TSTP Solution File: SEV304^5 by cocATP---0.2.0
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- Process Solution
%------------------------------------------------------------------------------
% File : cocATP---0.2.0
% Problem : SEV304^5 : TPTP v6.2.0. Bugfixed v6.2.0.
% Transfm : none
% Format : tptp:raw
% Command : python CASC.py /export/starexec/sandbox2/benchmark/theBenchmark.p
% Computer : n133.star.cs.uiowa.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2609 0 2.40GHz
% Memory : 32286.75MB
% OS : Linux 2.6.32-504.8.1.el6.x86_64
% CPULimit : 300s
% DateTime : Wed May 6 14:27:25 EDT 2015
% Result : Unknown 266.23s
% Output : None
% Verified :
% SZS Type : None (Parsing solution fails)
% Syntax : Number of formulae : 0
% Comments :
%------------------------------------------------------------------------------
%----No solution output by system
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.02 % Problem : SEV304^5 : TPTP v6.2.0. Bugfixed v6.2.0.
% 0.01/0.03 % Command : python CASC.py /export/starexec/sandbox2/benchmark/theBenchmark.p
% 0.03/1.08 % Computer : n133.star.cs.uiowa.edu
% 0.03/1.08 % Model : x86_64 x86_64
% 0.03/1.08 % CPU : Intel(R) Xeon(R) CPU E5-2609 0 @ 2.40GHz
% 0.03/1.08 % Memory : 32286.75MB
% 0.03/1.08 % OS : Linux 2.6.32-504.8.1.el6.x86_64
% 0.03/1.08 % CPULimit : 300
% 0.03/1.08 % DateTime : Thu Apr 16 12:31:37 CDT 2015
% 0.03/1.08 % CPUTime :
% 0.03/1.09 Python 2.7.5
% 0.05/1.36 Using paths ['/home/cristobal/cocATP/CASC/TPTP/', '/export/starexec/sandbox2/benchmark/', '/export/starexec/sandbox2/benchmark/']
% 0.05/1.36 FOF formula (<kernel.Constant object at 0x1991dd0>, <kernel.DependentProduct object at 0x1991488>) of role type named cONE_type
% 0.05/1.36 Using role type
% 0.05/1.36 Declaring cONE:((fofType->Prop)->Prop)
% 0.05/1.36 FOF formula (<kernel.Constant object at 0x19917e8>, <kernel.DependentProduct object at 0x1991320>) of role type named cSUCC_type
% 0.05/1.36 Using role type
% 0.05/1.36 Declaring cSUCC:(((fofType->Prop)->Prop)->((fofType->Prop)->Prop))
% 0.05/1.36 FOF formula (<kernel.Constant object at 0x1991830>, <kernel.DependentProduct object at 0x1991dd0>) of role type named cZERO_type
% 0.05/1.36 Using role type
% 0.05/1.36 Declaring cZERO:((fofType->Prop)->Prop)
% 0.05/1.36 FOF formula (((eq ((fofType->Prop)->Prop)) cZERO) (fun (Xp:(fofType->Prop))=> (((ex fofType) (fun (Xx:fofType)=> (Xp Xx)))->False))) of role definition named cZERO_def
% 0.05/1.36 A new definition: (((eq ((fofType->Prop)->Prop)) cZERO) (fun (Xp:(fofType->Prop))=> (((ex fofType) (fun (Xx:fofType)=> (Xp Xx)))->False)))
% 0.05/1.36 Defined: cZERO:=(fun (Xp:(fofType->Prop))=> (((ex fofType) (fun (Xx:fofType)=> (Xp Xx)))->False))
% 0.05/1.36 FOF formula (((eq (((fofType->Prop)->Prop)->((fofType->Prop)->Prop))) cSUCC) (fun (Xn:((fofType->Prop)->Prop)) (Xp:(fofType->Prop))=> ((ex fofType) (fun (Xx:fofType)=> ((and (Xp Xx)) (Xn (fun (Xt:fofType)=> ((and (not (((eq fofType) Xt) Xx))) (Xp Xt))))))))) of role definition named cSUCC_def
% 0.05/1.36 A new definition: (((eq (((fofType->Prop)->Prop)->((fofType->Prop)->Prop))) cSUCC) (fun (Xn:((fofType->Prop)->Prop)) (Xp:(fofType->Prop))=> ((ex fofType) (fun (Xx:fofType)=> ((and (Xp Xx)) (Xn (fun (Xt:fofType)=> ((and (not (((eq fofType) Xt) Xx))) (Xp Xt)))))))))
% 0.05/1.36 Defined: cSUCC:=(fun (Xn:((fofType->Prop)->Prop)) (Xp:(fofType->Prop))=> ((ex fofType) (fun (Xx:fofType)=> ((and (Xp Xx)) (Xn (fun (Xt:fofType)=> ((and (not (((eq fofType) Xt) Xx))) (Xp Xt))))))))
% 0.05/1.36 FOF formula (((eq ((fofType->Prop)->Prop)) cONE) (cSUCC cZERO)) of role definition named cONE_def
% 0.05/1.36 A new definition: (((eq ((fofType->Prop)->Prop)) cONE) (cSUCC cZERO))
% 0.05/1.36 Defined: cONE:=(cSUCC cZERO)
% 0.05/1.36 FOF formula (forall (K:(fofType->(fofType->Prop))) (S:(fofType->Prop)), (((and ((ex (fofType->(fofType->Prop))) (fun (Xs:(fofType->(fofType->Prop)))=> ((and (forall (Xx:fofType), ((S Xx)->((cSUCC (cSUCC (cSUCC (cSUCC (cSUCC cONE))))) (Xs Xx))))) (forall (Xy:(fofType->Prop)), (((cSUCC (cSUCC (cSUCC (cSUCC (cSUCC cONE))))) Xy)->((ex fofType) (fun (Xy0:fofType)=> (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and (S Xx)) (((eq (fofType->Prop)) Xy) (Xs Xx))))) ((fun (Xx:fofType) (Xy:fofType)=> (((eq fofType) Xx) Xy)) Xy0)))))))))) (forall (Xx:fofType) (Xy:fofType), (((K Xx) Xy)->((K Xy) Xx))))->((ex fofType) (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S Xx)) (S Xy))) (S Xz))) (not (((eq fofType) Xx) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) Xx)))) ((or ((and ((and ((K Xx) Xy)) ((K Xy) Xz))) ((K Xx) Xz))) ((and ((and (((K Xx) Xy)->False)) (((K Xy) Xz)->False))) (((K Xx) Xz)->False)))))))))))) of role
% conjecture named cSIXPEOPLE_pme
% 0.05/1.36 Conjecture to prove = (forall (K:(fofType->(fofType->Prop))) (S:(fofType->Prop)), (((and ((ex (fofType->(fofType->Prop))) (fun (Xs:(fofType->(fofType->Prop)))=> ((and (forall (Xx:fofType), ((S Xx)->((cSUCC (cSUCC (cSUCC (cSUCC (cSUCC cONE))))) (Xs Xx))))) (forall (Xy:(fofType->Prop)), (((cSUCC (cSUCC (cSUCC (cSUCC (cSUCC cONE))))) Xy)->((ex fofType) (fun (Xy0:fofType)=> (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and (S Xx)) (((eq (fofType->Prop)) Xy) (Xs Xx))))) ((fun (Xx:fofType) (Xy:fofType)=> (((eq fofType) Xx) Xy)) Xy0)))))))))) (forall (Xx:fofType) (Xy:fofType), (((K Xx) Xy)->((K Xy) Xx))))->((ex fofType) (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S Xx)) (S Xy))) (S Xz))) (not (((eq fofType) Xx) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) Xx)))) ((or ((and ((and ((K Xx) Xy)) ((K Xy) Xz))) ((K Xx) Xz))) ((and ((and (((K Xx) Xy)->False)) (((K Xy) Xz)->False))) (((K Xx)
% Xz)->False)))))))))))):Prop
% 4.84/5.98 Parameter fofType_DUMMY:fofType.
% 4.84/5.98 We need to prove ['(forall (K:(fofType->(fofType->Prop))) (S:(fofType->Prop)), (((and ((ex (fofType->(fofType->Prop))) (fun (Xs:(fofType->(fofType->Prop)))=> ((and (forall (Xx:fofType), ((S Xx)->((cSUCC (cSUCC (cSUCC (cSUCC (cSUCC cONE))))) (Xs Xx))))) (forall (Xy:(fofType->Prop)), (((cSUCC (cSUCC (cSUCC (cSUCC (cSUCC cONE))))) Xy)->((ex fofType) (fun (Xy0:fofType)=> (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and (S Xx)) (((eq (fofType->Prop)) Xy) (Xs Xx))))) ((fun (Xx:fofType) (Xy:fofType)=> (((eq fofType) Xx) Xy)) Xy0)))))))))) (forall (Xx:fofType) (Xy:fofType), (((K Xx) Xy)->((K Xy) Xx))))->((ex fofType) (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S Xx)) (S Xy))) (S Xz))) (not (((eq fofType) Xx) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) Xx)))) ((or ((and ((and ((K Xx) Xy)) ((K Xy) Xz))) ((K Xx) Xz))) ((and ((and (((K Xx) Xy)->False)) (((K Xy) Xz)->False))) (((K Xx) Xz)->False))))))))))))']
% 4.84/5.98 Parameter fofType:Type.
% 4.84/5.98 Definition cONE:=(cSUCC cZERO):((fofType->Prop)->Prop).
% 4.84/5.98 Definition cSUCC:=(fun (Xn:((fofType->Prop)->Prop)) (Xp:(fofType->Prop))=> ((ex fofType) (fun (Xx:fofType)=> ((and (Xp Xx)) (Xn (fun (Xt:fofType)=> ((and (not (((eq fofType) Xt) Xx))) (Xp Xt)))))))):(((fofType->Prop)->Prop)->((fofType->Prop)->Prop)).
% 4.84/5.98 Definition cZERO:=(fun (Xp:(fofType->Prop))=> (((ex fofType) (fun (Xx:fofType)=> (Xp Xx)))->False)):((fofType->Prop)->Prop).
% 4.84/5.98 Trying to prove (forall (K:(fofType->(fofType->Prop))) (S:(fofType->Prop)), (((and ((ex (fofType->(fofType->Prop))) (fun (Xs:(fofType->(fofType->Prop)))=> ((and (forall (Xx:fofType), ((S Xx)->((cSUCC (cSUCC (cSUCC (cSUCC (cSUCC cONE))))) (Xs Xx))))) (forall (Xy:(fofType->Prop)), (((cSUCC (cSUCC (cSUCC (cSUCC (cSUCC cONE))))) Xy)->((ex fofType) (fun (Xy0:fofType)=> (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and (S Xx)) (((eq (fofType->Prop)) Xy) (Xs Xx))))) ((fun (Xx:fofType) (Xy:fofType)=> (((eq fofType) Xx) Xy)) Xy0)))))))))) (forall (Xx:fofType) (Xy:fofType), (((K Xx) Xy)->((K Xy) Xx))))->((ex fofType) (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S Xx)) (S Xy))) (S Xz))) (not (((eq fofType) Xx) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) Xx)))) ((or ((and ((and ((K Xx) Xy)) ((K Xy) Xz))) ((K Xx) Xz))) ((and ((and (((K Xx) Xy)->False)) (((K Xy) Xz)->False))) (((K Xx) Xz)->False))))))))))))
% 4.84/5.98 Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S Xx)) (S Xy))) (S Xz))) (not (((eq fofType) Xx) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) Xx)))) ((or ((and ((and ((K Xx) Xy)) ((K Xy) Xz))) ((K Xx) Xz))) ((and ((and (((K Xx) Xy)->False)) (((K Xy) Xz)->False))) (((K Xx) Xz)->False)))))))))):(((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S Xx)) (S Xy))) (S Xz))) (not (((eq fofType) Xx) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) Xx)))) ((or ((and ((and ((K Xx) Xy)) ((K Xy) Xz))) ((K Xx) Xz))) ((and ((and (((K Xx) Xy)->False)) (((K Xy) Xz)->False))) (((K Xx) Xz)->False)))))))))) (fun (x:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x)) (S Xy))) (S Xz))) (not (((eq
% fofType) x) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x)))) ((or ((and ((and ((K x) Xy)) ((K Xy) Xz))) ((K x) Xz))) ((and ((and (((K x) Xy)->False)) (((K Xy) Xz)->False))) (((K x) Xz)->False))))))))))
% 4.84/5.98 Found (eta_expansion_dep00 (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S Xx)) (S Xy))) (S Xz))) (not (((eq fofType) Xx) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) Xx)))) ((or ((and ((and ((K Xx) Xy)) ((K Xy) Xz))) ((K Xx) Xz))) ((and ((and (((K Xx) Xy)->False)) (((K Xy) Xz)->False))) (((K Xx) Xz)->False)))))))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S Xx)) (S Xy))) (S Xz))) (not (((eq fofType) Xx) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) Xx)))) ((or ((and ((and ((K Xx) Xy)) ((K Xy) Xz))) ((K Xx) Xz))) ((and ((and (((K Xx) Xy)->False)) (((K Xy) Xz)->False))) (((K Xx) Xz)->False)))))))))) b)
% 7.04/8.18 Found ((eta_expansion_dep0 (fun (x1:fofType)=> Prop)) (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S Xx)) (S Xy))) (S Xz))) (not (((eq fofType) Xx) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) Xx)))) ((or ((and ((and ((K Xx) Xy)) ((K Xy) Xz))) ((K Xx) Xz))) ((and ((and (((K Xx) Xy)->False)) (((K Xy) Xz)->False))) (((K Xx) Xz)->False)))))))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S Xx)) (S Xy))) (S Xz))) (not (((eq fofType) Xx) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) Xx)))) ((or ((and ((and ((K Xx) Xy)) ((K Xy) Xz))) ((K Xx) Xz))) ((and ((and (((K Xx) Xy)->False)) (((K Xy) Xz)->False))) (((K Xx) Xz)->False)))))))))) b)
% 7.04/8.18 Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S Xx)) (S Xy))) (S Xz))) (not (((eq fofType) Xx) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) Xx)))) ((or ((and ((and ((K Xx) Xy)) ((K Xy) Xz))) ((K Xx) Xz))) ((and ((and (((K Xx) Xy)->False)) (((K Xy) Xz)->False))) (((K Xx) Xz)->False)))))))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S Xx)) (S Xy))) (S Xz))) (not (((eq fofType) Xx) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) Xx)))) ((or ((and ((and ((K Xx) Xy)) ((K Xy) Xz))) ((K Xx) Xz))) ((and ((and (((K Xx) Xy)->False)) (((K Xy) Xz)->False))) (((K Xx) Xz)->False)))))))))) b)
% 7.04/8.18 Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S Xx)) (S Xy))) (S Xz))) (not (((eq fofType) Xx) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) Xx)))) ((or ((and ((and ((K Xx) Xy)) ((K Xy) Xz))) ((K Xx) Xz))) ((and ((and (((K Xx) Xy)->False)) (((K Xy) Xz)->False))) (((K Xx) Xz)->False)))))))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S Xx)) (S Xy))) (S Xz))) (not (((eq fofType) Xx) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) Xx)))) ((or ((and ((and ((K Xx) Xy)) ((K Xy) Xz))) ((K Xx) Xz))) ((and ((and (((K Xx) Xy)->False)) (((K Xy) Xz)->False))) (((K Xx) Xz)->False)))))))))) b)
% 7.04/8.18 Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S Xx)) (S Xy))) (S Xz))) (not (((eq fofType) Xx) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) Xx)))) ((or ((and ((and ((K Xx) Xy)) ((K Xy) Xz))) ((K Xx) Xz))) ((and ((and (((K Xx) Xy)->False)) (((K Xy) Xz)->False))) (((K Xx) Xz)->False)))))))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S Xx)) (S Xy))) (S Xz))) (not (((eq fofType) Xx) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) Xx)))) ((or ((and ((and ((K Xx) Xy)) ((K Xy) Xz))) ((K Xx) Xz))) ((and ((and (((K Xx) Xy)->False)) (((K Xy) Xz)->False))) (((K Xx) Xz)->False)))))))))) b)
% 7.04/8.18 Found eq_ref00:=(eq_ref0 (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S Xx)) (S Xy))) (S Xz))) (not (((eq fofType) Xx) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) Xx)))) ((or ((and ((and ((K Xx) Xy)) ((K Xy) Xz))) ((K Xx) Xz))) ((and ((and (((K Xx) Xy)->False)) (((K Xy) Xz)->False))) (((K Xx) Xz)->False)))))))))):(((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S Xx)) (S Xy))) (S Xz))) (not (((eq fofType) Xx) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) Xx)))) ((or ((and ((and ((K Xx) Xy)) ((K Xy) Xz))) ((K Xx) Xz))) ((and ((and (((K Xx) Xy)->False)) (((K Xy) Xz)->False))) (((K Xx) Xz)->False)))))))))) (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S Xx)) (S Xy))) (S Xz))) (not (((eq fofType) Xx) Xy))))
% (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) Xx)))) ((or ((and ((and ((K Xx) Xy)) ((K Xy) Xz))) ((K Xx) Xz))) ((and ((and (((K Xx) Xy)->False)) (((K Xy) Xz)->False))) (((K Xx) Xz)->False))))))))))
% 7.04/8.19 Found (eq_ref0 (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S Xx)) (S Xy))) (S Xz))) (not (((eq fofType) Xx) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) Xx)))) ((or ((and ((and ((K Xx) Xy)) ((K Xy) Xz))) ((K Xx) Xz))) ((and ((and (((K Xx) Xy)->False)) (((K Xy) Xz)->False))) (((K Xx) Xz)->False)))))))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S Xx)) (S Xy))) (S Xz))) (not (((eq fofType) Xx) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) Xx)))) ((or ((and ((and ((K Xx) Xy)) ((K Xy) Xz))) ((K Xx) Xz))) ((and ((and (((K Xx) Xy)->False)) (((K Xy) Xz)->False))) (((K Xx) Xz)->False)))))))))) b)
% 7.04/8.19 Found ((eq_ref (fofType->Prop)) (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S Xx)) (S Xy))) (S Xz))) (not (((eq fofType) Xx) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) Xx)))) ((or ((and ((and ((K Xx) Xy)) ((K Xy) Xz))) ((K Xx) Xz))) ((and ((and (((K Xx) Xy)->False)) (((K Xy) Xz)->False))) (((K Xx) Xz)->False)))))))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S Xx)) (S Xy))) (S Xz))) (not (((eq fofType) Xx) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) Xx)))) ((or ((and ((and ((K Xx) Xy)) ((K Xy) Xz))) ((K Xx) Xz))) ((and ((and (((K Xx) Xy)->False)) (((K Xy) Xz)->False))) (((K Xx) Xz)->False)))))))))) b)
