))))), true, implies(implies(implies(implies(a, c), implies(implies(a, implies(a, c)), implies(a, c))), implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, implies(a, c)), implies(a, c)))), implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, implies(a, c)), implies(a, c))))), true, implies(implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(a, c)), implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, implies(a, c)), implies(a, c))))), true, implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(a, c)), implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, implies(a, c)), implies(a, c)))), true, implies(implies(a, implies(a, c)), implies(a, c))), true, implies(implies(implies(implies(a, implies(a, c)), implies(a, c)), implies(implies(implies(b, c), implies(a, c)), implies(a, c))), implies(implies(implies(b, c), implies(a, c)), implies(a, c)))), true, implies(implies(implies(implies(b, c), implies(a, c)), implies(a, implies(a, c))), implies(implies(implies(b, c), implies(a, c)), implies(a, c)))), true, implies(implies(a, implies(b, c)), implies(implies(implies(b, c), implies(a, c)), implies(a, c)))), true, implies(implies(implies(b, c), implies(a, c)), implies(implies(a, implies(b, c)), implies(a, c)))), true, implies(implies(a, b), implies(implies(a, implies(b, c)), implies(a, c)))), true, implies(implies(a, implies(b, c)), implies(implies(a, b), implies(a, c))))
% 109.53/14.41 = { by axiom 2 (condensed_detachment) }
% 109.53/14.41 fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh(fresh2(fresh2(fresh2(is_a_theorem(implies(implies(a, c), implies(implies(a, implies(a, c)), implies(a, c)))), true, implies(not(implies(implies(implies(implies(a, c), implies(implies(a, implies(a, c)), implies(a, c))), implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, implies(a, c)), implies(a, c)))), implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, implies(a, c)), implies(a, c))))), implies(implies(implies(implies(a, c), implies(implies(a, implies(a, c)), implies(a, c))), implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, implies(a, c)), implies(a, c)))), implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, implies(a, c)), implies(a, c)))))), true, implies(implies(implies(implies(a, c), implies(implies(a, implies(a, c)), implies(a, c))), implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, implies(a, c)), implies(a, c)))), implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, implies(a, c)), implies(a, c))))), true, implies(implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(a, c)), implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, implies(a, c)), implies(a, c))))), true, implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(a, c)), implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, implies(a, c)), implies(a, c)))), true, implies(implies(a, implies(a, c)), implies(a, c))), true, implies(implies(implies(implies(a, implies(a, c)), implies(a, c)), implies(implies(implies(b, c), implies(a, c)), implies(a, c))), implies(implies(implies(b, c), implies(a, c)), implies(a, c)))), true, implies(implies(implies(implies(b, c), implies(a, c)), implies(a, implies(a, c))), implies(implies(implies(b, c), implies(a, c)), implies(a, c)))), true, implies(implies(a, implies(b, c)), implies(implies(implies(b, c), implies(a, c)), implies(a, c)))), true, implies(implies(implies(b, c), implies(a, c)), implies(implies(a, implies(b, c)), implies(a, c)))), true, implies(implies(a, b), implies(implies(a, implies(b, c)), implies(a, c)))), true, implies(implies(a, implies(b, c)), implies(implies(a, b), implies(a, c))))
% 109.53/14.41 = { by lemma 19 }
% 109.53/14.42 fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh(fresh2(fresh2(is_a_theorem(implies(not(implies(implies(implies(implies(a, c), implies(implies(a, implies(a, c)), implies(a, c))), implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, implies(a, c)), implies(a, c)))), implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, implies(a, c)), implies(a, c))))), implies(implies(implies(implies(a, c), implies(implies(a, implies(a, c)), implies(a, c))), implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, implies(a, c)), implies(a, c)))), implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, implies(a, c)), implies(a, c)))))), true, implies(implies(implies(implies(a, c), implies(implies(a, implies(a, c)), implies(a, c))), implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, implies(a, c)), implies(a, c)))), implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, implies(a, c)), implies(a, c))))), true, implies(implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(a, c)), implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, implies(a, c)), implies(a, c))))), true, implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(a, c)), implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, implies(a, c)), implies(a, c)))), true, implies(implies(a, implies(a, c)), implies(a, c))), true, implies(implies(implies(implies(a, implies(a, c)), implies(a, c)), implies(implies(implies(b, c), implies(a, c)), implies(a, c))), implies(implies(implies(b, c), implies(a, c)), implies(a, c)))), true, implies(implies(implies(implies(b, c), implies(a, c)), implies(a, implies(a, c))), implies(implies(implies(b, c), implies(a, c)), implies(a, c)))), true, implies(implies(a, implies(b, c)), implies(implies(implies(b, c), implies(a, c)), implies(a, c)))), true, implies(implies(implies(b, c), implies(a, c)), implies(implies(a, implies(b, c)), implies(a, c)))), true, implies(implies(a, b), implies(implies(a, implies(b, c)), implies(a, c)))), true, implies(implies(a, implies(b, c)), implies(implies(a, b), implies(a, c))))
% 109.53/14.42 = { by lemma 17 }
% 109.53/14.42 fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh(fresh2(is_a_theorem(implies(implies(implies(implies(a, c), implies(implies(a, implies(a, c)), implies(a, c))), implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, implies(a, c)), implies(a, c)))), implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, implies(a, c)), implies(a, c))))), true, implies(implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(a, c)), implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, implies(a, c)), implies(a, c))))), true, implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(a, c)), implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, implies(a, c)), implies(a, c)))), true, implies(implies(a, implies(a, c)), implies(a, c))), true, implies(implies(implies(implies(a, implies(a, c)), implies(a, c)), implies(implies(implies(b, c), implies(a, c)), implies(a, c))), implies(implies(implies(b, c), implies(a, c)), implies(a, c)))), true, implies(implies(implies(implies(b, c), implies(a, c)), implies(a, implies(a, c))), implies(implies(implies(b, c), implies(a, c)), implies(a, c)))), true, implies(implies(a, implies(b, c)), implies(implies(implies(b, c), implies(a, c)), implies(a, c)))), true, implies(implies(implies(b, c), implies(a, c)), implies(implies(a, implies(b, c)), implies(a, c)))), true, implies(implies(a, b), implies(implies(a, implies(b, c)), implies(a, c)))), true, implies(implies(a, implies(b, c)), implies(implies(a, b), implies(a, c))))
% 109.53/14.42 = { by lemma 10 }
% 109.53/14.42 fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh(is_a_theorem(implies(implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(a, c)), implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, implies(a, c)), implies(a, c))))), true, implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(a, c)), implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, implies(a, c)), implies(a, c)))), true, implies(implies(a, implies(a, c)), implies(a, c))), true, implies(implies(implies(implies(a, implies(a, c)), implies(a, c)), implies(implies(implies(b, c), implies(a, c)), implies(a, c))), implies(implies(implies(b, c), implies(a, c)), implies(a, c)))), true, implies(implies(implies(implies(b, c), implies(a, c)), implies(a, implies(a, c))), implies(implies(implies(b, c), implies(a, c)), implies(a, c)))), true, implies(implies(a, implies(b, c)), implies(implies(implies(b, c), implies(a, c)), implies(a, c)))), true, implies(implies(implies(b, c), implies(a, c)), implies(implies(a, implies(b, c)), implies(a, c)))), true, implies(implies(a, b), implies(implies(a, implies(b, c)), implies(a, c)))), true, implies(implies(a, implies(b, c)), implies(implies(a, b), implies(a, c))))
% 109.53/14.42 = { by axiom 5 (condensed_detachment) }
