TSTP Solution File: NUM831^5 by cocATP---0.2.0
View Problem
- Process Solution
%------------------------------------------------------------------------------
% File : cocATP---0.2.0
% Problem : NUM831^5 : TPTP v7.0.0. Bugfixed v5.2.0.
% Transfm : none
% Format : tptp:raw
% Command : python CASC.py /export/starexec/sandbox2/benchmark/theBenchmark.p
% Computer : n136.star.cs.uiowa.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2609 0 2.40GHz
% Memory : 32218.625MB
% OS : Linux 3.10.0-693.2.2.el7.x86_64
% CPULimit : 300s
% DateTime : Mon Jan 8 13:11:51 EST 2018
% Result : Timeout 286.71s
% Output : None
% Verified :
% SZS Type : None (Parsing solution fails)
% Syntax : Number of formulae : 0
% Comments :
%------------------------------------------------------------------------------
%----No solution output by system
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.03 % Problem : NUM831^5 : TPTP v7.0.0. Bugfixed v5.2.0.
% 0.02/0.04 % Command : python CASC.py /export/starexec/sandbox2/benchmark/theBenchmark.p
% 0.02/0.23 % Computer : n136.star.cs.uiowa.edu
% 0.02/0.23 % Model : x86_64 x86_64
% 0.02/0.23 % CPU : Intel(R) Xeon(R) CPU E5-2609 0 @ 2.40GHz
% 0.02/0.23 % Memory : 32218.625MB
% 0.02/0.23 % OS : Linux 3.10.0-693.2.2.el7.x86_64
% 0.02/0.23 % CPULimit : 300
% 0.02/0.23 % DateTime : Fri Jan 5 14:41:05 CST 2018
% 0.02/0.23 % CPUTime :
% 0.02/0.25 Python 2.7.13
% 7.95/8.16 Using paths ['/home/cristobal/cocATP/CASC/TPTP/', '/export/starexec/sandbox2/benchmark/', '/export/starexec/sandbox2/benchmark/']
% 7.95/8.16 FOF formula (<kernel.Constant object at 0x2ad913f99050>, <kernel.Constant object at 0x2ad913f99680>) of role type named c0_type
% 7.95/8.16 Using role type
% 7.95/8.16 Declaring c0:fofType
% 7.95/8.16 FOF formula (<kernel.Constant object at 0x2ad914402c68>, <kernel.DependentProduct object at 0x2ad913fa9170>) of role type named cS_type
% 7.95/8.16 Using role type
% 7.95/8.16 Declaring cS:(fofType->fofType)
% 7.95/8.16 FOF formula (<kernel.Constant object at 0x2ad913f99050>, <kernel.Sort object at 0x2ad9143f85f0>) of role type named cIND_type
% 7.95/8.16 Using role type
% 7.95/8.16 Declaring cIND:Prop
% 7.95/8.16 FOF formula (((eq Prop) cIND) (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xx:fofType), ((Xp Xx)->(Xp (cS Xx)))))->(forall (Xx:fofType), (Xp Xx))))) of role definition named cIND_def
% 7.95/8.16 A new definition: (((eq Prop) cIND) (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xx:fofType), ((Xp Xx)->(Xp (cS Xx)))))->(forall (Xx:fofType), (Xp Xx)))))
% 7.95/8.16 Defined: cIND:=(forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xx:fofType), ((Xp Xx)->(Xp (cS Xx)))))->(forall (Xx:fofType), (Xp Xx))))
% 7.95/8.16 FOF formula (((and ((and cIND) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (cS Xx)) (cS Xy))->(((eq fofType) Xx) Xy))))) (forall (Xn:fofType), (not (((eq fofType) (cS Xn)) c0))))->((ex (fofType->(fofType->Prop))) (fun (Xd:(fofType->(fofType->Prop)))=> ((and ((and ((Xd c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((Xd Xx) Xy)->((Xd (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((Xd Xx) X)) (forall (Y:fofType), (((Xd Xx) Y)->(((eq fofType) X) Y))))))))))) of role conjecture named cTHM606_pme
% 7.95/8.16 Conjecture to prove = (((and ((and cIND) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (cS Xx)) (cS Xy))->(((eq fofType) Xx) Xy))))) (forall (Xn:fofType), (not (((eq fofType) (cS Xn)) c0))))->((ex (fofType->(fofType->Prop))) (fun (Xd:(fofType->(fofType->Prop)))=> ((and ((and ((Xd c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((Xd Xx) Xy)->((Xd (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((Xd Xx) X)) (forall (Y:fofType), (((Xd Xx) Y)->(((eq fofType) X) Y))))))))))):Prop
% 7.95/8.16 We need to prove ['(((and ((and cIND) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (cS Xx)) (cS Xy))->(((eq fofType) Xx) Xy))))) (forall (Xn:fofType), (not (((eq fofType) (cS Xn)) c0))))->((ex (fofType->(fofType->Prop))) (fun (Xd:(fofType->(fofType->Prop)))=> ((and ((and ((Xd c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((Xd Xx) Xy)->((Xd (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((Xd Xx) X)) (forall (Y:fofType), (((Xd Xx) Y)->(((eq fofType) X) Y)))))))))))']
% 7.95/8.16 Parameter fofType:Type.
% 7.95/8.16 Parameter c0:fofType.
% 7.95/8.16 Parameter cS:(fofType->fofType).
% 7.95/8.16 Definition cIND:=(forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xx:fofType), ((Xp Xx)->(Xp (cS Xx)))))->(forall (Xx:fofType), (Xp Xx)))):Prop.
% 7.95/8.16 Trying to prove (((and ((and cIND) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (cS Xx)) (cS Xy))->(((eq fofType) Xx) Xy))))) (forall (Xn:fofType), (not (((eq fofType) (cS Xn)) c0))))->((ex (fofType->(fofType->Prop))) (fun (Xd:(fofType->(fofType->Prop)))=> ((and ((and ((Xd c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((Xd Xx) Xy)->((Xd (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((Xd Xx) X)) (forall (Y:fofType), (((Xd Xx) Y)->(((eq fofType) X) Y)))))))))))
% 7.95/8.16 Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xd:(fofType->(fofType->Prop)))=> ((and ((and ((Xd c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((Xd Xx) Xy)->((Xd (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((Xd Xx) X)) (forall (Y:fofType), (((Xd Xx) Y)->(((eq fofType) X) Y)))))))))):(((eq ((fofType->(fofType->Prop))->Prop)) (fun (Xd:(fofType->(fofType->Prop)))=> ((and ((and ((Xd c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((Xd Xx) Xy)->((Xd (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((Xd Xx) X)) (forall (Y:fofType), (((Xd Xx) Y)->(((eq fofType) X) Y)))))))))) (fun (x:(fofType->(fofType->Prop)))=> ((and ((and ((x c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x Xx) Xy)->((x (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x Xx) X)) (forall (Y:fofType), (((x Xx) Y)->(((eq fofType) X) Y))))))))))
% 9.42/9.60 Found (eta_expansion_dep00 (fun (Xd:(fofType->(fofType->Prop)))=> ((and ((and ((Xd c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((Xd Xx) Xy)->((Xd (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((Xd Xx) X)) (forall (Y:fofType), (((Xd Xx) Y)->(((eq fofType) X) Y)))))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Xd:(fofType->(fofType->Prop)))=> ((and ((and ((Xd c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((Xd Xx) Xy)->((Xd (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((Xd Xx) X)) (forall (Y:fofType), (((Xd Xx) Y)->(((eq fofType) X) Y)))))))))) b)
% 9.42/9.60 Found ((eta_expansion_dep0 (fun (x1:(fofType->(fofType->Prop)))=> Prop)) (fun (Xd:(fofType->(fofType->Prop)))=> ((and ((and ((Xd c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((Xd Xx) Xy)->((Xd (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((Xd Xx) X)) (forall (Y:fofType), (((Xd Xx) Y)->(((eq fofType) X) Y)))))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Xd:(fofType->(fofType->Prop)))=> ((and ((and ((Xd c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((Xd Xx) Xy)->((Xd (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((Xd Xx) X)) (forall (Y:fofType), (((Xd Xx) Y)->(((eq fofType) X) Y)))))))))) b)
% 9.42/9.60 Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) (fun (Xd:(fofType->(fofType->Prop)))=> ((and ((and ((Xd c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((Xd Xx) Xy)->((Xd (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((Xd Xx) X)) (forall (Y:fofType), (((Xd Xx) Y)->(((eq fofType) X) Y)))))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Xd:(fofType->(fofType->Prop)))=> ((and ((and ((Xd c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((Xd Xx) Xy)->((Xd (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((Xd Xx) X)) (forall (Y:fofType), (((Xd Xx) Y)->(((eq fofType) X) Y)))))))))) b)
% 9.42/9.60 Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) (fun (Xd:(fofType->(fofType->Prop)))=> ((and ((and ((Xd c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((Xd Xx) Xy)->((Xd (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((Xd Xx) X)) (forall (Y:fofType), (((Xd Xx) Y)->(((eq fofType) X) Y)))))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Xd:(fofType->(fofType->Prop)))=> ((and ((and ((Xd c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((Xd Xx) Xy)->((Xd (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((Xd Xx) X)) (forall (Y:fofType), (((Xd Xx) Y)->(((eq fofType) X) Y)))))))))) b)
% 9.42/9.60 Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) (fun (Xd:(fofType->(fofType->Prop)))=> ((and ((and ((Xd c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((Xd Xx) Xy)->((Xd (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((Xd Xx) X)) (forall (Y:fofType), (((Xd Xx) Y)->(((eq fofType) X) Y)))))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Xd:(fofType->(fofType->Prop)))=> ((and ((and ((Xd c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((Xd Xx) Xy)->((Xd (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((Xd Xx) X)) (forall (Y:fofType), (((Xd Xx) Y)->(((eq fofType) X) Y)))))))))) b)
% 9.42/9.60 Found eta_expansion000:=(eta_expansion00 (fun (Xd:(fofType->(fofType->Prop)))=> ((and ((and ((Xd c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((Xd Xx) Xy)->((Xd (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (Xd Xx))))))):(((eq ((fofType->(fofType->Prop))->Prop)) (fun (Xd:(fofType->(fofType->Prop)))=> ((and ((and ((Xd c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((Xd Xx) Xy)->((Xd (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (Xd Xx))))))) (fun (x:(fofType->(fofType->Prop)))=> ((and ((and ((x c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x Xx) Xy)->((x (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x Xx)))))))
% 11.63/11.82 Found (eta_expansion00 (fun (Xd:(fofType->(fofType->Prop)))=> ((and ((and ((Xd c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((Xd Xx) Xy)->((Xd (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (Xd Xx))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Xd:(fofType->(fofType->Prop)))=> ((and ((and ((Xd c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((Xd Xx) Xy)->((Xd (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (Xd Xx))))))) b)
% 11.63/11.82 Found ((eta_expansion0 Prop) (fun (Xd:(fofType->(fofType->Prop)))=> ((and ((and ((Xd c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((Xd Xx) Xy)->((Xd (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (Xd Xx))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Xd:(fofType->(fofType->Prop)))=> ((and ((and ((Xd c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((Xd Xx) Xy)->((Xd (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (Xd Xx))))))) b)
% 11.63/11.82 Found (((eta_expansion (fofType->(fofType->Prop))) Prop) (fun (Xd:(fofType->(fofType->Prop)))=> ((and ((and ((Xd c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((Xd Xx) Xy)->((Xd (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (Xd Xx))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Xd:(fofType->(fofType->Prop)))=> ((and ((and ((Xd c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((Xd Xx) Xy)->((Xd (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (Xd Xx))))))) b)
% 11.63/11.82 Found (((eta_expansion (fofType->(fofType->Prop))) Prop) (fun (Xd:(fofType->(fofType->Prop)))=> ((and ((and ((Xd c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((Xd Xx) Xy)->((Xd (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (Xd Xx))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Xd:(fofType->(fofType->Prop)))=> ((and ((and ((Xd c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((Xd Xx) Xy)->((Xd (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (Xd Xx))))))) b)
% 11.63/11.82 Found (((eta_expansion (fofType->(fofType->Prop))) Prop) (fun (Xd:(fofType->(fofType->Prop)))=> ((and ((and ((Xd c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((Xd Xx) Xy)->((Xd (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (Xd Xx))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Xd:(fofType->(fofType->Prop)))=> ((and ((and ((Xd c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((Xd Xx) Xy)->((Xd (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (Xd Xx))))))) b)
% 11.63/11.82 Found eq_ref00:=(eq_ref0 (fun (Xd:(fofType->(fofType->Prop)))=> ((and ((and ((Xd c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((Xd Xx) Xy)->((Xd (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((Xd Xx) X)) (forall (Y:fofType), (((Xd Xx) Y)->(((eq fofType) X) Y)))))))))):(((eq ((fofType->(fofType->Prop))->Prop)) (fun (Xd:(fofType->(fofType->Prop)))=> ((and ((and ((Xd c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((Xd Xx) Xy)->((Xd (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((Xd Xx) X)) (forall (Y:fofType), (((Xd Xx) Y)->(((eq fofType) X) Y)))))))))) (fun (Xd:(fofType->(fofType->Prop)))=> ((and ((and ((Xd c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((Xd Xx) Xy)->((Xd (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((Xd Xx) X)) (forall (Y:fofType), (((Xd Xx) Y)->(((eq fofType) X) Y))))))))))
% 11.63/11.82 Found (eq_ref0 (fun (Xd:(fofType->(fofType->Prop)))=> ((and ((and ((Xd c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((Xd Xx) Xy)->((Xd (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((Xd Xx) X)) (forall (Y:fofType), (((Xd Xx) Y)->(((eq fofType) X) Y)))))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Xd:(fofType->(fofType->Prop)))=> ((and ((and ((Xd c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((Xd Xx) Xy)->((Xd (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((Xd Xx) X)) (forall (Y:fofType), (((Xd Xx) Y)->(((eq fofType) X) Y)))))))))) b)
% 13.36/13.61 Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) (fun (Xd:(fofType->(fofType->Prop)))=> ((and ((and ((Xd c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((Xd Xx) Xy)->((Xd (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((Xd Xx) X)) (forall (Y:fofType), (((Xd Xx) Y)->(((eq fofType) X) Y)))))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Xd:(fofType->(fofType->Prop)))=> ((and ((and ((Xd c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((Xd Xx) Xy)->((Xd (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((Xd Xx) X)) (forall (Y:fofType), (((Xd Xx) Y)->(((eq fofType) X) Y)))))))))) b)
% 13.36/13.61 Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) (fun (Xd:(fofType->(fofType->Prop)))=> ((and ((and ((Xd c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((Xd Xx) Xy)->((Xd (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((Xd Xx) X)) (forall (Y:fofType), (((Xd Xx) Y)->(((eq fofType) X) Y)))))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Xd:(fofType->(fofType->Prop)))=> ((and ((and ((Xd c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((Xd Xx) Xy)->((Xd (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((Xd Xx) X)) (forall (Y:fofType), (((Xd Xx) Y)->(((eq fofType) X) Y)))))))))) b)
% 13.36/13.61 Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) (fun (Xd:(fofType->(fofType->Prop)))=> ((and ((and ((Xd c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((Xd Xx) Xy)->((Xd (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((Xd Xx) X)) (forall (Y:fofType), (((Xd Xx) Y)->(((eq fofType) X) Y)))))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Xd:(fofType->(fofType->Prop)))=> ((and ((and ((Xd c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((Xd Xx) Xy)->((Xd (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((Xd Xx) X)) (forall (Y:fofType), (((Xd Xx) Y)->(((eq fofType) X) Y)))))))))) b)
% 13.36/13.61 Found eta_expansion000:=(eta_expansion00 (fun (Xd:(fofType->(fofType->Prop)))=> ((and ((and ((Xd c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((Xd Xx) Xy)->((Xd (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (Xd Xx))))))):(((eq ((fofType->(fofType->Prop))->Prop)) (fun (Xd:(fofType->(fofType->Prop)))=> ((and ((and ((Xd c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((Xd Xx) Xy)->((Xd (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (Xd Xx))))))) (fun (x:(fofType->(fofType->Prop)))=> ((and ((and ((x c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x Xx) Xy)->((x (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x Xx)))))))
% 13.36/13.61 Found (eta_expansion00 (fun (Xd:(fofType->(fofType->Prop)))=> ((and ((and ((Xd c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((Xd Xx) Xy)->((Xd (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (Xd Xx))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Xd:(fofType->(fofType->Prop)))=> ((and ((and ((Xd c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((Xd Xx) Xy)->((Xd (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (Xd Xx))))))) b)
% 13.36/13.61 Found ((eta_expansion0 Prop) (fun (Xd:(fofType->(fofType->Prop)))=> ((and ((and ((Xd c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((Xd Xx) Xy)->((Xd (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (Xd Xx))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Xd:(fofType->(fofType->Prop)))=> ((and ((and ((Xd c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((Xd Xx) Xy)->((Xd (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (Xd Xx))))))) b)
% 17.49/17.67 Found (((eta_expansion (fofType->(fofType->Prop))) Prop) (fun (Xd:(fofType->(fofType->Prop)))=> ((and ((and ((Xd c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((Xd Xx) Xy)->((Xd (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (Xd Xx))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Xd:(fofType->(fofType->Prop)))=> ((and ((and ((Xd c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((Xd Xx) Xy)->((Xd (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (Xd Xx))))))) b)
% 17.49/17.67 Found (((eta_expansion (fofType->(fofType->Prop))) Prop) (fun (Xd:(fofType->(fofType->Prop)))=> ((and ((and ((Xd c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((Xd Xx) Xy)->((Xd (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (Xd Xx))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Xd:(fofType->(fofType->Prop)))=> ((and ((and ((Xd c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((Xd Xx) Xy)->((Xd (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (Xd Xx))))))) b)
% 17.49/17.67 Found (((eta_expansion (fofType->(fofType->Prop))) Prop) (fun (Xd:(fofType->(fofType->Prop)))=> ((and ((and ((Xd c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((Xd Xx) Xy)->((Xd (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (Xd Xx))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Xd:(fofType->(fofType->Prop)))=> ((and ((and ((Xd c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((Xd Xx) Xy)->((Xd (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (Xd Xx))))))) b)
% 17.49/17.67 Found eta_expansion000:=(eta_expansion00 (fun (Xd:(fofType->(fofType->Prop)))=> ((and ((and ((Xd c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((Xd Xx) Xy)->((Xd (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((Xd Xx) X)) (forall (Y:fofType), (((Xd Xx) Y)->(((eq fofType) X) Y)))))))))):(((eq ((fofType->(fofType->Prop))->Prop)) (fun (Xd:(fofType->(fofType->Prop)))=> ((and ((and ((Xd c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((Xd Xx) Xy)->((Xd (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((Xd Xx) X)) (forall (Y:fofType), (((Xd Xx) Y)->(((eq fofType) X) Y)))))))))) (fun (x:(fofType->(fofType->Prop)))=> ((and ((and ((x c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x Xx) Xy)->((x (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x Xx) X)) (forall (Y:fofType), (((x Xx) Y)->(((eq fofType) X) Y))))))))))
% 17.49/17.67 Found (eta_expansion00 (fun (Xd:(fofType->(fofType->Prop)))=> ((and ((and ((Xd c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((Xd Xx) Xy)->((Xd (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((Xd Xx) X)) (forall (Y:fofType), (((Xd Xx) Y)->(((eq fofType) X) Y)))))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Xd:(fofType->(fofType->Prop)))=> ((and ((and ((Xd c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((Xd Xx) Xy)->((Xd (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((Xd Xx) X)) (forall (Y:fofType), (((Xd Xx) Y)->(((eq fofType) X) Y)))))))))) b)
% 17.49/17.67 Found ((eta_expansion0 Prop) (fun (Xd:(fofType->(fofType->Prop)))=> ((and ((and ((Xd c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((Xd Xx) Xy)->((Xd (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((Xd Xx) X)) (forall (Y:fofType), (((Xd Xx) Y)->(((eq fofType) X) Y)))))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Xd:(fofType->(fofType->Prop)))=> ((and ((and ((Xd c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((Xd Xx) Xy)->((Xd (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((Xd Xx) X)) (forall (Y:fofType), (((Xd Xx) Y)->(((eq fofType) X) Y)))))))))) b)
% 19.96/20.14 Found (((eta_expansion (fofType->(fofType->Prop))) Prop) (fun (Xd:(fofType->(fofType->Prop)))=> ((and ((and ((Xd c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((Xd Xx) Xy)->((Xd (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((Xd Xx) X)) (forall (Y:fofType), (((Xd Xx) Y)->(((eq fofType) X) Y)))))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Xd:(fofType->(fofType->Prop)))=> ((and ((and ((Xd c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((Xd Xx) Xy)->((Xd (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((Xd Xx) X)) (forall (Y:fofType), (((Xd Xx) Y)->(((eq fofType) X) Y)))))))))) b)
% 19.96/20.14 Found (((eta_expansion (fofType->(fofType->Prop))) Prop) (fun (Xd:(fofType->(fofType->Prop)))=> ((and ((and ((Xd c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((Xd Xx) Xy)->((Xd (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((Xd Xx) X)) (forall (Y:fofType), (((Xd Xx) Y)->(((eq fofType) X) Y)))))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Xd:(fofType->(fofType->Prop)))=> ((and ((and ((Xd c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((Xd Xx) Xy)->((Xd (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((Xd Xx) X)) (forall (Y:fofType), (((Xd Xx) Y)->(((eq fofType) X) Y)))))))))) b)