% 7.04/8.19 Found ((eq_ref (fofType->Prop)) (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S Xx)) (S Xy))) (S Xz))) (not (((eq fofType) Xx) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) Xx)))) ((or ((and ((and ((K Xx) Xy)) ((K Xy) Xz))) ((K Xx) Xz))) ((and ((and (((K Xx) Xy)->False)) (((K Xy) Xz)->False))) (((K Xx) Xz)->False)))))))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S Xx)) (S Xy))) (S Xz))) (not (((eq fofType) Xx) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) Xx)))) ((or ((and ((and ((K Xx) Xy)) ((K Xy) Xz))) ((K Xx) Xz))) ((and ((and (((K Xx) Xy)->False)) (((K Xy) Xz)->False))) (((K Xx) Xz)->False)))))))))) b)
% 7.04/8.19 Found ((eq_ref (fofType->Prop)) (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S Xx)) (S Xy))) (S Xz))) (not (((eq fofType) Xx) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) Xx)))) ((or ((and ((and ((K Xx) Xy)) ((K Xy) Xz))) ((K Xx) Xz))) ((and ((and (((K Xx) Xy)->False)) (((K Xy) Xz)->False))) (((K Xx) Xz)->False)))))))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S Xx)) (S Xy))) (S Xz))) (not (((eq fofType) Xx) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) Xx)))) ((or ((and ((and ((K Xx) Xy)) ((K Xy) Xz))) ((K Xx) Xz))) ((and ((and (((K Xx) Xy)->False)) (((K Xy) Xz)->False))) (((K Xx) Xz)->False)))))))))) b)
% 11.74/12.82 Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S Xx)) (S Xy))) (S Xz))) (not (((eq fofType) Xx) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) Xx)))) ((or ((and ((and ((K Xx) Xy)) ((K Xy) Xz))) ((K Xx) Xz))) ((and ((and (((K Xx) Xy)->False)) (((K Xy) Xz)->False))) (((K Xx) Xz)->False)))))))))):(((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S Xx)) (S Xy))) (S Xz))) (not (((eq fofType) Xx) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) Xx)))) ((or ((and ((and ((K Xx) Xy)) ((K Xy) Xz))) ((K Xx) Xz))) ((and ((and (((K Xx) Xy)->False)) (((K Xy) Xz)->False))) (((K Xx) Xz)->False)))))))))) (fun (x:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x)) (S Xy))) (S Xz))) (not (((eq
% fofType) x) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x)))) ((or ((and ((and ((K x) Xy)) ((K Xy) Xz))) ((K x) Xz))) ((and ((and (((K x) Xy)->False)) (((K Xy) Xz)->False))) (((K x) Xz)->False))))))))))
% 11.74/12.82 Found (eta_expansion_dep00 (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S Xx)) (S Xy))) (S Xz))) (not (((eq fofType) Xx) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) Xx)))) ((or ((and ((and ((K Xx) Xy)) ((K Xy) Xz))) ((K Xx) Xz))) ((and ((and (((K Xx) Xy)->False)) (((K Xy) Xz)->False))) (((K Xx) Xz)->False)))))))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S Xx)) (S Xy))) (S Xz))) (not (((eq fofType) Xx) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) Xx)))) ((or ((and ((and ((K Xx) Xy)) ((K Xy) Xz))) ((K Xx) Xz))) ((and ((and (((K Xx) Xy)->False)) (((K Xy) Xz)->False))) (((K Xx) Xz)->False)))))))))) b)
% 11.74/12.82 Found ((eta_expansion_dep0 (fun (x5:fofType)=> Prop)) (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S Xx)) (S Xy))) (S Xz))) (not (((eq fofType) Xx) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) Xx)))) ((or ((and ((and ((K Xx) Xy)) ((K Xy) Xz))) ((K Xx) Xz))) ((and ((and (((K Xx) Xy)->False)) (((K Xy) Xz)->False))) (((K Xx) Xz)->False)))))))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S Xx)) (S Xy))) (S Xz))) (not (((eq fofType) Xx) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) Xx)))) ((or ((and ((and ((K Xx) Xy)) ((K Xy) Xz))) ((K Xx) Xz))) ((and ((and (((K Xx) Xy)->False)) (((K Xy) Xz)->False))) (((K Xx) Xz)->False)))))))))) b)
% 11.74/12.82 Found (((eta_expansion_dep fofType) (fun (x5:fofType)=> Prop)) (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S Xx)) (S Xy))) (S Xz))) (not (((eq fofType) Xx) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) Xx)))) ((or ((and ((and ((K Xx) Xy)) ((K Xy) Xz))) ((K Xx) Xz))) ((and ((and (((K Xx) Xy)->False)) (((K Xy) Xz)->False))) (((K Xx) Xz)->False)))))))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S Xx)) (S Xy))) (S Xz))) (not (((eq fofType) Xx) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) Xx)))) ((or ((and ((and ((K Xx) Xy)) ((K Xy) Xz))) ((K Xx) Xz))) ((and ((and (((K Xx) Xy)->False)) (((K Xy) Xz)->False))) (((K Xx) Xz)->False)))))))))) b)
% 21.45/22.50 Found (((eta_expansion_dep fofType) (fun (x5:fofType)=> Prop)) (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S Xx)) (S Xy))) (S Xz))) (not (((eq fofType) Xx) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) Xx)))) ((or ((and ((and ((K Xx) Xy)) ((K Xy) Xz))) ((K Xx) Xz))) ((and ((and (((K Xx) Xy)->False)) (((K Xy) Xz)->False))) (((K Xx) Xz)->False)))))))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S Xx)) (S Xy))) (S Xz))) (not (((eq fofType) Xx) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) Xx)))) ((or ((and ((and ((K Xx) Xy)) ((K Xy) Xz))) ((K Xx) Xz))) ((and ((and (((K Xx) Xy)->False)) (((K Xy) Xz)->False))) (((K Xx) Xz)->False)))))))))) b)
% 21.45/22.50 Found (((eta_expansion_dep fofType) (fun (x5:fofType)=> Prop)) (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S Xx)) (S Xy))) (S Xz))) (not (((eq fofType) Xx) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) Xx)))) ((or ((and ((and ((K Xx) Xy)) ((K Xy) Xz))) ((K Xx) Xz))) ((and ((and (((K Xx) Xy)->False)) (((K Xy) Xz)->False))) (((K Xx) Xz)->False)))))))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S Xx)) (S Xy))) (S Xz))) (not (((eq fofType) Xx) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) Xx)))) ((or ((and ((and ((K Xx) Xy)) ((K Xy) Xz))) ((K Xx) Xz))) ((and ((and (((K Xx) Xy)->False)) (((K Xy) Xz)->False))) (((K Xx) Xz)->False)))))))))) b)
% 21.45/22.50 Found eta_expansion000:=(eta_expansion00 (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S Xx)) (S Xy))) (S Xz))) (not (((eq fofType) Xx) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) Xx)))) ((or ((and ((and ((K Xx) Xy)) ((K Xy) Xz))) ((K Xx) Xz))) ((and ((and (((K Xx) Xy)->False)) (((K Xy) Xz)->False))) (((K Xx) Xz)->False)))))))))):(((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S Xx)) (S Xy))) (S Xz))) (not (((eq fofType) Xx) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) Xx)))) ((or ((and ((and ((K Xx) Xy)) ((K Xy) Xz))) ((K Xx) Xz))) ((and ((and (((K Xx) Xy)->False)) (((K Xy) Xz)->False))) (((K Xx) Xz)->False)))))))))) (fun (x:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x)) (S Xy))) (S Xz))) (not (((eq fofType)
% x) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x)))) ((or ((and ((and ((K x) Xy)) ((K Xy) Xz))) ((K x) Xz))) ((and ((and (((K x) Xy)->False)) (((K Xy) Xz)->False))) (((K x) Xz)->False))))))))))
% 21.45/22.50 Found (eta_expansion00 (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S Xx)) (S Xy))) (S Xz))) (not (((eq fofType) Xx) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) Xx)))) ((or ((and ((and ((K Xx) Xy)) ((K Xy) Xz))) ((K Xx) Xz))) ((and ((and (((K Xx) Xy)->False)) (((K Xy) Xz)->False))) (((K Xx) Xz)->False)))))))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S Xx)) (S Xy))) (S Xz))) (not (((eq fofType) Xx) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) Xx)))) ((or ((and ((and ((K Xx) Xy)) ((K Xy) Xz))) ((K Xx) Xz))) ((and ((and (((K Xx) Xy)->False)) (((K Xy) Xz)->False))) (((K Xx) Xz)->False)))))))))) b)
% 21.45/22.50 Found ((eta_expansion0 Prop) (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S Xx)) (S Xy))) (S Xz))) (not (((eq fofType) Xx) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) Xx)))) ((or ((and ((and ((K Xx) Xy)) ((K Xy) Xz))) ((K Xx) Xz))) ((and ((and (((K Xx) Xy)->False)) (((K Xy) Xz)->False))) (((K Xx) Xz)->False)))))))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S Xx)) (S Xy))) (S Xz))) (not (((eq fofType) Xx) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) Xx)))) ((or ((and ((and ((K Xx) Xy)) ((K Xy) Xz))) ((K Xx) Xz))) ((and ((and (((K Xx) Xy)->False)) (((K Xy) Xz)->False))) (((K Xx) Xz)->False)))))))))) b)
% 64.86/65.96 Found (((eta_expansion fofType) Prop) (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S Xx)) (S Xy))) (S Xz))) (not (((eq fofType) Xx) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) Xx)))) ((or ((and ((and ((K Xx) Xy)) ((K Xy) Xz))) ((K Xx) Xz))) ((and ((and (((K Xx) Xy)->False)) (((K Xy) Xz)->False))) (((K Xx) Xz)->False)))))))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S Xx)) (S Xy))) (S Xz))) (not (((eq fofType) Xx) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) Xx)))) ((or ((and ((and ((K Xx) Xy)) ((K Xy) Xz))) ((K Xx) Xz))) ((and ((and (((K Xx) Xy)->False)) (((K Xy) Xz)->False))) (((K Xx) Xz)->False)))))))))) b)
% 64.86/65.96 Found (((eta_expansion fofType) Prop) (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S Xx)) (S Xy))) (S Xz))) (not (((eq fofType) Xx) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) Xx)))) ((or ((and ((and ((K Xx) Xy)) ((K Xy) Xz))) ((K Xx) Xz))) ((and ((and (((K Xx) Xy)->False)) (((K Xy) Xz)->False))) (((K Xx) Xz)->False)))))))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S Xx)) (S Xy))) (S Xz))) (not (((eq fofType) Xx) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) Xx)))) ((or ((and ((and ((K Xx) Xy)) ((K Xy) Xz))) ((K Xx) Xz))) ((and ((and (((K Xx) Xy)->False)) (((K Xy) Xz)->False))) (((K Xx) Xz)->False)))))))))) b)
% 64.86/65.96 Found (((eta_expansion fofType) Prop) (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S Xx)) (S Xy))) (S Xz))) (not (((eq fofType) Xx) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) Xx)))) ((or ((and ((and ((K Xx) Xy)) ((K Xy) Xz))) ((K Xx) Xz))) ((and ((and (((K Xx) Xy)->False)) (((K Xy) Xz)->False))) (((K Xx) Xz)->False)))))))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S Xx)) (S Xy))) (S Xz))) (not (((eq fofType) Xx) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) Xx)))) ((or ((and ((and ((K Xx) Xy)) ((K Xy) Xz))) ((K Xx) Xz))) ((and ((and (((K Xx) Xy)->False)) (((K Xy) Xz)->False))) (((K Xx) Xz)->False)))))))))) b)
% 64.86/65.96 Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% 64.86/65.96 Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S Xx)) (S Xy))) (S Xz))) (not (((eq fofType) Xx) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) Xx)))) ((or ((and ((and ((K Xx) Xy)) ((K Xy) Xz))) ((K Xx) Xz))) ((and ((and (((K Xx) Xy)->False)) (((K Xy) Xz)->False))) (((K Xx) Xz)->False))))))))))
% 64.86/65.96 Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S Xx)) (S Xy))) (S Xz))) (not (((eq fofType) Xx) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) Xx)))) ((or ((and ((and ((K Xx) Xy)) ((K Xy) Xz))) ((K Xx) Xz))) ((and ((and (((K Xx) Xy)->False)) (((K Xy) Xz)->False))) (((K Xx) Xz)->False))))))))))
% 74.77/75.84 Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S Xx)) (S Xy))) (S Xz))) (not (((eq fofType) Xx) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) Xx)))) ((or ((and ((and ((K Xx) Xy)) ((K Xy) Xz))) ((K Xx) Xz))) ((and ((and (((K Xx) Xy)->False)) (((K Xy) Xz)->False))) (((K Xx) Xz)->False))))))))))
% 74.77/75.84 Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S Xx)) (S Xy))) (S Xz))) (not (((eq fofType) Xx) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) Xx)))) ((or ((and ((and ((K Xx) Xy)) ((K Xy) Xz))) ((K Xx) Xz))) ((and ((and (((K Xx) Xy)->False)) (((K Xy) Xz)->False))) (((K Xx) Xz)->False))))))))))
% 74.77/75.84 Found eta_expansion000:=(eta_expansion00 a):(((eq (fofType->Prop)) a) (fun (x:fofType)=> (a x)))
% 74.77/75.84 Found (eta_expansion00 a) as proof of (((eq (fofType->Prop)) a) b)
% 74.77/75.84 Found ((eta_expansion0 Prop) a) as proof of (((eq (fofType->Prop)) a) b)
% 74.77/75.84 Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) b)
% 74.77/75.84 Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) b)
% 74.77/75.84 Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) b)
% 74.77/75.84 Found eq_ref00:=(eq_ref0 a):(((eq (fofType->Prop)) a) a)
% 74.77/75.84 Found (eq_ref0 a) as proof of (((eq (fofType->Prop)) a) b)
% 74.77/75.84 Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) b)
% 74.77/75.84 Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) b)
% 74.77/75.84 Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) b)
% 74.77/75.84 Found eq_ref00:=(eq_ref0 (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))):(((eq (fofType->Prop)) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz)))
% ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False))))))))
% 74.77/75.84 Found (eq_ref0 (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))) as proof of (((eq (fofType->Prop)) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))) b)
% 74.77/75.84 Found ((eq_ref (fofType->Prop)) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))) as proof of (((eq (fofType->Prop)) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))) b)
% 83.96/85.04 Found ((eq_ref (fofType->Prop)) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))) as proof of (((eq (fofType->Prop)) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))) b)
% 83.96/85.04 Found ((eq_ref (fofType->Prop)) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))) as proof of (((eq (fofType->Prop)) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))) b)
% 83.96/85.04 Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% 83.96/85.04 Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S Xx)) (S Xy))) (S Xz))) (not (((eq fofType) Xx) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) Xx)))) ((or ((and ((and ((K Xx) Xy)) ((K Xy) Xz))) ((K Xx) Xz))) ((and ((and (((K Xx) Xy)->False)) (((K Xy) Xz)->False))) (((K Xx) Xz)->False))))))))))
% 83.96/85.04 Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S Xx)) (S Xy))) (S Xz))) (not (((eq fofType) Xx) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) Xx)))) ((or ((and ((and ((K Xx) Xy)) ((K Xy) Xz))) ((K Xx) Xz))) ((and ((and (((K Xx) Xy)->False)) (((K Xy) Xz)->False))) (((K Xx) Xz)->False))))))))))
% 83.96/85.04 Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S Xx)) (S Xy))) (S Xz))) (not (((eq fofType) Xx) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) Xx)))) ((or ((and ((and ((K Xx) Xy)) ((K Xy) Xz))) ((K Xx) Xz))) ((and ((and (((K Xx) Xy)->False)) (((K Xy) Xz)->False))) (((K Xx) Xz)->False))))))))))
% 83.96/85.04 Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S Xx)) (S Xy))) (S Xz))) (not (((eq fofType) Xx) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) Xx)))) ((or ((and ((and ((K Xx) Xy)) ((K Xy) Xz))) ((K Xx) Xz))) ((and ((and (((K Xx) Xy)->False)) (((K Xy) Xz)->False))) (((K Xx) Xz)->False))))))))))