% 109.53/14.42 fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(is_a_theorem(implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(a, c))), true, implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, implies(a, c)), implies(a, c)))), true, implies(implies(a, implies(a, c)), implies(a, c))), true, implies(implies(implies(implies(a, implies(a, c)), implies(a, c)), implies(implies(implies(b, c), implies(a, c)), implies(a, c))), implies(implies(implies(b, c), implies(a, c)), implies(a, c)))), true, implies(implies(implies(implies(b, c), implies(a, c)), implies(a, implies(a, c))), implies(implies(implies(b, c), implies(a, c)), implies(a, c)))), true, implies(implies(a, implies(b, c)), implies(implies(implies(b, c), implies(a, c)), implies(a, c)))), true, implies(implies(implies(b, c), implies(a, c)), implies(implies(a, implies(b, c)), implies(a, c)))), true, implies(implies(a, b), implies(implies(a, implies(b, c)), implies(a, c)))), true, implies(implies(a, implies(b, c)), implies(implies(a, b), implies(a, c))))
% 109.53/14.42 = { by lemma 23 R->L }
% 109.53/14.42 fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(is_a_theorem(implies(implies(implies(not(implies(implies(a, implies(a, c)), implies(a, c))), c), c), implies(a, c))), true, implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(a, c))), true, implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, implies(a, c)), implies(a, c)))), true, implies(implies(a, implies(a, c)), implies(a, c))), true, implies(implies(implies(implies(a, implies(a, c)), implies(a, c)), implies(implies(implies(b, c), implies(a, c)), implies(a, c))), implies(implies(implies(b, c), implies(a, c)), implies(a, c)))), true, implies(implies(implies(implies(b, c), implies(a, c)), implies(a, implies(a, c))), implies(implies(implies(b, c), implies(a, c)), implies(a, c)))), true, implies(implies(a, implies(b, c)), implies(implies(implies(b, c), implies(a, c)), implies(a, c)))), true, implies(implies(implies(b, c), implies(a, c)), implies(implies(a, implies(b, c)), implies(a, c)))), true, implies(implies(a, b), implies(implies(a, implies(b, c)), implies(a, c)))), true, implies(implies(a, implies(b, c)), implies(implies(a, b), implies(a, c))))
% 109.53/14.42 = { by lemma 7 R->L }
% 109.53/14.42 fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(is_a_theorem(implies(a, implies(not(implies(implies(a, implies(a, c)), implies(a, c))), c))), true, implies(implies(implies(not(implies(implies(a, implies(a, c)), implies(a, c))), c), c), implies(a, c))), true, implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(a, c))), true, implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, implies(a, c)), implies(a, c)))), true, implies(implies(a, implies(a, c)), implies(a, c))), true, implies(implies(implies(implies(a, implies(a, c)), implies(a, c)), implies(implies(implies(b, c), implies(a, c)), implies(a, c))), implies(implies(implies(b, c), implies(a, c)), implies(a, c)))), true, implies(implies(implies(implies(b, c), implies(a, c)), implies(a, implies(a, c))), implies(implies(implies(b, c), implies(a, c)), implies(a, c)))), true, implies(implies(a, implies(b, c)), implies(implies(implies(b, c), implies(a, c)), implies(a, c)))), true, implies(implies(implies(b, c), implies(a, c)), implies(implies(a, implies(b, c)), implies(a, c)))), true, implies(implies(a, b), implies(implies(a, implies(b, c)), implies(a, c)))), true, implies(implies(a, implies(b, c)), implies(implies(a, b), implies(a, c))))
% 109.53/14.42 = { by lemma 23 R->L }
% 109.53/14.42 fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(is_a_theorem(implies(implies(implies(a, implies(a, c)), implies(a, c)), implies(not(implies(implies(a, implies(a, c)), implies(a, c))), c))), true, implies(a, implies(not(implies(implies(a, implies(a, c)), implies(a, c))), c))), true, implies(implies(implies(not(implies(implies(a, implies(a, c)), implies(a, c))), c), c), implies(a, c))), true, implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(a, c))), true, implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, implies(a, c)), implies(a, c)))), true, implies(implies(a, implies(a, c)), implies(a, c))), true, implies(implies(implies(implies(a, implies(a, c)), implies(a, c)), implies(implies(implies(b, c), implies(a, c)), implies(a, c))), implies(implies(implies(b, c), implies(a, c)), implies(a, c)))), true, implies(implies(implies(implies(b, c), implies(a, c)), implies(a, implies(a, c))), implies(implies(implies(b, c), implies(a, c)), implies(a, c)))), true, implies(implies(a, implies(b, c)), implies(implies(implies(b, c), implies(a, c)), implies(a, c)))), true, implies(implies(implies(b, c), implies(a, c)), implies(implies(a, implies(b, c)), implies(a, c)))), true, implies(implies(a, b), implies(implies(a, implies(b, c)), implies(a, c)))), true, implies(implies(a, implies(b, c)), implies(implies(a, b), implies(a, c))))