% 19.96/20.14 Found (((eta_expansion (fofType->(fofType->Prop))) Prop) (fun (Xd:(fofType->(fofType->Prop)))=> ((and ((and ((Xd c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((Xd Xx) Xy)->((Xd (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((Xd Xx) X)) (forall (Y:fofType), (((Xd Xx) Y)->(((eq fofType) X) Y)))))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Xd:(fofType->(fofType->Prop)))=> ((and ((and ((Xd c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((Xd Xx) Xy)->((Xd (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((Xd Xx) X)) (forall (Y:fofType), (((Xd Xx) Y)->(((eq fofType) X) Y)))))))))) b)
% 19.96/20.14 Found eta_expansion000:=(eta_expansion00 (fun (Xd:(fofType->(fofType->Prop)))=> ((and ((and ((Xd c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((Xd Xx) Xy)->((Xd (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (Xd Xx))))))):(((eq ((fofType->(fofType->Prop))->Prop)) (fun (Xd:(fofType->(fofType->Prop)))=> ((and ((and ((Xd c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((Xd Xx) Xy)->((Xd (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (Xd Xx))))))) (fun (x:(fofType->(fofType->Prop)))=> ((and ((and ((x c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x Xx) Xy)->((x (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x Xx)))))))
% 19.96/20.14 Found (eta_expansion00 (fun (Xd:(fofType->(fofType->Prop)))=> ((and ((and ((Xd c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((Xd Xx) Xy)->((Xd (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (Xd Xx))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Xd:(fofType->(fofType->Prop)))=> ((and ((and ((Xd c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((Xd Xx) Xy)->((Xd (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (Xd Xx))))))) b)
% 19.96/20.14 Found ((eta_expansion0 Prop) (fun (Xd:(fofType->(fofType->Prop)))=> ((and ((and ((Xd c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((Xd Xx) Xy)->((Xd (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (Xd Xx))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Xd:(fofType->(fofType->Prop)))=> ((and ((and ((Xd c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((Xd Xx) Xy)->((Xd (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (Xd Xx))))))) b)
% 19.96/20.14 Found (((eta_expansion (fofType->(fofType->Prop))) Prop) (fun (Xd:(fofType->(fofType->Prop)))=> ((and ((and ((Xd c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((Xd Xx) Xy)->((Xd (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (Xd Xx))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Xd:(fofType->(fofType->Prop)))=> ((and ((and ((Xd c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((Xd Xx) Xy)->((Xd (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (Xd Xx))))))) b)
% 41.25/41.50 Found (((eta_expansion (fofType->(fofType->Prop))) Prop) (fun (Xd:(fofType->(fofType->Prop)))=> ((and ((and ((Xd c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((Xd Xx) Xy)->((Xd (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (Xd Xx))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Xd:(fofType->(fofType->Prop)))=> ((and ((and ((Xd c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((Xd Xx) Xy)->((Xd (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (Xd Xx))))))) b)
% 41.25/41.50 Found (((eta_expansion (fofType->(fofType->Prop))) Prop) (fun (Xd:(fofType->(fofType->Prop)))=> ((and ((and ((Xd c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((Xd Xx) Xy)->((Xd (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (Xd Xx))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Xd:(fofType->(fofType->Prop)))=> ((and ((and ((Xd c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((Xd Xx) Xy)->((Xd (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (Xd Xx))))))) b)
% 41.25/41.50 Found eq_substitution0000:=(fun (a:fofType)=> ((eq_substitution000 a) (fun (x4:fofType)=> (cS Xx)))):(forall (a:fofType), ((((eq fofType) a) (cS Xy))->(((eq fofType) (cS Xx)) (cS Xx))))
% 41.25/41.50 Instantiate: x0:=(fun (x2:fofType) (x10:fofType)=> (forall (a:fofType), ((((eq fofType) a) (cS Xy))->(((eq fofType) x2) x2)))):(fofType->(fofType->Prop))
% 41.25/41.50 Found (fun (a:fofType)=> ((eq_substitution000 a) (fun (x4:fofType)=> (cS Xx)))) as proof of ((x0 (cS Xx)) (cS (cS Xy)))
% 41.25/41.50 Found (fun (a:fofType)=> (((fun (a:fofType)=> ((eq_substitution00 a) (cS Xy))) a) (fun (x4:fofType)=> (cS Xx)))) as proof of ((x0 (cS Xx)) (cS (cS Xy)))
% 41.25/41.50 Found (fun (a:fofType)=> (((fun (a:fofType)=> (((eq_substitution0 fofType) a) (cS Xy))) a) (fun (x4:fofType)=> (cS Xx)))) as proof of ((x0 (cS Xx)) (cS (cS Xy)))
% 41.25/41.50 Found (fun (a:fofType)=> (((fun (a:fofType)=> ((((eq_substitution fofType) fofType) a) (cS Xy))) a) (fun (x4:fofType)=> (cS Xx)))) as proof of ((x0 (cS Xx)) (cS (cS Xy)))
% 41.25/41.50 Found (fun (x00:((x0 Xx) Xy)) (a:fofType)=> (((fun (a:fofType)=> ((((eq_substitution fofType) fofType) a) (cS Xy))) a) (fun (x4:fofType)=> (cS Xx)))) as proof of ((x0 (cS Xx)) (cS (cS Xy)))
% 41.25/41.50 Found eq_substitution0000:=(fun (a:fofType)=> ((eq_substitution000 a) (fun (x4:fofType)=> (cS Xx)))):(forall (a:fofType), ((((eq fofType) a) (cS Xy))->(((eq fofType) (cS Xx)) (cS Xx))))
% 41.25/41.50 Instantiate: x0:=(fun (x2:fofType) (x10:fofType)=> (forall (a:fofType), ((((eq fofType) a) (cS Xy))->(((eq fofType) x2) x2)))):(fofType->(fofType->Prop))
% 41.25/41.50 Found (fun (a:fofType)=> ((eq_substitution000 a) (fun (x4:fofType)=> (cS Xx)))) as proof of ((x0 (cS Xx)) (cS (cS Xy)))
% 41.25/41.50 Found (fun (a:fofType)=> (((fun (a:fofType)=> ((eq_substitution00 a) (cS Xy))) a) (fun (x4:fofType)=> (cS Xx)))) as proof of ((x0 (cS Xx)) (cS (cS Xy)))
% 41.25/41.50 Found (fun (a:fofType)=> (((fun (a:fofType)=> (((eq_substitution0 fofType) a) (cS Xy))) a) (fun (x4:fofType)=> (cS Xx)))) as proof of ((x0 (cS Xx)) (cS (cS Xy)))
% 41.25/41.50 Found (fun (a:fofType)=> (((fun (a:fofType)=> ((((eq_substitution fofType) fofType) a) (cS Xy))) a) (fun (x4:fofType)=> (cS Xx)))) as proof of ((x0 (cS Xx)) (cS (cS Xy)))
% 41.25/41.50 Found (fun (x00:((x0 Xx) Xy)) (a:fofType)=> (((fun (a:fofType)=> ((((eq_substitution fofType) fofType) a) (cS Xy))) a) (fun (x4:fofType)=> (cS Xx)))) as proof of ((x0 (cS Xx)) (cS (cS Xy)))
% 41.25/41.50 Found eq_substitution0000:=(fun (a:fofType)=> ((eq_substitution000 a) (fun (x6:fofType)=> (cS Xx)))):(forall (a:fofType), ((((eq fofType) a) (cS Xy))->(((eq fofType) (cS Xx)) (cS Xx))))
% 41.25/41.50 Instantiate: x2:=(fun (x4:fofType) (x30:fofType)=> (forall (a:fofType), ((((eq fofType) a) (cS Xy))->(((eq fofType) x4) x4)))):(fofType->(fofType->Prop))
% 41.25/41.50 Found (fun (a:fofType)=> ((eq_substitution000 a) (fun (x6:fofType)=> (cS Xx)))) as proof of ((x2 (cS Xx)) (cS (cS Xy)))
% 48.91/49.15 Found (fun (a:fofType)=> (((fun (a:fofType)=> ((eq_substitution00 a) (cS Xy))) a) (fun (x6:fofType)=> (cS Xx)))) as proof of ((x2 (cS Xx)) (cS (cS Xy)))
% 48.91/49.15 Found (fun (a:fofType)=> (((fun (a:fofType)=> (((eq_substitution0 fofType) a) (cS Xy))) a) (fun (x6:fofType)=> (cS Xx)))) as proof of ((x2 (cS Xx)) (cS (cS Xy)))
% 48.91/49.15 Found (fun (a:fofType)=> (((fun (a:fofType)=> ((((eq_substitution fofType) fofType) a) (cS Xy))) a) (fun (x6:fofType)=> (cS Xx)))) as proof of ((x2 (cS Xx)) (cS (cS Xy)))
% 48.91/49.15 Found (fun (x00:((x2 Xx) Xy)) (a:fofType)=> (((fun (a:fofType)=> ((((eq_substitution fofType) fofType) a) (cS Xy))) a) (fun (x6:fofType)=> (cS Xx)))) as proof of ((x2 (cS Xx)) (cS (cS Xy)))
% 48.91/49.15 Found eq_substitution0000:=(fun (a:fofType)=> ((eq_substitution000 a) (fun (x6:fofType)=> (cS Xx)))):(forall (a:fofType), ((((eq fofType) a) (cS Xy))->(((eq fofType) (cS Xx)) (cS Xx))))
% 48.91/49.15 Instantiate: x0:=(fun (x4:fofType) (x30:fofType)=> (forall (a:fofType), ((((eq fofType) a) (cS Xy))->(((eq fofType) x4) x4)))):(fofType->(fofType->Prop))
% 48.91/49.15 Found (fun (a:fofType)=> ((eq_substitution000 a) (fun (x6:fofType)=> (cS Xx)))) as proof of ((x0 (cS Xx)) (cS (cS Xy)))
% 48.91/49.15 Found (fun (a:fofType)=> (((fun (a:fofType)=> ((eq_substitution00 a) (cS Xy))) a) (fun (x6:fofType)=> (cS Xx)))) as proof of ((x0 (cS Xx)) (cS (cS Xy)))
% 48.91/49.15 Found (fun (a:fofType)=> (((fun (a:fofType)=> (((eq_substitution0 fofType) a) (cS Xy))) a) (fun (x6:fofType)=> (cS Xx)))) as proof of ((x0 (cS Xx)) (cS (cS Xy)))
% 48.91/49.15 Found (fun (a:fofType)=> (((fun (a:fofType)=> ((((eq_substitution fofType) fofType) a) (cS Xy))) a) (fun (x6:fofType)=> (cS Xx)))) as proof of ((x0 (cS Xx)) (cS (cS Xy)))
% 48.91/49.15 Found (fun (x00:((x0 Xx) Xy)) (a:fofType)=> (((fun (a:fofType)=> ((((eq_substitution fofType) fofType) a) (cS Xy))) a) (fun (x6:fofType)=> (cS Xx)))) as proof of ((x0 (cS Xx)) (cS (cS Xy)))
% 48.91/49.15 Found eq_substitution0000:=(fun (a:fofType)=> ((eq_substitution000 a) (fun (x6:fofType)=> (cS Xx)))):(forall (a:fofType), ((((eq fofType) a) (cS Xy))->(((eq fofType) (cS Xx)) (cS Xx))))
% 48.91/49.15 Instantiate: x2:=(fun (x4:fofType) (x30:fofType)=> (forall (a:fofType), ((((eq fofType) a) (cS Xy))->(((eq fofType) x4) x4)))):(fofType->(fofType->Prop))
% 48.91/49.15 Found (fun (a:fofType)=> ((eq_substitution000 a) (fun (x6:fofType)=> (cS Xx)))) as proof of ((x2 (cS Xx)) (cS (cS Xy)))
% 48.91/49.15 Found (fun (a:fofType)=> (((fun (a:fofType)=> ((eq_substitution00 a) (cS Xy))) a) (fun (x6:fofType)=> (cS Xx)))) as proof of ((x2 (cS Xx)) (cS (cS Xy)))
% 48.91/49.15 Found (fun (a:fofType)=> (((fun (a:fofType)=> (((eq_substitution0 fofType) a) (cS Xy))) a) (fun (x6:fofType)=> (cS Xx)))) as proof of ((x2 (cS Xx)) (cS (cS Xy)))
% 48.91/49.15 Found (fun (a:fofType)=> (((fun (a:fofType)=> ((((eq_substitution fofType) fofType) a) (cS Xy))) a) (fun (x6:fofType)=> (cS Xx)))) as proof of ((x2 (cS Xx)) (cS (cS Xy)))
% 48.91/49.15 Found (fun (x00:((x2 Xx) Xy)) (a:fofType)=> (((fun (a:fofType)=> ((((eq_substitution fofType) fofType) a) (cS Xy))) a) (fun (x6:fofType)=> (cS Xx)))) as proof of ((x2 (cS Xx)) (cS (cS Xy)))
% 48.91/49.15 Found eq_substitution0000:=(fun (a:fofType)=> ((eq_substitution000 a) (fun (x6:fofType)=> (cS Xx)))):(forall (a:fofType), ((((eq fofType) a) (cS Xy))->(((eq fofType) (cS Xx)) (cS Xx))))
% 48.91/49.15 Instantiate: x0:=(fun (x4:fofType) (x30:fofType)=> (forall (a:fofType), ((((eq fofType) a) (cS Xy))->(((eq fofType) x4) x4)))):(fofType->(fofType->Prop))
% 48.91/49.15 Found (fun (a:fofType)=> ((eq_substitution000 a) (fun (x6:fofType)=> (cS Xx)))) as proof of ((x0 (cS Xx)) (cS (cS Xy)))
% 48.91/49.15 Found (fun (a:fofType)=> (((fun (a:fofType)=> ((eq_substitution00 a) (cS Xy))) a) (fun (x6:fofType)=> (cS Xx)))) as proof of ((x0 (cS Xx)) (cS (cS Xy)))
% 48.91/49.15 Found (fun (a:fofType)=> (((fun (a:fofType)=> (((eq_substitution0 fofType) a) (cS Xy))) a) (fun (x6:fofType)=> (cS Xx)))) as proof of ((x0 (cS Xx)) (cS (cS Xy)))
% 48.91/49.15 Found (fun (a:fofType)=> (((fun (a:fofType)=> ((((eq_substitution fofType) fofType) a) (cS Xy))) a) (fun (x6:fofType)=> (cS Xx)))) as proof of ((x0 (cS Xx)) (cS (cS Xy)))
% 48.91/49.15 Found (fun (x00:((x0 Xx) Xy)) (a:fofType)=> (((fun (a:fofType)=> ((((eq_substitution fofType) fofType) a) (cS Xy))) a) (fun (x6:fofType)=> (cS Xx)))) as proof of ((x0 (cS Xx)) (cS (cS Xy)))
% 57.12/57.30 Found eq_substitution0000:=(fun (a:fofType)=> ((eq_substitution000 a) (fun (x8:fofType)=> (cS Xx)))):(forall (a:fofType), ((((eq fofType) a) (cS Xy))->(((eq fofType) (cS Xx)) (cS Xx))))
% 57.12/57.30 Instantiate: x4:=(fun (x6:fofType) (x50:fofType)=> (forall (a:fofType), ((((eq fofType) a) (cS Xy))->(((eq fofType) x6) x6)))):(fofType->(fofType->Prop))
% 57.12/57.30 Found (fun (a:fofType)=> ((eq_substitution000 a) (fun (x8:fofType)=> (cS Xx)))) as proof of ((x4 (cS Xx)) (cS (cS Xy)))
% 57.12/57.30 Found (fun (a:fofType)=> (((fun (a:fofType)=> ((eq_substitution00 a) (cS Xy))) a) (fun (x8:fofType)=> (cS Xx)))) as proof of ((x4 (cS Xx)) (cS (cS Xy)))
% 57.12/57.30 Found (fun (a:fofType)=> (((fun (a:fofType)=> (((eq_substitution0 fofType) a) (cS Xy))) a) (fun (x8:fofType)=> (cS Xx)))) as proof of ((x4 (cS Xx)) (cS (cS Xy)))
% 57.12/57.30 Found (fun (a:fofType)=> (((fun (a:fofType)=> ((((eq_substitution fofType) fofType) a) (cS Xy))) a) (fun (x8:fofType)=> (cS Xx)))) as proof of ((x4 (cS Xx)) (cS (cS Xy)))
% 57.12/57.30 Found (fun (x00:((x4 Xx) Xy)) (a:fofType)=> (((fun (a:fofType)=> ((((eq_substitution fofType) fofType) a) (cS Xy))) a) (fun (x8:fofType)=> (cS Xx)))) as proof of ((x4 (cS Xx)) (cS (cS Xy)))
% 57.12/57.30 Found eq_sym000:=(eq_sym00 Y):((((eq fofType) x1) Y)->(((eq fofType) Y) x1))
% 57.12/57.30 Found (eq_sym00 Y) as proof of (((x0 Xx) Y)->(((eq fofType) x1) Y))
% 57.12/57.30 Found ((eq_sym0 x1) Y) as proof of (((x0 Xx) Y)->(((eq fofType) x1) Y))
% 57.12/57.30 Found (((eq_sym fofType) x1) Y) as proof of (((x0 Xx) Y)->(((eq fofType) x1) Y))
% 57.12/57.30 Found (((eq_sym fofType) x1) Y) as proof of (((x0 Xx) Y)->(((eq fofType) x1) Y))
% 57.12/57.30 Found (((eq_sym fofType) x1) Y) as proof of (((x0 Xx) Y)->(((eq fofType) x1) Y))
% 57.12/57.30 Found eq_ref00:=(eq_ref0 x1):(((eq fofType) x1) x1)
% 57.12/57.30 Found (eq_ref0 x1) as proof of (((eq fofType) x1) Y)
% 57.12/57.30 Found ((eq_ref fofType) x1) as proof of (((eq fofType) x1) Y)
% 57.12/57.30 Found ((eq_ref fofType) x1) as proof of (((eq fofType) x1) Y)
% 57.12/57.30 Found (fun (x00:((x0 Xx) Y))=> ((eq_ref fofType) x1)) as proof of (((eq fofType) x1) Y)
% 57.12/57.30 Found x000:=(x00 x1):(((eq fofType) x1) Y)
% 57.12/57.30 Found (x00 x1) as proof of (((eq fofType) x1) Y)
% 57.12/57.30 Found (x00 x1) as proof of (((eq fofType) x1) Y)
% 57.12/57.30 Found (fun (x00:((x0 Xx) Y))=> (x00 x1)) as proof of (((eq fofType) x1) Y)
% 57.12/57.30 Found (fun (Y:fofType) (x00:((x0 Xx) Y))=> (x00 x1)) as proof of (((x0 Xx) Y)->(((eq fofType) x1) Y))
% 57.12/57.30 Found (fun (Y:fofType) (x00:((x0 Xx) Y))=> (x00 x1)) as proof of (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) x1) Y)))
% 57.12/57.30 Found eq_ref00:=(eq_ref0 x10):(((eq fofType) x10) x10)
% 57.12/57.30 Found (eq_ref0 x10) as proof of (((eq fofType) x10) x1)
% 57.12/57.30 Found ((eq_ref fofType) x10) as proof of (((eq fofType) x10) x1)
% 57.12/57.30 Found ((eq_ref fofType) x10) as proof of (((eq fofType) x10) x1)
% 57.12/57.30 Found x2:(P x10)
% 57.12/57.30 Instantiate: x1:=x10:fofType
% 57.12/57.30 Found (fun (x2:(P x10))=> x2) as proof of (P x1)
% 57.12/57.30 Found (fun (P:(fofType->Prop)) (x2:(P x10))=> x2) as proof of ((P x10)->(P x1))
% 57.12/57.30 Found eq_ref000:=(eq_ref00 P):((P x10)->(P x10))
% 57.12/57.30 Found (eq_ref00 P) as proof of ((P x10)->(P x1))
% 57.12/57.30 Found ((eq_ref0 x10) P) as proof of ((P x10)->(P x1))
% 57.12/57.30 Found (((eq_ref fofType) x10) P) as proof of ((P x10)->(P x1))
% 57.12/57.30 Found (((eq_ref fofType) x10) P) as proof of ((P x10)->(P x1))
% 57.12/57.30 Found (fun (P:(fofType->Prop))=> (((eq_ref fofType) x10) P)) as proof of ((P x10)->(P x1))
% 57.12/57.30 Found eq_substitution0000:=(fun (a:fofType)=> ((eq_substitution000 a) (fun (x6:fofType)=> (cS Xx)))):(forall (a:fofType), ((((eq fofType) a) (cS Xy))->(((eq fofType) (cS Xx)) (cS Xx))))
% 57.12/57.30 Instantiate: x0:=(fun (x4:fofType) (x30:fofType)=> (forall (a:fofType), ((((eq fofType) a) (cS Xy))->(((eq fofType) x4) x4)))):(fofType->(fofType->Prop))
% 57.12/57.30 Found (fun (a:fofType)=> ((eq_substitution000 a) (fun (x6:fofType)=> (cS Xx)))) as proof of ((x0 (cS Xx)) (cS (cS Xy)))
% 57.12/57.30 Found (fun (a:fofType)=> (((fun (a:fofType)=> ((eq_substitution00 a) (cS Xy))) a) (fun (x6:fofType)=> (cS Xx)))) as proof of ((x0 (cS Xx)) (cS (cS Xy)))
% 57.12/57.30 Found (fun (a:fofType)=> (((fun (a:fofType)=> (((eq_substitution0 fofType) a) (cS Xy))) a) (fun (x6:fofType)=> (cS Xx)))) as proof of ((x0 (cS Xx)) (cS (cS Xy)))
% 57.12/57.30 Found (fun (a:fofType)=> (((fun (a:fofType)=> ((((eq_substitution fofType) fofType) a) (cS Xy))) a) (fun (x6:fofType)=> (cS Xx)))) as proof of ((x0 (cS Xx)) (cS (cS Xy)))
% 60.15/60.35 Found (fun (x00:((x0 Xx) Xy)) (a:fofType)=> (((fun (a:fofType)=> ((((eq_substitution fofType) fofType) a) (cS Xy))) a) (fun (x6:fofType)=> (cS Xx)))) as proof of ((x0 (cS Xx)) (cS (cS Xy)))
% 60.15/60.35 Found eq_substitution0000:=(fun (a:fofType)=> ((eq_substitution000 a) (fun (x8:fofType)=> (cS Xx)))):(forall (a:fofType), ((((eq fofType) a) (cS Xy))->(((eq fofType) (cS Xx)) (cS Xx))))
% 60.15/60.35 Instantiate: x0:=(fun (x6:fofType) (x50:fofType)=> (forall (a:fofType), ((((eq fofType) a) (cS Xy))->(((eq fofType) x6) x6)))):(fofType->(fofType->Prop))
% 60.15/60.35 Found (fun (a:fofType)=> ((eq_substitution000 a) (fun (x8:fofType)=> (cS Xx)))) as proof of ((x0 (cS Xx)) (cS (cS Xy)))
% 60.15/60.35 Found (fun (a:fofType)=> (((fun (a:fofType)=> ((eq_substitution00 a) (cS Xy))) a) (fun (x8:fofType)=> (cS Xx)))) as proof of ((x0 (cS Xx)) (cS (cS Xy)))
% 60.15/60.35 Found (fun (a:fofType)=> (((fun (a:fofType)=> (((eq_substitution0 fofType) a) (cS Xy))) a) (fun (x8:fofType)=> (cS Xx)))) as proof of ((x0 (cS Xx)) (cS (cS Xy)))
% 60.15/60.35 Found (fun (a:fofType)=> (((fun (a:fofType)=> ((((eq_substitution fofType) fofType) a) (cS Xy))) a) (fun (x8:fofType)=> (cS Xx)))) as proof of ((x0 (cS Xx)) (cS (cS Xy)))
% 60.15/60.35 Found (fun (x00:((x0 Xx) Xy)) (a:fofType)=> (((fun (a:fofType)=> ((((eq_substitution fofType) fofType) a) (cS Xy))) a) (fun (x8:fofType)=> (cS Xx)))) as proof of ((x0 (cS Xx)) (cS (cS Xy)))
% 60.15/60.35 Found eq_substitution0000:=(fun (a:fofType)=> ((eq_substitution000 a) (fun (x8:fofType)=> (cS Xx)))):(forall (a:fofType), ((((eq fofType) a) (cS Xy))->(((eq fofType) (cS Xx)) (cS Xx))))
% 60.15/60.35 Instantiate: x2:=(fun (x6:fofType) (x50:fofType)=> (forall (a:fofType), ((((eq fofType) a) (cS Xy))->(((eq fofType) x6) x6)))):(fofType->(fofType->Prop))
% 60.15/60.35 Found (fun (a:fofType)=> ((eq_substitution000 a) (fun (x8:fofType)=> (cS Xx)))) as proof of ((x2 (cS Xx)) (cS (cS Xy)))
% 60.15/60.35 Found (fun (a:fofType)=> (((fun (a:fofType)=> ((eq_substitution00 a) (cS Xy))) a) (fun (x8:fofType)=> (cS Xx)))) as proof of ((x2 (cS Xx)) (cS (cS Xy)))
% 60.15/60.35 Found (fun (a:fofType)=> (((fun (a:fofType)=> (((eq_substitution0 fofType) a) (cS Xy))) a) (fun (x8:fofType)=> (cS Xx)))) as proof of ((x2 (cS Xx)) (cS (cS Xy)))
% 60.15/60.35 Found (fun (a:fofType)=> (((fun (a:fofType)=> ((((eq_substitution fofType) fofType) a) (cS Xy))) a) (fun (x8:fofType)=> (cS Xx)))) as proof of ((x2 (cS Xx)) (cS (cS Xy)))
% 60.15/60.35 Found (fun (x00:((x2 Xx) Xy)) (a:fofType)=> (((fun (a:fofType)=> ((((eq_substitution fofType) fofType) a) (cS Xy))) a) (fun (x8:fofType)=> (cS Xx)))) as proof of ((x2 (cS Xx)) (cS (cS Xy)))
% 60.15/60.35 Found eq_sym000:=(eq_sym00 x'):((((eq fofType) x1) x')->(((eq fofType) x') x1))
% 60.15/60.35 Found (eq_sym00 x') as proof of (((x0 Xx) x')->(((eq fofType) x1) x'))
% 60.15/60.35 Found ((eq_sym0 x1) x') as proof of (((x0 Xx) x')->(((eq fofType) x1) x'))
% 60.15/60.35 Found (((eq_sym fofType) x1) x') as proof of (((x0 Xx) x')->(((eq fofType) x1) x'))
% 60.15/60.35 Found (((eq_sym fofType) x1) x') as proof of (((x0 Xx) x')->(((eq fofType) x1) x'))
% 60.15/60.35 Found (((eq_sym fofType) x1) x') as proof of (((x0 Xx) x')->(((eq fofType) x1) x'))
% 60.15/60.35 Found x000:=(x00 x1):(((eq fofType) x1) x')
% 60.15/60.35 Found (x00 x1) as proof of (((eq fofType) x1) x')
% 60.15/60.35 Found (x00 x1) as proof of (((eq fofType) x1) x')
% 60.15/60.35 Found (fun (x00:((x0 Xx) x'))=> (x00 x1)) as proof of (((eq fofType) x1) x')
% 60.15/60.35 Found (fun (x':fofType) (x00:((x0 Xx) x'))=> (x00 x1)) as proof of (((x0 Xx) x')->(((eq fofType) x1) x'))
% 60.15/60.35 Found (fun (x':fofType) (x00:((x0 Xx) x'))=> (x00 x1)) as proof of (forall (x':fofType), (((x0 Xx) x')->(((eq fofType) x1) x')))
% 60.15/60.35 Found eq_ref00:=(eq_ref0 x10):(((eq fofType) x10) x10)
% 60.15/60.35 Found (eq_ref0 x10) as proof of (((eq fofType) x10) x1)
% 60.15/60.35 Found ((eq_ref fofType) x10) as proof of (((eq fofType) x10) x1)
% 60.15/60.35 Found ((eq_ref fofType) x10) as proof of (((eq fofType) x10) x1)
% 60.15/60.35 Found x2:(P x10)
% 60.15/60.35 Instantiate: x1:=x10:fofType
% 60.15/60.35 Found (fun (x2:(P x10))=> x2) as proof of (P x1)
% 60.15/60.35 Found (fun (P:(fofType->Prop)) (x2:(P x10))=> x2) as proof of ((P x10)->(P x1))
% 60.15/60.35 Found eq_ref000:=(eq_ref00 P):((P x10)->(P x10))
% 60.15/60.35 Found (eq_ref00 P) as proof of ((P x10)->(P x1))
% 60.15/60.35 Found ((eq_ref0 x10) P) as proof of ((P x10)->(P x1))
% 60.15/60.35 Found (((eq_ref fofType) x10) P) as proof of ((P x10)->(P x1))
% 60.15/60.35 Found (((eq_ref fofType) x10) P) as proof of ((P x10)->(P x1))
% 64.51/64.71 Found (fun (P:(fofType->Prop))=> (((eq_ref fofType) x10) P)) as proof of ((P x10)->(P x1))
% 64.51/64.71 Found eq_substitution0000:=(fun (a:fofType)=> ((eq_substitution000 a) (fun (x8:fofType)=> (cS Xx)))):(forall (a:fofType), ((((eq fofType) a) (cS Xy))->(((eq fofType) (cS Xx)) (cS Xx))))
% 64.51/64.71 Instantiate: x4:=(fun (x6:fofType) (x50:fofType)=> (forall (a:fofType), ((((eq fofType) a) (cS Xy))->(((eq fofType) x6) x6)))):(fofType->(fofType->Prop))
% 64.51/64.71 Found (fun (a:fofType)=> ((eq_substitution000 a) (fun (x8:fofType)=> (cS Xx)))) as proof of ((x4 (cS Xx)) (cS (cS Xy)))
% 64.51/64.71 Found (fun (a:fofType)=> (((fun (a:fofType)=> ((eq_substitution00 a) (cS Xy))) a) (fun (x8:fofType)=> (cS Xx)))) as proof of ((x4 (cS Xx)) (cS (cS Xy)))
% 64.51/64.71 Found (fun (a:fofType)=> (((fun (a:fofType)=> (((eq_substitution0 fofType) a) (cS Xy))) a) (fun (x8:fofType)=> (cS Xx)))) as proof of ((x4 (cS Xx)) (cS (cS Xy)))
% 64.51/64.71 Found (fun (a:fofType)=> (((fun (a:fofType)=> ((((eq_substitution fofType) fofType) a) (cS Xy))) a) (fun (x8:fofType)=> (cS Xx)))) as proof of ((x4 (cS Xx)) (cS (cS Xy)))
% 64.51/64.71 Found (fun (x00:((x4 Xx) Xy)) (a:fofType)=> (((fun (a:fofType)=> ((((eq_substitution fofType) fofType) a) (cS Xy))) a) (fun (x8:fofType)=> (cS Xx)))) as proof of ((x4 (cS Xx)) (cS (cS Xy)))
% 64.51/64.71 Found eq_substitution0000:=(fun (a:fofType)=> ((eq_substitution000 a) (fun (x6:fofType)=> (cS Xx)))):(forall (a:fofType), ((((eq fofType) a) (cS Xy))->(((eq fofType) (cS Xx)) (cS Xx))))
% 64.51/64.71 Instantiate: x0:=(fun (x4:fofType) (x30:fofType)=> (forall (a:fofType), ((((eq fofType) a) (cS Xy))->(((eq fofType) x4) x4)))):(fofType->(fofType->Prop))
% 64.51/64.71 Found (fun (a:fofType)=> ((eq_substitution000 a) (fun (x6:fofType)=> (cS Xx)))) as proof of ((x0 (cS Xx)) (cS (cS Xy)))
% 64.51/64.71 Found (fun (a:fofType)=> (((fun (a:fofType)=> ((eq_substitution00 a) (cS Xy))) a) (fun (x6:fofType)=> (cS Xx)))) as proof of ((x0 (cS Xx)) (cS (cS Xy)))