% 83.96/85.04 Found eta_expansion_dep000:=(eta_expansion_dep00 a):(((eq (fofType->Prop)) a) (fun (x:fofType)=> (a x)))
% 83.96/85.04 Found (eta_expansion_dep00 a) as proof of (((eq (fofType->Prop)) a) b)
% 83.96/85.04 Found ((eta_expansion_dep0 (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) b)
% 83.96/85.04 Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) b)
% 97.17/98.21 Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) b)
% 97.17/98.21 Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) b)
% 97.17/98.21 Found eq_ref00:=(eq_ref0 a):(((eq (fofType->Prop)) a) a)
% 97.17/98.21 Found (eq_ref0 a) as proof of (((eq (fofType->Prop)) a) b)
% 97.17/98.21 Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) b)
% 97.17/98.21 Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) b)
% 97.17/98.21 Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) b)
% 97.17/98.21 Found eta_expansion000:=(eta_expansion00 (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x2)) (S Xy))) (S Xz))) (not (((eq fofType) x2) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x2)))) ((or ((and ((and ((K x2) Xy)) ((K Xy) Xz))) ((K x2) Xz))) ((and ((and (((K x2) Xy)->False)) (((K Xy) Xz)->False))) (((K x2) Xz)->False)))))))):(((eq (fofType->Prop)) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x2)) (S Xy))) (S Xz))) (not (((eq fofType) x2) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x2)))) ((or ((and ((and ((K x2) Xy)) ((K Xy) Xz))) ((K x2) Xz))) ((and ((and (((K x2) Xy)->False)) (((K Xy) Xz)->False))) (((K x2) Xz)->False)))))))) (fun (x:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x2)) (S x))) (S Xz))) (not (((eq fofType) x2) x)))) (not (((eq fofType) x) Xz)))) (not (((eq fofType) Xz) x2)))) ((or ((and ((and ((K x2) x)) ((K
% x) Xz))) ((K x2) Xz))) ((and ((and (((K x2) x)->False)) (((K x) Xz)->False))) (((K x2) Xz)->False))))))))
% 97.17/98.21 Found (eta_expansion00 (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x2)) (S Xy))) (S Xz))) (not (((eq fofType) x2) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x2)))) ((or ((and ((and ((K x2) Xy)) ((K Xy) Xz))) ((K x2) Xz))) ((and ((and (((K x2) Xy)->False)) (((K Xy) Xz)->False))) (((K x2) Xz)->False)))))))) as proof of (((eq (fofType->Prop)) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x2)) (S Xy))) (S Xz))) (not (((eq fofType) x2) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x2)))) ((or ((and ((and ((K x2) Xy)) ((K Xy) Xz))) ((K x2) Xz))) ((and ((and (((K x2) Xy)->False)) (((K Xy) Xz)->False))) (((K x2) Xz)->False)))))))) b)
% 97.17/98.21 Found ((eta_expansion0 Prop) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x2)) (S Xy))) (S Xz))) (not (((eq fofType) x2) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x2)))) ((or ((and ((and ((K x2) Xy)) ((K Xy) Xz))) ((K x2) Xz))) ((and ((and (((K x2) Xy)->False)) (((K Xy) Xz)->False))) (((K x2) Xz)->False)))))))) as proof of (((eq (fofType->Prop)) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x2)) (S Xy))) (S Xz))) (not (((eq fofType) x2) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x2)))) ((or ((and ((and ((K x2) Xy)) ((K Xy) Xz))) ((K x2) Xz))) ((and ((and (((K x2) Xy)->False)) (((K Xy) Xz)->False))) (((K x2) Xz)->False)))))))) b)
% 97.17/98.21 Found (((eta_expansion fofType) Prop) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x2)) (S Xy))) (S Xz))) (not (((eq fofType) x2) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x2)))) ((or ((and ((and ((K x2) Xy)) ((K Xy) Xz))) ((K x2) Xz))) ((and ((and (((K x2) Xy)->False)) (((K Xy) Xz)->False))) (((K x2) Xz)->False)))))))) as proof of (((eq (fofType->Prop)) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x2)) (S Xy))) (S Xz))) (not (((eq fofType) x2) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x2)))) ((or ((and ((and ((K x2) Xy)) ((K Xy) Xz))) ((K x2) Xz))) ((and ((and (((K x2) Xy)->False)) (((K Xy) Xz)->False))) (((K x2) Xz)->False)))))))) b)
% 97.17/98.21 Found (((eta_expansion fofType) Prop) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x2)) (S Xy))) (S Xz))) (not (((eq fofType) x2) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x2)))) ((or ((and ((and ((K x2) Xy)) ((K Xy) Xz))) ((K x2) Xz))) ((and ((and (((K x2) Xy)->False)) (((K Xy) Xz)->False))) (((K x2) Xz)->False)))))))) as proof of (((eq (fofType->Prop)) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x2)) (S Xy))) (S Xz))) (not (((eq fofType) x2) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x2)))) ((or ((and ((and ((K x2) Xy)) ((K Xy) Xz))) ((K x2) Xz))) ((and ((and (((K x2) Xy)->False)) (((K Xy) Xz)->False))) (((K x2) Xz)->False)))))))) b)
% 105.58/106.65 Found (((eta_expansion fofType) Prop) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x2)) (S Xy))) (S Xz))) (not (((eq fofType) x2) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x2)))) ((or ((and ((and ((K x2) Xy)) ((K Xy) Xz))) ((K x2) Xz))) ((and ((and (((K x2) Xy)->False)) (((K Xy) Xz)->False))) (((K x2) Xz)->False)))))))) as proof of (((eq (fofType->Prop)) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x2)) (S Xy))) (S Xz))) (not (((eq fofType) x2) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x2)))) ((or ((and ((and ((K x2) Xy)) ((K Xy) Xz))) ((K x2) Xz))) ((and ((and (((K x2) Xy)->False)) (((K Xy) Xz)->False))) (((K x2) Xz)->False)))))))) b)
% 105.58/106.65 Found eta_expansion000:=(eta_expansion00 (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))):(((eq (fofType->Prop)) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))) (fun (x:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S x))) (S Xz))) (not (((eq fofType) x0) x)))) (not (((eq fofType) x) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) x)) ((K
% x) Xz))) ((K x0) Xz))) ((and ((and (((K x0) x)->False)) (((K x) Xz)->False))) (((K x0) Xz)->False))))))))
% 105.58/106.65 Found (eta_expansion00 (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))) as proof of (((eq (fofType->Prop)) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))) b)
% 105.58/106.65 Found ((eta_expansion0 Prop) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))) as proof of (((eq (fofType->Prop)) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))) b)
% 114.27/115.40 Found (((eta_expansion fofType) Prop) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))) as proof of (((eq (fofType->Prop)) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))) b)
% 114.27/115.40 Found (((eta_expansion fofType) Prop) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))) as proof of (((eq (fofType->Prop)) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))) b)
% 114.27/115.40 Found (((eta_expansion fofType) Prop) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))) as proof of (((eq (fofType->Prop)) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))) b)
% 114.27/115.40 Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% 114.27/115.40 Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S Xx)) (S Xy))) (S Xz))) (not (((eq fofType) Xx) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) Xx)))) ((or ((and ((and ((K Xx) Xy)) ((K Xy) Xz))) ((K Xx) Xz))) ((and ((and (((K Xx) Xy)->False)) (((K Xy) Xz)->False))) (((K Xx) Xz)->False))))))))))
% 114.27/115.40 Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S Xx)) (S Xy))) (S Xz))) (not (((eq fofType) Xx) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) Xx)))) ((or ((and ((and ((K Xx) Xy)) ((K Xy) Xz))) ((K Xx) Xz))) ((and ((and (((K Xx) Xy)->False)) (((K Xy) Xz)->False))) (((K Xx) Xz)->False))))))))))
% 114.27/115.40 Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S Xx)) (S Xy))) (S Xz))) (not (((eq fofType) Xx) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) Xx)))) ((or ((and ((and ((K Xx) Xy)) ((K Xy) Xz))) ((K Xx) Xz))) ((and ((and (((K Xx) Xy)->False)) (((K Xy) Xz)->False))) (((K Xx) Xz)->False))))))))))
% 114.27/115.40 Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S Xx)) (S Xy))) (S Xz))) (not (((eq fofType) Xx) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) Xx)))) ((or ((and ((and ((K Xx) Xy)) ((K Xy) Xz))) ((K Xx) Xz))) ((and ((and (((K Xx) Xy)->False)) (((K Xy) Xz)->False))) (((K Xx) Xz)->False))))))))))
% 140.08/141.14 Found eta_expansion000:=(eta_expansion00 a):(((eq (fofType->Prop)) a) (fun (x:fofType)=> (a x)))
% 140.08/141.14 Found (eta_expansion00 a) as proof of (((eq (fofType->Prop)) a) b)
% 140.08/141.14 Found ((eta_expansion0 Prop) a) as proof of (((eq (fofType->Prop)) a) b)
% 140.08/141.14 Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) b)
% 140.08/141.14 Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) b)
% 140.08/141.14 Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) b)
% 140.08/141.14 Found eq_ref00:=(eq_ref0 a):(((eq (fofType->Prop)) a) a)
% 140.08/141.14 Found (eq_ref0 a) as proof of (((eq (fofType->Prop)) a) b)
% 140.08/141.14 Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) b)
% 140.08/141.14 Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) b)
% 140.08/141.14 Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) b)
% 140.08/141.14 Found eq_ref00:=(eq_ref0 (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x4)) (S Xy))) (S Xz))) (not (((eq fofType) x4) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x4)))) ((or ((and ((and ((K x4) Xy)) ((K Xy) Xz))) ((K x4) Xz))) ((and ((and (((K x4) Xy)->False)) (((K Xy) Xz)->False))) (((K x4) Xz)->False)))))))):(((eq (fofType->Prop)) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x4)) (S Xy))) (S Xz))) (not (((eq fofType) x4) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x4)))) ((or ((and ((and ((K x4) Xy)) ((K Xy) Xz))) ((K x4) Xz))) ((and ((and (((K x4) Xy)->False)) (((K Xy) Xz)->False))) (((K x4) Xz)->False)))))))) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x4)) (S Xy))) (S Xz))) (not (((eq fofType) x4) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x4)))) ((or ((and ((and ((K x4) Xy)) ((K Xy) Xz)))
% ((K x4) Xz))) ((and ((and (((K x4) Xy)->False)) (((K Xy) Xz)->False))) (((K x4) Xz)->False))))))))
% 140.08/141.14 Found (eq_ref0 (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x4)) (S Xy))) (S Xz))) (not (((eq fofType) x4) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x4)))) ((or ((and ((and ((K x4) Xy)) ((K Xy) Xz))) ((K x4) Xz))) ((and ((and (((K x4) Xy)->False)) (((K Xy) Xz)->False))) (((K x4) Xz)->False)))))))) as proof of (((eq (fofType->Prop)) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x4)) (S Xy))) (S Xz))) (not (((eq fofType) x4) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x4)))) ((or ((and ((and ((K x4) Xy)) ((K Xy) Xz))) ((K x4) Xz))) ((and ((and (((K x4) Xy)->False)) (((K Xy) Xz)->False))) (((K x4) Xz)->False)))))))) b)
% 140.08/141.14 Found ((eq_ref (fofType->Prop)) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x4)) (S Xy))) (S Xz))) (not (((eq fofType) x4) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x4)))) ((or ((and ((and ((K x4) Xy)) ((K Xy) Xz))) ((K x4) Xz))) ((and ((and (((K x4) Xy)->False)) (((K Xy) Xz)->False))) (((K x4) Xz)->False)))))))) as proof of (((eq (fofType->Prop)) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x4)) (S Xy))) (S Xz))) (not (((eq fofType) x4) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x4)))) ((or ((and ((and ((K x4) Xy)) ((K Xy) Xz))) ((K x4) Xz))) ((and ((and (((K x4) Xy)->False)) (((K Xy) Xz)->False))) (((K x4) Xz)->False)))))))) b)
% 140.08/141.14 Found ((eq_ref (fofType->Prop)) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x4)) (S Xy))) (S Xz))) (not (((eq fofType) x4) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x4)))) ((or ((and ((and ((K x4) Xy)) ((K Xy) Xz))) ((K x4) Xz))) ((and ((and (((K x4) Xy)->False)) (((K Xy) Xz)->False))) (((K x4) Xz)->False)))))))) as proof of (((eq (fofType->Prop)) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x4)) (S Xy))) (S Xz))) (not (((eq fofType) x4) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x4)))) ((or ((and ((and ((K x4) Xy)) ((K Xy) Xz))) ((K x4) Xz))) ((and ((and (((K x4) Xy)->False)) (((K Xy) Xz)->False))) (((K x4) Xz)->False)))))))) b)
% 145.88/146.95 Found ((eq_ref (fofType->Prop)) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x4)) (S Xy))) (S Xz))) (not (((eq fofType) x4) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x4)))) ((or ((and ((and ((K x4) Xy)) ((K Xy) Xz))) ((K x4) Xz))) ((and ((and (((K x4) Xy)->False)) (((K Xy) Xz)->False))) (((K x4) Xz)->False)))))))) as proof of (((eq (fofType->Prop)) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x4)) (S Xy))) (S Xz))) (not (((eq fofType) x4) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x4)))) ((or ((and ((and ((K x4) Xy)) ((K Xy) Xz))) ((K x4) Xz))) ((and ((and (((K x4) Xy)->False)) (((K Xy) Xz)->False))) (((K x4) Xz)->False)))))))) b)
% 145.88/146.95 Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% 145.88/146.95 Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) b)
% 145.88/146.95 Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% 145.88/146.95 Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% 145.88/146.95 Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% 145.88/146.95 Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% 145.88/146.95 Found (eq_ref0 b) as proof of (((eq Prop) b) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))))
% 145.88/146.95 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))))
% 145.88/146.95 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))))
% 145.88/146.95 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))))
% 145.88/146.95 Found ((eq_trans0000 ((eq_ref Prop) (f x0))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f x0)) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))))
% 145.88/146.95 Found (((eq_trans000 ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False))))))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f x0)) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))))
% 145.88/146.96 Found ((((eq_trans00 ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False))))))))) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False))))))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType)
% Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))))) as proof of (((eq Prop) (f x0)) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))))
% 145.88/146.96 Found (((((eq_trans0 (f x0)) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False))))))))) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False))))))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq
% fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))))) as proof of (((eq Prop) (f x0)) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))))
% 145.88/146.96 Found ((((((eq_trans Prop) (f x0)) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False))))))))) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False))))))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not
% (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))))) as proof of (((eq Prop) (f x0)) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))))
% 146.08/147.18 Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% 146.08/147.18 Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) b)
% 146.08/147.18 Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% 146.08/147.18 Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% 146.08/147.18 Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% 146.08/147.18 Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% 146.08/147.18 Found (eq_ref0 b) as proof of (((eq Prop) b) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))))
% 146.08/147.18 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))))
% 146.08/147.18 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))))
% 146.08/147.18 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))))
% 146.08/147.18 Found ((eq_trans0000 ((eq_ref Prop) (f x0))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f x0)) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))))
% 146.08/147.18 Found (((eq_trans000 ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False))))))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f x0)) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))))
% 146.08/147.19 Found ((((eq_trans00 ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False))))))))) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False))))))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType)
% Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))))) as proof of (((eq Prop) (f x0)) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))))
% 146.08/147.19 Found (((((eq_trans0 (f x0)) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False))))))))) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False))))))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq
% fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))))) as proof of (((eq Prop) (f x0)) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))))
% 146.08/147.19 Found ((((((eq_trans Prop) (f x0)) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False))))))))) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False))))))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not
% (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))))) as proof of (((eq Prop) (f x0)) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))))
% 153.69/154.70 Found eq_ref000:=(eq_ref00 P0):((P0 (f x0))->(P0 (f x0)))
% 153.69/154.70 Found (eq_ref00 P0) as proof of (P1 (f x0))
% 153.69/154.70 Found ((eq_ref0 (f x0)) P0) as proof of (P1 (f x0))
% 153.69/154.70 Found (((eq_ref Prop) (f x0)) P0) as proof of (P1 (f x0))
% 153.69/154.70 Found (((eq_ref Prop) (f x0)) P0) as proof of (P1 (f x0))
% 153.69/154.70 Found eq_ref000:=(eq_ref00 P0):((P0 (f x0))->(P0 (f x0)))
% 153.69/154.70 Found (eq_ref00 P0) as proof of (P1 (f x0))
% 153.69/154.70 Found ((eq_ref0 (f x0)) P0) as proof of (P1 (f x0))
% 153.69/154.70 Found (((eq_ref Prop) (f x0)) P0) as proof of (P1 (f x0))
% 153.69/154.70 Found (((eq_ref Prop) (f x0)) P0) as proof of (P1 (f x0))
% 153.69/154.70 Found eq_ref00:=(eq_ref0 (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))):(((eq (fofType->Prop)) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz)))
% ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False))))))))
% 153.69/154.70 Found (eq_ref0 (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))) as proof of (((eq (fofType->Prop)) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))) b)
% 153.69/154.70 Found ((eq_ref (fofType->Prop)) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))) as proof of (((eq (fofType->Prop)) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))) b)
% 153.69/154.70 Found ((eq_ref (fofType->Prop)) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))) as proof of (((eq (fofType->Prop)) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))) b)
% 153.99/155.05 Found ((eq_ref (fofType->Prop)) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))) as proof of (((eq (fofType->Prop)) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))) b)
% 153.99/155.05 Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x2)) (S Xy))) (S Xz))) (not (((eq fofType) x2) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x2)))) ((or ((and ((and ((K x2) Xy)) ((K Xy) Xz))) ((K x2) Xz))) ((and ((and (((K x2) Xy)->False)) (((K Xy) Xz)->False))) (((K x2) Xz)->False)))))))):(((eq (fofType->Prop)) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x2)) (S Xy))) (S Xz))) (not (((eq fofType) x2) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x2)))) ((or ((and ((and ((K x2) Xy)) ((K Xy) Xz))) ((K x2) Xz))) ((and ((and (((K x2) Xy)->False)) (((K Xy) Xz)->False))) (((K x2) Xz)->False)))))))) (fun (x:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x2)) (S x))) (S Xz))) (not (((eq fofType) x2) x)))) (not (((eq fofType) x) Xz)))) (not (((eq fofType) Xz) x2)))) ((or ((and ((and ((K x2)
% x)) ((K x) Xz))) ((K x2) Xz))) ((and ((and (((K x2) x)->False)) (((K x) Xz)->False))) (((K x2) Xz)->False))))))))
% 153.99/155.05 Found (eta_expansion_dep00 (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x2)) (S Xy))) (S Xz))) (not (((eq fofType) x2) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x2)))) ((or ((and ((and ((K x2) Xy)) ((K Xy) Xz))) ((K x2) Xz))) ((and ((and (((K x2) Xy)->False)) (((K Xy) Xz)->False))) (((K x2) Xz)->False)))))))) as proof of (((eq (fofType->Prop)) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x2)) (S Xy))) (S Xz))) (not (((eq fofType) x2) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x2)))) ((or ((and ((and ((K x2) Xy)) ((K Xy) Xz))) ((K x2) Xz))) ((and ((and (((K x2) Xy)->False)) (((K Xy) Xz)->False))) (((K x2) Xz)->False)))))))) b)
% 153.99/155.05 Found ((eta_expansion_dep0 (fun (x6:fofType)=> Prop)) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x2)) (S Xy))) (S Xz))) (not (((eq fofType) x2) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x2)))) ((or ((and ((and ((K x2) Xy)) ((K Xy) Xz))) ((K x2) Xz))) ((and ((and (((K x2) Xy)->False)) (((K Xy) Xz)->False))) (((K x2) Xz)->False)))))))) as proof of (((eq (fofType->Prop)) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x2)) (S Xy))) (S Xz))) (not (((eq fofType) x2) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x2)))) ((or ((and ((and ((K x2) Xy)) ((K Xy) Xz))) ((K x2) Xz))) ((and ((and (((K x2) Xy)->False)) (((K Xy) Xz)->False))) (((K x2) Xz)->False)))))))) b)
% 160.39/161.49 Found (((eta_expansion_dep fofType) (fun (x6:fofType)=> Prop)) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x2)) (S Xy))) (S Xz))) (not (((eq fofType) x2) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x2)))) ((or ((and ((and ((K x2) Xy)) ((K Xy) Xz))) ((K x2) Xz))) ((and ((and (((K x2) Xy)->False)) (((K Xy) Xz)->False))) (((K x2) Xz)->False)))))))) as proof of (((eq (fofType->Prop)) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x2)) (S Xy))) (S Xz))) (not (((eq fofType) x2) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x2)))) ((or ((and ((and ((K x2) Xy)) ((K Xy) Xz))) ((K x2) Xz))) ((and ((and (((K x2) Xy)->False)) (((K Xy) Xz)->False))) (((K x2) Xz)->False)))))))) b)
% 160.39/161.49 Found (((eta_expansion_dep fofType) (fun (x6:fofType)=> Prop)) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x2)) (S Xy))) (S Xz))) (not (((eq fofType) x2) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x2)))) ((or ((and ((and ((K x2) Xy)) ((K Xy) Xz))) ((K x2) Xz))) ((and ((and (((K x2) Xy)->False)) (((K Xy) Xz)->False))) (((K x2) Xz)->False)))))))) as proof of (((eq (fofType->Prop)) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x2)) (S Xy))) (S Xz))) (not (((eq fofType) x2) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x2)))) ((or ((and ((and ((K x2) Xy)) ((K Xy) Xz))) ((K x2) Xz))) ((and ((and (((K x2) Xy)->False)) (((K Xy) Xz)->False))) (((K x2) Xz)->False)))))))) b)
% 160.39/161.49 Found (((eta_expansion_dep fofType) (fun (x6:fofType)=> Prop)) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x2)) (S Xy))) (S Xz))) (not (((eq fofType) x2) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x2)))) ((or ((and ((and ((K x2) Xy)) ((K Xy) Xz))) ((K x2) Xz))) ((and ((and (((K x2) Xy)->False)) (((K Xy) Xz)->False))) (((K x2) Xz)->False)))))))) as proof of (((eq (fofType->Prop)) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x2)) (S Xy))) (S Xz))) (not (((eq fofType) x2) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x2)))) ((or ((and ((and ((K x2) Xy)) ((K Xy) Xz))) ((K x2) Xz))) ((and ((and (((K x2) Xy)->False)) (((K Xy) Xz)->False))) (((K x2) Xz)->False)))))))) b)
% 160.39/161.49 Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% 160.39/161.49 Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S Xx)) (S Xy))) (S Xz))) (not (((eq fofType) Xx) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) Xx)))) ((or ((and ((and ((K Xx) Xy)) ((K Xy) Xz))) ((K Xx) Xz))) ((and ((and (((K Xx) Xy)->False)) (((K Xy) Xz)->False))) (((K Xx) Xz)->False))))))))))
% 160.39/161.49 Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S Xx)) (S Xy))) (S Xz))) (not (((eq fofType) Xx) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) Xx)))) ((or ((and ((and ((K Xx) Xy)) ((K Xy) Xz))) ((K Xx) Xz))) ((and ((and (((K Xx) Xy)->False)) (((K Xy) Xz)->False))) (((K Xx) Xz)->False))))))))))
% 160.39/161.49 Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S Xx)) (S Xy))) (S Xz))) (not (((eq fofType) Xx) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) Xx)))) ((or ((and ((and ((K Xx) Xy)) ((K Xy) Xz))) ((K Xx) Xz))) ((and ((and (((K Xx) Xy)->False)) (((K Xy) Xz)->False))) (((K Xx) Xz)->False))))))))))
% 160.39/161.49 Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S Xx)) (S Xy))) (S Xz))) (not (((eq fofType) Xx) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) Xx)))) ((or ((and ((and ((K Xx) Xy)) ((K Xy) Xz))) ((K Xx) Xz))) ((and ((and (((K Xx) Xy)->False)) (((K Xy) Xz)->False))) (((K Xx) Xz)->False))))))))))
% 190.81/191.83 Found eq_ref00:=(eq_ref0 a):(((eq (fofType->Prop)) a) a)
% 190.81/191.83 Found (eq_ref0 a) as proof of (((eq (fofType->Prop)) a) b)
% 190.81/191.83 Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) b)
% 190.81/191.83 Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) b)
% 190.81/191.83 Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) b)
% 190.81/191.83 Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% 190.81/191.83 Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) b)
% 190.81/191.83 Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) b)
% 190.81/191.83 Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) b)
% 190.81/191.83 Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) b)
% 190.81/191.83 Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% 190.81/191.83 Found (eq_ref0 b) as proof of (((eq Prop) b) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x2)) (S Xy))) (S Xz))) (not (((eq fofType) x2) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x2)))) ((or ((and ((and ((K x2) Xy)) ((K Xy) Xz))) ((K x2) Xz))) ((and ((and (((K x2) Xy)->False)) (((K Xy) Xz)->False))) (((K x2) Xz)->False)))))))))
% 190.81/191.83 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x2)) (S Xy))) (S Xz))) (not (((eq fofType) x2) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x2)))) ((or ((and ((and ((K x2) Xy)) ((K Xy) Xz))) ((K x2) Xz))) ((and ((and (((K x2) Xy)->False)) (((K Xy) Xz)->False))) (((K x2) Xz)->False)))))))))
% 190.81/191.83 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x2)) (S Xy))) (S Xz))) (not (((eq fofType) x2) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x2)))) ((or ((and ((and ((K x2) Xy)) ((K Xy) Xz))) ((K x2) Xz))) ((and ((and (((K x2) Xy)->False)) (((K Xy) Xz)->False))) (((K x2) Xz)->False)))))))))
% 190.81/191.83 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x2)) (S Xy))) (S Xz))) (not (((eq fofType) x2) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x2)))) ((or ((and ((and ((K x2) Xy)) ((K Xy) Xz))) ((K x2) Xz))) ((and ((and (((K x2) Xy)->False)) (((K Xy) Xz)->False))) (((K x2) Xz)->False)))))))))
% 190.81/191.83 Found ((eq_trans0000 ((eq_ref Prop) (f x2))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f x2)) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x2)) (S Xy))) (S Xz))) (not (((eq fofType) x2) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x2)))) ((or ((and ((and ((K x2) Xy)) ((K Xy) Xz))) ((K x2) Xz))) ((and ((and (((K x2) Xy)->False)) (((K Xy) Xz)->False))) (((K x2) Xz)->False)))))))))
% 190.81/191.83 Found (((eq_trans000 ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x2)) (S Xy))) (S Xz))) (not (((eq fofType) x2) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x2)))) ((or ((and ((and ((K x2) Xy)) ((K Xy) Xz))) ((K x2) Xz))) ((and ((and (((K x2) Xy)->False)) (((K Xy) Xz)->False))) (((K x2) Xz)->False))))))))) ((eq_ref Prop) (f x2))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f x2)) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x2)) (S Xy))) (S Xz))) (not (((eq fofType) x2) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x2)))) ((or ((and ((and ((K x2) Xy)) ((K Xy) Xz))) ((K x2) Xz))) ((and ((and (((K x2) Xy)->False)) (((K Xy) Xz)->False))) (((K x2) Xz)->False)))))))))
% 190.81/191.83 Found ((((eq_trans00 ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x2)) (S Xy))) (S Xz))) (not (((eq fofType) x2) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x2)))) ((or ((and ((and ((K x2) Xy)) ((K Xy) Xz))) ((K x2) Xz))) ((and ((and (((K x2) Xy)->False)) (((K Xy) Xz)->False))) (((K x2) Xz)->False))))))))) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x2)) (S Xy))) (S Xz))) (not (((eq fofType) x2) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x2)))) ((or ((and ((and ((K x2) Xy)) ((K Xy) Xz))) ((K x2) Xz))) ((and ((and (((K x2) Xy)->False)) (((K Xy) Xz)->False))) (((K x2) Xz)->False))))))))) ((eq_ref Prop) (f x2))) ((eq_ref Prop) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x2)) (S Xy))) (S Xz))) (not (((eq fofType) x2) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType)