% 109.53/14.42 = { by axiom 3 (cn_3) }
% 109.53/14.42 fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(true, true, implies(a, implies(not(implies(implies(a, implies(a, c)), implies(a, c))), c))), true, implies(implies(implies(not(implies(implies(a, implies(a, c)), implies(a, c))), c), c), implies(a, c))), true, implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(a, c))), true, implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, implies(a, c)), implies(a, c)))), true, implies(implies(a, implies(a, c)), implies(a, c))), true, implies(implies(implies(implies(a, implies(a, c)), implies(a, c)), implies(implies(implies(b, c), implies(a, c)), implies(a, c))), implies(implies(implies(b, c), implies(a, c)), implies(a, c)))), true, implies(implies(implies(implies(b, c), implies(a, c)), implies(a, implies(a, c))), implies(implies(implies(b, c), implies(a, c)), implies(a, c)))), true, implies(implies(a, implies(b, c)), implies(implies(implies(b, c), implies(a, c)), implies(a, c)))), true, implies(implies(implies(b, c), implies(a, c)), implies(implies(a, implies(b, c)), implies(a, c)))), true, implies(implies(a, b), implies(implies(a, implies(b, c)), implies(a, c)))), true, implies(implies(a, implies(b, c)), implies(implies(a, b), implies(a, c))))
% 109.53/14.42 = { by axiom 1 (condensed_detachment) }
% 109.53/14.42 fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(true, true, implies(implies(implies(not(implies(implies(a, implies(a, c)), implies(a, c))), c), c), implies(a, c))), true, implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(a, c))), true, implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, implies(a, c)), implies(a, c)))), true, implies(implies(a, implies(a, c)), implies(a, c))), true, implies(implies(implies(implies(a, implies(a, c)), implies(a, c)), implies(implies(implies(b, c), implies(a, c)), implies(a, c))), implies(implies(implies(b, c), implies(a, c)), implies(a, c)))), true, implies(implies(implies(implies(b, c), implies(a, c)), implies(a, implies(a, c))), implies(implies(implies(b, c), implies(a, c)), implies(a, c)))), true, implies(implies(a, implies(b, c)), implies(implies(implies(b, c), implies(a, c)), implies(a, c)))), true, implies(implies(implies(b, c), implies(a, c)), implies(implies(a, implies(b, c)), implies(a, c)))), true, implies(implies(a, b), implies(implies(a, implies(b, c)), implies(a, c)))), true, implies(implies(a, implies(b, c)), implies(implies(a, b), implies(a, c))))
% 109.53/14.42 = { by axiom 1 (condensed_detachment) }
% 109.53/14.42 fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(true, true, implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(a, c))), true, implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, implies(a, c)), implies(a, c)))), true, implies(implies(a, implies(a, c)), implies(a, c))), true, implies(implies(implies(implies(a, implies(a, c)), implies(a, c)), implies(implies(implies(b, c), implies(a, c)), implies(a, c))), implies(implies(implies(b, c), implies(a, c)), implies(a, c)))), true, implies(implies(implies(implies(b, c), implies(a, c)), implies(a, implies(a, c))), implies(implies(implies(b, c), implies(a, c)), implies(a, c)))), true, implies(implies(a, implies(b, c)), implies(implies(implies(b, c), implies(a, c)), implies(a, c)))), true, implies(implies(implies(b, c), implies(a, c)), implies(implies(a, implies(b, c)), implies(a, c)))), true, implies(implies(a, b), implies(implies(a, implies(b, c)), implies(a, c)))), true, implies(implies(a, implies(b, c)), implies(implies(a, b), implies(a, c))))
% 109.53/14.42 = { by axiom 1 (condensed_detachment) }
% 109.53/14.43 fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(true, true, implies(not(implies(implies(a, implies(a, c)), implies(a, c))), implies(implies(a, implies(a, c)), implies(a, c)))), true, implies(implies(a, implies(a, c)), implies(a, c))), true, implies(implies(implies(implies(a, implies(a, c)), implies(a, c)), implies(implies(implies(b, c), implies(a, c)), implies(a, c))), implies(implies(implies(b, c), implies(a, c)), implies(a, c)))), true, implies(implies(implies(implies(b, c), implies(a, c)), implies(a, implies(a, c))), implies(implies(implies(b, c), implies(a, c)), implies(a, c)))), true, implies(implies(a, implies(b, c)), implies(implies(implies(b, c), implies(a, c)), implies(a, c)))), true, implies(implies(implies(b, c), implies(a, c)), implies(implies(a, implies(b, c)), implies(a, c)))), true, implies(implies(a, b), implies(implies(a, implies(b, c)), implies(a, c)))), true, implies(implies(a, implies(b, c)), implies(implies(a, b), implies(a, c))))