% 64.51/64.71 Found (fun (a:fofType)=> (((fun (a:fofType)=> (((eq_substitution0 fofType) a) (cS Xy))) a) (fun (x6:fofType)=> (cS Xx)))) as proof of ((x0 (cS Xx)) (cS (cS Xy)))
% 64.51/64.71 Found (fun (a:fofType)=> (((fun (a:fofType)=> ((((eq_substitution fofType) fofType) a) (cS Xy))) a) (fun (x6:fofType)=> (cS Xx)))) as proof of ((x0 (cS Xx)) (cS (cS Xy)))
% 64.51/64.71 Found (fun (x00:((x0 Xx) Xy)) (a:fofType)=> (((fun (a:fofType)=> ((((eq_substitution fofType) fofType) a) (cS Xy))) a) (fun (x6:fofType)=> (cS Xx)))) as proof of ((x0 (cS Xx)) (cS (cS Xy)))
% 64.51/64.71 Found eq_substitution0000:=(fun (a:fofType)=> ((eq_substitution000 a) (fun (x8:fofType)=> (cS Xx)))):(forall (a:fofType), ((((eq fofType) a) (cS Xy))->(((eq fofType) (cS Xx)) (cS Xx))))
% 64.51/64.71 Instantiate: x0:=(fun (x6:fofType) (x50:fofType)=> (forall (a:fofType), ((((eq fofType) a) (cS Xy))->(((eq fofType) x6) x6)))):(fofType->(fofType->Prop))
% 64.51/64.71 Found (fun (a:fofType)=> ((eq_substitution000 a) (fun (x8:fofType)=> (cS Xx)))) as proof of ((x0 (cS Xx)) (cS (cS Xy)))
% 64.51/64.71 Found (fun (a:fofType)=> (((fun (a:fofType)=> ((eq_substitution00 a) (cS Xy))) a) (fun (x8:fofType)=> (cS Xx)))) as proof of ((x0 (cS Xx)) (cS (cS Xy)))
% 64.51/64.71 Found (fun (a:fofType)=> (((fun (a:fofType)=> (((eq_substitution0 fofType) a) (cS Xy))) a) (fun (x8:fofType)=> (cS Xx)))) as proof of ((x0 (cS Xx)) (cS (cS Xy)))
% 64.51/64.71 Found (fun (a:fofType)=> (((fun (a:fofType)=> ((((eq_substitution fofType) fofType) a) (cS Xy))) a) (fun (x8:fofType)=> (cS Xx)))) as proof of ((x0 (cS Xx)) (cS (cS Xy)))
% 64.51/64.71 Found (fun (x00:((x0 Xx) Xy)) (a:fofType)=> (((fun (a:fofType)=> ((((eq_substitution fofType) fofType) a) (cS Xy))) a) (fun (x8:fofType)=> (cS Xx)))) as proof of ((x0 (cS Xx)) (cS (cS Xy)))
% 64.51/64.71 Found eq_substitution0000:=(fun (a:fofType)=> ((eq_substitution000 a) (fun (x8:fofType)=> (cS Xx)))):(forall (a:fofType), ((((eq fofType) a) (cS Xy))->(((eq fofType) (cS Xx)) (cS Xx))))
% 64.51/64.71 Instantiate: x2:=(fun (x6:fofType) (x50:fofType)=> (forall (a:fofType), ((((eq fofType) a) (cS Xy))->(((eq fofType) x6) x6)))):(fofType->(fofType->Prop))
% 64.51/64.71 Found (fun (a:fofType)=> ((eq_substitution000 a) (fun (x8:fofType)=> (cS Xx)))) as proof of ((x2 (cS Xx)) (cS (cS Xy)))
% 64.51/64.71 Found (fun (a:fofType)=> (((fun (a:fofType)=> ((eq_substitution00 a) (cS Xy))) a) (fun (x8:fofType)=> (cS Xx)))) as proof of ((x2 (cS Xx)) (cS (cS Xy)))
% 72.91/73.09 Found (fun (a:fofType)=> (((fun (a:fofType)=> (((eq_substitution0 fofType) a) (cS Xy))) a) (fun (x8:fofType)=> (cS Xx)))) as proof of ((x2 (cS Xx)) (cS (cS Xy)))
% 72.91/73.09 Found (fun (a:fofType)=> (((fun (a:fofType)=> ((((eq_substitution fofType) fofType) a) (cS Xy))) a) (fun (x8:fofType)=> (cS Xx)))) as proof of ((x2 (cS Xx)) (cS (cS Xy)))
% 72.91/73.09 Found (fun (x00:((x2 Xx) Xy)) (a:fofType)=> (((fun (a:fofType)=> ((((eq_substitution fofType) fofType) a) (cS Xy))) a) (fun (x8:fofType)=> (cS Xx)))) as proof of ((x2 (cS Xx)) (cS (cS Xy)))
% 72.91/73.09 Found eq_ref00:=(eq_ref0 b):(((eq ((fofType->(fofType->Prop))->Prop)) b) b)
% 72.91/73.09 Found (eq_ref0 b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (Xd:(fofType->(fofType->Prop)))=> ((and ((and ((Xd c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((Xd Xx) Xy)->((Xd (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((Xd Xx) X)) (forall (Y:fofType), (((Xd Xx) Y)->(((eq fofType) X) Y))))))))))
% 72.91/73.09 Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (Xd:(fofType->(fofType->Prop)))=> ((and ((and ((Xd c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((Xd Xx) Xy)->((Xd (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((Xd Xx) X)) (forall (Y:fofType), (((Xd Xx) Y)->(((eq fofType) X) Y))))))))))
% 72.91/73.09 Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (Xd:(fofType->(fofType->Prop)))=> ((and ((and ((Xd c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((Xd Xx) Xy)->((Xd (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((Xd Xx) X)) (forall (Y:fofType), (((Xd Xx) Y)->(((eq fofType) X) Y))))))))))
% 72.91/73.09 Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (Xd:(fofType->(fofType->Prop)))=> ((and ((and ((Xd c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((Xd Xx) Xy)->((Xd (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((Xd Xx) X)) (forall (Y:fofType), (((Xd Xx) Y)->(((eq fofType) X) Y))))))))))
% 72.91/73.09 Found eq_ref00:=(eq_ref0 a):(((eq ((fofType->(fofType->Prop))->Prop)) a) a)
% 72.91/73.09 Found (eq_ref0 a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b)
% 72.91/73.09 Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b)
% 72.91/73.09 Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b)
% 72.91/73.09 Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b)
% 72.91/73.09 Found eq_sym000:=(eq_sym00 Y):((((eq fofType) x3) Y)->(((eq fofType) Y) x3))
% 72.91/73.09 Found (eq_sym00 Y) as proof of (((x2 Xx) Y)->(((eq fofType) x3) Y))
% 72.91/73.09 Found ((eq_sym0 x3) Y) as proof of (((x2 Xx) Y)->(((eq fofType) x3) Y))
% 72.91/73.09 Found (((eq_sym fofType) x3) Y) as proof of (((x2 Xx) Y)->(((eq fofType) x3) Y))
% 72.91/73.09 Found (((eq_sym fofType) x3) Y) as proof of (((x2 Xx) Y)->(((eq fofType) x3) Y))
% 72.91/73.09 Found (((eq_sym fofType) x3) Y) as proof of (((x2 Xx) Y)->(((eq fofType) x3) Y))
% 72.91/73.09 Found eq_ref00:=(eq_ref0 x3):(((eq fofType) x3) x3)
% 72.91/73.09 Found (eq_ref0 x3) as proof of (((eq fofType) x3) Y)
% 72.91/73.09 Found ((eq_ref fofType) x3) as proof of (((eq fofType) x3) Y)
% 72.91/73.09 Found ((eq_ref fofType) x3) as proof of (((eq fofType) x3) Y)
% 72.91/73.09 Found (fun (x00:((x2 Xx) Y))=> ((eq_ref fofType) x3)) as proof of (((eq fofType) x3) Y)
% 72.91/73.09 Found x000:=(x00 x3):(((eq fofType) x3) Y)
% 72.91/73.09 Found (x00 x3) as proof of (((eq fofType) x3) Y)
% 72.91/73.09 Found (x00 x3) as proof of (((eq fofType) x3) Y)
% 72.91/73.09 Found (fun (x00:((x2 Xx) Y))=> (x00 x3)) as proof of (((eq fofType) x3) Y)
% 72.91/73.09 Found (fun (Y:fofType) (x00:((x2 Xx) Y))=> (x00 x3)) as proof of (((x2 Xx) Y)->(((eq fofType) x3) Y))
% 72.91/73.09 Found (fun (Y:fofType) (x00:((x2 Xx) Y))=> (x00 x3)) as proof of (forall (Y:fofType), (((x2 Xx) Y)->(((eq fofType) x3) Y)))
% 72.91/73.09 Found eq_ref00:=(eq_ref0 x30):(((eq fofType) x30) x30)
% 72.91/73.09 Found (eq_ref0 x30) as proof of (((eq fofType) x30) x3)
% 72.91/73.09 Found ((eq_ref fofType) x30) as proof of (((eq fofType) x30) x3)
% 76.43/76.67 Found ((eq_ref fofType) x30) as proof of (((eq fofType) x30) x3)
% 76.43/76.67 Found x4:(P x30)
% 76.43/76.67 Instantiate: x3:=x30:fofType
% 76.43/76.67 Found (fun (x4:(P x30))=> x4) as proof of (P x3)
% 76.43/76.67 Found (fun (P:(fofType->Prop)) (x4:(P x30))=> x4) as proof of ((P x30)->(P x3))
% 76.43/76.67 Found eq_ref000:=(eq_ref00 P):((P x30)->(P x30))
% 76.43/76.67 Found (eq_ref00 P) as proof of ((P x30)->(P x3))
% 76.43/76.67 Found ((eq_ref0 x30) P) as proof of ((P x30)->(P x3))
% 76.43/76.67 Found (((eq_ref fofType) x30) P) as proof of ((P x30)->(P x3))
% 76.43/76.67 Found (((eq_ref fofType) x30) P) as proof of ((P x30)->(P x3))
% 76.43/76.67 Found (fun (P:(fofType->Prop))=> (((eq_ref fofType) x30) P)) as proof of ((P x30)->(P x3))
% 76.43/76.67 Found eq_ref00:=(eq_ref0 x1):(((eq fofType) x1) x1)
% 76.43/76.67 Found (eq_ref0 x1) as proof of (((eq fofType) x1) x10)
% 76.43/76.67 Found ((eq_ref fofType) x1) as proof of (((eq fofType) x1) x10)
% 76.43/76.67 Found ((eq_ref fofType) x1) as proof of (((eq fofType) x1) x10)
% 76.43/76.67 Found ((eq_ref fofType) x1) as proof of (((eq fofType) x1) x10)
% 76.43/76.67 Found (eq_sym000 ((eq_ref fofType) x1)) as proof of (((eq fofType) x10) x1)
% 76.43/76.67 Found ((eq_sym00 x10) ((eq_ref fofType) x1)) as proof of (((eq fofType) x10) x1)
% 76.43/76.67 Found (((eq_sym0 x1) x10) ((eq_ref fofType) x1)) as proof of (((eq fofType) x10) x1)
% 76.43/76.67 Found ((((eq_sym fofType) x1) x10) ((eq_ref fofType) x1)) as proof of (((eq fofType) x10) x1)
% 76.43/76.67 Found eq_ref00:=(eq_ref0 (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y)))))))):(((eq Prop) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y)))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y))))))))
% 76.43/76.67 Found (eq_ref0 (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y)))))))) as proof of (((eq Prop) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y)))))))) b)
% 76.43/76.67 Found ((eq_ref Prop) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y)))))))) as proof of (((eq Prop) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y)))))))) b)
% 76.43/76.67 Found ((eq_ref Prop) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y)))))))) as proof of (((eq Prop) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y)))))))) b)
% 76.43/76.67 Found ((eq_ref Prop) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y)))))))) as proof of (((eq Prop) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y)))))))) b)
% 76.43/76.67 Found eq_ref00:=(eq_ref0 b):(((eq ((fofType->(fofType->Prop))->Prop)) b) b)
% 76.43/76.67 Found (eq_ref0 b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (Xd:(fofType->(fofType->Prop)))=> ((and ((and ((Xd c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((Xd Xx) Xy)->((Xd (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (Xd Xx)))))))
% 76.43/76.67 Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (Xd:(fofType->(fofType->Prop)))=> ((and ((and ((Xd c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((Xd Xx) Xy)->((Xd (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (Xd Xx)))))))
% 76.43/76.67 Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (Xd:(fofType->(fofType->Prop)))=> ((and ((and ((Xd c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((Xd Xx) Xy)->((Xd (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (Xd Xx)))))))
% 76.43/76.67 Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (Xd:(fofType->(fofType->Prop)))=> ((and ((and ((Xd c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((Xd Xx) Xy)->((Xd (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (Xd Xx)))))))
% 80.73/80.94 Found eq_ref00:=(eq_ref0 a):(((eq ((fofType->(fofType->Prop))->Prop)) a) a)
% 80.73/80.94 Found (eq_ref0 a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b)
% 80.73/80.94 Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b)
% 80.73/80.94 Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b)
% 80.73/80.94 Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b)
% 80.73/80.94 Found eq_sym000:=(eq_sym00 Y):((((eq fofType) x3) Y)->(((eq fofType) Y) x3))
% 80.73/80.94 Found (eq_sym00 Y) as proof of (((x0 Xx) Y)->(((eq fofType) x3) Y))
% 80.73/80.94 Found ((eq_sym0 x3) Y) as proof of (((x0 Xx) Y)->(((eq fofType) x3) Y))
% 80.73/80.94 Found (((eq_sym fofType) x3) Y) as proof of (((x0 Xx) Y)->(((eq fofType) x3) Y))
% 80.73/80.94 Found (((eq_sym fofType) x3) Y) as proof of (((x0 Xx) Y)->(((eq fofType) x3) Y))
% 80.73/80.94 Found (((eq_sym fofType) x3) Y) as proof of (((x0 Xx) Y)->(((eq fofType) x3) Y))
% 80.73/80.94 Found eq_ref00:=(eq_ref0 x3):(((eq fofType) x3) x3)
% 80.73/80.94 Found (eq_ref0 x3) as proof of (((eq fofType) x3) Y)
% 80.73/80.94 Found ((eq_ref fofType) x3) as proof of (((eq fofType) x3) Y)
% 80.73/80.94 Found ((eq_ref fofType) x3) as proof of (((eq fofType) x3) Y)
% 80.73/80.94 Found (fun (x00:((x0 Xx) Y))=> ((eq_ref fofType) x3)) as proof of (((eq fofType) x3) Y)
% 80.73/80.94 Found eq_substitution0000:=(fun (a:fofType)=> ((eq_substitution000 a) (fun (x8:fofType)=> (cS Xx)))):(forall (a:fofType), ((((eq fofType) a) (cS Xy))->(((eq fofType) (cS Xx)) (cS Xx))))
% 80.73/80.94 Instantiate: x0:=(fun (x6:fofType) (x50:fofType)=> (forall (a:fofType), ((((eq fofType) a) (cS Xy))->(((eq fofType) x6) x6)))):(fofType->(fofType->Prop))
% 80.73/80.94 Found (fun (a:fofType)=> ((eq_substitution000 a) (fun (x8:fofType)=> (cS Xx)))) as proof of ((x0 (cS Xx)) (cS (cS Xy)))
% 80.73/80.94 Found (fun (a:fofType)=> (((fun (a:fofType)=> ((eq_substitution00 a) (cS Xy))) a) (fun (x8:fofType)=> (cS Xx)))) as proof of ((x0 (cS Xx)) (cS (cS Xy)))
% 80.73/80.94 Found (fun (a:fofType)=> (((fun (a:fofType)=> (((eq_substitution0 fofType) a) (cS Xy))) a) (fun (x8:fofType)=> (cS Xx)))) as proof of ((x0 (cS Xx)) (cS (cS Xy)))
% 80.73/80.94 Found (fun (a:fofType)=> (((fun (a:fofType)=> ((((eq_substitution fofType) fofType) a) (cS Xy))) a) (fun (x8:fofType)=> (cS Xx)))) as proof of ((x0 (cS Xx)) (cS (cS Xy)))
% 80.73/80.94 Found (fun (x00:((x0 Xx) Xy)) (a:fofType)=> (((fun (a:fofType)=> ((((eq_substitution fofType) fofType) a) (cS Xy))) a) (fun (x8:fofType)=> (cS Xx)))) as proof of ((x0 (cS Xx)) (cS (cS Xy)))
% 80.73/80.94 Found eq_sym000:=(eq_sym00 x'):((((eq fofType) x3) x')->(((eq fofType) x') x3))
% 80.73/80.94 Found (eq_sym00 x') as proof of (((x2 Xx) x')->(((eq fofType) x3) x'))
% 80.73/80.94 Found ((eq_sym0 x3) x') as proof of (((x2 Xx) x')->(((eq fofType) x3) x'))
% 80.73/80.94 Found (((eq_sym fofType) x3) x') as proof of (((x2 Xx) x')->(((eq fofType) x3) x'))
% 80.73/80.94 Found (((eq_sym fofType) x3) x') as proof of (((x2 Xx) x')->(((eq fofType) x3) x'))
% 80.73/80.94 Found (((eq_sym fofType) x3) x') as proof of (((x2 Xx) x')->(((eq fofType) x3) x'))
% 80.73/80.94 Found eq_substitution0000:=(fun (a:fofType)=> ((eq_substitution000 a) (fun (x8:fofType)=> (cS Xx)))):(forall (a:fofType), ((((eq fofType) a) (cS Xy))->(((eq fofType) (cS Xx)) (cS Xx))))
% 80.73/80.94 Instantiate: x2:=(fun (x6:fofType) (x50:fofType)=> (forall (a:fofType), ((((eq fofType) a) (cS Xy))->(((eq fofType) x6) x6)))):(fofType->(fofType->Prop))
% 80.73/80.94 Found (fun (a:fofType)=> ((eq_substitution000 a) (fun (x8:fofType)=> (cS Xx)))) as proof of ((x2 (cS Xx)) (cS (cS Xy)))
% 80.73/80.94 Found (fun (a:fofType)=> (((fun (a:fofType)=> ((eq_substitution00 a) (cS Xy))) a) (fun (x8:fofType)=> (cS Xx)))) as proof of ((x2 (cS Xx)) (cS (cS Xy)))
% 80.73/80.94 Found (fun (a:fofType)=> (((fun (a:fofType)=> (((eq_substitution0 fofType) a) (cS Xy))) a) (fun (x8:fofType)=> (cS Xx)))) as proof of ((x2 (cS Xx)) (cS (cS Xy)))
% 80.73/80.94 Found (fun (a:fofType)=> (((fun (a:fofType)=> ((((eq_substitution fofType) fofType) a) (cS Xy))) a) (fun (x8:fofType)=> (cS Xx)))) as proof of ((x2 (cS Xx)) (cS (cS Xy)))
% 80.73/80.94 Found (fun (x00:((x2 Xx) Xy)) (a:fofType)=> (((fun (a:fofType)=> ((((eq_substitution fofType) fofType) a) (cS Xy))) a) (fun (x8:fofType)=> (cS Xx)))) as proof of ((x2 (cS Xx)) (cS (cS Xy)))
% 83.14/83.34 Found eq_ref00:=(eq_ref0 x3):(((eq fofType) x3) x3)
% 83.14/83.34 Found (eq_ref0 x3) as proof of (((eq fofType) x3) x')
% 83.14/83.34 Found ((eq_ref fofType) x3) as proof of (((eq fofType) x3) x')
% 83.14/83.34 Found ((eq_ref fofType) x3) as proof of (((eq fofType) x3) x')
% 83.14/83.34 Found (fun (x00:((x2 Xx) x'))=> ((eq_ref fofType) x3)) as proof of (((eq fofType) x3) x')
% 83.14/83.34 Found x000:=(x00 x3):(((eq fofType) x3) x')
% 83.14/83.34 Found (x00 x3) as proof of (((eq fofType) x3) x')
% 83.14/83.34 Found (x00 x3) as proof of (((eq fofType) x3) x')
% 83.14/83.34 Found (fun (x00:((x2 Xx) x'))=> (x00 x3)) as proof of (((eq fofType) x3) x')
% 83.14/83.34 Found (fun (x':fofType) (x00:((x2 Xx) x'))=> (x00 x3)) as proof of (((x2 Xx) x')->(((eq fofType) x3) x'))
% 83.14/83.34 Found (fun (x':fofType) (x00:((x2 Xx) x'))=> (x00 x3)) as proof of (forall (x':fofType), (((x2 Xx) x')->(((eq fofType) x3) x')))
% 83.14/83.34 Found eq_ref00:=(eq_ref0 x30):(((eq fofType) x30) x30)
% 83.14/83.34 Found (eq_ref0 x30) as proof of (((eq fofType) x30) x3)
% 83.14/83.34 Found ((eq_ref fofType) x30) as proof of (((eq fofType) x30) x3)
% 83.14/83.34 Found ((eq_ref fofType) x30) as proof of (((eq fofType) x30) x3)
% 83.14/83.34 Found x4:(P x30)
% 83.14/83.34 Instantiate: x3:=x30:fofType
% 83.14/83.34 Found (fun (x4:(P x30))=> x4) as proof of (P x3)
% 83.14/83.34 Found (fun (P:(fofType->Prop)) (x4:(P x30))=> x4) as proof of ((P x30)->(P x3))
% 83.14/83.34 Found eq_ref000:=(eq_ref00 P):((P x30)->(P x30))
% 83.14/83.34 Found (eq_ref00 P) as proof of ((P x30)->(P x3))
% 83.14/83.34 Found ((eq_ref0 x30) P) as proof of ((P x30)->(P x3))
% 83.14/83.34 Found (((eq_ref fofType) x30) P) as proof of ((P x30)->(P x3))
% 83.14/83.34 Found (((eq_ref fofType) x30) P) as proof of ((P x30)->(P x3))
% 83.14/83.34 Found (fun (P:(fofType->Prop))=> (((eq_ref fofType) x30) P)) as proof of ((P x30)->(P x3))
% 83.14/83.34 Found eq_ref00:=(eq_ref0 x1):(((eq fofType) x1) x1)
% 83.14/83.34 Found (eq_ref0 x1) as proof of (((eq fofType) x1) x10)
% 83.14/83.34 Found ((eq_ref fofType) x1) as proof of (((eq fofType) x1) x10)
% 83.14/83.34 Found ((eq_ref fofType) x1) as proof of (((eq fofType) x1) x10)
% 83.14/83.34 Found ((eq_ref fofType) x1) as proof of (((eq fofType) x1) x10)
% 83.14/83.34 Found (eq_sym000 ((eq_ref fofType) x1)) as proof of (((eq fofType) x10) x1)
% 83.14/83.34 Found ((eq_sym00 x10) ((eq_ref fofType) x1)) as proof of (((eq fofType) x10) x1)
% 83.14/83.34 Found (((eq_sym0 x1) x10) ((eq_ref fofType) x1)) as proof of (((eq fofType) x10) x1)
% 83.14/83.34 Found ((((eq_sym fofType) x1) x10) ((eq_ref fofType) x1)) as proof of (((eq fofType) x10) x1)
% 83.14/83.34 Found eq_ref00:=(eq_ref0 (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx))))):(((eq Prop) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx)))))
% 83.14/83.34 Found (eq_ref0 (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx))))) as proof of (((eq Prop) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx))))) b)
% 83.14/83.34 Found ((eq_ref Prop) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx))))) as proof of (((eq Prop) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx))))) b)
% 83.14/83.34 Found ((eq_ref Prop) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx))))) as proof of (((eq Prop) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx))))) b)
% 83.14/83.34 Found ((eq_ref Prop) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx))))) as proof of (((eq Prop) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx))))) b)
% 83.14/83.34 Found eq_substitution0000:=(fun (a:fofType)=> ((eq_substitution000 a) (fun (x6:fofType)=> (cS Xx)))):(forall (a:fofType), ((((eq fofType) a) (cS Xy))->(((eq fofType) (cS Xx)) (cS Xx))))
% 83.14/83.34 Instantiate: x0:=(fun (x4:fofType) (x30:fofType)=> (forall (a:fofType), ((((eq fofType) a) (cS Xy))->(((eq fofType) x4) x4)))):(fofType->(fofType->Prop))
% 83.14/83.34 Found (fun (a:fofType)=> ((eq_substitution000 a) (fun (x6:fofType)=> (cS Xx)))) as proof of ((x0 (cS Xx)) (cS (cS Xy)))
% 83.14/83.34 Found (fun (a:fofType)=> (((fun (a:fofType)=> ((eq_substitution00 a) (cS Xy))) a) (fun (x6:fofType)=> (cS Xx)))) as proof of ((x0 (cS Xx)) (cS (cS Xy)))
% 83.14/83.34 Found (fun (a:fofType)=> (((fun (a:fofType)=> (((eq_substitution0 fofType) a) (cS Xy))) a) (fun (x6:fofType)=> (cS Xx)))) as proof of ((x0 (cS Xx)) (cS (cS Xy)))
% 83.14/83.34 Found (fun (a:fofType)=> (((fun (a:fofType)=> ((((eq_substitution fofType) fofType) a) (cS Xy))) a) (fun (x6:fofType)=> (cS Xx)))) as proof of ((x0 (cS Xx)) (cS (cS Xy)))
% 84.70/84.91 Found (fun (x2:(forall (Xn:fofType), (not (((eq fofType) (cS Xn)) c0)))) (a:fofType)=> (((fun (a:fofType)=> ((((eq_substitution fofType) fofType) a) (cS Xy))) a) (fun (x6:fofType)=> (cS Xx)))) as proof of ((x0 (cS Xx)) (cS (cS Xy)))
% 84.70/84.91 Found (fun (x1:((and cIND) (forall (Xx0:fofType) (Xy0:fofType), ((((eq fofType) (cS Xx0)) (cS Xy0))->(((eq fofType) Xx0) Xy0))))) (x2:(forall (Xn:fofType), (not (((eq fofType) (cS Xn)) c0)))) (a:fofType)=> (((fun (a:fofType)=> ((((eq_substitution fofType) fofType) a) (cS Xy))) a) (fun (x6:fofType)=> (cS Xx)))) as proof of ((forall (Xn:fofType), (not (((eq fofType) (cS Xn)) c0)))->((x0 (cS Xx)) (cS (cS Xy))))
% 84.70/84.91 Found (fun (x1:((and cIND) (forall (Xx0:fofType) (Xy0:fofType), ((((eq fofType) (cS Xx0)) (cS Xy0))->(((eq fofType) Xx0) Xy0))))) (x2:(forall (Xn:fofType), (not (((eq fofType) (cS Xn)) c0)))) (a:fofType)=> (((fun (a:fofType)=> ((((eq_substitution fofType) fofType) a) (cS Xy))) a) (fun (x6:fofType)=> (cS Xx)))) as proof of (((and cIND) (forall (Xx0:fofType) (Xy0:fofType), ((((eq fofType) (cS Xx0)) (cS Xy0))->(((eq fofType) Xx0) Xy0))))->((forall (Xn:fofType), (not (((eq fofType) (cS Xn)) c0)))->((x0 (cS Xx)) (cS (cS Xy)))))
% 84.70/84.91 Found (and_rect00 (fun (x1:((and cIND) (forall (Xx0:fofType) (Xy0:fofType), ((((eq fofType) (cS Xx0)) (cS Xy0))->(((eq fofType) Xx0) Xy0))))) (x2:(forall (Xn:fofType), (not (((eq fofType) (cS Xn)) c0)))) (a:fofType)=> (((fun (a:fofType)=> ((((eq_substitution fofType) fofType) a) (cS Xy))) a) (fun (x6:fofType)=> (cS Xx))))) as proof of ((x0 (cS Xx)) (cS (cS Xy)))
% 84.70/84.91 Found ((and_rect0 ((x0 (cS Xx)) (cS (cS Xy)))) (fun (x1:((and cIND) (forall (Xx0:fofType) (Xy0:fofType), ((((eq fofType) (cS Xx0)) (cS Xy0))->(((eq fofType) Xx0) Xy0))))) (x2:(forall (Xn:fofType), (not (((eq fofType) (cS Xn)) c0)))) (a:fofType)=> (((fun (a:fofType)=> ((((eq_substitution fofType) fofType) a) (cS Xy))) a) (fun (x6:fofType)=> (cS Xx))))) as proof of ((x0 (cS Xx)) (cS (cS Xy)))
% 84.70/84.91 Found (((fun (P:Type) (x1:(((and cIND) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (cS Xx)) (cS Xy))->(((eq fofType) Xx) Xy))))->((forall (Xn:fofType), (not (((eq fofType) (cS Xn)) c0)))->P)))=> (((((and_rect ((and cIND) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (cS Xx)) (cS Xy))->(((eq fofType) Xx) Xy))))) (forall (Xn:fofType), (not (((eq fofType) (cS Xn)) c0)))) P) x1) x)) ((x0 (cS Xx)) (cS (cS Xy)))) (fun (x1:((and cIND) (forall (Xx0:fofType) (Xy0:fofType), ((((eq fofType) (cS Xx0)) (cS Xy0))->(((eq fofType) Xx0) Xy0))))) (x2:(forall (Xn:fofType), (not (((eq fofType) (cS Xn)) c0)))) (a:fofType)=> (((fun (a:fofType)=> ((((eq_substitution fofType) fofType) a) (cS Xy))) a) (fun (x6:fofType)=> (cS Xx))))) as proof of ((x0 (cS Xx)) (cS (cS Xy)))
% 84.70/84.91 Found (fun (x00:((x0 Xx) Xy))=> (((fun (P:Type) (x1:(((and cIND) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (cS Xx)) (cS Xy))->(((eq fofType) Xx) Xy))))->((forall (Xn:fofType), (not (((eq fofType) (cS Xn)) c0)))->P)))=> (((((and_rect ((and cIND) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (cS Xx)) (cS Xy))->(((eq fofType) Xx) Xy))))) (forall (Xn:fofType), (not (((eq fofType) (cS Xn)) c0)))) P) x1) x)) ((x0 (cS Xx)) (cS (cS Xy)))) (fun (x1:((and cIND) (forall (Xx0:fofType) (Xy0:fofType), ((((eq fofType) (cS Xx0)) (cS Xy0))->(((eq fofType) Xx0) Xy0))))) (x2:(forall (Xn:fofType), (not (((eq fofType) (cS Xn)) c0)))) (a:fofType)=> (((fun (a:fofType)=> ((((eq_substitution fofType) fofType) a) (cS Xy))) a) (fun (x6:fofType)=> (cS Xx)))))) as proof of ((x0 (cS Xx)) (cS (cS Xy)))