% Xz) x2)))) ((or ((and ((and ((K x2) Xy)) ((K Xy) Xz))) ((K x2) Xz))) ((and ((and (((K x2) Xy)->False)) (((K Xy) Xz)->False))) (((K x2) Xz)->False)))))))))) as proof of (((eq Prop) (f x2)) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x2)) (S Xy))) (S Xz))) (not (((eq fofType) x2) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x2)))) ((or ((and ((and ((K x2) Xy)) ((K Xy) Xz))) ((K x2) Xz))) ((and ((and (((K x2) Xy)->False)) (((K Xy) Xz)->False))) (((K x2) Xz)->False)))))))))
% 190.81/191.83 Found (((((eq_trans0 (f x2)) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x2)) (S Xy))) (S Xz))) (not (((eq fofType) x2) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x2)))) ((or ((and ((and ((K x2) Xy)) ((K Xy) Xz))) ((K x2) Xz))) ((and ((and (((K x2) Xy)->False)) (((K Xy) Xz)->False))) (((K x2) Xz)->False))))))))) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x2)) (S Xy))) (S Xz))) (not (((eq fofType) x2) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x2)))) ((or ((and ((and ((K x2) Xy)) ((K Xy) Xz))) ((K x2) Xz))) ((and ((and (((K x2) Xy)->False)) (((K Xy) Xz)->False))) (((K x2) Xz)->False))))))))) ((eq_ref Prop) (f x2))) ((eq_ref Prop) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x2)) (S Xy))) (S Xz))) (not (((eq fofType) x2) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq
% fofType) Xz) x2)))) ((or ((and ((and ((K x2) Xy)) ((K Xy) Xz))) ((K x2) Xz))) ((and ((and (((K x2) Xy)->False)) (((K Xy) Xz)->False))) (((K x2) Xz)->False)))))))))) as proof of (((eq Prop) (f x2)) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x2)) (S Xy))) (S Xz))) (not (((eq fofType) x2) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x2)))) ((or ((and ((and ((K x2) Xy)) ((K Xy) Xz))) ((K x2) Xz))) ((and ((and (((K x2) Xy)->False)) (((K Xy) Xz)->False))) (((K x2) Xz)->False)))))))))
% 190.81/191.83 Found ((((((eq_trans Prop) (f x2)) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x2)) (S Xy))) (S Xz))) (not (((eq fofType) x2) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x2)))) ((or ((and ((and ((K x2) Xy)) ((K Xy) Xz))) ((K x2) Xz))) ((and ((and (((K x2) Xy)->False)) (((K Xy) Xz)->False))) (((K x2) Xz)->False))))))))) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x2)) (S Xy))) (S Xz))) (not (((eq fofType) x2) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x2)))) ((or ((and ((and ((K x2) Xy)) ((K Xy) Xz))) ((K x2) Xz))) ((and ((and (((K x2) Xy)->False)) (((K Xy) Xz)->False))) (((K x2) Xz)->False))))))))) ((eq_ref Prop) (f x2))) ((eq_ref Prop) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x2)) (S Xy))) (S Xz))) (not (((eq fofType) x2) Xy)))) (not (((eq fofType) Xy) Xz)))) (not
% (((eq fofType) Xz) x2)))) ((or ((and ((and ((K x2) Xy)) ((K Xy) Xz))) ((K x2) Xz))) ((and ((and (((K x2) Xy)->False)) (((K Xy) Xz)->False))) (((K x2) Xz)->False)))))))))) as proof of (((eq Prop) (f x2)) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x2)) (S Xy))) (S Xz))) (not (((eq fofType) x2) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x2)))) ((or ((and ((and ((K x2) Xy)) ((K Xy) Xz))) ((K x2) Xz))) ((and ((and (((K x2) Xy)->False)) (((K Xy) Xz)->False))) (((K x2) Xz)->False)))))))))
% 191.00/192.09 Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% 191.00/192.09 Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) b)
% 191.00/192.09 Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) b)
% 191.00/192.09 Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) b)
% 191.00/192.09 Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) b)
% 191.00/192.09 Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% 191.00/192.09 Found (eq_ref0 b) as proof of (((eq Prop) b) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x2)) (S Xy))) (S Xz))) (not (((eq fofType) x2) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x2)))) ((or ((and ((and ((K x2) Xy)) ((K Xy) Xz))) ((K x2) Xz))) ((and ((and (((K x2) Xy)->False)) (((K Xy) Xz)->False))) (((K x2) Xz)->False)))))))))
% 191.00/192.09 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x2)) (S Xy))) (S Xz))) (not (((eq fofType) x2) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x2)))) ((or ((and ((and ((K x2) Xy)) ((K Xy) Xz))) ((K x2) Xz))) ((and ((and (((K x2) Xy)->False)) (((K Xy) Xz)->False))) (((K x2) Xz)->False)))))))))
% 191.00/192.09 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x2)) (S Xy))) (S Xz))) (not (((eq fofType) x2) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x2)))) ((or ((and ((and ((K x2) Xy)) ((K Xy) Xz))) ((K x2) Xz))) ((and ((and (((K x2) Xy)->False)) (((K Xy) Xz)->False))) (((K x2) Xz)->False)))))))))
% 191.00/192.09 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x2)) (S Xy))) (S Xz))) (not (((eq fofType) x2) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x2)))) ((or ((and ((and ((K x2) Xy)) ((K Xy) Xz))) ((K x2) Xz))) ((and ((and (((K x2) Xy)->False)) (((K Xy) Xz)->False))) (((K x2) Xz)->False)))))))))
% 191.00/192.09 Found ((eq_trans0000 ((eq_ref Prop) (f x2))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f x2)) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x2)) (S Xy))) (S Xz))) (not (((eq fofType) x2) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x2)))) ((or ((and ((and ((K x2) Xy)) ((K Xy) Xz))) ((K x2) Xz))) ((and ((and (((K x2) Xy)->False)) (((K Xy) Xz)->False))) (((K x2) Xz)->False)))))))))
% 191.00/192.09 Found (((eq_trans000 ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x2)) (S Xy))) (S Xz))) (not (((eq fofType) x2) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x2)))) ((or ((and ((and ((K x2) Xy)) ((K Xy) Xz))) ((K x2) Xz))) ((and ((and (((K x2) Xy)->False)) (((K Xy) Xz)->False))) (((K x2) Xz)->False))))))))) ((eq_ref Prop) (f x2))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f x2)) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x2)) (S Xy))) (S Xz))) (not (((eq fofType) x2) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x2)))) ((or ((and ((and ((K x2) Xy)) ((K Xy) Xz))) ((K x2) Xz))) ((and ((and (((K x2) Xy)->False)) (((K Xy) Xz)->False))) (((K x2) Xz)->False)))))))))
% 191.00/192.09 Found ((((eq_trans00 ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x2)) (S Xy))) (S Xz))) (not (((eq fofType) x2) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x2)))) ((or ((and ((and ((K x2) Xy)) ((K Xy) Xz))) ((K x2) Xz))) ((and ((and (((K x2) Xy)->False)) (((K Xy) Xz)->False))) (((K x2) Xz)->False))))))))) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x2)) (S Xy))) (S Xz))) (not (((eq fofType) x2) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x2)))) ((or ((and ((and ((K x2) Xy)) ((K Xy) Xz))) ((K x2) Xz))) ((and ((and (((K x2) Xy)->False)) (((K Xy) Xz)->False))) (((K x2) Xz)->False))))))))) ((eq_ref Prop) (f x2))) ((eq_ref Prop) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x2)) (S Xy))) (S Xz))) (not (((eq fofType) x2) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType)
% Xz) x2)))) ((or ((and ((and ((K x2) Xy)) ((K Xy) Xz))) ((K x2) Xz))) ((and ((and (((K x2) Xy)->False)) (((K Xy) Xz)->False))) (((K x2) Xz)->False)))))))))) as proof of (((eq Prop) (f x2)) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x2)) (S Xy))) (S Xz))) (not (((eq fofType) x2) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x2)))) ((or ((and ((and ((K x2) Xy)) ((K Xy) Xz))) ((K x2) Xz))) ((and ((and (((K x2) Xy)->False)) (((K Xy) Xz)->False))) (((K x2) Xz)->False)))))))))
% 191.00/192.09 Found (((((eq_trans0 (f x2)) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x2)) (S Xy))) (S Xz))) (not (((eq fofType) x2) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x2)))) ((or ((and ((and ((K x2) Xy)) ((K Xy) Xz))) ((K x2) Xz))) ((and ((and (((K x2) Xy)->False)) (((K Xy) Xz)->False))) (((K x2) Xz)->False))))))))) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x2)) (S Xy))) (S Xz))) (not (((eq fofType) x2) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x2)))) ((or ((and ((and ((K x2) Xy)) ((K Xy) Xz))) ((K x2) Xz))) ((and ((and (((K x2) Xy)->False)) (((K Xy) Xz)->False))) (((K x2) Xz)->False))))))))) ((eq_ref Prop) (f x2))) ((eq_ref Prop) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x2)) (S Xy))) (S Xz))) (not (((eq fofType) x2) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq
% fofType) Xz) x2)))) ((or ((and ((and ((K x2) Xy)) ((K Xy) Xz))) ((K x2) Xz))) ((and ((and (((K x2) Xy)->False)) (((K Xy) Xz)->False))) (((K x2) Xz)->False)))))))))) as proof of (((eq Prop) (f x2)) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x2)) (S Xy))) (S Xz))) (not (((eq fofType) x2) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x2)))) ((or ((and ((and ((K x2) Xy)) ((K Xy) Xz))) ((K x2) Xz))) ((and ((and (((K x2) Xy)->False)) (((K Xy) Xz)->False))) (((K x2) Xz)->False)))))))))
% 191.00/192.09 Found ((((((eq_trans Prop) (f x2)) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x2)) (S Xy))) (S Xz))) (not (((eq fofType) x2) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x2)))) ((or ((and ((and ((K x2) Xy)) ((K Xy) Xz))) ((K x2) Xz))) ((and ((and (((K x2) Xy)->False)) (((K Xy) Xz)->False))) (((K x2) Xz)->False))))))))) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x2)) (S Xy))) (S Xz))) (not (((eq fofType) x2) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x2)))) ((or ((and ((and ((K x2) Xy)) ((K Xy) Xz))) ((K x2) Xz))) ((and ((and (((K x2) Xy)->False)) (((K Xy) Xz)->False))) (((K x2) Xz)->False))))))))) ((eq_ref Prop) (f x2))) ((eq_ref Prop) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x2)) (S Xy))) (S Xz))) (not (((eq fofType) x2) Xy)))) (not (((eq fofType) Xy) Xz)))) (not
% (((eq fofType) Xz) x2)))) ((or ((and ((and ((K x2) Xy)) ((K Xy) Xz))) ((K x2) Xz))) ((and ((and (((K x2) Xy)->False)) (((K Xy) Xz)->False))) (((K x2) Xz)->False)))))))))) as proof of (((eq Prop) (f x2)) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x2)) (S Xy))) (S Xz))) (not (((eq fofType) x2) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x2)))) ((or ((and ((and ((K x2) Xy)) ((K Xy) Xz))) ((K x2) Xz))) ((and ((and (((K x2) Xy)->False)) (((K Xy) Xz)->False))) (((K x2) Xz)->False)))))))))
% 198.61/199.62 Found eq_ref000:=(eq_ref00 P0):((P0 (f x2))->(P0 (f x2)))
% 198.61/199.62 Found (eq_ref00 P0) as proof of (P1 (f x2))
% 198.61/199.62 Found ((eq_ref0 (f x2)) P0) as proof of (P1 (f x2))
% 198.61/199.62 Found (((eq_ref Prop) (f x2)) P0) as proof of (P1 (f x2))
% 198.61/199.62 Found (((eq_ref Prop) (f x2)) P0) as proof of (P1 (f x2))
% 198.61/199.62 Found eq_ref000:=(eq_ref00 P0):((P0 (f x2))->(P0 (f x2)))
% 198.61/199.62 Found (eq_ref00 P0) as proof of (P1 (f x2))
% 198.61/199.62 Found ((eq_ref0 (f x2)) P0) as proof of (P1 (f x2))
% 198.61/199.62 Found (((eq_ref Prop) (f x2)) P0) as proof of (P1 (f x2))
% 198.61/199.62 Found (((eq_ref Prop) (f x2)) P0) as proof of (P1 (f x2))
% 198.61/199.62 Found eq_ref00:=(eq_ref0 (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x6)) (S Xy))) (S Xz))) (not (((eq fofType) x6) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x6)))) ((or ((and ((and ((K x6) Xy)) ((K Xy) Xz))) ((K x6) Xz))) ((and ((and (((K x6) Xy)->False)) (((K Xy) Xz)->False))) (((K x6) Xz)->False)))))))):(((eq (fofType->Prop)) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x6)) (S Xy))) (S Xz))) (not (((eq fofType) x6) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x6)))) ((or ((and ((and ((K x6) Xy)) ((K Xy) Xz))) ((K x6) Xz))) ((and ((and (((K x6) Xy)->False)) (((K Xy) Xz)->False))) (((K x6) Xz)->False)))))))) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x6)) (S Xy))) (S Xz))) (not (((eq fofType) x6) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x6)))) ((or ((and ((and ((K x6) Xy)) ((K Xy) Xz)))
% ((K x6) Xz))) ((and ((and (((K x6) Xy)->False)) (((K Xy) Xz)->False))) (((K x6) Xz)->False))))))))
% 198.61/199.62 Found (eq_ref0 (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x6)) (S Xy))) (S Xz))) (not (((eq fofType) x6) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x6)))) ((or ((and ((and ((K x6) Xy)) ((K Xy) Xz))) ((K x6) Xz))) ((and ((and (((K x6) Xy)->False)) (((K Xy) Xz)->False))) (((K x6) Xz)->False)))))))) as proof of (((eq (fofType->Prop)) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x6)) (S Xy))) (S Xz))) (not (((eq fofType) x6) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x6)))) ((or ((and ((and ((K x6) Xy)) ((K Xy) Xz))) ((K x6) Xz))) ((and ((and (((K x6) Xy)->False)) (((K Xy) Xz)->False))) (((K x6) Xz)->False)))))))) b)
% 198.61/199.62 Found ((eq_ref (fofType->Prop)) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x6)) (S Xy))) (S Xz))) (not (((eq fofType) x6) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x6)))) ((or ((and ((and ((K x6) Xy)) ((K Xy) Xz))) ((K x6) Xz))) ((and ((and (((K x6) Xy)->False)) (((K Xy) Xz)->False))) (((K x6) Xz)->False)))))))) as proof of (((eq (fofType->Prop)) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x6)) (S Xy))) (S Xz))) (not (((eq fofType) x6) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x6)))) ((or ((and ((and ((K x6) Xy)) ((K Xy) Xz))) ((K x6) Xz))) ((and ((and (((K x6) Xy)->False)) (((K Xy) Xz)->False))) (((K x6) Xz)->False)))))))) b)
% 198.61/199.62 Found ((eq_ref (fofType->Prop)) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x6)) (S Xy))) (S Xz))) (not (((eq fofType) x6) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x6)))) ((or ((and ((and ((K x6) Xy)) ((K Xy) Xz))) ((K x6) Xz))) ((and ((and (((K x6) Xy)->False)) (((K Xy) Xz)->False))) (((K x6) Xz)->False)))))))) as proof of (((eq (fofType->Prop)) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x6)) (S Xy))) (S Xz))) (not (((eq fofType) x6) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x6)))) ((or ((and ((and ((K x6) Xy)) ((K Xy) Xz))) ((K x6) Xz))) ((and ((and (((K x6) Xy)->False)) (((K Xy) Xz)->False))) (((K x6) Xz)->False)))))))) b)
% 210.51/211.59 Found ((eq_ref (fofType->Prop)) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x6)) (S Xy))) (S Xz))) (not (((eq fofType) x6) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x6)))) ((or ((and ((and ((K x6) Xy)) ((K Xy) Xz))) ((K x6) Xz))) ((and ((and (((K x6) Xy)->False)) (((K Xy) Xz)->False))) (((K x6) Xz)->False)))))))) as proof of (((eq (fofType->Prop)) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x6)) (S Xy))) (S Xz))) (not (((eq fofType) x6) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x6)))) ((or ((and ((and ((K x6) Xy)) ((K Xy) Xz))) ((K x6) Xz))) ((and ((and (((K x6) Xy)->False)) (((K Xy) Xz)->False))) (((K x6) Xz)->False)))))))) b)