% 109.53/14.43 = { by axiom 1 (condensed_detachment) }
% 109.53/14.43 fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(true, true, implies(implies(a, implies(a, c)), implies(a, c))), true, implies(implies(implies(implies(a, implies(a, c)), implies(a, c)), implies(implies(implies(b, c), implies(a, c)), implies(a, c))), implies(implies(implies(b, c), implies(a, c)), implies(a, c)))), true, implies(implies(implies(implies(b, c), implies(a, c)), implies(a, implies(a, c))), implies(implies(implies(b, c), implies(a, c)), implies(a, c)))), true, implies(implies(a, implies(b, c)), implies(implies(implies(b, c), implies(a, c)), implies(a, c)))), true, implies(implies(implies(b, c), implies(a, c)), implies(implies(a, implies(b, c)), implies(a, c)))), true, implies(implies(a, b), implies(implies(a, implies(b, c)), implies(a, c)))), true, implies(implies(a, implies(b, c)), implies(implies(a, b), implies(a, c))))
% 109.53/14.43 = { by axiom 1 (condensed_detachment) }
% 109.53/14.43 fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(true, true, implies(implies(implies(implies(a, implies(a, c)), implies(a, c)), implies(implies(implies(b, c), implies(a, c)), implies(a, c))), implies(implies(implies(b, c), implies(a, c)), implies(a, c)))), true, implies(implies(implies(implies(b, c), implies(a, c)), implies(a, implies(a, c))), implies(implies(implies(b, c), implies(a, c)), implies(a, c)))), true, implies(implies(a, implies(b, c)), implies(implies(implies(b, c), implies(a, c)), implies(a, c)))), true, implies(implies(implies(b, c), implies(a, c)), implies(implies(a, implies(b, c)), implies(a, c)))), true, implies(implies(a, b), implies(implies(a, implies(b, c)), implies(a, c)))), true, implies(implies(a, implies(b, c)), implies(implies(a, b), implies(a, c))))
% 109.53/14.43 = { by axiom 1 (condensed_detachment) }
% 109.53/14.43 fresh2(fresh2(fresh2(fresh2(fresh2(true, true, implies(implies(implies(implies(b, c), implies(a, c)), implies(a, implies(a, c))), implies(implies(implies(b, c), implies(a, c)), implies(a, c)))), true, implies(implies(a, implies(b, c)), implies(implies(implies(b, c), implies(a, c)), implies(a, c)))), true, implies(implies(implies(b, c), implies(a, c)), implies(implies(a, implies(b, c)), implies(a, c)))), true, implies(implies(a, b), implies(implies(a, implies(b, c)), implies(a, c)))), true, implies(implies(a, implies(b, c)), implies(implies(a, b), implies(a, c))))
% 109.53/14.43 = { by axiom 1 (condensed_detachment) }
% 109.53/14.43 fresh2(fresh2(fresh2(fresh2(true, true, implies(implies(a, implies(b, c)), implies(implies(implies(b, c), implies(a, c)), implies(a, c)))), true, implies(implies(implies(b, c), implies(a, c)), implies(implies(a, implies(b, c)), implies(a, c)))), true, implies(implies(a, b), implies(implies(a, implies(b, c)), implies(a, c)))), true, implies(implies(a, implies(b, c)), implies(implies(a, b), implies(a, c))))
% 109.53/14.43 = { by axiom 1 (condensed_detachment) }
% 109.53/14.43 fresh2(fresh2(fresh2(true, true, implies(implies(implies(b, c), implies(a, c)), implies(implies(a, implies(b, c)), implies(a, c)))), true, implies(implies(a, b), implies(implies(a, implies(b, c)), implies(a, c)))), true, implies(implies(a, implies(b, c)), implies(implies(a, b), implies(a, c))))
% 109.53/14.43 = { by axiom 1 (condensed_detachment) }
% 109.53/14.43 fresh2(fresh2(true, true, implies(implies(a, b), implies(implies(a, implies(b, c)), implies(a, c)))), true, implies(implies(a, implies(b, c)), implies(implies(a, b), implies(a, c))))
% 109.53/14.43 = { by axiom 1 (condensed_detachment) }
% 109.53/14.43 fresh2(true, true, implies(implies(a, implies(b, c)), implies(implies(a, b), implies(a, c))))
% 109.53/14.43 = { by axiom 1 (condensed_detachment) }
% 109.53/14.43 true
% 109.53/14.43 % SZS output end Proof
% 109.53/14.43
% 109.53/14.43 RESULT: Unsatisfiable (the axioms are contradictory).
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