% 84.70/84.91 Found eq_ref00:=(eq_ref0 b):(((eq ((fofType->(fofType->Prop))->Prop)) b) b)
% 84.70/84.91 Found (eq_ref0 b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (Xd:(fofType->(fofType->Prop)))=> ((and ((and ((Xd c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((Xd Xx) Xy)->((Xd (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((Xd Xx) X)) (forall (Y:fofType), (((Xd Xx) Y)->(((eq fofType) X) Y))))))))))
% 84.70/84.91 Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (Xd:(fofType->(fofType->Prop)))=> ((and ((and ((Xd c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((Xd Xx) Xy)->((Xd (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((Xd Xx) X)) (forall (Y:fofType), (((Xd Xx) Y)->(((eq fofType) X) Y))))))))))
% 90.86/91.03 Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (Xd:(fofType->(fofType->Prop)))=> ((and ((and ((Xd c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((Xd Xx) Xy)->((Xd (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((Xd Xx) X)) (forall (Y:fofType), (((Xd Xx) Y)->(((eq fofType) X) Y))))))))))
% 90.86/91.03 Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (Xd:(fofType->(fofType->Prop)))=> ((and ((and ((Xd c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((Xd Xx) Xy)->((Xd (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((Xd Xx) X)) (forall (Y:fofType), (((Xd Xx) Y)->(((eq fofType) X) Y))))))))))
% 90.86/91.03 Found eta_expansion000:=(eta_expansion00 a):(((eq ((fofType->(fofType->Prop))->Prop)) a) (fun (x:(fofType->(fofType->Prop)))=> (a x)))
% 90.86/91.03 Found (eta_expansion00 a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b)
% 90.86/91.03 Found ((eta_expansion0 Prop) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b)
% 90.86/91.03 Found (((eta_expansion (fofType->(fofType->Prop))) Prop) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b)
% 90.86/91.03 Found (((eta_expansion (fofType->(fofType->Prop))) Prop) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b)
% 90.86/91.03 Found (((eta_expansion (fofType->(fofType->Prop))) Prop) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b)
% 90.86/91.03 Found eq_ref00:=(eq_ref0 a):(((eq ((fofType->(fofType->Prop))->Prop)) a) a)
% 90.86/91.03 Found (eq_ref0 a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b)
% 90.86/91.03 Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b)
% 90.86/91.03 Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b)
% 90.86/91.03 Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b)
% 90.86/91.03 Found eq_sym000:=(eq_sym00 x'):((((eq fofType) x3) x')->(((eq fofType) x') x3))
% 90.86/91.03 Found (eq_sym00 x') as proof of (((x0 Xx) x')->(((eq fofType) x3) x'))
% 90.86/91.03 Found ((eq_sym0 x3) x') as proof of (((x0 Xx) x')->(((eq fofType) x3) x'))
% 90.86/91.03 Found (((eq_sym fofType) x3) x') as proof of (((x0 Xx) x')->(((eq fofType) x3) x'))
% 90.86/91.03 Found (((eq_sym fofType) x3) x') as proof of (((x0 Xx) x')->(((eq fofType) x3) x'))
% 90.86/91.03 Found (((eq_sym fofType) x3) x') as proof of (((x0 Xx) x')->(((eq fofType) x3) x'))
% 90.86/91.03 Found eq_ref00:=(eq_ref0 x3):(((eq fofType) x3) x3)
% 90.86/91.03 Found (eq_ref0 x3) as proof of (((eq fofType) x3) x')
% 90.86/91.03 Found ((eq_ref fofType) x3) as proof of (((eq fofType) x3) x')
% 90.86/91.03 Found ((eq_ref fofType) x3) as proof of (((eq fofType) x3) x')
% 90.86/91.03 Found (fun (x00:((x0 Xx) x'))=> ((eq_ref fofType) x3)) as proof of (((eq fofType) x3) x')
% 90.86/91.03 Found iff_sym:=(fun (A:Prop) (B:Prop) (H:((iff A) B))=> ((((conj (B->A)) (A->B)) (((proj2 (A->B)) (B->A)) H)) (((proj1 (A->B)) (B->A)) H))):(forall (A:Prop) (B:Prop), (((iff A) B)->((iff B) A)))
% 90.86/91.03 Instantiate: b:=(forall (A:Prop) (B:Prop), (((iff A) B)->((iff B) A))):Prop
% 90.86/91.03 Found iff_sym as proof of b
% 90.86/91.03 Found eq_substitution0000:=(fun (a:fofType)=> ((eq_substitution000 a) (fun (x8:fofType)=> (cS Xx)))):(forall (a:fofType), ((((eq fofType) a) (cS Xy))->(((eq fofType) (cS Xx)) (cS Xx))))
% 90.86/91.03 Instantiate: x0:=(fun (x6:fofType) (x50:fofType)=> (forall (a:fofType), ((((eq fofType) a) (cS Xy))->(((eq fofType) x6) x6)))):(fofType->(fofType->Prop))
% 90.86/91.03 Found (fun (a:fofType)=> ((eq_substitution000 a) (fun (x8:fofType)=> (cS Xx)))) as proof of ((x0 (cS Xx)) (cS (cS Xy)))
% 90.86/91.03 Found (fun (a:fofType)=> (((fun (a:fofType)=> ((eq_substitution00 a) (cS Xy))) a) (fun (x8:fofType)=> (cS Xx)))) as proof of ((x0 (cS Xx)) (cS (cS Xy)))
% 90.86/91.03 Found (fun (a:fofType)=> (((fun (a:fofType)=> (((eq_substitution0 fofType) a) (cS Xy))) a) (fun (x8:fofType)=> (cS Xx)))) as proof of ((x0 (cS Xx)) (cS (cS Xy)))
% 92.97/93.18 Found (fun (a:fofType)=> (((fun (a:fofType)=> ((((eq_substitution fofType) fofType) a) (cS Xy))) a) (fun (x8:fofType)=> (cS Xx)))) as proof of ((x0 (cS Xx)) (cS (cS Xy)))
% 92.97/93.18 Found (fun (x00:((x0 Xx) Xy)) (a:fofType)=> (((fun (a:fofType)=> ((((eq_substitution fofType) fofType) a) (cS Xy))) a) (fun (x8:fofType)=> (cS Xx)))) as proof of ((x0 (cS Xx)) (cS (cS Xy)))
% 92.97/93.18 Found eq_substitution0000:=(fun (a:fofType)=> ((eq_substitution000 a) (fun (x8:fofType)=> (cS Xx)))):(forall (a:fofType), ((((eq fofType) a) (cS Xy))->(((eq fofType) (cS Xx)) (cS Xx))))
% 92.97/93.18 Instantiate: x2:=(fun (x6:fofType) (x50:fofType)=> (forall (a:fofType), ((((eq fofType) a) (cS Xy))->(((eq fofType) x6) x6)))):(fofType->(fofType->Prop))
% 92.97/93.18 Found (fun (a:fofType)=> ((eq_substitution000 a) (fun (x8:fofType)=> (cS Xx)))) as proof of ((x2 (cS Xx)) (cS (cS Xy)))
% 92.97/93.18 Found (fun (a:fofType)=> (((fun (a:fofType)=> ((eq_substitution00 a) (cS Xy))) a) (fun (x8:fofType)=> (cS Xx)))) as proof of ((x2 (cS Xx)) (cS (cS Xy)))
% 92.97/93.18 Found (fun (a:fofType)=> (((fun (a:fofType)=> (((eq_substitution0 fofType) a) (cS Xy))) a) (fun (x8:fofType)=> (cS Xx)))) as proof of ((x2 (cS Xx)) (cS (cS Xy)))
% 92.97/93.18 Found (fun (a:fofType)=> (((fun (a:fofType)=> ((((eq_substitution fofType) fofType) a) (cS Xy))) a) (fun (x8:fofType)=> (cS Xx)))) as proof of ((x2 (cS Xx)) (cS (cS Xy)))
% 92.97/93.18 Found (fun (x00:((x2 Xx) Xy)) (a:fofType)=> (((fun (a:fofType)=> ((((eq_substitution fofType) fofType) a) (cS Xy))) a) (fun (x8:fofType)=> (cS Xx)))) as proof of ((x2 (cS Xx)) (cS (cS Xy)))
% 92.97/93.18 Found eq_ref000:=(eq_ref00 P):((P x3)->(P x3))
% 92.97/93.18 Found (eq_ref00 P) as proof of ((P x3)->(P Y))
% 92.97/93.18 Found ((eq_ref0 x3) P) as proof of ((P x3)->(P Y))
% 92.97/93.18 Found (((eq_ref fofType) x3) P) as proof of ((P x3)->(P Y))
% 92.97/93.18 Found (((eq_ref fofType) x3) P) as proof of ((P x3)->(P Y))
% 92.97/93.18 Found (fun (P:(fofType->Prop))=> (((eq_ref fofType) x3) P)) as proof of ((P x3)->(P Y))
% 92.97/93.18 Found (fun (x00:((x0 Xx) Y)) (P:(fofType->Prop))=> (((eq_ref fofType) x3) P)) as proof of (((eq fofType) x3) Y)
% 92.97/93.18 Found x4:(P x3)
% 92.97/93.18 Instantiate: x3:=Y:fofType
% 92.97/93.18 Found (fun (x4:(P x3))=> x4) as proof of (P Y)
% 92.97/93.18 Found (fun (P:(fofType->Prop)) (x4:(P x3))=> x4) as proof of ((P x3)->(P Y))
% 92.97/93.18 Found (fun (x00:((x0 Xx) Y)) (P:(fofType->Prop)) (x4:(P x3))=> x4) as proof of (((eq fofType) x3) Y)
% 92.97/93.18 Found eq_ref00:=(eq_ref0 x3):(((eq fofType) x3) x3)
% 92.97/93.18 Found (eq_ref0 x3) as proof of (((eq fofType) x3) Y)
% 92.97/93.18 Found ((eq_ref fofType) x3) as proof of (((eq fofType) x3) Y)
% 92.97/93.18 Found ((eq_ref fofType) x3) as proof of (((eq fofType) x3) Y)
% 92.97/93.18 Found ((eq_ref fofType) x3) as proof of (((eq fofType) x3) Y)
% 92.97/93.18 Found (fun (x00:((x0 Xx) Y))=> ((eq_ref fofType) x3)) as proof of (((eq fofType) x3) Y)
% 92.97/93.18 Found eq_ref00:=(eq_ref0 b):(((eq ((fofType->(fofType->Prop))->Prop)) b) b)
% 92.97/93.18 Found (eq_ref0 b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (Xd:(fofType->(fofType->Prop)))=> ((and ((and ((Xd c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((Xd Xx) Xy)->((Xd (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (Xd Xx)))))))
% 92.97/93.18 Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (Xd:(fofType->(fofType->Prop)))=> ((and ((and ((Xd c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((Xd Xx) Xy)->((Xd (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (Xd Xx)))))))
% 92.97/93.18 Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (Xd:(fofType->(fofType->Prop)))=> ((and ((and ((Xd c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((Xd Xx) Xy)->((Xd (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (Xd Xx)))))))
% 92.97/93.18 Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (Xd:(fofType->(fofType->Prop)))=> ((and ((and ((Xd c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((Xd Xx) Xy)->((Xd (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (Xd Xx)))))))
% 92.97/93.18 Found eq_ref00:=(eq_ref0 a):(((eq ((fofType->(fofType->Prop))->Prop)) a) a)
% 93.26/93.47 Found (eq_ref0 a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b)
% 93.26/93.47 Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b)
% 93.26/93.47 Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b)
% 93.26/93.47 Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b)
% 93.26/93.47 Found eq_substitution0000:=(fun (a:fofType)=> ((eq_substitution000 a) (fun (x6:fofType)=> (cS Xx)))):(forall (a:fofType), ((((eq fofType) a) (cS Xy))->(((eq fofType) (cS Xx)) (cS Xx))))
% 93.26/93.47 Instantiate: x0:=(fun (x4:fofType) (x30:fofType)=> (forall (a:fofType), ((((eq fofType) a) (cS Xy))->(((eq fofType) x4) x4)))):(fofType->(fofType->Prop))
% 93.26/93.47 Found (fun (a:fofType)=> ((eq_substitution000 a) (fun (x6:fofType)=> (cS Xx)))) as proof of ((x0 (cS Xx)) (cS (cS Xy)))
% 93.26/93.47 Found (fun (a:fofType)=> (((fun (a:fofType)=> ((eq_substitution00 a) (cS Xy))) a) (fun (x6:fofType)=> (cS Xx)))) as proof of ((x0 (cS Xx)) (cS (cS Xy)))
% 93.26/93.47 Found (fun (a:fofType)=> (((fun (a:fofType)=> (((eq_substitution0 fofType) a) (cS Xy))) a) (fun (x6:fofType)=> (cS Xx)))) as proof of ((x0 (cS Xx)) (cS (cS Xy)))
% 93.26/93.47 Found (fun (a:fofType)=> (((fun (a:fofType)=> ((((eq_substitution fofType) fofType) a) (cS Xy))) a) (fun (x6:fofType)=> (cS Xx)))) as proof of ((x0 (cS Xx)) (cS (cS Xy)))
% 93.26/93.47 Found (fun (x2:(forall (Xn:fofType), (not (((eq fofType) (cS Xn)) c0)))) (a:fofType)=> (((fun (a:fofType)=> ((((eq_substitution fofType) fofType) a) (cS Xy))) a) (fun (x6:fofType)=> (cS Xx)))) as proof of ((x0 (cS Xx)) (cS (cS Xy)))
% 93.26/93.47 Found (fun (x1:((and cIND) (forall (Xx0:fofType) (Xy0:fofType), ((((eq fofType) (cS Xx0)) (cS Xy0))->(((eq fofType) Xx0) Xy0))))) (x2:(forall (Xn:fofType), (not (((eq fofType) (cS Xn)) c0)))) (a:fofType)=> (((fun (a:fofType)=> ((((eq_substitution fofType) fofType) a) (cS Xy))) a) (fun (x6:fofType)=> (cS Xx)))) as proof of ((forall (Xn:fofType), (not (((eq fofType) (cS Xn)) c0)))->((x0 (cS Xx)) (cS (cS Xy))))
% 93.26/93.47 Found (fun (x1:((and cIND) (forall (Xx0:fofType) (Xy0:fofType), ((((eq fofType) (cS Xx0)) (cS Xy0))->(((eq fofType) Xx0) Xy0))))) (x2:(forall (Xn:fofType), (not (((eq fofType) (cS Xn)) c0)))) (a:fofType)=> (((fun (a:fofType)=> ((((eq_substitution fofType) fofType) a) (cS Xy))) a) (fun (x6:fofType)=> (cS Xx)))) as proof of (((and cIND) (forall (Xx0:fofType) (Xy0:fofType), ((((eq fofType) (cS Xx0)) (cS Xy0))->(((eq fofType) Xx0) Xy0))))->((forall (Xn:fofType), (not (((eq fofType) (cS Xn)) c0)))->((x0 (cS Xx)) (cS (cS Xy)))))
% 93.26/93.47 Found (and_rect00 (fun (x1:((and cIND) (forall (Xx0:fofType) (Xy0:fofType), ((((eq fofType) (cS Xx0)) (cS Xy0))->(((eq fofType) Xx0) Xy0))))) (x2:(forall (Xn:fofType), (not (((eq fofType) (cS Xn)) c0)))) (a:fofType)=> (((fun (a:fofType)=> ((((eq_substitution fofType) fofType) a) (cS Xy))) a) (fun (x6:fofType)=> (cS Xx))))) as proof of ((x0 (cS Xx)) (cS (cS Xy)))
% 93.26/93.47 Found ((and_rect0 ((x0 (cS Xx)) (cS (cS Xy)))) (fun (x1:((and cIND) (forall (Xx0:fofType) (Xy0:fofType), ((((eq fofType) (cS Xx0)) (cS Xy0))->(((eq fofType) Xx0) Xy0))))) (x2:(forall (Xn:fofType), (not (((eq fofType) (cS Xn)) c0)))) (a:fofType)=> (((fun (a:fofType)=> ((((eq_substitution fofType) fofType) a) (cS Xy))) a) (fun (x6:fofType)=> (cS Xx))))) as proof of ((x0 (cS Xx)) (cS (cS Xy)))
% 93.26/93.47 Found (((fun (P:Type) (x1:(((and cIND) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (cS Xx)) (cS Xy))->(((eq fofType) Xx) Xy))))->((forall (Xn:fofType), (not (((eq fofType) (cS Xn)) c0)))->P)))=> (((((and_rect ((and cIND) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (cS Xx)) (cS Xy))->(((eq fofType) Xx) Xy))))) (forall (Xn:fofType), (not (((eq fofType) (cS Xn)) c0)))) P) x1) x)) ((x0 (cS Xx)) (cS (cS Xy)))) (fun (x1:((and cIND) (forall (Xx0:fofType) (Xy0:fofType), ((((eq fofType) (cS Xx0)) (cS Xy0))->(((eq fofType) Xx0) Xy0))))) (x2:(forall (Xn:fofType), (not (((eq fofType) (cS Xn)) c0)))) (a:fofType)=> (((fun (a:fofType)=> ((((eq_substitution fofType) fofType) a) (cS Xy))) a) (fun (x6:fofType)=> (cS Xx))))) as proof of ((x0 (cS Xx)) (cS (cS Xy)))
% 98.21/98.44 Found (fun (x00:((x0 Xx) Xy))=> (((fun (P:Type) (x1:(((and cIND) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (cS Xx)) (cS Xy))->(((eq fofType) Xx) Xy))))->((forall (Xn:fofType), (not (((eq fofType) (cS Xn)) c0)))->P)))=> (((((and_rect ((and cIND) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (cS Xx)) (cS Xy))->(((eq fofType) Xx) Xy))))) (forall (Xn:fofType), (not (((eq fofType) (cS Xn)) c0)))) P) x1) x)) ((x0 (cS Xx)) (cS (cS Xy)))) (fun (x1:((and cIND) (forall (Xx0:fofType) (Xy0:fofType), ((((eq fofType) (cS Xx0)) (cS Xy0))->(((eq fofType) Xx0) Xy0))))) (x2:(forall (Xn:fofType), (not (((eq fofType) (cS Xn)) c0)))) (a:fofType)=> (((fun (a:fofType)=> ((((eq_substitution fofType) fofType) a) (cS Xy))) a) (fun (x6:fofType)=> (cS Xx)))))) as proof of ((x0 (cS Xx)) (cS (cS Xy)))
% 98.21/98.44 Found eq_ref00:=(eq_ref0 (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x2 Xx) X)) (forall (Y:fofType), (((x2 Xx) Y)->(((eq fofType) X) Y)))))))):(((eq Prop) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x2 Xx) X)) (forall (Y:fofType), (((x2 Xx) Y)->(((eq fofType) X) Y)))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x2 Xx) X)) (forall (Y:fofType), (((x2 Xx) Y)->(((eq fofType) X) Y))))))))
% 98.21/98.44 Found (eq_ref0 (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x2 Xx) X)) (forall (Y:fofType), (((x2 Xx) Y)->(((eq fofType) X) Y)))))))) as proof of (((eq Prop) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x2 Xx) X)) (forall (Y:fofType), (((x2 Xx) Y)->(((eq fofType) X) Y)))))))) b)
% 98.21/98.44 Found ((eq_ref Prop) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x2 Xx) X)) (forall (Y:fofType), (((x2 Xx) Y)->(((eq fofType) X) Y)))))))) as proof of (((eq Prop) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x2 Xx) X)) (forall (Y:fofType), (((x2 Xx) Y)->(((eq fofType) X) Y)))))))) b)
% 98.21/98.44 Found ((eq_ref Prop) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x2 Xx) X)) (forall (Y:fofType), (((x2 Xx) Y)->(((eq fofType) X) Y)))))))) as proof of (((eq Prop) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x2 Xx) X)) (forall (Y:fofType), (((x2 Xx) Y)->(((eq fofType) X) Y)))))))) b)
% 98.21/98.44 Found ((eq_ref Prop) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x2 Xx) X)) (forall (Y:fofType), (((x2 Xx) Y)->(((eq fofType) X) Y)))))))) as proof of (((eq Prop) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x2 Xx) X)) (forall (Y:fofType), (((x2 Xx) Y)->(((eq fofType) X) Y)))))))) b)
% 98.21/98.44 Found eq_ref00:=(eq_ref0 x10):(((eq fofType) x10) x10)
% 98.21/98.44 Found (eq_ref0 x10) as proof of (((eq fofType) x10) x1)
% 98.21/98.44 Found ((eq_ref fofType) x10) as proof of (((eq fofType) x10) x1)
% 98.21/98.44 Found ((eq_ref fofType) x10) as proof of (((eq fofType) x10) x1)
% 98.21/98.44 Found eq_ref00:=(eq_ref0 x10):(((eq fofType) x10) x10)
% 98.21/98.44 Found (eq_ref0 x10) as proof of (((eq fofType) x10) x1)
% 98.21/98.44 Found ((eq_ref fofType) x10) as proof of (((eq fofType) x10) x1)
% 98.21/98.44 Found ((eq_ref fofType) x10) as proof of (((eq fofType) x10) x1)
% 98.21/98.44 Found (fun (x3:(forall (Xn:fofType), (not (((eq fofType) (cS Xn)) c0))))=> ((eq_ref fofType) x10)) as proof of (((eq fofType) x10) x1)
% 98.21/98.44 Found (fun (x2:((and cIND) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (cS Xx)) (cS Xy))->(((eq fofType) Xx) Xy))))) (x3:(forall (Xn:fofType), (not (((eq fofType) (cS Xn)) c0))))=> ((eq_ref fofType) x10)) as proof of ((forall (Xn:fofType), (not (((eq fofType) (cS Xn)) c0)))->(((eq fofType) x10) x1))
% 98.21/98.44 Found (fun (x2:((and cIND) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (cS Xx)) (cS Xy))->(((eq fofType) Xx) Xy))))) (x3:(forall (Xn:fofType), (not (((eq fofType) (cS Xn)) c0))))=> ((eq_ref fofType) x10)) as proof of (((and cIND) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (cS Xx)) (cS Xy))->(((eq fofType) Xx) Xy))))->((forall (Xn:fofType), (not (((eq fofType) (cS Xn)) c0)))->(((eq fofType) x10) x1)))
% 98.21/98.44 Found (and_rect00 (fun (x2:((and cIND) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (cS Xx)) (cS Xy))->(((eq fofType) Xx) Xy))))) (x3:(forall (Xn:fofType), (not (((eq fofType) (cS Xn)) c0))))=> ((eq_ref fofType) x10))) as proof of (((eq fofType) x10) x1)
% 102.42/102.60 Found ((and_rect0 (((eq fofType) x10) x1)) (fun (x2:((and cIND) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (cS Xx)) (cS Xy))->(((eq fofType) Xx) Xy))))) (x3:(forall (Xn:fofType), (not (((eq fofType) (cS Xn)) c0))))=> ((eq_ref fofType) x10))) as proof of (((eq fofType) x10) x1)
% 102.42/102.60 Found (((fun (P:Type) (x2:(((and cIND) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (cS Xx)) (cS Xy))->(((eq fofType) Xx) Xy))))->((forall (Xn:fofType), (not (((eq fofType) (cS Xn)) c0)))->P)))=> (((((and_rect ((and cIND) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (cS Xx)) (cS Xy))->(((eq fofType) Xx) Xy))))) (forall (Xn:fofType), (not (((eq fofType) (cS Xn)) c0)))) P) x2) x)) (((eq fofType) x10) x1)) (fun (x2:((and cIND) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (cS Xx)) (cS Xy))->(((eq fofType) Xx) Xy))))) (x3:(forall (Xn:fofType), (not (((eq fofType) (cS Xn)) c0))))=> ((eq_ref fofType) x10))) as proof of (((eq fofType) x10) x1)
% 102.42/102.60 Found iff_sym:=(fun (A:Prop) (B:Prop) (H:((iff A) B))=> ((((conj (B->A)) (A->B)) (((proj2 (A->B)) (B->A)) H)) (((proj1 (A->B)) (B->A)) H))):(forall (A:Prop) (B:Prop), (((iff A) B)->((iff B) A)))
% 102.42/102.60 Instantiate: b:=(forall (A:Prop) (B:Prop), (((iff A) B)->((iff B) A))):Prop
% 102.42/102.60 Found iff_sym as proof of b
% 102.42/102.60 Found eq_ref000:=(eq_ref00 P):((P x3)->(P x3))
% 102.42/102.60 Found (eq_ref00 P) as proof of ((P x3)->(P x'))
% 102.42/102.60 Found ((eq_ref0 x3) P) as proof of ((P x3)->(P x'))
% 102.42/102.60 Found (((eq_ref fofType) x3) P) as proof of ((P x3)->(P x'))
% 102.42/102.60 Found (((eq_ref fofType) x3) P) as proof of ((P x3)->(P x'))
% 102.42/102.60 Found (fun (P:(fofType->Prop))=> (((eq_ref fofType) x3) P)) as proof of ((P x3)->(P x'))
% 102.42/102.60 Found (fun (x00:((x0 Xx) x')) (P:(fofType->Prop))=> (((eq_ref fofType) x3) P)) as proof of (((eq fofType) x3) x')
% 102.42/102.60 Found x4:(P x3)
% 102.42/102.60 Instantiate: x3:=x':fofType
% 102.42/102.60 Found (fun (x4:(P x3))=> x4) as proof of (P x')
% 102.42/102.60 Found (fun (P:(fofType->Prop)) (x4:(P x3))=> x4) as proof of ((P x3)->(P x'))
% 102.42/102.60 Found (fun (x00:((x0 Xx) x')) (P:(fofType->Prop)) (x4:(P x3))=> x4) as proof of (((eq fofType) x3) x')
% 102.42/102.60 Found eq_ref00:=(eq_ref0 x3):(((eq fofType) x3) x3)
% 102.42/102.60 Found (eq_ref0 x3) as proof of (((eq fofType) x3) x')
% 102.42/102.60 Found ((eq_ref fofType) x3) as proof of (((eq fofType) x3) x')
% 102.42/102.60 Found ((eq_ref fofType) x3) as proof of (((eq fofType) x3) x')
% 102.42/102.60 Found ((eq_ref fofType) x3) as proof of (((eq fofType) x3) x')
% 102.42/102.60 Found (fun (x00:((x0 Xx) x'))=> ((eq_ref fofType) x3)) as proof of (((eq fofType) x3) x')
% 102.42/102.60 Found eq_ref00:=(eq_ref0 (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y)))))))):(((eq Prop) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y)))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y))))))))
% 102.42/102.60 Found (eq_ref0 (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y)))))))) as proof of (((eq Prop) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y)))))))) b)
% 102.42/102.60 Found ((eq_ref Prop) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y)))))))) as proof of (((eq Prop) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y)))))))) b)
% 102.42/102.60 Found ((eq_ref Prop) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y)))))))) as proof of (((eq Prop) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y)))))))) b)
% 102.42/102.60 Found ((eq_ref Prop) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y)))))))) as proof of (((eq Prop) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y)))))))) b)
% 105.56/105.80 Found eq_ref00:=(eq_ref0 x3):(((eq fofType) x3) x3)
% 105.56/105.80 Found (eq_ref0 x3) as proof of (((eq fofType) x3) x30)
% 105.56/105.80 Found ((eq_ref fofType) x3) as proof of (((eq fofType) x3) x30)
% 105.56/105.80 Found ((eq_ref fofType) x3) as proof of (((eq fofType) x3) x30)
% 105.56/105.80 Found ((eq_ref fofType) x3) as proof of (((eq fofType) x3) x30)
% 105.56/105.80 Found (eq_sym000 ((eq_ref fofType) x3)) as proof of (((eq fofType) x30) x3)
% 105.56/105.80 Found ((eq_sym00 x30) ((eq_ref fofType) x3)) as proof of (((eq fofType) x30) x3)
% 105.56/105.80 Found (((eq_sym0 x3) x30) ((eq_ref fofType) x3)) as proof of (((eq fofType) x30) x3)
% 105.56/105.80 Found ((((eq_sym fofType) x3) x30) ((eq_ref fofType) x3)) as proof of (((eq fofType) x30) x3)
% 105.56/105.80 Found eq_sym000:=(eq_sym00 Y):((((eq fofType) x5) Y)->(((eq fofType) Y) x5))
% 105.56/105.80 Found (eq_sym00 Y) as proof of (((x4 Xx) Y)->(((eq fofType) x5) Y))