% 210.51/211.59 Found eq_ref00:=(eq_ref0 (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x2)) (S Xy))) (S Xz))) (not (((eq fofType) x2) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x2)))) ((or ((and ((and ((K x2) Xy)) ((K Xy) Xz))) ((K x2) Xz))) ((and ((and (((K x2) Xy)->False)) (((K Xy) Xz)->False))) (((K x2) Xz)->False)))))))):(((eq (fofType->Prop)) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x2)) (S Xy))) (S Xz))) (not (((eq fofType) x2) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x2)))) ((or ((and ((and ((K x2) Xy)) ((K Xy) Xz))) ((K x2) Xz))) ((and ((and (((K x2) Xy)->False)) (((K Xy) Xz)->False))) (((K x2) Xz)->False)))))))) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x2)) (S Xy))) (S Xz))) (not (((eq fofType) x2) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x2)))) ((or ((and ((and ((K x2) Xy)) ((K Xy) Xz)))
% ((K x2) Xz))) ((and ((and (((K x2) Xy)->False)) (((K Xy) Xz)->False))) (((K x2) Xz)->False))))))))
% 210.51/211.59 Found (eq_ref0 (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x2)) (S Xy))) (S Xz))) (not (((eq fofType) x2) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x2)))) ((or ((and ((and ((K x2) Xy)) ((K Xy) Xz))) ((K x2) Xz))) ((and ((and (((K x2) Xy)->False)) (((K Xy) Xz)->False))) (((K x2) Xz)->False)))))))) as proof of (((eq (fofType->Prop)) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x2)) (S Xy))) (S Xz))) (not (((eq fofType) x2) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x2)))) ((or ((and ((and ((K x2) Xy)) ((K Xy) Xz))) ((K x2) Xz))) ((and ((and (((K x2) Xy)->False)) (((K Xy) Xz)->False))) (((K x2) Xz)->False)))))))) b)
% 210.51/211.59 Found ((eq_ref (fofType->Prop)) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x2)) (S Xy))) (S Xz))) (not (((eq fofType) x2) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x2)))) ((or ((and ((and ((K x2) Xy)) ((K Xy) Xz))) ((K x2) Xz))) ((and ((and (((K x2) Xy)->False)) (((K Xy) Xz)->False))) (((K x2) Xz)->False)))))))) as proof of (((eq (fofType->Prop)) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x2)) (S Xy))) (S Xz))) (not (((eq fofType) x2) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x2)))) ((or ((and ((and ((K x2) Xy)) ((K Xy) Xz))) ((K x2) Xz))) ((and ((and (((K x2) Xy)->False)) (((K Xy) Xz)->False))) (((K x2) Xz)->False)))))))) b)
% 210.51/211.59 Found ((eq_ref (fofType->Prop)) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x2)) (S Xy))) (S Xz))) (not (((eq fofType) x2) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x2)))) ((or ((and ((and ((K x2) Xy)) ((K Xy) Xz))) ((K x2) Xz))) ((and ((and (((K x2) Xy)->False)) (((K Xy) Xz)->False))) (((K x2) Xz)->False)))))))) as proof of (((eq (fofType->Prop)) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x2)) (S Xy))) (S Xz))) (not (((eq fofType) x2) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x2)))) ((or ((and ((and ((K x2) Xy)) ((K Xy) Xz))) ((K x2) Xz))) ((and ((and (((K x2) Xy)->False)) (((K Xy) Xz)->False))) (((K x2) Xz)->False)))))))) b)
% 210.91/211.99 Found ((eq_ref (fofType->Prop)) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x2)) (S Xy))) (S Xz))) (not (((eq fofType) x2) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x2)))) ((or ((and ((and ((K x2) Xy)) ((K Xy) Xz))) ((K x2) Xz))) ((and ((and (((K x2) Xy)->False)) (((K Xy) Xz)->False))) (((K x2) Xz)->False)))))))) as proof of (((eq (fofType->Prop)) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x2)) (S Xy))) (S Xz))) (not (((eq fofType) x2) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x2)))) ((or ((and ((and ((K x2) Xy)) ((K Xy) Xz))) ((K x2) Xz))) ((and ((and (((K x2) Xy)->False)) (((K Xy) Xz)->False))) (((K x2) Xz)->False)))))))) b)
% 210.91/211.99 Found eta_expansion000:=(eta_expansion00 (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x4)) (S Xy))) (S Xz))) (not (((eq fofType) x4) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x4)))) ((or ((and ((and ((K x4) Xy)) ((K Xy) Xz))) ((K x4) Xz))) ((and ((and (((K x4) Xy)->False)) (((K Xy) Xz)->False))) (((K x4) Xz)->False)))))))):(((eq (fofType->Prop)) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x4)) (S Xy))) (S Xz))) (not (((eq fofType) x4) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x4)))) ((or ((and ((and ((K x4) Xy)) ((K Xy) Xz))) ((K x4) Xz))) ((and ((and (((K x4) Xy)->False)) (((K Xy) Xz)->False))) (((K x4) Xz)->False)))))))) (fun (x:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x4)) (S x))) (S Xz))) (not (((eq fofType) x4) x)))) (not (((eq fofType) x) Xz)))) (not (((eq fofType) Xz) x4)))) ((or ((and ((and ((K x4) x)) ((K
% x) Xz))) ((K x4) Xz))) ((and ((and (((K x4) x)->False)) (((K x) Xz)->False))) (((K x4) Xz)->False))))))))
% 210.91/211.99 Found (eta_expansion00 (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x4)) (S Xy))) (S Xz))) (not (((eq fofType) x4) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x4)))) ((or ((and ((and ((K x4) Xy)) ((K Xy) Xz))) ((K x4) Xz))) ((and ((and (((K x4) Xy)->False)) (((K Xy) Xz)->False))) (((K x4) Xz)->False)))))))) as proof of (((eq (fofType->Prop)) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x4)) (S Xy))) (S Xz))) (not (((eq fofType) x4) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x4)))) ((or ((and ((and ((K x4) Xy)) ((K Xy) Xz))) ((K x4) Xz))) ((and ((and (((K x4) Xy)->False)) (((K Xy) Xz)->False))) (((K x4) Xz)->False)))))))) b)
% 210.91/211.99 Found ((eta_expansion0 Prop) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x4)) (S Xy))) (S Xz))) (not (((eq fofType) x4) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x4)))) ((or ((and ((and ((K x4) Xy)) ((K Xy) Xz))) ((K x4) Xz))) ((and ((and (((K x4) Xy)->False)) (((K Xy) Xz)->False))) (((K x4) Xz)->False)))))))) as proof of (((eq (fofType->Prop)) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x4)) (S Xy))) (S Xz))) (not (((eq fofType) x4) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x4)))) ((or ((and ((and ((K x4) Xy)) ((K Xy) Xz))) ((K x4) Xz))) ((and ((and (((K x4) Xy)->False)) (((K Xy) Xz)->False))) (((K x4) Xz)->False)))))))) b)
% 210.91/211.99 Found (((eta_expansion fofType) Prop) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x4)) (S Xy))) (S Xz))) (not (((eq fofType) x4) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x4)))) ((or ((and ((and ((K x4) Xy)) ((K Xy) Xz))) ((K x4) Xz))) ((and ((and (((K x4) Xy)->False)) (((K Xy) Xz)->False))) (((K x4) Xz)->False)))))))) as proof of (((eq (fofType->Prop)) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x4)) (S Xy))) (S Xz))) (not (((eq fofType) x4) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x4)))) ((or ((and ((and ((K x4) Xy)) ((K Xy) Xz))) ((K x4) Xz))) ((and ((and (((K x4) Xy)->False)) (((K Xy) Xz)->False))) (((K x4) Xz)->False)))))))) b)
% 211.33/212.37 Found (((eta_expansion fofType) Prop) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x4)) (S Xy))) (S Xz))) (not (((eq fofType) x4) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x4)))) ((or ((and ((and ((K x4) Xy)) ((K Xy) Xz))) ((K x4) Xz))) ((and ((and (((K x4) Xy)->False)) (((K Xy) Xz)->False))) (((K x4) Xz)->False)))))))) as proof of (((eq (fofType->Prop)) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x4)) (S Xy))) (S Xz))) (not (((eq fofType) x4) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x4)))) ((or ((and ((and ((K x4) Xy)) ((K Xy) Xz))) ((K x4) Xz))) ((and ((and (((K x4) Xy)->False)) (((K Xy) Xz)->False))) (((K x4) Xz)->False)))))))) b)
% 211.33/212.37 Found (((eta_expansion fofType) Prop) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x4)) (S Xy))) (S Xz))) (not (((eq fofType) x4) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x4)))) ((or ((and ((and ((K x4) Xy)) ((K Xy) Xz))) ((K x4) Xz))) ((and ((and (((K x4) Xy)->False)) (((K Xy) Xz)->False))) (((K x4) Xz)->False)))))))) as proof of (((eq (fofType->Prop)) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x4)) (S Xy))) (S Xz))) (not (((eq fofType) x4) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x4)))) ((or ((and ((and ((K x4) Xy)) ((K Xy) Xz))) ((K x4) Xz))) ((and ((and (((K x4) Xy)->False)) (((K Xy) Xz)->False))) (((K x4) Xz)->False)))))))) b)
% 211.33/212.37 Found eta_expansion000:=(eta_expansion00 (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))):(((eq (fofType->Prop)) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))) (fun (x:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S x))) (S Xz))) (not (((eq fofType) x0) x)))) (not (((eq fofType) x) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) x)) ((K
% x) Xz))) ((K x0) Xz))) ((and ((and (((K x0) x)->False)) (((K x) Xz)->False))) (((K x0) Xz)->False))))))))
% 211.33/212.37 Found (eta_expansion00 (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))) as proof of (((eq (fofType->Prop)) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))) b)
% 211.33/212.37 Found ((eta_expansion0 Prop) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))) as proof of (((eq (fofType->Prop)) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))) b)
% 245.02/246.04 Found (((eta_expansion fofType) Prop) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))) as proof of (((eq (fofType->Prop)) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))) b)
% 245.02/246.04 Found (((eta_expansion fofType) Prop) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))) as proof of (((eq (fofType->Prop)) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))) b)
% 245.02/246.04 Found (((eta_expansion fofType) Prop) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))) as proof of (((eq (fofType->Prop)) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))) b)
% 245.02/246.04 Found eq_ref00:=(eq_ref0 ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False))))))))):(((eq Prop) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False))))))))) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and
% ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))))
% 245.02/246.04 Found (eq_ref0 ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False))))))))) as proof of (((eq Prop) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False))))))))) b)
% 245.92/246.90 Found ((eq_ref Prop) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False))))))))) as proof of (((eq Prop) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False))))))))) b)
% 245.92/246.90 Found ((eq_ref Prop) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False))))))))) as proof of (((eq Prop) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False))))))))) b)
% 245.92/246.90 Found ((eq_ref Prop) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False))))))))) as proof of (((eq Prop) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False))))))))) b)
% 245.92/246.90 Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% 245.92/246.90 Found (eq_ref0 b) as proof of (((eq Prop) b) (f x0))
% 245.92/246.90 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% 245.92/246.90 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% 245.92/246.90 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% 245.92/246.90 Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% 245.92/246.90 Found (eq_ref0 b) as proof of (((eq Prop) b) (f x0))
% 245.92/246.90 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% 245.92/246.90 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% 245.92/246.90 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% 245.92/246.90 Found eq_ref00:=(eq_ref0 ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False))))))))):(((eq Prop) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False))))))))) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and
% ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))))
% 248.53/249.51 Found (eq_ref0 ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False))))))))) as proof of (((eq Prop) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False))))))))) b)
% 248.53/249.51 Found ((eq_ref Prop) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False))))))))) as proof of (((eq Prop) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False))))))))) b)
% 248.53/249.51 Found ((eq_ref Prop) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False))))))))) as proof of (((eq Prop) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False))))))))) b)
% 248.53/249.51 Found ((eq_ref Prop) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False))))))))) as proof of (((eq Prop) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False))))))))) b)
% 248.53/249.51 Found eq_ref00:=(eq_ref0 (f x4)):(((eq Prop) (f x4)) (f x4))
% 248.53/249.51 Found (eq_ref0 (f x4)) as proof of (((eq Prop) (f x4)) b)
% 248.53/249.51 Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) b)
% 248.53/249.51 Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) b)
% 248.53/249.51 Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) b)
% 248.53/249.51 Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% 248.53/249.51 Found (eq_ref0 b) as proof of (((eq Prop) b) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x4)) (S Xy))) (S Xz))) (not (((eq fofType) x4) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x4)))) ((or ((and ((and ((K x4) Xy)) ((K Xy) Xz))) ((K x4) Xz))) ((and ((and (((K x4) Xy)->False)) (((K Xy) Xz)->False))) (((K x4) Xz)->False)))))))))
% 248.53/249.53 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x4)) (S Xy))) (S Xz))) (not (((eq fofType) x4) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x4)))) ((or ((and ((and ((K x4) Xy)) ((K Xy) Xz))) ((K x4) Xz))) ((and ((and (((K x4) Xy)->False)) (((K Xy) Xz)->False))) (((K x4) Xz)->False)))))))))
% 248.53/249.53 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x4)) (S Xy))) (S Xz))) (not (((eq fofType) x4) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x4)))) ((or ((and ((and ((K x4) Xy)) ((K Xy) Xz))) ((K x4) Xz))) ((and ((and (((K x4) Xy)->False)) (((K Xy) Xz)->False))) (((K x4) Xz)->False)))))))))
% 248.53/249.53 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x4)) (S Xy))) (S Xz))) (not (((eq fofType) x4) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x4)))) ((or ((and ((and ((K x4) Xy)) ((K Xy) Xz))) ((K x4) Xz))) ((and ((and (((K x4) Xy)->False)) (((K Xy) Xz)->False))) (((K x4) Xz)->False)))))))))
% 248.53/249.53 Found ((eq_trans0000 ((eq_ref Prop) (f x4))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f x4)) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x4)) (S Xy))) (S Xz))) (not (((eq fofType) x4) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x4)))) ((or ((and ((and ((K x4) Xy)) ((K Xy) Xz))) ((K x4) Xz))) ((and ((and (((K x4) Xy)->False)) (((K Xy) Xz)->False))) (((K x4) Xz)->False)))))))))