% 105.56/105.80 Found ((eq_sym0 x5) Y) as proof of (((x4 Xx) Y)->(((eq fofType) x5) Y))
% 105.56/105.80 Found (((eq_sym fofType) x5) Y) as proof of (((x4 Xx) Y)->(((eq fofType) x5) Y))
% 105.56/105.80 Found (((eq_sym fofType) x5) Y) as proof of (((x4 Xx) Y)->(((eq fofType) x5) Y))
% 105.56/105.80 Found (((eq_sym fofType) x5) Y) as proof of (((x4 Xx) Y)->(((eq fofType) x5) Y))
% 105.56/105.80 Found eq_ref00:=(eq_ref0 (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x2 Xx))))):(((eq Prop) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x2 Xx))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x2 Xx)))))
% 105.56/105.80 Found (eq_ref0 (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x2 Xx))))) as proof of (((eq Prop) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x2 Xx))))) b)
% 105.56/105.80 Found ((eq_ref Prop) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x2 Xx))))) as proof of (((eq Prop) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x2 Xx))))) b)
% 105.56/105.80 Found ((eq_ref Prop) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x2 Xx))))) as proof of (((eq Prop) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x2 Xx))))) b)
% 105.56/105.80 Found ((eq_ref Prop) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x2 Xx))))) as proof of (((eq Prop) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x2 Xx))))) b)
% 105.56/105.80 Found eq_ref00:=(eq_ref0 x5):(((eq fofType) x5) x5)
% 105.56/105.80 Found (eq_ref0 x5) as proof of (((eq fofType) x5) Y)
% 105.56/105.80 Found ((eq_ref fofType) x5) as proof of (((eq fofType) x5) Y)
% 105.56/105.80 Found ((eq_ref fofType) x5) as proof of (((eq fofType) x5) Y)
% 105.56/105.80 Found (fun (x00:((x4 Xx) Y))=> ((eq_ref fofType) x5)) as proof of (((eq fofType) x5) Y)
% 105.56/105.80 Found x000:=(x00 x5):(((eq fofType) x5) Y)
% 105.56/105.80 Found (x00 x5) as proof of (((eq fofType) x5) Y)
% 105.56/105.80 Found (x00 x5) as proof of (((eq fofType) x5) Y)
% 105.56/105.80 Found (fun (x00:((x4 Xx) Y))=> (x00 x5)) as proof of (((eq fofType) x5) Y)
% 105.56/105.80 Found (fun (Y:fofType) (x00:((x4 Xx) Y))=> (x00 x5)) as proof of (((x4 Xx) Y)->(((eq fofType) x5) Y))
% 105.56/105.80 Found (fun (Y:fofType) (x00:((x4 Xx) Y))=> (x00 x5)) as proof of (forall (Y:fofType), (((x4 Xx) Y)->(((eq fofType) x5) Y)))
% 105.56/105.80 Found eq_ref00:=(eq_ref0 x50):(((eq fofType) x50) x50)
% 105.56/105.80 Found (eq_ref0 x50) as proof of (((eq fofType) x50) x5)
% 105.56/105.80 Found ((eq_ref fofType) x50) as proof of (((eq fofType) x50) x5)
% 105.56/105.80 Found ((eq_ref fofType) x50) as proof of (((eq fofType) x50) x5)
% 105.56/105.80 Found x6:(P x50)
% 105.56/105.80 Instantiate: x5:=x50:fofType
% 105.56/105.80 Found (fun (x6:(P x50))=> x6) as proof of (P x5)
% 105.56/105.80 Found (fun (P:(fofType->Prop)) (x6:(P x50))=> x6) as proof of ((P x50)->(P x5))
% 105.56/105.80 Found eq_ref000:=(eq_ref00 P):((P x50)->(P x50))
% 105.56/105.80 Found (eq_ref00 P) as proof of ((P x50)->(P x5))
% 105.56/105.80 Found ((eq_ref0 x50) P) as proof of ((P x50)->(P x5))
% 105.56/105.80 Found (((eq_ref fofType) x50) P) as proof of ((P x50)->(P x5))
% 105.56/105.80 Found (((eq_ref fofType) x50) P) as proof of ((P x50)->(P x5))
% 105.56/105.80 Found (fun (P:(fofType->Prop))=> (((eq_ref fofType) x50) P)) as proof of ((P x50)->(P x5))
% 105.56/105.80 Found x4:(P x10)
% 105.56/105.80 Instantiate: x1:=x10:fofType
% 105.56/105.80 Found (fun (x4:(P x10))=> x4) as proof of (P x1)
% 105.56/105.80 Found (fun (P:(fofType->Prop)) (x4:(P x10))=> x4) as proof of ((P x10)->(P x1))
% 105.56/105.80 Found eq_ref000:=(eq_ref00 P):((P x10)->(P x10))
% 105.56/105.80 Found (eq_ref00 P) as proof of ((P x10)->(P x1))
% 105.56/105.80 Found ((eq_ref0 x10) P) as proof of ((P x10)->(P x1))
% 105.56/105.80 Found (((eq_ref fofType) x10) P) as proof of ((P x10)->(P x1))
% 105.56/105.80 Found (((eq_ref fofType) x10) P) as proof of ((P x10)->(P x1))
% 107.73/107.98 Found (fun (P:(fofType->Prop))=> (((eq_ref fofType) x10) P)) as proof of ((P x10)->(P x1))
% 107.73/107.98 Found eq_substitution0000:=(fun (a:fofType)=> ((eq_substitution000 a) (fun (x8:fofType)=> (cS Xx)))):(forall (a:fofType), ((((eq fofType) a) (cS Xy))->(((eq fofType) (cS Xx)) (cS Xx))))
% 107.73/107.98 Instantiate: x0:=(fun (x6:fofType) (x50:fofType)=> (forall (a:fofType), ((((eq fofType) a) (cS Xy))->(((eq fofType) x6) x6)))):(fofType->(fofType->Prop))
% 107.73/107.98 Found (fun (a:fofType)=> ((eq_substitution000 a) (fun (x8:fofType)=> (cS Xx)))) as proof of ((x0 (cS Xx)) (cS (cS Xy)))
% 107.73/107.98 Found (fun (a:fofType)=> (((fun (a:fofType)=> ((eq_substitution00 a) (cS Xy))) a) (fun (x8:fofType)=> (cS Xx)))) as proof of ((x0 (cS Xx)) (cS (cS Xy)))
% 107.73/107.98 Found (fun (a:fofType)=> (((fun (a:fofType)=> (((eq_substitution0 fofType) a) (cS Xy))) a) (fun (x8:fofType)=> (cS Xx)))) as proof of ((x0 (cS Xx)) (cS (cS Xy)))
% 107.73/107.98 Found (fun (a:fofType)=> (((fun (a:fofType)=> ((((eq_substitution fofType) fofType) a) (cS Xy))) a) (fun (x8:fofType)=> (cS Xx)))) as proof of ((x0 (cS Xx)) (cS (cS Xy)))
% 107.73/107.98 Found (fun (x00:((x0 Xx) Xy)) (a:fofType)=> (((fun (a:fofType)=> ((((eq_substitution fofType) fofType) a) (cS Xy))) a) (fun (x8:fofType)=> (cS Xx)))) as proof of ((x0 (cS Xx)) (cS (cS Xy)))
% 107.73/107.98 Found x4:(P x3)
% 107.73/107.98 Instantiate: x3:=Y:fofType
% 107.73/107.98 Found (fun (x4:(P x3))=> x4) as proof of (P Y)
% 107.73/107.98 Found (fun (P:(fofType->Prop)) (x4:(P x3))=> x4) as proof of ((P x3)->(P Y))
% 107.73/107.98 Found (fun (P:(fofType->Prop)) (x4:(P x3))=> x4) as proof of (((eq fofType) x3) Y)
% 107.73/107.98 Found (fun (x00:((x0 Xx) Y)) (P:(fofType->Prop)) (x4:(P x3))=> x4) as proof of (((eq fofType) x3) Y)
% 107.73/107.98 Found eq_ref000:=(eq_ref00 P):((P x3)->(P x3))
% 107.73/107.98 Found (eq_ref00 P) as proof of ((P x3)->(P Y))
% 107.73/107.98 Found ((eq_ref0 x3) P) as proof of ((P x3)->(P Y))
% 107.73/107.98 Found (((eq_ref fofType) x3) P) as proof of ((P x3)->(P Y))
% 107.73/107.98 Found (((eq_ref fofType) x3) P) as proof of ((P x3)->(P Y))
% 107.73/107.98 Found (fun (P:(fofType->Prop))=> (((eq_ref fofType) x3) P)) as proof of ((P x3)->(P Y))
% 107.73/107.98 Found (fun (P:(fofType->Prop))=> (((eq_ref fofType) x3) P)) as proof of (((eq fofType) x3) Y)
% 107.73/107.98 Found (fun (x00:((x0 Xx) Y)) (P:(fofType->Prop))=> (((eq_ref fofType) x3) P)) as proof of (((eq fofType) x3) Y)
% 107.73/107.98 Found eq_ref00:=(eq_ref0 x10):(((eq fofType) x10) x10)
% 107.73/107.98 Found (eq_ref0 x10) as proof of (((eq fofType) x10) x1)
% 107.73/107.98 Found ((eq_ref fofType) x10) as proof of (((eq fofType) x10) x1)
% 107.73/107.98 Found ((eq_ref fofType) x10) as proof of (((eq fofType) x10) x1)
% 107.73/107.98 Found eq_ref00:=(eq_ref0 b):(((eq ((fofType->(fofType->Prop))->Prop)) b) b)
% 107.73/107.98 Found (eq_ref0 b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (Xd:(fofType->(fofType->Prop)))=> ((and ((and ((Xd c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((Xd Xx) Xy)->((Xd (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((Xd Xx) X)) (forall (Y:fofType), (((Xd Xx) Y)->(((eq fofType) X) Y))))))))))
% 107.73/107.98 Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (Xd:(fofType->(fofType->Prop)))=> ((and ((and ((Xd c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((Xd Xx) Xy)->((Xd (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((Xd Xx) X)) (forall (Y:fofType), (((Xd Xx) Y)->(((eq fofType) X) Y))))))))))
% 107.73/107.98 Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (Xd:(fofType->(fofType->Prop)))=> ((and ((and ((Xd c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((Xd Xx) Xy)->((Xd (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((Xd Xx) X)) (forall (Y:fofType), (((Xd Xx) Y)->(((eq fofType) X) Y))))))))))
% 107.73/107.98 Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (Xd:(fofType->(fofType->Prop)))=> ((and ((and ((Xd c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((Xd Xx) Xy)->((Xd (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((Xd Xx) X)) (forall (Y:fofType), (((Xd Xx) Y)->(((eq fofType) X) Y))))))))))
% 107.73/107.98 Found eta_expansion000:=(eta_expansion00 a):(((eq ((fofType->(fofType->Prop))->Prop)) a) (fun (x:(fofType->(fofType->Prop)))=> (a x)))
% 109.25/109.43 Found (eta_expansion00 a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b)
% 109.25/109.43 Found ((eta_expansion0 Prop) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b)
% 109.25/109.43 Found (((eta_expansion (fofType->(fofType->Prop))) Prop) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b)
% 109.25/109.43 Found (((eta_expansion (fofType->(fofType->Prop))) Prop) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b)
% 109.25/109.43 Found (((eta_expansion (fofType->(fofType->Prop))) Prop) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b)
% 109.25/109.43 Found eq_ref00:=(eq_ref0 a):(((eq ((fofType->(fofType->Prop))->Prop)) a) a)
% 109.25/109.43 Found (eq_ref0 a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b)
% 109.25/109.43 Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b)
% 109.25/109.43 Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b)
% 109.25/109.43 Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b)
% 109.25/109.43 Found eq_ref00:=(eq_ref0 x10):(((eq fofType) x10) x10)
% 109.25/109.43 Found (eq_ref0 x10) as proof of (((eq fofType) x10) x1)
% 109.25/109.43 Found ((eq_ref fofType) x10) as proof of (((eq fofType) x10) x1)
% 109.25/109.43 Found ((eq_ref fofType) x10) as proof of (((eq fofType) x10) x1)
% 109.25/109.43 Found (fun (x3:(forall (Xn:fofType), (not (((eq fofType) (cS Xn)) c0))))=> ((eq_ref fofType) x10)) as proof of (((eq fofType) x10) x1)
% 109.25/109.43 Found (fun (x2:((and cIND) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (cS Xx)) (cS Xy))->(((eq fofType) Xx) Xy))))) (x3:(forall (Xn:fofType), (not (((eq fofType) (cS Xn)) c0))))=> ((eq_ref fofType) x10)) as proof of ((forall (Xn:fofType), (not (((eq fofType) (cS Xn)) c0)))->(((eq fofType) x10) x1))
% 109.25/109.43 Found (fun (x2:((and cIND) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (cS Xx)) (cS Xy))->(((eq fofType) Xx) Xy))))) (x3:(forall (Xn:fofType), (not (((eq fofType) (cS Xn)) c0))))=> ((eq_ref fofType) x10)) as proof of (((and cIND) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (cS Xx)) (cS Xy))->(((eq fofType) Xx) Xy))))->((forall (Xn:fofType), (not (((eq fofType) (cS Xn)) c0)))->(((eq fofType) x10) x1)))
% 109.25/109.43 Found (and_rect00 (fun (x2:((and cIND) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (cS Xx)) (cS Xy))->(((eq fofType) Xx) Xy))))) (x3:(forall (Xn:fofType), (not (((eq fofType) (cS Xn)) c0))))=> ((eq_ref fofType) x10))) as proof of (((eq fofType) x10) x1)
% 109.25/109.43 Found ((and_rect0 (((eq fofType) x10) x1)) (fun (x2:((and cIND) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (cS Xx)) (cS Xy))->(((eq fofType) Xx) Xy))))) (x3:(forall (Xn:fofType), (not (((eq fofType) (cS Xn)) c0))))=> ((eq_ref fofType) x10))) as proof of (((eq fofType) x10) x1)
% 109.25/109.43 Found (((fun (P:Type) (x2:(((and cIND) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (cS Xx)) (cS Xy))->(((eq fofType) Xx) Xy))))->((forall (Xn:fofType), (not (((eq fofType) (cS Xn)) c0)))->P)))=> (((((and_rect ((and cIND) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (cS Xx)) (cS Xy))->(((eq fofType) Xx) Xy))))) (forall (Xn:fofType), (not (((eq fofType) (cS Xn)) c0)))) P) x2) x)) (((eq fofType) x10) x1)) (fun (x2:((and cIND) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (cS Xx)) (cS Xy))->(((eq fofType) Xx) Xy))))) (x3:(forall (Xn:fofType), (not (((eq fofType) (cS Xn)) c0))))=> ((eq_ref fofType) x10))) as proof of (((eq fofType) x10) x1)
% 109.25/109.43 Found x00000:=(x0000 x3):(((eq fofType) x3) Y)
% 109.25/109.43 Found (x0000 x3) as proof of (((eq fofType) x3) Y)
% 109.25/109.43 Found ((fun (x30:fofType)=> ((x000 x30) x2)) x3) as proof of (((eq fofType) x3) Y)
% 109.25/109.43 Found ((fun (x30:fofType)=> (((x00 x1) x30) x2)) x3) as proof of (((eq fofType) x3) Y)
% 109.25/109.43 Found ((fun (x30:fofType)=> (((x00 x1) x30) x2)) x3) as proof of (((eq fofType) x3) Y)
% 109.25/109.43 Found (fun (x00:((x0 Xx) Y))=> ((fun (x30:fofType)=> (((x00 x1) x30) x2)) x3)) as proof of (((eq fofType) x3) Y)
% 109.25/109.43 Found (fun (Y:fofType) (x00:((x0 Xx) Y))=> ((fun (x30:fofType)=> (((x00 x1) x30) x2)) x3)) as proof of (((x0 Xx) Y)->(((eq fofType) x3) Y))
% 109.25/109.43 Found (fun (Y:fofType) (x00:((x0 Xx) Y))=> ((fun (x30:fofType)=> (((x00 x1) x30) x2)) x3)) as proof of (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) x3) Y)))
% 111.64/111.86 Found eq_ref00:=(eq_ref0 x30):(((eq fofType) x30) x30)
% 111.64/111.86 Found (eq_ref0 x30) as proof of (((eq fofType) x30) x3)
% 111.64/111.86 Found ((eq_ref fofType) x30) as proof of (((eq fofType) x30) x3)
% 111.64/111.86 Found ((eq_ref fofType) x30) as proof of (((eq fofType) x30) x3)
% 111.64/111.86 Found (fun (x20:(forall (Xn:fofType), (not (((eq fofType) (cS Xn)) c0))))=> ((eq_ref fofType) x30)) as proof of (((eq fofType) x30) x3)
% 111.64/111.86 Found x4:(P x30)
% 111.64/111.86 Instantiate: x3:=x30:fofType
% 111.64/111.86 Found (fun (x4:(P x30))=> x4) as proof of (P x3)
% 111.64/111.86 Found (fun (P:(fofType->Prop)) (x4:(P x30))=> x4) as proof of ((P x30)->(P x3))
% 111.64/111.86 Found (fun (x20:(forall (Xn:fofType), (not (((eq fofType) (cS Xn)) c0)))) (P:(fofType->Prop)) (x4:(P x30))=> x4) as proof of (((eq fofType) x30) x3)
% 111.64/111.86 Found eq_ref000:=(eq_ref00 P):((P x30)->(P x30))
% 111.64/111.86 Found (eq_ref00 P) as proof of ((P x30)->(P x3))
% 111.64/111.86 Found ((eq_ref0 x30) P) as proof of ((P x30)->(P x3))
% 111.64/111.86 Found (((eq_ref fofType) x30) P) as proof of ((P x30)->(P x3))
% 111.64/111.86 Found (((eq_ref fofType) x30) P) as proof of ((P x30)->(P x3))
% 111.64/111.86 Found (fun (P:(fofType->Prop))=> (((eq_ref fofType) x30) P)) as proof of ((P x30)->(P x3))
% 111.64/111.86 Found (fun (x20:(forall (Xn:fofType), (not (((eq fofType) (cS Xn)) c0)))) (P:(fofType->Prop))=> (((eq_ref fofType) x30) P)) as proof of (((eq fofType) x30) x3)
% 111.64/111.86 Found eq_sym000:=(eq_sym00 Y):((((eq fofType) x3) Y)->(((eq fofType) Y) x3))
% 111.64/111.86 Found (eq_sym00 Y) as proof of (((x0 Xx) Y)->(((eq fofType) x3) Y))
% 111.64/111.86 Found ((eq_sym0 x3) Y) as proof of (((x0 Xx) Y)->(((eq fofType) x3) Y))
% 111.64/111.86 Found (((eq_sym fofType) x3) Y) as proof of (((x0 Xx) Y)->(((eq fofType) x3) Y))
% 111.64/111.86 Found (((eq_sym fofType) x3) Y) as proof of (((x0 Xx) Y)->(((eq fofType) x3) Y))
% 111.64/111.86 Found (((eq_sym fofType) x3) Y) as proof of (((x0 Xx) Y)->(((eq fofType) x3) Y))
% 111.64/111.86 Found eq_sym000:=(eq_sym00 Y):((((eq fofType) x5) Y)->(((eq fofType) Y) x5))
% 111.64/111.86 Found (eq_sym00 Y) as proof of (((x0 Xx) Y)->(((eq fofType) x5) Y))
% 111.64/111.86 Found ((eq_sym0 x5) Y) as proof of (((x0 Xx) Y)->(((eq fofType) x5) Y))
% 111.64/111.86 Found (((eq_sym fofType) x5) Y) as proof of (((x0 Xx) Y)->(((eq fofType) x5) Y))
% 111.64/111.86 Found (((eq_sym fofType) x5) Y) as proof of (((x0 Xx) Y)->(((eq fofType) x5) Y))
% 111.64/111.86 Found (((eq_sym fofType) x5) Y) as proof of (((x0 Xx) Y)->(((eq fofType) x5) Y))
% 111.64/111.86 Found eq_sym000:=(eq_sym00 Y):((((eq fofType) x5) Y)->(((eq fofType) Y) x5))
% 111.64/111.86 Found (eq_sym00 Y) as proof of (((x2 Xx) Y)->(((eq fofType) x5) Y))
% 111.64/111.86 Found ((eq_sym0 x5) Y) as proof of (((x2 Xx) Y)->(((eq fofType) x5) Y))
% 111.64/111.86 Found (((eq_sym fofType) x5) Y) as proof of (((x2 Xx) Y)->(((eq fofType) x5) Y))
% 111.64/111.86 Found (((eq_sym fofType) x5) Y) as proof of (((x2 Xx) Y)->(((eq fofType) x5) Y))
% 111.64/111.86 Found (((eq_sym fofType) x5) Y) as proof of (((x2 Xx) Y)->(((eq fofType) x5) Y))
% 111.64/111.86 Found eq_ref00:=(eq_ref0 x3):(((eq fofType) x3) x3)
% 111.64/111.86 Found (eq_ref0 x3) as proof of (((eq fofType) x3) Y)
% 111.64/111.86 Found ((eq_ref fofType) x3) as proof of (((eq fofType) x3) Y)
% 111.64/111.86 Found ((eq_ref fofType) x3) as proof of (((eq fofType) x3) Y)
% 111.64/111.86 Found (fun (x00:((x0 Xx) Y))=> ((eq_ref fofType) x3)) as proof of (((eq fofType) x3) Y)
% 111.64/111.86 Found eq_ref00:=(eq_ref0 (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx))))):(((eq Prop) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx)))))
% 111.64/111.86 Found (eq_ref0 (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx))))) as proof of (((eq Prop) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx))))) b)
% 111.64/111.86 Found ((eq_ref Prop) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx))))) as proof of (((eq Prop) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx))))) b)
% 111.64/111.86 Found ((eq_ref Prop) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx))))) as proof of (((eq Prop) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx))))) b)
% 111.64/111.86 Found ((eq_ref Prop) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx))))) as proof of (((eq Prop) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx))))) b)
% 111.64/111.86 Found eq_ref00:=(eq_ref0 x5):(((eq fofType) x5) x5)
% 111.64/111.86 Found (eq_ref0 x5) as proof of (((eq fofType) x5) Y)
% 111.64/111.86 Found ((eq_ref fofType) x5) as proof of (((eq fofType) x5) Y)
% 115.32/115.52 Found ((eq_ref fofType) x5) as proof of (((eq fofType) x5) Y)
% 115.32/115.52 Found (fun (x00:((x0 Xx) Y))=> ((eq_ref fofType) x5)) as proof of (((eq fofType) x5) Y)
% 115.32/115.52 Found eq_ref00:=(eq_ref0 x5):(((eq fofType) x5) x5)
% 115.32/115.52 Found (eq_ref0 x5) as proof of (((eq fofType) x5) Y)
% 115.32/115.52 Found ((eq_ref fofType) x5) as proof of (((eq fofType) x5) Y)
% 115.32/115.52 Found ((eq_ref fofType) x5) as proof of (((eq fofType) x5) Y)
% 115.32/115.52 Found (fun (x00:((x2 Xx) Y))=> ((eq_ref fofType) x5)) as proof of (((eq fofType) x5) Y)
% 115.32/115.52 Found ex_intro0:=(ex_intro (fofType->(fofType->Prop))):(forall (P:((fofType->(fofType->Prop))->Prop)) (x:(fofType->(fofType->Prop))), ((P x)->((ex (fofType->(fofType->Prop))) P)))
% 115.32/115.52 Instantiate: b:=(forall (P:((fofType->(fofType->Prop))->Prop)) (x:(fofType->(fofType->Prop))), ((P x)->((ex (fofType->(fofType->Prop))) P))):Prop
% 115.32/115.52 Found ex_intro0 as proof of b
% 115.32/115.52 Found eq_ref00:=(eq_ref0 x3):(((eq fofType) x3) x3)
% 115.32/115.52 Found (eq_ref0 x3) as proof of (((eq fofType) x3) x30)
% 115.32/115.52 Found ((eq_ref fofType) x3) as proof of (((eq fofType) x3) x30)
% 115.32/115.52 Found ((eq_ref fofType) x3) as proof of (((eq fofType) x3) x30)
% 115.32/115.52 Found ((eq_ref fofType) x3) as proof of (((eq fofType) x3) x30)
% 115.32/115.52 Found (eq_sym000 ((eq_ref fofType) x3)) as proof of (((eq fofType) x30) x3)
% 115.32/115.52 Found ((eq_sym00 x30) ((eq_ref fofType) x3)) as proof of (((eq fofType) x30) x3)
% 115.32/115.52 Found (((eq_sym0 x3) x30) ((eq_ref fofType) x3)) as proof of (((eq fofType) x30) x3)
% 115.32/115.52 Found ((((eq_sym fofType) x3) x30) ((eq_ref fofType) x3)) as proof of (((eq fofType) x30) x3)
% 115.32/115.52 Found eq_sym000:=(eq_sym00 x'):((((eq fofType) x5) x')->(((eq fofType) x') x5))
% 115.32/115.52 Found (eq_sym00 x') as proof of (((x4 Xx) x')->(((eq fofType) x5) x'))
% 115.32/115.52 Found ((eq_sym0 x5) x') as proof of (((x4 Xx) x')->(((eq fofType) x5) x'))
% 115.32/115.52 Found (((eq_sym fofType) x5) x') as proof of (((x4 Xx) x')->(((eq fofType) x5) x'))
% 115.32/115.52 Found (((eq_sym fofType) x5) x') as proof of (((x4 Xx) x')->(((eq fofType) x5) x'))
% 115.32/115.52 Found (((eq_sym fofType) x5) x') as proof of (((x4 Xx) x')->(((eq fofType) x5) x'))
% 115.32/115.52 Found x4:(P x10)
% 115.32/115.52 Instantiate: x1:=x10:fofType
% 115.32/115.52 Found (fun (x4:(P x10))=> x4) as proof of (P x1)
% 115.32/115.52 Found (fun (P:(fofType->Prop)) (x4:(P x10))=> x4) as proof of ((P x10)->(P x1))
% 115.32/115.52 Found eq_ref000:=(eq_ref00 P):((P x10)->(P x10))
% 115.32/115.52 Found (eq_ref00 P) as proof of ((P x10)->(P x1))
% 115.32/115.52 Found ((eq_ref0 x10) P) as proof of ((P x10)->(P x1))
% 115.32/115.52 Found (((eq_ref fofType) x10) P) as proof of ((P x10)->(P x1))
% 115.32/115.52 Found (((eq_ref fofType) x10) P) as proof of ((P x10)->(P x1))
% 115.32/115.52 Found (fun (P:(fofType->Prop))=> (((eq_ref fofType) x10) P)) as proof of ((P x10)->(P x1))
% 115.32/115.52 Found eq_ref00:=(eq_ref0 x5):(((eq fofType) x5) x5)
% 115.32/115.52 Found (eq_ref0 x5) as proof of (((eq fofType) x5) x')
% 115.32/115.52 Found ((eq_ref fofType) x5) as proof of (((eq fofType) x5) x')
% 115.32/115.52 Found ((eq_ref fofType) x5) as proof of (((eq fofType) x5) x')
% 115.32/115.52 Found (fun (x00:((x4 Xx) x'))=> ((eq_ref fofType) x5)) as proof of (((eq fofType) x5) x')
% 115.32/115.52 Found eq_substitution0000:=(fun (a:fofType)=> ((eq_substitution000 a) (fun (x8:fofType)=> (cS Xx)))):(forall (a:fofType), ((((eq fofType) a) (cS Xy))->(((eq fofType) (cS Xx)) (cS Xx))))
% 115.32/115.52 Instantiate: x0:=(fun (x6:fofType) (x50:fofType)=> (forall (a:fofType), ((((eq fofType) a) (cS Xy))->(((eq fofType) x6) x6)))):(fofType->(fofType->Prop))
% 115.32/115.52 Found (fun (a:fofType)=> ((eq_substitution000 a) (fun (x8:fofType)=> (cS Xx)))) as proof of ((x0 (cS Xx)) (cS (cS Xy)))
% 115.32/115.52 Found (fun (a:fofType)=> (((fun (a:fofType)=> ((eq_substitution00 a) (cS Xy))) a) (fun (x8:fofType)=> (cS Xx)))) as proof of ((x0 (cS Xx)) (cS (cS Xy)))
% 115.32/115.52 Found (fun (a:fofType)=> (((fun (a:fofType)=> (((eq_substitution0 fofType) a) (cS Xy))) a) (fun (x8:fofType)=> (cS Xx)))) as proof of ((x0 (cS Xx)) (cS (cS Xy)))
% 115.32/115.52 Found (fun (a:fofType)=> (((fun (a:fofType)=> ((((eq_substitution fofType) fofType) a) (cS Xy))) a) (fun (x8:fofType)=> (cS Xx)))) as proof of ((x0 (cS Xx)) (cS (cS Xy)))
% 115.32/115.52 Found (fun (x4:(forall (Xx0:fofType) (Xy0:fofType), ((((eq fofType) (cS Xx0)) (cS Xy0))->(((eq fofType) Xx0) Xy0)))) (a:fofType)=> (((fun (a:fofType)=> ((((eq_substitution fofType) fofType) a) (cS Xy))) a) (fun (x8:fofType)=> (cS Xx)))) as proof of ((x0 (cS Xx)) (cS (cS Xy)))
% 115.32/115.52 Found (fun (x3:cIND) (x4:(forall (Xx0:fofType) (Xy0:fofType), ((((eq fofType) (cS Xx0)) (cS Xy0))->(((eq fofType) Xx0) Xy0)))) (a:fofType)=> (((fun (a:fofType)=> ((((eq_substitution fofType) fofType) a) (cS Xy))) a) (fun (x8:fofType)=> (cS Xx)))) as proof of ((forall (Xx0:fofType) (Xy0:fofType), ((((eq fofType) (cS Xx0)) (cS Xy0))->(((eq fofType) Xx0) Xy0)))->((x0 (cS Xx)) (cS (cS Xy))))
% 115.76/115.99 Found (fun (x3:cIND) (x4:(forall (Xx0:fofType) (Xy0:fofType), ((((eq fofType) (cS Xx0)) (cS Xy0))->(((eq fofType) Xx0) Xy0)))) (a:fofType)=> (((fun (a:fofType)=> ((((eq_substitution fofType) fofType) a) (cS Xy))) a) (fun (x8:fofType)=> (cS Xx)))) as proof of (cIND->((forall (Xx0:fofType) (Xy0:fofType), ((((eq fofType) (cS Xx0)) (cS Xy0))->(((eq fofType) Xx0) Xy0)))->((x0 (cS Xx)) (cS (cS Xy)))))
% 115.76/115.99 Found (and_rect10 (fun (x3:cIND) (x4:(forall (Xx0:fofType) (Xy0:fofType), ((((eq fofType) (cS Xx0)) (cS Xy0))->(((eq fofType) Xx0) Xy0)))) (a:fofType)=> (((fun (a:fofType)=> ((((eq_substitution fofType) fofType) a) (cS Xy))) a) (fun (x8:fofType)=> (cS Xx))))) as proof of ((x0 (cS Xx)) (cS (cS Xy)))
% 115.76/115.99 Found ((and_rect1 ((x0 (cS Xx)) (cS (cS Xy)))) (fun (x3:cIND) (x4:(forall (Xx0:fofType) (Xy0:fofType), ((((eq fofType) (cS Xx0)) (cS Xy0))->(((eq fofType) Xx0) Xy0)))) (a:fofType)=> (((fun (a:fofType)=> ((((eq_substitution fofType) fofType) a) (cS Xy))) a) (fun (x8:fofType)=> (cS Xx))))) as proof of ((x0 (cS Xx)) (cS (cS Xy)))
% 115.76/115.99 Found (((fun (P:Type) (x3:(cIND->((forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (cS Xx)) (cS Xy))->(((eq fofType) Xx) Xy)))->P)))=> (((((and_rect cIND) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (cS Xx)) (cS Xy))->(((eq fofType) Xx) Xy)))) P) x3) x1)) ((x0 (cS Xx)) (cS (cS Xy)))) (fun (x3:cIND) (x4:(forall (Xx0:fofType) (Xy0:fofType), ((((eq fofType) (cS Xx0)) (cS Xy0))->(((eq fofType) Xx0) Xy0)))) (a:fofType)=> (((fun (a:fofType)=> ((((eq_substitution fofType) fofType) a) (cS Xy))) a) (fun (x8:fofType)=> (cS Xx))))) as proof of ((x0 (cS Xx)) (cS (cS Xy)))
% 115.76/115.99 Found (fun (x00:((x0 Xx) Xy))=> (((fun (P:Type) (x3:(cIND->((forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (cS Xx)) (cS Xy))->(((eq fofType) Xx) Xy)))->P)))=> (((((and_rect cIND) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (cS Xx)) (cS Xy))->(((eq fofType) Xx) Xy)))) P) x3) x1)) ((x0 (cS Xx)) (cS (cS Xy)))) (fun (x3:cIND) (x4:(forall (Xx0:fofType) (Xy0:fofType), ((((eq fofType) (cS Xx0)) (cS Xy0))->(((eq fofType) Xx0) Xy0)))) (a:fofType)=> (((fun (a:fofType)=> ((((eq_substitution fofType) fofType) a) (cS Xy))) a) (fun (x8:fofType)=> (cS Xx)))))) as proof of ((x0 (cS Xx)) (cS (cS Xy)))
% 115.76/115.99 Found eq_substitution0000:=(fun (a:fofType)=> ((eq_substitution000 a) (fun (x8:fofType)=> (cS Xx)))):(forall (a:fofType), ((((eq fofType) a) (cS Xy))->(((eq fofType) (cS Xx)) (cS Xx))))
% 115.76/115.99 Instantiate: x2:=(fun (x6:fofType) (x50:fofType)=> (forall (a:fofType), ((((eq fofType) a) (cS Xy))->(((eq fofType) x6) x6)))):(fofType->(fofType->Prop))
% 115.76/115.99 Found (fun (a:fofType)=> ((eq_substitution000 a) (fun (x8:fofType)=> (cS Xx)))) as proof of ((x2 (cS Xx)) (cS (cS Xy)))
% 115.76/115.99 Found (fun (a:fofType)=> (((fun (a:fofType)=> ((eq_substitution00 a) (cS Xy))) a) (fun (x8:fofType)=> (cS Xx)))) as proof of ((x2 (cS Xx)) (cS (cS Xy)))
% 115.76/115.99 Found (fun (a:fofType)=> (((fun (a:fofType)=> (((eq_substitution0 fofType) a) (cS Xy))) a) (fun (x8:fofType)=> (cS Xx)))) as proof of ((x2 (cS Xx)) (cS (cS Xy)))
% 115.76/115.99 Found (fun (a:fofType)=> (((fun (a:fofType)=> ((((eq_substitution fofType) fofType) a) (cS Xy))) a) (fun (x8:fofType)=> (cS Xx)))) as proof of ((x2 (cS Xx)) (cS (cS Xy)))
% 115.76/115.99 Found (fun (x4:(forall (Xx0:fofType) (Xy0:fofType), ((((eq fofType) (cS Xx0)) (cS Xy0))->(((eq fofType) Xx0) Xy0)))) (a:fofType)=> (((fun (a:fofType)=> ((((eq_substitution fofType) fofType) a) (cS Xy))) a) (fun (x8:fofType)=> (cS Xx)))) as proof of ((x2 (cS Xx)) (cS (cS Xy)))
% 115.76/115.99 Found (fun (x3:cIND) (x4:(forall (Xx0:fofType) (Xy0:fofType), ((((eq fofType) (cS Xx0)) (cS Xy0))->(((eq fofType) Xx0) Xy0)))) (a:fofType)=> (((fun (a:fofType)=> ((((eq_substitution fofType) fofType) a) (cS Xy))) a) (fun (x8:fofType)=> (cS Xx)))) as proof of ((forall (Xx0:fofType) (Xy0:fofType), ((((eq fofType) (cS Xx0)) (cS Xy0))->(((eq fofType) Xx0) Xy0)))->((x2 (cS Xx)) (cS (cS Xy))))
% 116.00/116.19 Found (fun (x3:cIND) (x4:(forall (Xx0:fofType) (Xy0:fofType), ((((eq fofType) (cS Xx0)) (cS Xy0))->(((eq fofType) Xx0) Xy0)))) (a:fofType)=> (((fun (a:fofType)=> ((((eq_substitution fofType) fofType) a) (cS Xy))) a) (fun (x8:fofType)=> (cS Xx)))) as proof of (cIND->((forall (Xx0:fofType) (Xy0:fofType), ((((eq fofType) (cS Xx0)) (cS Xy0))->(((eq fofType) Xx0) Xy0)))->((x2 (cS Xx)) (cS (cS Xy)))))
% 116.00/116.19 Found (and_rect10 (fun (x3:cIND) (x4:(forall (Xx0:fofType) (Xy0:fofType), ((((eq fofType) (cS Xx0)) (cS Xy0))->(((eq fofType) Xx0) Xy0)))) (a:fofType)=> (((fun (a:fofType)=> ((((eq_substitution fofType) fofType) a) (cS Xy))) a) (fun (x8:fofType)=> (cS Xx))))) as proof of ((x2 (cS Xx)) (cS (cS Xy)))
% 116.00/116.19 Found ((and_rect1 ((x2 (cS Xx)) (cS (cS Xy)))) (fun (x3:cIND) (x4:(forall (Xx0:fofType) (Xy0:fofType), ((((eq fofType) (cS Xx0)) (cS Xy0))->(((eq fofType) Xx0) Xy0)))) (a:fofType)=> (((fun (a:fofType)=> ((((eq_substitution fofType) fofType) a) (cS Xy))) a) (fun (x8:fofType)=> (cS Xx))))) as proof of ((x2 (cS Xx)) (cS (cS Xy)))
% 116.00/116.20 Found (((fun (P:Type) (x3:(cIND->((forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (cS Xx)) (cS Xy))->(((eq fofType) Xx) Xy)))->P)))=> (((((and_rect cIND) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (cS Xx)) (cS Xy))->(((eq fofType) Xx) Xy)))) P) x3) x0)) ((x2 (cS Xx)) (cS (cS Xy)))) (fun (x3:cIND) (x4:(forall (Xx0:fofType) (Xy0:fofType), ((((eq fofType) (cS Xx0)) (cS Xy0))->(((eq fofType) Xx0) Xy0)))) (a:fofType)=> (((fun (a:fofType)=> ((((eq_substitution fofType) fofType) a) (cS Xy))) a) (fun (x8:fofType)=> (cS Xx))))) as proof of ((x2 (cS Xx)) (cS (cS Xy)))
% 116.00/116.20 Found (fun (x00:((x2 Xx) Xy))=> (((fun (P:Type) (x3:(cIND->((forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (cS Xx)) (cS Xy))->(((eq fofType) Xx) Xy)))->P)))=> (((((and_rect cIND) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (cS Xx)) (cS Xy))->(((eq fofType) Xx) Xy)))) P) x3) x0)) ((x2 (cS Xx)) (cS (cS Xy)))) (fun (x3:cIND) (x4:(forall (Xx0:fofType) (Xy0:fofType), ((((eq fofType) (cS Xx0)) (cS Xy0))->(((eq fofType) Xx0) Xy0)))) (a:fofType)=> (((fun (a:fofType)=> ((((eq_substitution fofType) fofType) a) (cS Xy))) a) (fun (x8:fofType)=> (cS Xx)))))) as proof of ((x2 (cS Xx)) (cS (cS Xy)))
% 116.00/116.20 Found x4:(P x3)
% 116.00/116.20 Instantiate: x3:=x':fofType
% 116.00/116.20 Found (fun (x4:(P x3))=> x4) as proof of (P x')
% 116.00/116.20 Found (fun (P:(fofType->Prop)) (x4:(P x3))=> x4) as proof of ((P x3)->(P x'))
% 116.00/116.20 Found (fun (P:(fofType->Prop)) (x4:(P x3))=> x4) as proof of (((eq fofType) x3) x')
% 116.00/116.20 Found (fun (x00:((x0 Xx) x')) (P:(fofType->Prop)) (x4:(P x3))=> x4) as proof of (((eq fofType) x3) x')
% 116.00/116.20 Found eq_ref000:=(eq_ref00 P):((P x3)->(P x3))
% 116.00/116.20 Found (eq_ref00 P) as proof of ((P x3)->(P x'))
% 116.00/116.20 Found ((eq_ref0 x3) P) as proof of ((P x3)->(P x'))
% 116.00/116.20 Found (((eq_ref fofType) x3) P) as proof of ((P x3)->(P x'))
% 116.00/116.20 Found (((eq_ref fofType) x3) P) as proof of ((P x3)->(P x'))
% 116.00/116.20 Found (fun (P:(fofType->Prop))=> (((eq_ref fofType) x3) P)) as proof of ((P x3)->(P x'))
% 116.00/116.20 Found (fun (P:(fofType->Prop))=> (((eq_ref fofType) x3) P)) as proof of (((eq fofType) x3) x')
% 116.00/116.20 Found (fun (x00:((x0 Xx) x')) (P:(fofType->Prop))=> (((eq_ref fofType) x3) P)) as proof of (((eq fofType) x3) x')
% 116.00/116.20 Found eq_substitution0000:=(fun (a:fofType)=> ((eq_substitution000 a) (fun (x8:fofType)=> (cS Xx)))):(forall (a:fofType), ((((eq fofType) a) (cS Xy))->(((eq fofType) (cS Xx)) (cS Xx))))
% 116.00/116.20 Instantiate: x0:=(fun (x6:fofType) (x50:fofType)=> (forall (a:fofType), ((((eq fofType) a) (cS Xy))->(((eq fofType) x6) x6)))):(fofType->(fofType->Prop))
% 116.00/116.20 Found (fun (a:fofType)=> ((eq_substitution000 a) (fun (x8:fofType)=> (cS Xx)))) as proof of ((x0 (cS Xx)) (cS (cS Xy)))
% 116.00/116.20 Found (fun (a:fofType)=> (((fun (a:fofType)=> ((eq_substitution00 a) (cS Xy))) a) (fun (x8:fofType)=> (cS Xx)))) as proof of ((x0 (cS Xx)) (cS (cS Xy)))
% 116.00/116.20 Found (fun (a:fofType)=> (((fun (a:fofType)=> (((eq_substitution0 fofType) a) (cS Xy))) a) (fun (x8:fofType)=> (cS Xx)))) as proof of ((x0 (cS Xx)) (cS (cS Xy)))
% 116.00/116.20 Found (fun (a:fofType)=> (((fun (a:fofType)=> ((((eq_substitution fofType) fofType) a) (cS Xy))) a) (fun (x8:fofType)=> (cS Xx)))) as proof of ((x0 (cS Xx)) (cS (cS Xy)))
% 117.71/117.93 Found (fun (x00:((x0 Xx) Xy)) (a:fofType)=> (((fun (a:fofType)=> ((((eq_substitution fofType) fofType) a) (cS Xy))) a) (fun (x8:fofType)=> (cS Xx)))) as proof of ((x0 (cS Xx)) (cS (cS Xy)))
% 117.71/117.93 Found eq_sym000:=(eq_sym00 Y):((((eq fofType) x1) Y)->(((eq fofType) Y) x1))
% 117.71/117.93 Found (eq_sym00 Y) as proof of (((x0 Xx) Y)->(((eq fofType) x1) Y))
% 117.71/117.93 Found ((eq_sym0 x1) Y) as proof of (((x0 Xx) Y)->(((eq fofType) x1) Y))
% 117.71/117.93 Found (((eq_sym fofType) x1) Y) as proof of (((x0 Xx) Y)->(((eq fofType) x1) Y))
% 117.71/117.93 Found (((eq_sym fofType) x1) Y) as proof of (((x0 Xx) Y)->(((eq fofType) x1) Y))
% 117.71/117.93 Found (((eq_sym fofType) x1) Y) as proof of (((x0 Xx) Y)->(((eq fofType) x1) Y))
% 117.71/117.93 Found x000:=(x00 x5):(((eq fofType) x5) x')
% 117.71/117.93 Found (x00 x5) as proof of (((eq fofType) x5) x')
% 117.71/117.93 Found (x00 x5) as proof of (((eq fofType) x5) x')
% 117.71/117.93 Found (fun (x00:((x4 Xx) x'))=> (x00 x5)) as proof of (((eq fofType) x5) x')
% 117.71/117.93 Found (fun (x':fofType) (x00:((x4 Xx) x'))=> (x00 x5)) as proof of (((x4 Xx) x')->(((eq fofType) x5) x'))
% 117.71/117.93 Found (fun (x':fofType) (x00:((x4 Xx) x'))=> (x00 x5)) as proof of (forall (x':fofType), (((x4 Xx) x')->(((eq fofType) x5) x')))
% 117.71/117.93 Found eq_ref00:=(eq_ref0 x50):(((eq fofType) x50) x50)
% 117.71/117.93 Found (eq_ref0 x50) as proof of (((eq fofType) x50) x5)
% 117.71/117.93 Found ((eq_ref fofType) x50) as proof of (((eq fofType) x50) x5)
% 117.71/117.93 Found ((eq_ref fofType) x50) as proof of (((eq fofType) x50) x5)
% 117.71/117.93 Found x6:(P x50)
% 117.71/117.93 Instantiate: x5:=x50:fofType
% 117.71/117.93 Found (fun (x6:(P x50))=> x6) as proof of (P x5)
% 117.71/117.93 Found (fun (P:(fofType->Prop)) (x6:(P x50))=> x6) as proof of ((P x50)->(P x5))
% 117.71/117.93 Found eq_ref000:=(eq_ref00 P):((P x50)->(P x50))
% 117.71/117.93 Found (eq_ref00 P) as proof of ((P x50)->(P x5))
% 117.71/117.93 Found ((eq_ref0 x50) P) as proof of ((P x50)->(P x5))
% 117.71/117.93 Found (((eq_ref fofType) x50) P) as proof of ((P x50)->(P x5))
% 117.71/117.93 Found (((eq_ref fofType) x50) P) as proof of ((P x50)->(P x5))
% 117.71/117.93 Found (fun (P:(fofType->Prop))=> (((eq_ref fofType) x50) P)) as proof of ((P x50)->(P x5))
% 117.71/117.93 Found eq_ref00:=(eq_ref0 b):(((eq ((fofType->(fofType->Prop))->Prop)) b) b)
% 117.71/117.93 Found (eq_ref0 b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (Xd:(fofType->(fofType->Prop)))=> ((and ((and ((Xd c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((Xd Xx) Xy)->((Xd (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (Xd Xx)))))))
% 117.71/117.93 Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (Xd:(fofType->(fofType->Prop)))=> ((and ((and ((Xd c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((Xd Xx) Xy)->((Xd (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (Xd Xx)))))))
% 117.71/117.93 Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (Xd:(fofType->(fofType->Prop)))=> ((and ((and ((Xd c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((Xd Xx) Xy)->((Xd (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (Xd Xx)))))))
% 117.71/117.93 Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (Xd:(fofType->(fofType->Prop)))=> ((and ((and ((Xd c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((Xd Xx) Xy)->((Xd (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (Xd Xx)))))))
% 117.71/117.93 Found eta_expansion000:=(eta_expansion00 a):(((eq ((fofType->(fofType->Prop))->Prop)) a) (fun (x:(fofType->(fofType->Prop)))=> (a x)))
% 117.71/117.93 Found (eta_expansion00 a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b)
% 117.71/117.93 Found ((eta_expansion0 Prop) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b)
% 117.71/117.93 Found (((eta_expansion (fofType->(fofType->Prop))) Prop) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b)
% 117.71/117.93 Found (((eta_expansion (fofType->(fofType->Prop))) Prop) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b)
% 117.71/117.93 Found (((eta_expansion (fofType->(fofType->Prop))) Prop) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b)
% 117.71/117.93 Found eq_ref00:=(eq_ref0 a):(((eq ((fofType->(fofType->Prop))->Prop)) a) a)
% 117.71/117.93 Found (eq_ref0 a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b)
% 121.91/122.13 Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b)
% 121.91/122.13 Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b)
% 121.91/122.13 Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b)
% 121.91/122.13 Found eq_ref00:=(eq_ref0 x1):(((eq fofType) x1) x1)
% 121.91/122.13 Found (eq_ref0 x1) as proof of (((eq fofType) x1) Y)
% 121.91/122.13 Found ((eq_ref fofType) x1) as proof of (((eq fofType) x1) Y)
% 121.91/122.13 Found ((eq_ref fofType) x1) as proof of (((eq fofType) x1) Y)
% 121.91/122.13 Found (fun (x00:((x0 Xx) Y))=> ((eq_ref fofType) x1)) as proof of (((eq fofType) x1) Y)
% 121.91/122.13 Found x00000:=(x0000 x3):(((eq fofType) x3) x')
% 121.91/122.13 Found (x0000 x3) as proof of (((eq fofType) x3) x')
% 121.91/122.13 Found ((fun (x30:fofType)=> ((x000 x30) x2)) x3) as proof of (((eq fofType) x3) x')
% 121.91/122.13 Found ((fun (x30:fofType)=> (((x00 x1) x30) x2)) x3) as proof of (((eq fofType) x3) x')
% 121.91/122.13 Found ((fun (x30:fofType)=> (((x00 x1) x30) x2)) x3) as proof of (((eq fofType) x3) x')
% 121.91/122.13 Found (fun (x00:((x0 Xx) x'))=> ((fun (x30:fofType)=> (((x00 x1) x30) x2)) x3)) as proof of (((eq fofType) x3) x')
% 121.91/122.13 Found (fun (x':fofType) (x00:((x0 Xx) x'))=> ((fun (x30:fofType)=> (((x00 x1) x30) x2)) x3)) as proof of (((x0 Xx) x')->(((eq fofType) x3) x'))
% 121.91/122.13 Found (fun (x':fofType) (x00:((x0 Xx) x'))=> ((fun (x30:fofType)=> (((x00 x1) x30) x2)) x3)) as proof of (forall (x':fofType), (((x0 Xx) x')->(((eq fofType) x3) x')))
% 121.91/122.13 Found eq_ref00:=(eq_ref0 x30):(((eq fofType) x30) x30)
% 121.91/122.13 Found (eq_ref0 x30) as proof of (((eq fofType) x30) x3)
% 121.91/122.13 Found ((eq_ref fofType) x30) as proof of (((eq fofType) x30) x3)
% 121.91/122.13 Found ((eq_ref fofType) x30) as proof of (((eq fofType) x30) x3)
% 121.91/122.13 Found (fun (x20:(forall (Xn:fofType), (not (((eq fofType) (cS Xn)) c0))))=> ((eq_ref fofType) x30)) as proof of (((eq fofType) x30) x3)
% 121.91/122.13 Found x4:(P x30)
% 121.91/122.13 Instantiate: x3:=x30:fofType
% 121.91/122.13 Found (fun (x4:(P x30))=> x4) as proof of (P x3)
% 121.91/122.13 Found (fun (P:(fofType->Prop)) (x4:(P x30))=> x4) as proof of ((P x30)->(P x3))
% 121.91/122.13 Found (fun (x20:(forall (Xn:fofType), (not (((eq fofType) (cS Xn)) c0)))) (P:(fofType->Prop)) (x4:(P x30))=> x4) as proof of (((eq fofType) x30) x3)
% 121.91/122.13 Found eq_ref000:=(eq_ref00 P):((P x30)->(P x30))
% 121.91/122.13 Found (eq_ref00 P) as proof of ((P x30)->(P x3))
% 121.91/122.13 Found ((eq_ref0 x30) P) as proof of ((P x30)->(P x3))
% 121.91/122.13 Found (((eq_ref fofType) x30) P) as proof of ((P x30)->(P x3))
% 121.91/122.13 Found (((eq_ref fofType) x30) P) as proof of ((P x30)->(P x3))
% 121.91/122.13 Found (fun (P:(fofType->Prop))=> (((eq_ref fofType) x30) P)) as proof of ((P x30)->(P x3))
% 121.91/122.13 Found (fun (x20:(forall (Xn:fofType), (not (((eq fofType) (cS Xn)) c0)))) (P:(fofType->Prop))=> (((eq_ref fofType) x30) P)) as proof of (((eq fofType) x30) x3)
% 121.91/122.13 Found eq_ref00:=(eq_ref0 (cS x50)):(((eq fofType) (cS x50)) (cS x50))
% 121.91/122.13 Found (eq_ref0 (cS x50)) as proof of (((eq fofType) (cS x50)) (cS x5))
% 121.91/122.13 Found ((eq_ref fofType) (cS x50)) as proof of (((eq fofType) (cS x50)) (cS x5))
% 121.91/122.13 Found ((eq_ref fofType) (cS x50)) as proof of (((eq fofType) (cS x50)) (cS x5))
% 121.91/122.13 Found ((eq_ref fofType) (cS x50)) as proof of (((eq fofType) (cS x50)) (cS x5))
% 121.91/122.13 Found (x300 ((eq_ref fofType) (cS x50))) as proof of (((eq fofType) x50) x5)
% 121.91/122.13 Found ((x30 x5) ((eq_ref fofType) (cS x50))) as proof of (((eq fofType) x50) x5)
% 121.91/122.13 Found (((x3 x50) x5) ((eq_ref fofType) (cS x50))) as proof of (((eq fofType) x50) x5)
% 121.91/122.13 Found ex_intro0:=(ex_intro (fofType->(fofType->Prop))):(forall (P:((fofType->(fofType->Prop))->Prop)) (x:(fofType->(fofType->Prop))), ((P x)->((ex (fofType->(fofType->Prop))) P)))
% 121.91/122.13 Instantiate: b:=(forall (P:((fofType->(fofType->Prop))->Prop)) (x:(fofType->(fofType->Prop))), ((P x)->((ex (fofType->(fofType->Prop))) P))):Prop
% 121.91/122.13 Found ex_intro0 as proof of b
% 121.91/122.13 Found eq_sym000:=(eq_sym00 x'):((((eq fofType) x3) x')->(((eq fofType) x') x3))
% 121.91/122.13 Found (eq_sym00 x') as proof of (((x0 Xx) x')->(((eq fofType) x3) x'))
% 121.91/122.13 Found ((eq_sym0 x3) x') as proof of (((x0 Xx) x')->(((eq fofType) x3) x'))
% 121.91/122.13 Found (((eq_sym fofType) x3) x') as proof of (((x0 Xx) x')->(((eq fofType) x3) x'))
% 121.91/122.13 Found (((eq_sym fofType) x3) x') as proof of (((x0 Xx) x')->(((eq fofType) x3) x'))
% 121.91/122.13 Found (((eq_sym fofType) x3) x') as proof of (((x0 Xx) x')->(((eq fofType) x3) x'))
% 125.96/126.21 Found eq_ref00:=(eq_ref0 x3):(((eq fofType) x3) x3)
% 125.96/126.21 Found (eq_ref0 x3) as proof of (((eq fofType) x3) x')
% 125.96/126.21 Found ((eq_ref fofType) x3) as proof of (((eq fofType) x3) x')
% 125.96/126.21 Found ((eq_ref fofType) x3) as proof of (((eq fofType) x3) x')
% 125.96/126.21 Found (fun (x00:((x0 Xx) x'))=> ((eq_ref fofType) x3)) as proof of (((eq fofType) x3) x')
% 125.96/126.21 Found eq_sym000:=(eq_sym00 x'):((((eq fofType) x5) x')->(((eq fofType) x') x5))
% 125.96/126.21 Found (eq_sym00 x') as proof of (((x0 Xx) x')->(((eq fofType) x5) x'))
% 125.96/126.21 Found ((eq_sym0 x5) x') as proof of (((x0 Xx) x')->(((eq fofType) x5) x'))
% 125.96/126.21 Found (((eq_sym fofType) x5) x') as proof of (((x0 Xx) x')->(((eq fofType) x5) x'))
% 125.96/126.21 Found (((eq_sym fofType) x5) x') as proof of (((x0 Xx) x')->(((eq fofType) x5) x'))
% 125.96/126.21 Found (((eq_sym fofType) x5) x') as proof of (((x0 Xx) x')->(((eq fofType) x5) x'))
% 125.96/126.21 Found eq_ref00:=(eq_ref0 (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x4 Xx) X)) (forall (Y:fofType), (((x4 Xx) Y)->(((eq fofType) X) Y)))))))):(((eq Prop) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x4 Xx) X)) (forall (Y:fofType), (((x4 Xx) Y)->(((eq fofType) X) Y)))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x4 Xx) X)) (forall (Y:fofType), (((x4 Xx) Y)->(((eq fofType) X) Y))))))))
% 125.96/126.21 Found (eq_ref0 (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x4 Xx) X)) (forall (Y:fofType), (((x4 Xx) Y)->(((eq fofType) X) Y)))))))) as proof of (((eq Prop) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x4 Xx) X)) (forall (Y:fofType), (((x4 Xx) Y)->(((eq fofType) X) Y)))))))) b)
% 125.96/126.21 Found ((eq_ref Prop) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x4 Xx) X)) (forall (Y:fofType), (((x4 Xx) Y)->(((eq fofType) X) Y)))))))) as proof of (((eq Prop) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x4 Xx) X)) (forall (Y:fofType), (((x4 Xx) Y)->(((eq fofType) X) Y)))))))) b)
% 125.96/126.21 Found ((eq_ref Prop) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x4 Xx) X)) (forall (Y:fofType), (((x4 Xx) Y)->(((eq fofType) X) Y)))))))) as proof of (((eq Prop) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x4 Xx) X)) (forall (Y:fofType), (((x4 Xx) Y)->(((eq fofType) X) Y)))))))) b)
% 125.96/126.21 Found ((eq_ref Prop) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x4 Xx) X)) (forall (Y:fofType), (((x4 Xx) Y)->(((eq fofType) X) Y)))))))) as proof of (((eq Prop) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x4 Xx) X)) (forall (Y:fofType), (((x4 Xx) Y)->(((eq fofType) X) Y)))))))) b)
% 125.96/126.21 Found eq_sym000:=(eq_sym00 x'):((((eq fofType) x5) x')->(((eq fofType) x') x5))
% 125.96/126.21 Found (eq_sym00 x') as proof of (((x2 Xx) x')->(((eq fofType) x5) x'))
% 125.96/126.21 Found ((eq_sym0 x5) x') as proof of (((x2 Xx) x')->(((eq fofType) x5) x'))
% 125.96/126.21 Found (((eq_sym fofType) x5) x') as proof of (((x2 Xx) x')->(((eq fofType) x5) x'))
% 125.96/126.21 Found (((eq_sym fofType) x5) x') as proof of (((x2 Xx) x')->(((eq fofType) x5) x'))
% 125.96/126.21 Found (((eq_sym fofType) x5) x') as proof of (((x2 Xx) x')->(((eq fofType) x5) x'))
% 125.96/126.21 Found ex_intro0:=(ex_intro (fofType->(fofType->Prop))):(forall (P:((fofType->(fofType->Prop))->Prop)) (x:(fofType->(fofType->Prop))), ((P x)->((ex (fofType->(fofType->Prop))) P)))
% 125.96/126.21 Instantiate: b:=(forall (P:((fofType->(fofType->Prop))->Prop)) (x:(fofType->(fofType->Prop))), ((P x)->((ex (fofType->(fofType->Prop))) P))):Prop
% 125.96/126.21 Found ex_intro0 as proof of b
% 125.96/126.21 Found eq_ref00:=(eq_ref0 x5):(((eq fofType) x5) x5)
% 125.96/126.21 Found (eq_ref0 x5) as proof of (((eq fofType) x5) x')
% 125.96/126.21 Found ((eq_ref fofType) x5) as proof of (((eq fofType) x5) x')
% 125.96/126.21 Found ((eq_ref fofType) x5) as proof of (((eq fofType) x5) x')
% 125.96/126.21 Found (fun (x00:((x0 Xx) x'))=> ((eq_ref fofType) x5)) as proof of (((eq fofType) x5) x')
% 125.96/126.21 Found eq_ref00:=(eq_ref0 x5):(((eq fofType) x5) x5)
% 125.96/126.21 Found (eq_ref0 x5) as proof of (((eq fofType) x5) x')
% 125.96/126.21 Found ((eq_ref fofType) x5) as proof of (((eq fofType) x5) x')
% 125.96/126.21 Found ((eq_ref fofType) x5) as proof of (((eq fofType) x5) x')
% 125.96/126.21 Found (fun (x00:((x2 Xx) x'))=> ((eq_ref fofType) x5)) as proof of (((eq fofType) x5) x')
% 125.96/126.21 Found eq_ref000:=(eq_ref00 P):((P x3)->(P x3))