% 248.53/249.53 Found (((eq_trans000 ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x4)) (S Xy))) (S Xz))) (not (((eq fofType) x4) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x4)))) ((or ((and ((and ((K x4) Xy)) ((K Xy) Xz))) ((K x4) Xz))) ((and ((and (((K x4) Xy)->False)) (((K Xy) Xz)->False))) (((K x4) Xz)->False))))))))) ((eq_ref Prop) (f x4))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f x4)) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x4)) (S Xy))) (S Xz))) (not (((eq fofType) x4) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x4)))) ((or ((and ((and ((K x4) Xy)) ((K Xy) Xz))) ((K x4) Xz))) ((and ((and (((K x4) Xy)->False)) (((K Xy) Xz)->False))) (((K x4) Xz)->False)))))))))
% 248.53/249.53 Found ((((eq_trans00 ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x4)) (S Xy))) (S Xz))) (not (((eq fofType) x4) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x4)))) ((or ((and ((and ((K x4) Xy)) ((K Xy) Xz))) ((K x4) Xz))) ((and ((and (((K x4) Xy)->False)) (((K Xy) Xz)->False))) (((K x4) Xz)->False))))))))) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x4)) (S Xy))) (S Xz))) (not (((eq fofType) x4) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x4)))) ((or ((and ((and ((K x4) Xy)) ((K Xy) Xz))) ((K x4) Xz))) ((and ((and (((K x4) Xy)->False)) (((K Xy) Xz)->False))) (((K x4) Xz)->False))))))))) ((eq_ref Prop) (f x4))) ((eq_ref Prop) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x4)) (S Xy))) (S Xz))) (not (((eq fofType) x4) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType)
% Xz) x4)))) ((or ((and ((and ((K x4) Xy)) ((K Xy) Xz))) ((K x4) Xz))) ((and ((and (((K x4) Xy)->False)) (((K Xy) Xz)->False))) (((K x4) Xz)->False)))))))))) as proof of (((eq Prop) (f x4)) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x4)) (S Xy))) (S Xz))) (not (((eq fofType) x4) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x4)))) ((or ((and ((and ((K x4) Xy)) ((K Xy) Xz))) ((K x4) Xz))) ((and ((and (((K x4) Xy)->False)) (((K Xy) Xz)->False))) (((K x4) Xz)->False)))))))))
% 248.74/249.73 Found (((((eq_trans0 (f x4)) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x4)) (S Xy))) (S Xz))) (not (((eq fofType) x4) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x4)))) ((or ((and ((and ((K x4) Xy)) ((K Xy) Xz))) ((K x4) Xz))) ((and ((and (((K x4) Xy)->False)) (((K Xy) Xz)->False))) (((K x4) Xz)->False))))))))) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x4)) (S Xy))) (S Xz))) (not (((eq fofType) x4) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x4)))) ((or ((and ((and ((K x4) Xy)) ((K Xy) Xz))) ((K x4) Xz))) ((and ((and (((K x4) Xy)->False)) (((K Xy) Xz)->False))) (((K x4) Xz)->False))))))))) ((eq_ref Prop) (f x4))) ((eq_ref Prop) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x4)) (S Xy))) (S Xz))) (not (((eq fofType) x4) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq
% fofType) Xz) x4)))) ((or ((and ((and ((K x4) Xy)) ((K Xy) Xz))) ((K x4) Xz))) ((and ((and (((K x4) Xy)->False)) (((K Xy) Xz)->False))) (((K x4) Xz)->False)))))))))) as proof of (((eq Prop) (f x4)) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x4)) (S Xy))) (S Xz))) (not (((eq fofType) x4) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x4)))) ((or ((and ((and ((K x4) Xy)) ((K Xy) Xz))) ((K x4) Xz))) ((and ((and (((K x4) Xy)->False)) (((K Xy) Xz)->False))) (((K x4) Xz)->False)))))))))
% 248.74/249.73 Found ((((((eq_trans Prop) (f x4)) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x4)) (S Xy))) (S Xz))) (not (((eq fofType) x4) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x4)))) ((or ((and ((and ((K x4) Xy)) ((K Xy) Xz))) ((K x4) Xz))) ((and ((and (((K x4) Xy)->False)) (((K Xy) Xz)->False))) (((K x4) Xz)->False))))))))) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x4)) (S Xy))) (S Xz))) (not (((eq fofType) x4) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x4)))) ((or ((and ((and ((K x4) Xy)) ((K Xy) Xz))) ((K x4) Xz))) ((and ((and (((K x4) Xy)->False)) (((K Xy) Xz)->False))) (((K x4) Xz)->False))))))))) ((eq_ref Prop) (f x4))) ((eq_ref Prop) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x4)) (S Xy))) (S Xz))) (not (((eq fofType) x4) Xy)))) (not (((eq fofType) Xy) Xz)))) (not
% (((eq fofType) Xz) x4)))) ((or ((and ((and ((K x4) Xy)) ((K Xy) Xz))) ((K x4) Xz))) ((and ((and (((K x4) Xy)->False)) (((K Xy) Xz)->False))) (((K x4) Xz)->False)))))))))) as proof of (((eq Prop) (f x4)) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x4)) (S Xy))) (S Xz))) (not (((eq fofType) x4) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x4)))) ((or ((and ((and ((K x4) Xy)) ((K Xy) Xz))) ((K x4) Xz))) ((and ((and (((K x4) Xy)->False)) (((K Xy) Xz)->False))) (((K x4) Xz)->False)))))))))
% 248.74/249.73 Found eq_ref00:=(eq_ref0 (f x4)):(((eq Prop) (f x4)) (f x4))
% 248.74/249.73 Found (eq_ref0 (f x4)) as proof of (((eq Prop) (f x4)) b)
% 248.74/249.73 Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) b)
% 248.74/249.73 Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) b)
% 248.74/249.73 Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) b)
% 248.74/249.73 Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% 248.74/249.73 Found (eq_ref0 b) as proof of (((eq Prop) b) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x4)) (S Xy))) (S Xz))) (not (((eq fofType) x4) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x4)))) ((or ((and ((and ((K x4) Xy)) ((K Xy) Xz))) ((K x4) Xz))) ((and ((and (((K x4) Xy)->False)) (((K Xy) Xz)->False))) (((K x4) Xz)->False)))))))))
% 248.74/249.76 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x4)) (S Xy))) (S Xz))) (not (((eq fofType) x4) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x4)))) ((or ((and ((and ((K x4) Xy)) ((K Xy) Xz))) ((K x4) Xz))) ((and ((and (((K x4) Xy)->False)) (((K Xy) Xz)->False))) (((K x4) Xz)->False)))))))))
% 248.74/249.76 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x4)) (S Xy))) (S Xz))) (not (((eq fofType) x4) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x4)))) ((or ((and ((and ((K x4) Xy)) ((K Xy) Xz))) ((K x4) Xz))) ((and ((and (((K x4) Xy)->False)) (((K Xy) Xz)->False))) (((K x4) Xz)->False)))))))))
% 248.74/249.76 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x4)) (S Xy))) (S Xz))) (not (((eq fofType) x4) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x4)))) ((or ((and ((and ((K x4) Xy)) ((K Xy) Xz))) ((K x4) Xz))) ((and ((and (((K x4) Xy)->False)) (((K Xy) Xz)->False))) (((K x4) Xz)->False)))))))))
% 248.74/249.76 Found ((eq_trans0000 ((eq_ref Prop) (f x4))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f x4)) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x4)) (S Xy))) (S Xz))) (not (((eq fofType) x4) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x4)))) ((or ((and ((and ((K x4) Xy)) ((K Xy) Xz))) ((K x4) Xz))) ((and ((and (((K x4) Xy)->False)) (((K Xy) Xz)->False))) (((K x4) Xz)->False)))))))))
% 248.74/249.76 Found (((eq_trans000 ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x4)) (S Xy))) (S Xz))) (not (((eq fofType) x4) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x4)))) ((or ((and ((and ((K x4) Xy)) ((K Xy) Xz))) ((K x4) Xz))) ((and ((and (((K x4) Xy)->False)) (((K Xy) Xz)->False))) (((K x4) Xz)->False))))))))) ((eq_ref Prop) (f x4))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f x4)) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x4)) (S Xy))) (S Xz))) (not (((eq fofType) x4) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x4)))) ((or ((and ((and ((K x4) Xy)) ((K Xy) Xz))) ((K x4) Xz))) ((and ((and (((K x4) Xy)->False)) (((K Xy) Xz)->False))) (((K x4) Xz)->False)))))))))
% 248.74/249.76 Found ((((eq_trans00 ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x4)) (S Xy))) (S Xz))) (not (((eq fofType) x4) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x4)))) ((or ((and ((and ((K x4) Xy)) ((K Xy) Xz))) ((K x4) Xz))) ((and ((and (((K x4) Xy)->False)) (((K Xy) Xz)->False))) (((K x4) Xz)->False))))))))) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x4)) (S Xy))) (S Xz))) (not (((eq fofType) x4) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x4)))) ((or ((and ((and ((K x4) Xy)) ((K Xy) Xz))) ((K x4) Xz))) ((and ((and (((K x4) Xy)->False)) (((K Xy) Xz)->False))) (((K x4) Xz)->False))))))))) ((eq_ref Prop) (f x4))) ((eq_ref Prop) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x4)) (S Xy))) (S Xz))) (not (((eq fofType) x4) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType)
% Xz) x4)))) ((or ((and ((and ((K x4) Xy)) ((K Xy) Xz))) ((K x4) Xz))) ((and ((and (((K x4) Xy)->False)) (((K Xy) Xz)->False))) (((K x4) Xz)->False)))))))))) as proof of (((eq Prop) (f x4)) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x4)) (S Xy))) (S Xz))) (not (((eq fofType) x4) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x4)))) ((or ((and ((and ((K x4) Xy)) ((K Xy) Xz))) ((K x4) Xz))) ((and ((and (((K x4) Xy)->False)) (((K Xy) Xz)->False))) (((K x4) Xz)->False)))))))))
% 265.52/266.52 Found (((((eq_trans0 (f x4)) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x4)) (S Xy))) (S Xz))) (not (((eq fofType) x4) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x4)))) ((or ((and ((and ((K x4) Xy)) ((K Xy) Xz))) ((K x4) Xz))) ((and ((and (((K x4) Xy)->False)) (((K Xy) Xz)->False))) (((K x4) Xz)->False))))))))) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x4)) (S Xy))) (S Xz))) (not (((eq fofType) x4) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x4)))) ((or ((and ((and ((K x4) Xy)) ((K Xy) Xz))) ((K x4) Xz))) ((and ((and (((K x4) Xy)->False)) (((K Xy) Xz)->False))) (((K x4) Xz)->False))))))))) ((eq_ref Prop) (f x4))) ((eq_ref Prop) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x4)) (S Xy))) (S Xz))) (not (((eq fofType) x4) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq
% fofType) Xz) x4)))) ((or ((and ((and ((K x4) Xy)) ((K Xy) Xz))) ((K x4) Xz))) ((and ((and (((K x4) Xy)->False)) (((K Xy) Xz)->False))) (((K x4) Xz)->False)))))))))) as proof of (((eq Prop) (f x4)) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x4)) (S Xy))) (S Xz))) (not (((eq fofType) x4) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x4)))) ((or ((and ((and ((K x4) Xy)) ((K Xy) Xz))) ((K x4) Xz))) ((and ((and (((K x4) Xy)->False)) (((K Xy) Xz)->False))) (((K x4) Xz)->False)))))))))
% 265.52/266.52 Found ((((((eq_trans Prop) (f x4)) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x4)) (S Xy))) (S Xz))) (not (((eq fofType) x4) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x4)))) ((or ((and ((and ((K x4) Xy)) ((K Xy) Xz))) ((K x4) Xz))) ((and ((and (((K x4) Xy)->False)) (((K Xy) Xz)->False))) (((K x4) Xz)->False))))))))) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x4)) (S Xy))) (S Xz))) (not (((eq fofType) x4) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x4)))) ((or ((and ((and ((K x4) Xy)) ((K Xy) Xz))) ((K x4) Xz))) ((and ((and (((K x4) Xy)->False)) (((K Xy) Xz)->False))) (((K x4) Xz)->False))))))))) ((eq_ref Prop) (f x4))) ((eq_ref Prop) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x4)) (S Xy))) (S Xz))) (not (((eq fofType) x4) Xy)))) (not (((eq fofType) Xy) Xz)))) (not
% (((eq fofType) Xz) x4)))) ((or ((and ((and ((K x4) Xy)) ((K Xy) Xz))) ((K x4) Xz))) ((and ((and (((K x4) Xy)->False)) (((K Xy) Xz)->False))) (((K x4) Xz)->False)))))))))) as proof of (((eq Prop) (f x4)) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x4)) (S Xy))) (S Xz))) (not (((eq fofType) x4) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x4)))) ((or ((and ((and ((K x4) Xy)) ((K Xy) Xz))) ((K x4) Xz))) ((and ((and (((K x4) Xy)->False)) (((K Xy) Xz)->False))) (((K x4) Xz)->False)))))))))
% 265.52/266.52 Found eq_ref000:=(eq_ref00 P0):((P0 (f x4))->(P0 (f x4)))
% 265.52/266.52 Found (eq_ref00 P0) as proof of (P1 (f x4))
% 265.52/266.52 Found ((eq_ref0 (f x4)) P0) as proof of (P1 (f x4))
% 265.52/266.52 Found (((eq_ref Prop) (f x4)) P0) as proof of (P1 (f x4))
% 265.52/266.52 Found (((eq_ref Prop) (f x4)) P0) as proof of (P1 (f x4))
% 265.52/266.52 Found eq_ref000:=(eq_ref00 P0):((P0 (f x4))->(P0 (f x4)))
% 265.52/266.52 Found (eq_ref00 P0) as proof of (P1 (f x4))
% 265.52/266.52 Found ((eq_ref0 (f x4)) P0) as proof of (P1 (f x4))
% 265.52/266.52 Found (((eq_ref Prop) (f x4)) P0) as proof of (P1 (f x4))
% 265.52/266.52 Found (((eq_ref Prop) (f x4)) P0) as proof of (P1 (f x4))
% 265.52/266.52 Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% 265.52/266.52 Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) b)
% 265.52/266.52 Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% 265.52/266.52 Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% 265.52/266.52 Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% 265.72/266.73 Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% 265.72/266.73 Found (eq_ref0 b) as proof of (((eq Prop) b) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))))
% 265.72/266.73 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))))
% 265.72/266.73 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))))
% 265.72/266.73 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))))
% 265.72/266.73 Found ((eq_trans0000 ((eq_ref Prop) (f x0))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f x0)) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))))
% 265.72/266.73 Found (((eq_trans000 ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False))))))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f x0)) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))))
% 265.72/266.73 Found ((((eq_trans00 ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False))))))))) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False))))))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType)
% Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))))) as proof of (((eq Prop) (f x0)) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))))
% 265.72/266.74 Found (((((eq_trans0 (f x0)) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False))))))))) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False))))))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq
% fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))))) as proof of (((eq Prop) (f x0)) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))))
% 265.72/266.74 Found ((((((eq_trans Prop) (f x0)) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False))))))))) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False))))))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not