% 128.75/128.91 Found (eq_ref00 P) as proof of ((P x3)->(P Y))
% 128.75/128.91 Found ((eq_ref0 x3) P) as proof of ((P x3)->(P Y))
% 128.75/128.91 Found (((eq_ref fofType) x3) P) as proof of ((P x3)->(P Y))
% 128.75/128.91 Found (((eq_ref fofType) x3) P) as proof of ((P x3)->(P Y))
% 128.75/128.91 Found (fun (P:(fofType->Prop))=> (((eq_ref fofType) x3) P)) as proof of ((P x3)->(P Y))
% 128.75/128.91 Found (fun (x00:((x0 Xx) Y)) (P:(fofType->Prop))=> (((eq_ref fofType) x3) P)) as proof of (((eq fofType) x3) Y)
% 128.75/128.91 Found x4:(P x3)
% 128.75/128.91 Instantiate: x3:=Y:fofType
% 128.75/128.91 Found (fun (x4:(P x3))=> x4) as proof of (P Y)
% 128.75/128.91 Found (fun (P:(fofType->Prop)) (x4:(P x3))=> x4) as proof of ((P x3)->(P Y))
% 128.75/128.91 Found (fun (x00:((x0 Xx) Y)) (P:(fofType->Prop)) (x4:(P x3))=> x4) as proof of (((eq fofType) x3) Y)
% 128.75/128.91 Found eq_ref00:=(eq_ref0 x3):(((eq fofType) x3) x3)
% 128.75/128.91 Found (eq_ref0 x3) as proof of (((eq fofType) x3) Y)
% 128.75/128.91 Found ((eq_ref fofType) x3) as proof of (((eq fofType) x3) Y)
% 128.75/128.91 Found ((eq_ref fofType) x3) as proof of (((eq fofType) x3) Y)
% 128.75/128.91 Found ((eq_ref fofType) x3) as proof of (((eq fofType) x3) Y)
% 128.75/128.91 Found (fun (x00:((x0 Xx) Y))=> ((eq_ref fofType) x3)) as proof of (((eq fofType) x3) Y)
% 128.75/128.91 Found x6:(P x5)
% 128.75/128.91 Instantiate: x5:=Y:fofType
% 128.75/128.91 Found (fun (x6:(P x5))=> x6) as proof of (P Y)
% 128.75/128.91 Found (fun (P:(fofType->Prop)) (x6:(P x5))=> x6) as proof of ((P x5)->(P Y))
% 128.75/128.91 Found (fun (x00:((x0 Xx) Y)) (P:(fofType->Prop)) (x6:(P x5))=> x6) as proof of (((eq fofType) x5) Y)
% 128.75/128.91 Found eq_ref000:=(eq_ref00 P):((P x5)->(P x5))
% 128.75/128.91 Found (eq_ref00 P) as proof of ((P x5)->(P Y))
% 128.75/128.91 Found ((eq_ref0 x5) P) as proof of ((P x5)->(P Y))
% 128.75/128.91 Found (((eq_ref fofType) x5) P) as proof of ((P x5)->(P Y))
% 128.75/128.91 Found (((eq_ref fofType) x5) P) as proof of ((P x5)->(P Y))
% 128.75/128.91 Found (fun (P:(fofType->Prop))=> (((eq_ref fofType) x5) P)) as proof of ((P x5)->(P Y))
% 128.75/128.91 Found (fun (x00:((x0 Xx) Y)) (P:(fofType->Prop))=> (((eq_ref fofType) x5) P)) as proof of (((eq fofType) x5) Y)
% 128.75/128.91 Found x6:(P x5)
% 128.75/128.91 Instantiate: x5:=Y:fofType
% 128.75/128.91 Found (fun (x6:(P x5))=> x6) as proof of (P Y)
% 128.75/128.91 Found (fun (P:(fofType->Prop)) (x6:(P x5))=> x6) as proof of ((P x5)->(P Y))
% 128.75/128.91 Found (fun (x00:((x2 Xx) Y)) (P:(fofType->Prop)) (x6:(P x5))=> x6) as proof of (((eq fofType) x5) Y)
% 128.75/128.91 Found eq_ref000:=(eq_ref00 P):((P x5)->(P x5))
% 128.75/128.91 Found (eq_ref00 P) as proof of ((P x5)->(P Y))
% 128.75/128.91 Found ((eq_ref0 x5) P) as proof of ((P x5)->(P Y))
% 128.75/128.91 Found (((eq_ref fofType) x5) P) as proof of ((P x5)->(P Y))
% 128.75/128.91 Found (((eq_ref fofType) x5) P) as proof of ((P x5)->(P Y))
% 128.75/128.91 Found (fun (P:(fofType->Prop))=> (((eq_ref fofType) x5) P)) as proof of ((P x5)->(P Y))
% 128.75/128.91 Found (fun (x00:((x2 Xx) Y)) (P:(fofType->Prop))=> (((eq_ref fofType) x5) P)) as proof of (((eq fofType) x5) Y)
% 128.75/128.91 Found eq_ref00:=(eq_ref0 x5):(((eq fofType) x5) x5)
% 128.75/128.91 Found (eq_ref0 x5) as proof of (((eq fofType) x5) Y)
% 128.75/128.91 Found ((eq_ref fofType) x5) as proof of (((eq fofType) x5) Y)
% 128.75/128.91 Found ((eq_ref fofType) x5) as proof of (((eq fofType) x5) Y)
% 128.75/128.91 Found ((eq_ref fofType) x5) as proof of (((eq fofType) x5) Y)
% 128.75/128.91 Found (fun (x00:((x0 Xx) Y))=> ((eq_ref fofType) x5)) as proof of (((eq fofType) x5) Y)
% 128.75/128.91 Found eq_ref00:=(eq_ref0 x5):(((eq fofType) x5) x5)
% 128.75/128.91 Found (eq_ref0 x5) as proof of (((eq fofType) x5) Y)
% 128.75/128.91 Found ((eq_ref fofType) x5) as proof of (((eq fofType) x5) Y)
% 128.75/128.91 Found ((eq_ref fofType) x5) as proof of (((eq fofType) x5) Y)
% 128.75/128.91 Found ((eq_ref fofType) x5) as proof of (((eq fofType) x5) Y)
% 128.75/128.91 Found (fun (x00:((x2 Xx) Y))=> ((eq_ref fofType) x5)) as proof of (((eq fofType) x5) Y)
% 128.75/128.91 Found eq_substitution0000:=(fun (a:fofType)=> ((eq_substitution000 a) (fun (x8:fofType)=> (cS Xx)))):(forall (a:fofType), ((((eq fofType) a) (cS Xy))->(((eq fofType) (cS Xx)) (cS Xx))))
% 128.75/128.91 Instantiate: x0:=(fun (x6:fofType) (x50:fofType)=> (forall (a:fofType), ((((eq fofType) a) (cS Xy))->(((eq fofType) x6) x6)))):(fofType->(fofType->Prop))
% 128.75/128.91 Found (fun (a:fofType)=> ((eq_substitution000 a) (fun (x8:fofType)=> (cS Xx)))) as proof of ((x0 (cS Xx)) (cS (cS Xy)))
% 128.75/128.91 Found (fun (a:fofType)=> (((fun (a:fofType)=> ((eq_substitution00 a) (cS Xy))) a) (fun (x8:fofType)=> (cS Xx)))) as proof of ((x0 (cS Xx)) (cS (cS Xy)))
% 128.75/128.91 Found (fun (a:fofType)=> (((fun (a:fofType)=> (((eq_substitution0 fofType) a) (cS Xy))) a) (fun (x8:fofType)=> (cS Xx)))) as proof of ((x0 (cS Xx)) (cS (cS Xy)))
% 128.75/128.91 Found (fun (a:fofType)=> (((fun (a:fofType)=> ((((eq_substitution fofType) fofType) a) (cS Xy))) a) (fun (x8:fofType)=> (cS Xx)))) as proof of ((x0 (cS Xx)) (cS (cS Xy)))
% 128.97/129.20 Found (fun (x4:(forall (Xx0:fofType) (Xy0:fofType), ((((eq fofType) (cS Xx0)) (cS Xy0))->(((eq fofType) Xx0) Xy0)))) (a:fofType)=> (((fun (a:fofType)=> ((((eq_substitution fofType) fofType) a) (cS Xy))) a) (fun (x8:fofType)=> (cS Xx)))) as proof of ((x0 (cS Xx)) (cS (cS Xy)))
% 128.97/129.20 Found (fun (x3:cIND) (x4:(forall (Xx0:fofType) (Xy0:fofType), ((((eq fofType) (cS Xx0)) (cS Xy0))->(((eq fofType) Xx0) Xy0)))) (a:fofType)=> (((fun (a:fofType)=> ((((eq_substitution fofType) fofType) a) (cS Xy))) a) (fun (x8:fofType)=> (cS Xx)))) as proof of ((forall (Xx0:fofType) (Xy0:fofType), ((((eq fofType) (cS Xx0)) (cS Xy0))->(((eq fofType) Xx0) Xy0)))->((x0 (cS Xx)) (cS (cS Xy))))
% 128.97/129.20 Found (fun (x3:cIND) (x4:(forall (Xx0:fofType) (Xy0:fofType), ((((eq fofType) (cS Xx0)) (cS Xy0))->(((eq fofType) Xx0) Xy0)))) (a:fofType)=> (((fun (a:fofType)=> ((((eq_substitution fofType) fofType) a) (cS Xy))) a) (fun (x8:fofType)=> (cS Xx)))) as proof of (cIND->((forall (Xx0:fofType) (Xy0:fofType), ((((eq fofType) (cS Xx0)) (cS Xy0))->(((eq fofType) Xx0) Xy0)))->((x0 (cS Xx)) (cS (cS Xy)))))
% 128.97/129.20 Found (and_rect10 (fun (x3:cIND) (x4:(forall (Xx0:fofType) (Xy0:fofType), ((((eq fofType) (cS Xx0)) (cS Xy0))->(((eq fofType) Xx0) Xy0)))) (a:fofType)=> (((fun (a:fofType)=> ((((eq_substitution fofType) fofType) a) (cS Xy))) a) (fun (x8:fofType)=> (cS Xx))))) as proof of ((x0 (cS Xx)) (cS (cS Xy)))
% 128.97/129.20 Found ((and_rect1 ((x0 (cS Xx)) (cS (cS Xy)))) (fun (x3:cIND) (x4:(forall (Xx0:fofType) (Xy0:fofType), ((((eq fofType) (cS Xx0)) (cS Xy0))->(((eq fofType) Xx0) Xy0)))) (a:fofType)=> (((fun (a:fofType)=> ((((eq_substitution fofType) fofType) a) (cS Xy))) a) (fun (x8:fofType)=> (cS Xx))))) as proof of ((x0 (cS Xx)) (cS (cS Xy)))
% 128.97/129.20 Found (((fun (P:Type) (x3:(cIND->((forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (cS Xx)) (cS Xy))->(((eq fofType) Xx) Xy)))->P)))=> (((((and_rect cIND) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (cS Xx)) (cS Xy))->(((eq fofType) Xx) Xy)))) P) x3) x1)) ((x0 (cS Xx)) (cS (cS Xy)))) (fun (x3:cIND) (x4:(forall (Xx0:fofType) (Xy0:fofType), ((((eq fofType) (cS Xx0)) (cS Xy0))->(((eq fofType) Xx0) Xy0)))) (a:fofType)=> (((fun (a:fofType)=> ((((eq_substitution fofType) fofType) a) (cS Xy))) a) (fun (x8:fofType)=> (cS Xx))))) as proof of ((x0 (cS Xx)) (cS (cS Xy)))
% 128.97/129.20 Found (fun (x00:((x0 Xx) Xy))=> (((fun (P:Type) (x3:(cIND->((forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (cS Xx)) (cS Xy))->(((eq fofType) Xx) Xy)))->P)))=> (((((and_rect cIND) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (cS Xx)) (cS Xy))->(((eq fofType) Xx) Xy)))) P) x3) x1)) ((x0 (cS Xx)) (cS (cS Xy)))) (fun (x3:cIND) (x4:(forall (Xx0:fofType) (Xy0:fofType), ((((eq fofType) (cS Xx0)) (cS Xy0))->(((eq fofType) Xx0) Xy0)))) (a:fofType)=> (((fun (a:fofType)=> ((((eq_substitution fofType) fofType) a) (cS Xy))) a) (fun (x8:fofType)=> (cS Xx)))))) as proof of ((x0 (cS Xx)) (cS (cS Xy)))
% 128.97/129.20 Found eq_sym000:=(eq_sym00 x'):((((eq fofType) x1) x')->(((eq fofType) x') x1))
% 128.97/129.20 Found (eq_sym00 x') as proof of (((x0 Xx) x')->(((eq fofType) x1) x'))
% 128.97/129.20 Found ((eq_sym0 x1) x') as proof of (((x0 Xx) x')->(((eq fofType) x1) x'))
% 128.97/129.20 Found (((eq_sym fofType) x1) x') as proof of (((x0 Xx) x')->(((eq fofType) x1) x'))
% 128.97/129.20 Found (((eq_sym fofType) x1) x') as proof of (((x0 Xx) x')->(((eq fofType) x1) x'))
% 128.97/129.20 Found (((eq_sym fofType) x1) x') as proof of (((x0 Xx) x')->(((eq fofType) x1) x'))
% 128.97/129.20 Found eq_substitution0000:=(fun (a:fofType)=> ((eq_substitution000 a) (fun (x8:fofType)=> (cS Xx)))):(forall (a:fofType), ((((eq fofType) a) (cS Xy))->(((eq fofType) (cS Xx)) (cS Xx))))
% 128.97/129.20 Instantiate: x2:=(fun (x6:fofType) (x50:fofType)=> (forall (a:fofType), ((((eq fofType) a) (cS Xy))->(((eq fofType) x6) x6)))):(fofType->(fofType->Prop))
% 128.97/129.20 Found (fun (a:fofType)=> ((eq_substitution000 a) (fun (x8:fofType)=> (cS Xx)))) as proof of ((x2 (cS Xx)) (cS (cS Xy)))
% 128.97/129.20 Found (fun (a:fofType)=> (((fun (a:fofType)=> ((eq_substitution00 a) (cS Xy))) a) (fun (x8:fofType)=> (cS Xx)))) as proof of ((x2 (cS Xx)) (cS (cS Xy)))
% 130.53/130.74 Found (fun (a:fofType)=> (((fun (a:fofType)=> (((eq_substitution0 fofType) a) (cS Xy))) a) (fun (x8:fofType)=> (cS Xx)))) as proof of ((x2 (cS Xx)) (cS (cS Xy)))
% 130.53/130.74 Found (fun (a:fofType)=> (((fun (a:fofType)=> ((((eq_substitution fofType) fofType) a) (cS Xy))) a) (fun (x8:fofType)=> (cS Xx)))) as proof of ((x2 (cS Xx)) (cS (cS Xy)))
% 130.53/130.74 Found (fun (x4:(forall (Xx0:fofType) (Xy0:fofType), ((((eq fofType) (cS Xx0)) (cS Xy0))->(((eq fofType) Xx0) Xy0)))) (a:fofType)=> (((fun (a:fofType)=> ((((eq_substitution fofType) fofType) a) (cS Xy))) a) (fun (x8:fofType)=> (cS Xx)))) as proof of ((x2 (cS Xx)) (cS (cS Xy)))
% 130.53/130.74 Found (fun (x3:cIND) (x4:(forall (Xx0:fofType) (Xy0:fofType), ((((eq fofType) (cS Xx0)) (cS Xy0))->(((eq fofType) Xx0) Xy0)))) (a:fofType)=> (((fun (a:fofType)=> ((((eq_substitution fofType) fofType) a) (cS Xy))) a) (fun (x8:fofType)=> (cS Xx)))) as proof of ((forall (Xx0:fofType) (Xy0:fofType), ((((eq fofType) (cS Xx0)) (cS Xy0))->(((eq fofType) Xx0) Xy0)))->((x2 (cS Xx)) (cS (cS Xy))))
% 130.53/130.74 Found (fun (x3:cIND) (x4:(forall (Xx0:fofType) (Xy0:fofType), ((((eq fofType) (cS Xx0)) (cS Xy0))->(((eq fofType) Xx0) Xy0)))) (a:fofType)=> (((fun (a:fofType)=> ((((eq_substitution fofType) fofType) a) (cS Xy))) a) (fun (x8:fofType)=> (cS Xx)))) as proof of (cIND->((forall (Xx0:fofType) (Xy0:fofType), ((((eq fofType) (cS Xx0)) (cS Xy0))->(((eq fofType) Xx0) Xy0)))->((x2 (cS Xx)) (cS (cS Xy)))))
% 130.53/130.74 Found (and_rect10 (fun (x3:cIND) (x4:(forall (Xx0:fofType) (Xy0:fofType), ((((eq fofType) (cS Xx0)) (cS Xy0))->(((eq fofType) Xx0) Xy0)))) (a:fofType)=> (((fun (a:fofType)=> ((((eq_substitution fofType) fofType) a) (cS Xy))) a) (fun (x8:fofType)=> (cS Xx))))) as proof of ((x2 (cS Xx)) (cS (cS Xy)))
% 130.53/130.74 Found ((and_rect1 ((x2 (cS Xx)) (cS (cS Xy)))) (fun (x3:cIND) (x4:(forall (Xx0:fofType) (Xy0:fofType), ((((eq fofType) (cS Xx0)) (cS Xy0))->(((eq fofType) Xx0) Xy0)))) (a:fofType)=> (((fun (a:fofType)=> ((((eq_substitution fofType) fofType) a) (cS Xy))) a) (fun (x8:fofType)=> (cS Xx))))) as proof of ((x2 (cS Xx)) (cS (cS Xy)))
% 130.53/130.74 Found (((fun (P:Type) (x3:(cIND->((forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (cS Xx)) (cS Xy))->(((eq fofType) Xx) Xy)))->P)))=> (((((and_rect cIND) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (cS Xx)) (cS Xy))->(((eq fofType) Xx) Xy)))) P) x3) x0)) ((x2 (cS Xx)) (cS (cS Xy)))) (fun (x3:cIND) (x4:(forall (Xx0:fofType) (Xy0:fofType), ((((eq fofType) (cS Xx0)) (cS Xy0))->(((eq fofType) Xx0) Xy0)))) (a:fofType)=> (((fun (a:fofType)=> ((((eq_substitution fofType) fofType) a) (cS Xy))) a) (fun (x8:fofType)=> (cS Xx))))) as proof of ((x2 (cS Xx)) (cS (cS Xy)))
% 130.53/130.74 Found (fun (x00:((x2 Xx) Xy))=> (((fun (P:Type) (x3:(cIND->((forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (cS Xx)) (cS Xy))->(((eq fofType) Xx) Xy)))->P)))=> (((((and_rect cIND) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (cS Xx)) (cS Xy))->(((eq fofType) Xx) Xy)))) P) x3) x0)) ((x2 (cS Xx)) (cS (cS Xy)))) (fun (x3:cIND) (x4:(forall (Xx0:fofType) (Xy0:fofType), ((((eq fofType) (cS Xx0)) (cS Xy0))->(((eq fofType) Xx0) Xy0)))) (a:fofType)=> (((fun (a:fofType)=> ((((eq_substitution fofType) fofType) a) (cS Xy))) a) (fun (x8:fofType)=> (cS Xx)))))) as proof of ((x2 (cS Xx)) (cS (cS Xy)))
% 130.53/130.74 Found eq_ref00:=(eq_ref0 x1):(((eq fofType) x1) x1)
% 130.53/130.74 Found (eq_ref0 x1) as proof of (((eq fofType) x1) x')
% 130.53/130.74 Found ((eq_ref fofType) x1) as proof of (((eq fofType) x1) x')
% 130.53/130.74 Found ((eq_ref fofType) x1) as proof of (((eq fofType) x1) x')
% 130.53/130.74 Found (fun (x00:((x0 Xx) x'))=> ((eq_ref fofType) x1)) as proof of (((eq fofType) x1) x')
% 130.53/130.74 Found eq_ref00:=(eq_ref0 (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y)))))))):(((eq Prop) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y)))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y))))))))
% 130.53/130.74 Found (eq_ref0 (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y)))))))) as proof of (((eq Prop) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y)))))))) b)
% 132.05/132.26 Found ((eq_ref Prop) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y)))))))) as proof of (((eq Prop) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y)))))))) b)
% 132.05/132.26 Found ((eq_ref Prop) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y)))))))) as proof of (((eq Prop) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y)))))))) b)
% 132.05/132.26 Found ((eq_ref Prop) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y)))))))) as proof of (((eq Prop) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y)))))))) b)
% 132.05/132.26 Found eq_ref00:=(eq_ref0 (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x2 Xx) X)) (forall (Y:fofType), (((x2 Xx) Y)->(((eq fofType) X) Y)))))))):(((eq Prop) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x2 Xx) X)) (forall (Y:fofType), (((x2 Xx) Y)->(((eq fofType) X) Y)))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x2 Xx) X)) (forall (Y:fofType), (((x2 Xx) Y)->(((eq fofType) X) Y))))))))
% 132.05/132.26 Found (eq_ref0 (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x2 Xx) X)) (forall (Y:fofType), (((x2 Xx) Y)->(((eq fofType) X) Y)))))))) as proof of (((eq Prop) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x2 Xx) X)) (forall (Y:fofType), (((x2 Xx) Y)->(((eq fofType) X) Y)))))))) b)
% 132.05/132.26 Found ((eq_ref Prop) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x2 Xx) X)) (forall (Y:fofType), (((x2 Xx) Y)->(((eq fofType) X) Y)))))))) as proof of (((eq Prop) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x2 Xx) X)) (forall (Y:fofType), (((x2 Xx) Y)->(((eq fofType) X) Y)))))))) b)
% 132.05/132.26 Found ((eq_ref Prop) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x2 Xx) X)) (forall (Y:fofType), (((x2 Xx) Y)->(((eq fofType) X) Y)))))))) as proof of (((eq Prop) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x2 Xx) X)) (forall (Y:fofType), (((x2 Xx) Y)->(((eq fofType) X) Y)))))))) b)
% 132.05/132.26 Found ((eq_ref Prop) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x2 Xx) X)) (forall (Y:fofType), (((x2 Xx) Y)->(((eq fofType) X) Y)))))))) as proof of (((eq Prop) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x2 Xx) X)) (forall (Y:fofType), (((x2 Xx) Y)->(((eq fofType) X) Y)))))))) b)
% 132.05/132.26 Found ex_intro0:=(ex_intro (fofType->(fofType->Prop))):(forall (P:((fofType->(fofType->Prop))->Prop)) (x:(fofType->(fofType->Prop))), ((P x)->((ex (fofType->(fofType->Prop))) P)))
% 132.05/132.26 Instantiate: b:=(forall (P:((fofType->(fofType->Prop))->Prop)) (x:(fofType->(fofType->Prop))), ((P x)->((ex (fofType->(fofType->Prop))) P))):Prop
% 132.05/132.26 Found ex_intro0 as proof of b
% 132.05/132.26 Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% 132.05/132.26 Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) b)
% 132.05/132.26 Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% 132.05/132.26 Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% 132.05/132.26 Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% 132.05/132.26 Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% 132.05/132.26 Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y)))))))))
% 132.05/132.26 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y)))))))))
% 132.12/132.29 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y)))))))))
% 132.12/132.29 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y)))))))))
% 132.12/132.29 Found ((eq_trans0000 ((eq_ref Prop) (f x0))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f x0)) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y)))))))))
% 132.12/132.29 Found (((eq_trans000 ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y))))))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f x0)) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y)))))))))
% 132.12/132.29 Found ((((eq_trans00 ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y))))))))) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y))))))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y)))))))))) as proof of (((eq Prop) (f x0)) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y)))))))))
% 132.12/132.29 Found (((((eq_trans0 (f x0)) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y))))))))) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y))))))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y)))))))))) as proof of (((eq Prop) (f x0)) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y)))))))))
% 132.12/132.29 Found ((((((eq_trans Prop) (f x0)) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y))))))))) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y))))))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y)))))))))) as proof of (((eq Prop) (f x0)) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y)))))))))
% 132.26/132.49 Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% 132.26/132.49 Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) b)
% 132.26/132.49 Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% 132.26/132.49 Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% 132.26/132.49 Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% 132.26/132.49 Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% 132.26/132.49 Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y)))))))))
% 132.26/132.49 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y)))))))))
% 132.26/132.49 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y)))))))))
% 132.26/132.49 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y)))))))))
% 132.26/132.49 Found ((eq_trans0000 ((eq_ref Prop) (f x0))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f x0)) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y)))))))))
% 132.26/132.49 Found (((eq_trans000 ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y))))))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f x0)) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y)))))))))
% 132.26/132.49 Found ((((eq_trans00 ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y))))))))) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y))))))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y)))))))))) as proof of (((eq Prop) (f x0)) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y)))))))))
% 134.05/134.23 Found (((((eq_trans0 (f x0)) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y))))))))) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y))))))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y)))))))))) as proof of (((eq Prop) (f x0)) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y)))))))))
% 134.05/134.23 Found ((((((eq_trans Prop) (f x0)) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y))))))))) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y))))))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y)))))))))) as proof of (((eq Prop) (f x0)) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y)))))))))
% 134.05/134.23 Found eq_ref00:=(eq_ref0 x3):(((eq fofType) x3) x3)
% 134.05/134.23 Found (eq_ref0 x3) as proof of (((eq fofType) x3) x30)
% 134.05/134.23 Found ((eq_ref fofType) x3) as proof of (((eq fofType) x3) x30)
% 134.05/134.23 Found ((eq_ref fofType) x3) as proof of (((eq fofType) x3) x30)
% 134.05/134.23 Found ((eq_ref fofType) x3) as proof of (((eq fofType) x3) x30)
% 134.05/134.23 Found (eq_sym000 ((eq_ref fofType) x3)) as proof of (((eq fofType) x30) x3)
% 134.05/134.23 Found ((eq_sym00 x30) ((eq_ref fofType) x3)) as proof of (((eq fofType) x30) x3)
% 134.05/134.23 Found (((eq_sym0 x3) x30) ((eq_ref fofType) x3)) as proof of (((eq fofType) x30) x3)
% 134.05/134.23 Found ((((eq_sym fofType) x3) x30) ((eq_ref fofType) x3)) as proof of (((eq fofType) x30) x3)
% 134.05/134.23 Found (fun (x20:(forall (Xn:fofType), (not (((eq fofType) (cS Xn)) c0))))=> ((((eq_sym fofType) x3) x30) ((eq_ref fofType) x3))) as proof of (((eq fofType) x30) x3)
% 134.05/134.23 Found eq_ref00:=(eq_ref0 x30):(((eq fofType) x30) x30)
% 134.05/134.23 Found (eq_ref0 x30) as proof of (((eq fofType) x30) x3)
% 134.05/134.23 Found ((eq_ref fofType) x30) as proof of (((eq fofType) x30) x3)
% 134.05/134.23 Found ((eq_ref fofType) x30) as proof of (((eq fofType) x30) x3)
% 134.05/134.23 Found ex_intro0:=(ex_intro (fofType->(fofType->Prop))):(forall (P:((fofType->(fofType->Prop))->Prop)) (x:(fofType->(fofType->Prop))), ((P x)->((ex (fofType->(fofType->Prop))) P)))
% 134.05/134.23 Instantiate: b:=(forall (P:((fofType->(fofType->Prop))->Prop)) (x:(fofType->(fofType->Prop))), ((P x)->((ex (fofType->(fofType->Prop))) P))):Prop
% 134.05/134.23 Found ex_intro0 as proof of b
% 134.05/134.23 Found eq_ref00:=(eq_ref0 x30):(((eq fofType) x30) x30)
% 134.05/134.23 Found (eq_ref0 x30) as proof of (((eq fofType) x30) x3)