% (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))))) as proof of (((eq Prop) (f x0)) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))))
% 265.72/266.74 Found (fun (x2:(forall (Xx:fofType) (Xy:fofType), (((K Xx) Xy)->((K Xy) Xx))))=> ((((((eq_trans Prop) (f x0)) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False))))))))) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False))))))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S
% Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False))))))))))) as proof of (((eq Prop) (f x0)) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))))
% 265.72/266.75 Found (fun (x1:((ex (fofType->(fofType->Prop))) (fun (Xs:(fofType->(fofType->Prop)))=> ((and (forall (Xx:fofType), ((S Xx)->((cSUCC (cSUCC (cSUCC (cSUCC (cSUCC cONE))))) (Xs Xx))))) (forall (Xy:(fofType->Prop)), (((cSUCC (cSUCC (cSUCC (cSUCC (cSUCC cONE))))) Xy)->((ex fofType) (fun (Xy0:fofType)=> (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and (S Xx)) (((eq (fofType->Prop)) Xy) (Xs Xx))))) (fun (Xy:fofType)=> (((eq fofType) Xy0) Xy))))))))))) (x2:(forall (Xx:fofType) (Xy:fofType), (((K Xx) Xy)->((K Xy) Xx))))=> ((((((eq_trans Prop) (f x0)) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False))))))))) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and
% ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False))))))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False))))))))))) as proof of ((forall (Xx:fofType) (Xy:fofType), (((K Xx) Xy)->((K Xy) Xx)))->(((eq Prop) (f x0)) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K
% x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False))))))))))
% 265.72/266.75 Found (fun (x1:((ex (fofType->(fofType->Prop))) (fun (Xs:(fofType->(fofType->Prop)))=> ((and (forall (Xx:fofType), ((S Xx)->((cSUCC (cSUCC (cSUCC (cSUCC (cSUCC cONE))))) (Xs Xx))))) (forall (Xy:(fofType->Prop)), (((cSUCC (cSUCC (cSUCC (cSUCC (cSUCC cONE))))) Xy)->((ex fofType) (fun (Xy0:fofType)=> (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and (S Xx)) (((eq (fofType->Prop)) Xy) (Xs Xx))))) (fun (Xy:fofType)=> (((eq fofType) Xy0) Xy))))))))))) (x2:(forall (Xx:fofType) (Xy:fofType), (((K Xx) Xy)->((K Xy) Xx))))=> ((((((eq_trans Prop) (f x0)) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False))))))))) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and
% ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False))))))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False))))))))))) as proof of (((ex (fofType->(fofType->Prop))) (fun (Xs:(fofType->(fofType->Prop)))=> ((and (forall (Xx:fofType), ((S Xx)->((cSUCC (cSUCC (cSUCC (cSUCC (cSUCC cONE))))) (Xs Xx))))) (forall (Xy:(fofType->Prop)), (((cSUCC (cSUCC (cSUCC (cSUCC (cSUCC cONE))))) Xy)->((ex fofType) (fun (Xy0:fofType)=> (((eq (fofType->Prop)) (fun (Xx:fofType)=>
% ((and (S Xx)) (((eq (fofType->Prop)) Xy) (Xs Xx))))) (fun (Xy:fofType)=> (((eq fofType) Xy0) Xy))))))))))->((forall (Xx:fofType) (Xy:fofType), (((K Xx) Xy)->((K Xy) Xx)))->(((eq Prop) (f x0)) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))))))
% 265.72/266.79 Found (and_rect00 (fun (x1:((ex (fofType->(fofType->Prop))) (fun (Xs:(fofType->(fofType->Prop)))=> ((and (forall (Xx:fofType), ((S Xx)->((cSUCC (cSUCC (cSUCC (cSUCC (cSUCC cONE))))) (Xs Xx))))) (forall (Xy:(fofType->Prop)), (((cSUCC (cSUCC (cSUCC (cSUCC (cSUCC cONE))))) Xy)->((ex fofType) (fun (Xy0:fofType)=> (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and (S Xx)) (((eq (fofType->Prop)) Xy) (Xs Xx))))) (fun (Xy:fofType)=> (((eq fofType) Xy0) Xy))))))))))) (x2:(forall (Xx:fofType) (Xy:fofType), (((K Xx) Xy)->((K Xy) Xx))))=> ((((((eq_trans Prop) (f x0)) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False))))))))) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and
% ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False))))))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))))))) as proof of (((eq Prop) (f x0)) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and
% (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))))
% 265.72/266.79 Found ((and_rect0 (((eq Prop) (f x0)) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))))) (fun (x1:((ex (fofType->(fofType->Prop))) (fun (Xs:(fofType->(fofType->Prop)))=> ((and (forall (Xx:fofType), ((S Xx)->((cSUCC (cSUCC (cSUCC (cSUCC (cSUCC cONE))))) (Xs Xx))))) (forall (Xy:(fofType->Prop)), (((cSUCC (cSUCC (cSUCC (cSUCC (cSUCC cONE))))) Xy)->((ex fofType) (fun (Xy0:fofType)=> (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and (S Xx)) (((eq (fofType->Prop)) Xy) (Xs Xx))))) (fun (Xy:fofType)=> (((eq fofType) Xy0) Xy))))))))))) (x2:(forall (Xx:fofType) (Xy:fofType), (((K Xx) Xy)->((K Xy) Xx))))=> ((((((eq_trans Prop) (f x0)) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun
% (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False))))))))) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False))))))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0)
% Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))))))) as proof of (((eq Prop) (f x0)) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))))
% 265.82/266.80 Found (((fun (P0:Type) (x1:(((ex (fofType->(fofType->Prop))) (fun (Xs:(fofType->(fofType->Prop)))=> ((and (forall (Xx:fofType), ((S Xx)->((cSUCC (cSUCC (cSUCC (cSUCC (cSUCC cONE))))) (Xs Xx))))) (forall (Xy:(fofType->Prop)), (((cSUCC (cSUCC (cSUCC (cSUCC (cSUCC cONE))))) Xy)->((ex fofType) (fun (Xy0:fofType)=> (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and (S Xx)) (((eq (fofType->Prop)) Xy) (Xs Xx))))) (fun (Xy:fofType)=> (((eq fofType) Xy0) Xy))))))))))->((forall (Xx:fofType) (Xy:fofType), (((K Xx) Xy)->((K Xy) Xx)))->P0)))=> (((((and_rect ((ex (fofType->(fofType->Prop))) (fun (Xs:(fofType->(fofType->Prop)))=> ((and (forall (Xx:fofType), ((S Xx)->((cSUCC (cSUCC (cSUCC (cSUCC (cSUCC cONE))))) (Xs Xx))))) (forall (Xy:(fofType->Prop)), (((cSUCC (cSUCC (cSUCC (cSUCC (cSUCC cONE))))) Xy)->((ex fofType) (fun (Xy0:fofType)=> (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and (S Xx)) (((eq (fofType->Prop)) Xy) (Xs Xx))))) (fun (Xy:fofType)=> (((eq fofType) Xy0) Xy))))))))))) (forall (Xx:fofType)
% (Xy:fofType), (((K Xx) Xy)->((K Xy) Xx)))) P0) x1) x)) (((eq Prop) (f x0)) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))))) (fun (x1:((ex (fofType->(fofType->Prop))) (fun (Xs:(fofType->(fofType->Prop)))=> ((and (forall (Xx:fofType), ((S Xx)->((cSUCC (cSUCC (cSUCC (cSUCC (cSUCC cONE))))) (Xs Xx))))) (forall (Xy:(fofType->Prop)), (((cSUCC (cSUCC (cSUCC (cSUCC (cSUCC cONE))))) Xy)->((ex fofType) (fun (Xy0:fofType)=> (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and (S Xx)) (((eq (fofType->Prop)) Xy) (Xs Xx))))) (fun (Xy:fofType)=> (((eq fofType) Xy0) Xy))))))))))) (x2:(forall (Xx:fofType) (Xy:fofType), (((K Xx) Xy)->((K Xy) Xx))))=> ((((((eq_trans Prop) (f x0)) ((ex fofType) (fun (Xy:fofType)=> ((ex
% fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False))))))))) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False))))))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and
% ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))))))) as proof of (((eq Prop) (f x0)) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))))
% 266.12/267.16 Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% 266.12/267.16 Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) b)
% 266.12/267.16 Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% 266.12/267.16 Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% 266.12/267.16 Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% 266.12/267.16 Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% 266.12/267.16 Found (eq_ref0 b) as proof of (((eq Prop) b) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))))
% 266.12/267.16 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))))
% 266.12/267.16 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))))
% 266.12/267.16 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))))
% 266.12/267.16 Found ((eq_trans0000 ((eq_ref Prop) (f x0))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f x0)) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))))
% 266.12/267.17 Found (((eq_trans000 ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False))))))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f x0)) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))))
% 266.12/267.17 Found ((((eq_trans00 ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False))))))))) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False))))))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType)
% Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))))) as proof of (((eq Prop) (f x0)) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))))
% 266.12/267.17 Found (((((eq_trans0 (f x0)) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False))))))))) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False))))))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq
% fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))))) as proof of (((eq Prop) (f x0)) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))))
% 266.12/267.18 Found ((((((eq_trans Prop) (f x0)) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False))))))))) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False))))))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not
% (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))))) as proof of (((eq Prop) (f x0)) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))))
% 266.12/267.18 Found (fun (x2:(forall (Xx:fofType) (Xy:fofType), (((K Xx) Xy)->((K Xy) Xx))))=> ((((((eq_trans Prop) (f x0)) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False))))))))) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False))))))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S
% Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False))))))))))) as proof of (((eq Prop) (f x0)) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))))
% 266.12/267.18 Found (fun (x1:((ex (fofType->(fofType->Prop))) (fun (Xs:(fofType->(fofType->Prop)))=> ((and (forall (Xx:fofType), ((S Xx)->((cSUCC (cSUCC (cSUCC (cSUCC (cSUCC cONE))))) (Xs Xx))))) (forall (Xy:(fofType->Prop)), (((cSUCC (cSUCC (cSUCC (cSUCC (cSUCC cONE))))) Xy)->((ex fofType) (fun (Xy0:fofType)=> (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and (S Xx)) (((eq (fofType->Prop)) Xy) (Xs Xx))))) (fun (Xy:fofType)=> (((eq fofType) Xy0) Xy))))))))))) (x2:(forall (Xx:fofType) (Xy:fofType), (((K Xx) Xy)->((K Xy) Xx))))=> ((((((eq_trans Prop) (f x0)) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False))))))))) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and
% ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False))))))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False))))))))))) as proof of ((forall (Xx:fofType) (Xy:fofType), (((K Xx) Xy)->((K Xy) Xx)))->(((eq Prop) (f x0)) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K
% x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False))))))))))
% 266.23/267.22 Found (fun (x1:((ex (fofType->(fofType->Prop))) (fun (Xs:(fofType->(fofType->Prop)))=> ((and (forall (Xx:fofType), ((S Xx)->((cSUCC (cSUCC (cSUCC (cSUCC (cSUCC cONE))))) (Xs Xx))))) (forall (Xy:(fofType->Prop)), (((cSUCC (cSUCC (cSUCC (cSUCC (cSUCC cONE))))) Xy)->((ex fofType) (fun (Xy0:fofType)=> (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and (S Xx)) (((eq (fofType->Prop)) Xy) (Xs Xx))))) (fun (Xy:fofType)=> (((eq fofType) Xy0) Xy))))))))))) (x2:(forall (Xx:fofType) (Xy:fofType), (((K Xx) Xy)->((K Xy) Xx))))=> ((((((eq_trans Prop) (f x0)) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False))))))))) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and
% ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False))))))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False))))))))))) as proof of (((ex (fofType->(fofType->Prop))) (fun (Xs:(fofType->(fofType->Prop)))=> ((and (forall (Xx:fofType), ((S Xx)->((cSUCC (cSUCC (cSUCC (cSUCC (cSUCC cONE))))) (Xs Xx))))) (forall (Xy:(fofType->Prop)), (((cSUCC (cSUCC (cSUCC (cSUCC (cSUCC cONE))))) Xy)->((ex fofType) (fun (Xy0:fofType)=> (((eq (fofType->Prop)) (fun (Xx:fofType)=>
% ((and (S Xx)) (((eq (fofType->Prop)) Xy) (Xs Xx))))) (fun (Xy:fofType)=> (((eq fofType) Xy0) Xy))))))))))->((forall (Xx:fofType) (Xy:fofType), (((K Xx) Xy)->((K Xy) Xx)))->(((eq Prop) (f x0)) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))))))
% 266.23/267.22 Found (and_rect00 (fun (x1:((ex (fofType->(fofType->Prop))) (fun (Xs:(fofType->(fofType->Prop)))=> ((and (forall (Xx:fofType), ((S Xx)->((cSUCC (cSUCC (cSUCC (cSUCC (cSUCC cONE))))) (Xs Xx))))) (forall (Xy:(fofType->Prop)), (((cSUCC (cSUCC (cSUCC (cSUCC (cSUCC cONE))))) Xy)->((ex fofType) (fun (Xy0:fofType)=> (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and (S Xx)) (((eq (fofType->Prop)) Xy) (Xs Xx))))) (fun (Xy:fofType)=> (((eq fofType) Xy0) Xy))))))))))) (x2:(forall (Xx:fofType) (Xy:fofType), (((K Xx) Xy)->((K Xy) Xx))))=> ((((((eq_trans Prop) (f x0)) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False))))))))) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and
% ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False))))))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))))))) as proof of (((eq Prop) (f x0)) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and
% (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))))
% 266.23/267.23 Found ((and_rect0 (((eq Prop) (f x0)) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))))) (fun (x1:((ex (fofType->(fofType->Prop))) (fun (Xs:(fofType->(fofType->Prop)))=> ((and (forall (Xx:fofType), ((S Xx)->((cSUCC (cSUCC (cSUCC (cSUCC (cSUCC cONE))))) (Xs Xx))))) (forall (Xy:(fofType->Prop)), (((cSUCC (cSUCC (cSUCC (cSUCC (cSUCC cONE))))) Xy)->((ex fofType) (fun (Xy0:fofType)=> (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and (S Xx)) (((eq (fofType->Prop)) Xy) (Xs Xx))))) (fun (Xy:fofType)=> (((eq fofType) Xy0) Xy))))))))))) (x2:(forall (Xx:fofType) (Xy:fofType), (((K Xx) Xy)->((K Xy) Xx))))=> ((((((eq_trans Prop) (f x0)) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun
% (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False))))))))) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False))))))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0)
% Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))))))) as proof of (((eq Prop) (f x0)) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((
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