% 134.05/134.23 Found ((eq_ref fofType) x30) as proof of (((eq fofType) x30) x3)
% 134.05/134.23 Found ((eq_ref fofType) x30) as proof of (((eq fofType) x30) x3)
% 134.05/134.23 Found (fun (x5:(forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy))))=> ((eq_ref fofType) x30)) as proof of (((eq fofType) x30) x3)
% 135.12/135.33 Found (fun (x4:cIND) (x5:(forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy))))=> ((eq_ref fofType) x30)) as proof of ((forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy)))->(((eq fofType) x30) x3))
% 135.12/135.33 Found (fun (x4:cIND) (x5:(forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy))))=> ((eq_ref fofType) x30)) as proof of (cIND->((forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy)))->(((eq fofType) x30) x3)))
% 135.12/135.33 Found (and_rect10 (fun (x4:cIND) (x5:(forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy))))=> ((eq_ref fofType) x30))) as proof of (((eq fofType) x30) x3)
% 135.12/135.33 Found ((and_rect1 (((eq fofType) x30) x3)) (fun (x4:cIND) (x5:(forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy))))=> ((eq_ref fofType) x30))) as proof of (((eq fofType) x30) x3)
% 135.12/135.33 Found (((fun (P:Type) (x4:(cIND->((forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy)))->P)))=> (((((and_rect cIND) (forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy)))) P) x4) x10)) (((eq fofType) x30) x3)) (fun (x4:cIND) (x5:(forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy))))=> ((eq_ref fofType) x30))) as proof of (((eq fofType) x30) x3)
% 135.12/135.33 Found (fun (x20:(forall (Xn:fofType), (not (((eq fofType) (cS Xn)) c0))))=> (((fun (P:Type) (x4:(cIND->((forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy)))->P)))=> (((((and_rect cIND) (forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy)))) P) x4) x10)) (((eq fofType) x30) x3)) (fun (x4:cIND) (x5:(forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy))))=> ((eq_ref fofType) x30)))) as proof of (((eq fofType) x30) x3)
% 135.12/135.33 Found eq_ref000:=(eq_ref00 P):((P x1)->(P x1))
% 135.12/135.33 Found (eq_ref00 P) as proof of ((P x1)->(P Y))
% 135.12/135.33 Found ((eq_ref0 x1) P) as proof of ((P x1)->(P Y))
% 135.12/135.33 Found (((eq_ref fofType) x1) P) as proof of ((P x1)->(P Y))
% 135.12/135.33 Found (((eq_ref fofType) x1) P) as proof of ((P x1)->(P Y))
% 135.12/135.33 Found (fun (P:(fofType->Prop))=> (((eq_ref fofType) x1) P)) as proof of ((P x1)->(P Y))
% 135.12/135.33 Found (fun (x00:((x0 Xx) Y)) (P:(fofType->Prop))=> (((eq_ref fofType) x1) P)) as proof of (((eq fofType) x1) Y)
% 135.12/135.33 Found x4:(P x1)
% 135.12/135.33 Instantiate: x1:=Y:fofType
% 135.12/135.33 Found (fun (x4:(P x1))=> x4) as proof of (P Y)
% 135.12/135.33 Found (fun (P:(fofType->Prop)) (x4:(P x1))=> x4) as proof of ((P x1)->(P Y))
% 135.12/135.33 Found (fun (x00:((x0 Xx) Y)) (P:(fofType->Prop)) (x4:(P x1))=> x4) as proof of (((eq fofType) x1) Y)
% 135.12/135.33 Found eq_ref000:=(eq_ref00 P0):((P0 (f x0))->(P0 (f x0)))
% 135.12/135.33 Found (eq_ref00 P0) as proof of (P1 (f x0))
% 135.12/135.33 Found ((eq_ref0 (f x0)) P0) as proof of (P1 (f x0))
% 135.12/135.33 Found (((eq_ref Prop) (f x0)) P0) as proof of (P1 (f x0))
% 135.12/135.33 Found (((eq_ref Prop) (f x0)) P0) as proof of (P1 (f x0))
% 135.12/135.33 Found eq_ref000:=(eq_ref00 P0):((P0 (f x0))->(P0 (f x0)))
% 135.12/135.33 Found (eq_ref00 P0) as proof of (P1 (f x0))
% 135.12/135.33 Found ((eq_ref0 (f x0)) P0) as proof of (P1 (f x0))
% 135.12/135.33 Found (((eq_ref Prop) (f x0)) P0) as proof of (P1 (f x0))
% 135.12/135.33 Found (((eq_ref Prop) (f x0)) P0) as proof of (P1 (f x0))
% 135.12/135.33 Found eq_ref00:=(eq_ref0 x30):(((eq fofType) x30) x30)
% 135.12/135.33 Found (eq_ref0 x30) as proof of (((eq fofType) x30) x3)
% 135.12/135.33 Found ((eq_ref fofType) x30) as proof of (((eq fofType) x30) x3)
% 135.12/135.33 Found ((eq_ref fofType) x30) as proof of (((eq fofType) x30) x3)
% 135.12/135.33 Found (fun (x5:(forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (cS Xx)) (cS Xy))->(((eq fofType) Xx) Xy))))=> ((eq_ref fofType) x30)) as proof of (((eq fofType) x30) x3)
% 135.12/135.33 Found (fun (x4:cIND) (x5:(forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (cS Xx)) (cS Xy))->(((eq fofType) Xx) Xy))))=> ((eq_ref fofType) x30)) as proof of ((forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (cS Xx)) (cS Xy))->(((eq fofType) Xx) Xy)))->(((eq fofType) x30) x3))
% 139.42/139.61 Found (fun (x4:cIND) (x5:(forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (cS Xx)) (cS Xy))->(((eq fofType) Xx) Xy))))=> ((eq_ref fofType) x30)) as proof of (cIND->((forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (cS Xx)) (cS Xy))->(((eq fofType) Xx) Xy)))->(((eq fofType) x30) x3)))
% 139.42/139.61 Found (and_rect10 (fun (x4:cIND) (x5:(forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (cS Xx)) (cS Xy))->(((eq fofType) Xx) Xy))))=> ((eq_ref fofType) x30))) as proof of (((eq fofType) x30) x3)
% 139.42/139.61 Found ((and_rect1 (((eq fofType) x30) x3)) (fun (x4:cIND) (x5:(forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (cS Xx)) (cS Xy))->(((eq fofType) Xx) Xy))))=> ((eq_ref fofType) x30))) as proof of (((eq fofType) x30) x3)
% 139.42/139.61 Found (((fun (P:Type) (x4:(cIND->((forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (cS Xx)) (cS Xy))->(((eq fofType) Xx) Xy)))->P)))=> (((((and_rect cIND) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (cS Xx)) (cS Xy))->(((eq fofType) Xx) Xy)))) P) x4) x0)) (((eq fofType) x30) x3)) (fun (x4:cIND) (x5:(forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (cS Xx)) (cS Xy))->(((eq fofType) Xx) Xy))))=> ((eq_ref fofType) x30))) as proof of (((eq fofType) x30) x3)
% 139.42/139.61 Found eq_ref00:=(eq_ref0 (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x4 Xx))))):(((eq Prop) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x4 Xx))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x4 Xx)))))
% 139.42/139.61 Found (eq_ref0 (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x4 Xx))))) as proof of (((eq Prop) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x4 Xx))))) b)
% 139.42/139.61 Found ((eq_ref Prop) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x4 Xx))))) as proof of (((eq Prop) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x4 Xx))))) b)
% 139.42/139.61 Found ((eq_ref Prop) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x4 Xx))))) as proof of (((eq Prop) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x4 Xx))))) b)
% 139.42/139.61 Found ((eq_ref Prop) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x4 Xx))))) as proof of (((eq Prop) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x4 Xx))))) b)
% 139.42/139.61 Found eq_ref00:=(eq_ref0 x1):(((eq fofType) x1) x1)
% 139.42/139.61 Found (eq_ref0 x1) as proof of (((eq fofType) x1) Y)
% 139.42/139.61 Found ((eq_ref fofType) x1) as proof of (((eq fofType) x1) Y)
% 139.42/139.61 Found ((eq_ref fofType) x1) as proof of (((eq fofType) x1) Y)
% 139.42/139.61 Found ((eq_ref fofType) x1) as proof of (((eq fofType) x1) Y)
% 139.42/139.61 Found (fun (x00:((x0 Xx) Y))=> ((eq_ref fofType) x1)) as proof of (((eq fofType) x1) Y)
% 139.42/139.61 Found eq_ref00:=(eq_ref0 (cS x50)):(((eq fofType) (cS x50)) (cS x50))
% 139.42/139.61 Found (eq_ref0 (cS x50)) as proof of (((eq fofType) (cS x50)) (cS x5))
% 139.42/139.61 Found ((eq_ref fofType) (cS x50)) as proof of (((eq fofType) (cS x50)) (cS x5))
% 139.42/139.61 Found ((eq_ref fofType) (cS x50)) as proof of (((eq fofType) (cS x50)) (cS x5))
% 139.42/139.61 Found ((eq_ref fofType) (cS x50)) as proof of (((eq fofType) (cS x50)) (cS x5))
% 139.42/139.61 Found (x300 ((eq_ref fofType) (cS x50))) as proof of (((eq fofType) x50) x5)
% 139.42/139.61 Found ((x30 x5) ((eq_ref fofType) (cS x50))) as proof of (((eq fofType) x50) x5)
% 139.42/139.61 Found (((x3 x50) x5) ((eq_ref fofType) (cS x50))) as proof of (((eq fofType) x50) x5)
% 139.42/139.61 Found x4:(P x3)
% 139.42/139.61 Instantiate: x3:=x':fofType
% 139.42/139.61 Found (fun (x4:(P x3))=> x4) as proof of (P x')
% 139.42/139.61 Found (fun (P:(fofType->Prop)) (x4:(P x3))=> x4) as proof of ((P x3)->(P x'))
% 139.42/139.61 Found (fun (x00:((x0 Xx) x')) (P:(fofType->Prop)) (x4:(P x3))=> x4) as proof of (((eq fofType) x3) x')
% 139.42/139.61 Found eq_ref000:=(eq_ref00 P):((P x3)->(P x3))
% 139.42/139.61 Found (eq_ref00 P) as proof of ((P x3)->(P x'))
% 139.42/139.61 Found ((eq_ref0 x3) P) as proof of ((P x3)->(P x'))
% 139.42/139.61 Found (((eq_ref fofType) x3) P) as proof of ((P x3)->(P x'))
% 139.42/139.61 Found (((eq_ref fofType) x3) P) as proof of ((P x3)->(P x'))
% 139.42/139.61 Found (fun (P:(fofType->Prop))=> (((eq_ref fofType) x3) P)) as proof of ((P x3)->(P x'))
% 139.42/139.61 Found (fun (x00:((x0 Xx) x')) (P:(fofType->Prop))=> (((eq_ref fofType) x3) P)) as proof of (((eq fofType) x3) x')
% 139.42/139.61 Found eq_ref00:=(eq_ref0 x3):(((eq fofType) x3) x3)
% 139.42/139.61 Found (eq_ref0 x3) as proof of (((eq fofType) x3) x')
% 139.42/139.61 Found ((eq_ref fofType) x3) as proof of (((eq fofType) x3) x')
% 141.80/141.97 Found ((eq_ref fofType) x3) as proof of (((eq fofType) x3) x')
% 141.80/141.97 Found ((eq_ref fofType) x3) as proof of (((eq fofType) x3) x')
% 141.80/141.97 Found (fun (x00:((x0 Xx) x'))=> ((eq_ref fofType) x3)) as proof of (((eq fofType) x3) x')
% 141.80/141.97 Found x6:(P x5)
% 141.80/141.97 Instantiate: x5:=x':fofType
% 141.80/141.97 Found (fun (x6:(P x5))=> x6) as proof of (P x')
% 141.80/141.97 Found (fun (P:(fofType->Prop)) (x6:(P x5))=> x6) as proof of ((P x5)->(P x'))
% 141.80/141.97 Found (fun (x00:((x0 Xx) x')) (P:(fofType->Prop)) (x6:(P x5))=> x6) as proof of (((eq fofType) x5) x')
% 141.80/141.97 Found eq_ref000:=(eq_ref00 P):((P x5)->(P x5))
% 141.80/141.97 Found (eq_ref00 P) as proof of ((P x5)->(P x'))
% 141.80/141.97 Found ((eq_ref0 x5) P) as proof of ((P x5)->(P x'))
% 141.80/141.97 Found (((eq_ref fofType) x5) P) as proof of ((P x5)->(P x'))
% 141.80/141.97 Found (((eq_ref fofType) x5) P) as proof of ((P x5)->(P x'))
% 141.80/141.97 Found (fun (P:(fofType->Prop))=> (((eq_ref fofType) x5) P)) as proof of ((P x5)->(P x'))
% 141.80/141.97 Found (fun (x00:((x0 Xx) x')) (P:(fofType->Prop))=> (((eq_ref fofType) x5) P)) as proof of (((eq fofType) x5) x')
% 141.80/141.97 Found eq_ref000:=(eq_ref00 P):((P x5)->(P x5))
% 141.80/141.97 Found (eq_ref00 P) as proof of ((P x5)->(P x'))
% 141.80/141.97 Found ((eq_ref0 x5) P) as proof of ((P x5)->(P x'))
% 141.80/141.97 Found (((eq_ref fofType) x5) P) as proof of ((P x5)->(P x'))
% 141.80/141.97 Found (((eq_ref fofType) x5) P) as proof of ((P x5)->(P x'))
% 141.80/141.97 Found (fun (P:(fofType->Prop))=> (((eq_ref fofType) x5) P)) as proof of ((P x5)->(P x'))
% 141.80/141.97 Found (fun (x00:((x2 Xx) x')) (P:(fofType->Prop))=> (((eq_ref fofType) x5) P)) as proof of (((eq fofType) x5) x')
% 141.80/141.97 Found x6:(P x5)
% 141.80/141.97 Instantiate: x5:=x':fofType
% 141.80/141.97 Found (fun (x6:(P x5))=> x6) as proof of (P x')
% 141.80/141.97 Found (fun (P:(fofType->Prop)) (x6:(P x5))=> x6) as proof of ((P x5)->(P x'))
% 141.80/141.97 Found (fun (x00:((x2 Xx) x')) (P:(fofType->Prop)) (x6:(P x5))=> x6) as proof of (((eq fofType) x5) x')
% 141.80/141.97 Found eq_ref00:=(eq_ref0 x5):(((eq fofType) x5) x5)
% 141.80/141.97 Found (eq_ref0 x5) as proof of (((eq fofType) x5) x')
% 141.80/141.97 Found ((eq_ref fofType) x5) as proof of (((eq fofType) x5) x')
% 141.80/141.97 Found ((eq_ref fofType) x5) as proof of (((eq fofType) x5) x')
% 141.80/141.97 Found ((eq_ref fofType) x5) as proof of (((eq fofType) x5) x')
% 141.80/141.97 Found (fun (x00:((x0 Xx) x'))=> ((eq_ref fofType) x5)) as proof of (((eq fofType) x5) x')
% 141.80/141.97 Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% 141.80/141.97 Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) b)
% 141.80/141.97 Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% 141.80/141.97 Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% 141.80/141.97 Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% 141.80/141.97 Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% 141.80/141.97 Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx))))))
% 141.80/141.97 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx))))))
% 141.80/141.97 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx))))))
% 141.80/141.97 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx))))))
% 141.80/141.97 Found ((eq_trans0000 ((eq_ref Prop) (f x0))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f x0)) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx))))))
% 141.80/141.97 Found (((eq_trans000 ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx)))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f x0)) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx))))))
% 141.80/141.97 Found ((((eq_trans00 ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx)))))) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx)))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx))))))) as proof of (((eq Prop) (f x0)) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx))))))
% 142.15/142.33 Found (((((eq_trans0 (f x0)) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx)))))) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx)))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx))))))) as proof of (((eq Prop) (f x0)) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx))))))
% 142.15/142.33 Found ((((((eq_trans Prop) (f x0)) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx)))))) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx)))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx))))))) as proof of (((eq Prop) (f x0)) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx))))))
% 142.15/142.33 Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% 142.15/142.33 Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) b)
% 142.15/142.33 Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% 142.15/142.33 Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% 142.15/142.33 Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% 142.15/142.33 Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% 142.15/142.33 Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx))))))
% 142.15/142.33 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx))))))
% 142.15/142.33 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx))))))
% 142.15/142.33 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx))))))
% 142.15/142.33 Found ((eq_trans0000 ((eq_ref Prop) (f x0))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f x0)) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx))))))
% 142.15/142.33 Found (((eq_trans000 ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx)))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f x0)) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx))))))
% 142.84/143.03 Found ((((eq_trans00 ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx)))))) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx)))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx))))))) as proof of (((eq Prop) (f x0)) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx))))))
% 142.84/143.03 Found (((((eq_trans0 (f x0)) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx)))))) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx)))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx))))))) as proof of (((eq Prop) (f x0)) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx))))))
% 142.84/143.03 Found ((((((eq_trans Prop) (f x0)) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx)))))) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx)))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx))))))) as proof of (((eq Prop) (f x0)) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx))))))
% 142.84/143.03 Found eq_ref00:=(eq_ref0 x5):(((eq fofType) x5) x5)
% 142.84/143.03 Found (eq_ref0 x5) as proof of (((eq fofType) x5) x')
% 142.84/143.03 Found ((eq_ref fofType) x5) as proof of (((eq fofType) x5) x')
% 142.84/143.03 Found ((eq_ref fofType) x5) as proof of (((eq fofType) x5) x')
% 142.84/143.03 Found ((eq_ref fofType) x5) as proof of (((eq fofType) x5) x')
% 142.84/143.03 Found (fun (x00:((x2 Xx) x'))=> ((eq_ref fofType) x5)) as proof of (((eq fofType) x5) x')
% 142.84/143.03 Found eq_ref00:=(eq_ref0 (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx))))):(((eq Prop) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx)))))
% 142.84/143.03 Found (eq_ref0 (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx))))) as proof of (((eq Prop) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx))))) b)
% 142.84/143.03 Found ((eq_ref Prop) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx))))) as proof of (((eq Prop) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx))))) b)
% 142.84/143.03 Found ((eq_ref Prop) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx))))) as proof of (((eq Prop) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx))))) b)
% 142.84/143.03 Found ((eq_ref Prop) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx))))) as proof of (((eq Prop) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx))))) b)
% 142.84/143.03 Found eq_ref00:=(eq_ref0 (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x2 Xx))))):(((eq Prop) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x2 Xx))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x2 Xx)))))
% 146.65/146.89 Found (eq_ref0 (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x2 Xx))))) as proof of (((eq Prop) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x2 Xx))))) b)
% 146.65/146.89 Found ((eq_ref Prop) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x2 Xx))))) as proof of (((eq Prop) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x2 Xx))))) b)
% 146.65/146.89 Found ((eq_ref Prop) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x2 Xx))))) as proof of (((eq Prop) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x2 Xx))))) b)
% 146.65/146.89 Found ((eq_ref Prop) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x2 Xx))))) as proof of (((eq Prop) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x2 Xx))))) b)
% 146.65/146.89 Found eq_ref000:=(eq_ref00 P):((P x3)->(P x3))
% 146.65/146.89 Found (eq_ref00 P) as proof of ((P x3)->(P Y))
% 146.65/146.89 Found ((eq_ref0 x3) P) as proof of ((P x3)->(P Y))
% 146.65/146.89 Found (((eq_ref fofType) x3) P) as proof of ((P x3)->(P Y))
% 146.65/146.89 Found (((eq_ref fofType) x3) P) as proof of ((P x3)->(P Y))
% 146.65/146.89 Found (fun (P:(fofType->Prop))=> (((eq_ref fofType) x3) P)) as proof of ((P x3)->(P Y))
% 146.65/146.89 Found (fun (P:(fofType->Prop))=> (((eq_ref fofType) x3) P)) as proof of (((eq fofType) x3) Y)
% 146.65/146.89 Found (fun (x00:((x0 Xx) Y)) (P:(fofType->Prop))=> (((eq_ref fofType) x3) P)) as proof of (((eq fofType) x3) Y)
% 146.65/146.89 Found x4:(P x3)
% 146.65/146.89 Instantiate: x3:=Y:fofType
% 146.65/146.89 Found (fun (x4:(P x3))=> x4) as proof of (P Y)
% 146.65/146.89 Found (fun (P:(fofType->Prop)) (x4:(P x3))=> x4) as proof of ((P x3)->(P Y))
% 146.65/146.89 Found (fun (P:(fofType->Prop)) (x4:(P x3))=> x4) as proof of (((eq fofType) x3) Y)
% 146.65/146.89 Found (fun (x00:((x0 Xx) Y)) (P:(fofType->Prop)) (x4:(P x3))=> x4) as proof of (((eq fofType) x3) Y)
% 146.65/146.89 Found eq_ref000:=(eq_ref00 P0):((P0 (f x0))->(P0 (f x0)))
% 146.65/146.89 Found (eq_ref00 P0) as proof of (P1 (f x0))
% 146.65/146.89 Found ((eq_ref0 (f x0)) P0) as proof of (P1 (f x0))
% 146.65/146.89 Found (((eq_ref Prop) (f x0)) P0) as proof of (P1 (f x0))
% 146.65/146.89 Found (((eq_ref Prop) (f x0)) P0) as proof of (P1 (f x0))
% 146.65/146.89 Found eq_ref000:=(eq_ref00 P0):((P0 (f x0))->(P0 (f x0)))
% 146.65/146.89 Found (eq_ref00 P0) as proof of (P1 (f x0))
% 146.65/146.89 Found ((eq_ref0 (f x0)) P0) as proof of (P1 (f x0))
% 146.65/146.89 Found (((eq_ref Prop) (f x0)) P0) as proof of (P1 (f x0))
% 146.65/146.89 Found (((eq_ref Prop) (f x0)) P0) as proof of (P1 (f x0))
% 146.65/146.89 Found ex_intro0:=(ex_intro (fofType->(fofType->Prop))):(forall (P:((fofType->(fofType->Prop))->Prop)) (x:(fofType->(fofType->Prop))), ((P x)->((ex (fofType->(fofType->Prop))) P)))
% 146.65/146.89 Instantiate: b:=(forall (P:((fofType->(fofType->Prop))->Prop)) (x:(fofType->(fofType->Prop))), ((P x)->((ex (fofType->(fofType->Prop))) P))):Prop
% 146.65/146.89 Found ex_intro0 as proof of b
% 146.65/146.89 Found x6:(P x30)
% 146.65/146.89 Instantiate: x3:=x30:fofType
% 146.65/146.89 Found (fun (x6:(P x30))=> x6) as proof of (P x3)
% 146.65/146.89 Found (fun (P:(fofType->Prop)) (x6:(P x30))=> x6) as proof of ((P x30)->(P x3))
% 146.65/146.89 Found eq_ref000:=(eq_ref00 P):((P x30)->(P x30))
% 146.65/146.89 Found (eq_ref00 P) as proof of ((P x30)->(P x3))
% 146.65/146.89 Found ((eq_ref0 x30) P) as proof of ((P x30)->(P x3))
% 146.65/146.89 Found (((eq_ref fofType) x30) P) as proof of ((P x30)->(P x3))
% 146.65/146.89 Found (((eq_ref fofType) x30) P) as proof of ((P x30)->(P x3))
% 146.65/146.89 Found (fun (P:(fofType->Prop))=> (((eq_ref fofType) x30) P)) as proof of ((P x30)->(P x3))
% 146.65/146.89 Found eq_ref00:=(eq_ref0 x5):(((eq fofType) x5) x5)
% 146.65/146.89 Found (eq_ref0 x5) as proof of (((eq fofType) x5) x50)
% 146.65/146.89 Found ((eq_ref fofType) x5) as proof of (((eq fofType) x5) x50)
% 146.65/146.89 Found ((eq_ref fofType) x5) as proof of (((eq fofType) x5) x50)
% 146.65/146.89 Found ((eq_ref fofType) x5) as proof of (((eq fofType) x5) x50)
% 146.65/146.89 Found (eq_sym000 ((eq_ref fofType) x5)) as proof of (((eq fofType) x50) x5)
% 146.65/146.89 Found ((eq_sym00 x50) ((eq_ref fofType) x5)) as proof of (((eq fofType) x50) x5)
% 146.65/146.89 Found (((eq_sym0 x5) x50) ((eq_ref fofType) x5)) as proof of (((eq fofType) x50) x5)
% 146.65/146.89 Found ((((eq_sym fofType) x5) x50) ((eq_ref fofType) x5)) as proof of (((eq fofType) x50) x5)
% 146.65/146.89 Found eq_ref00:=(eq_ref0 x30):(((eq fofType) x30) x30)
% 146.65/146.89 Found (eq_ref0 x30) as proof of (((eq fofType) x30) x3)
% 146.65/146.89 Found ((eq_ref fofType) x30) as proof of (((eq fofType) x30) x3)
% 146.65/146.89 Found ((eq_ref fofType) x30) as proof of (((eq fofType) x30) x3)
% 146.65/146.89 Found eq_ref00:=(eq_ref0 x3):(((eq fofType) x3) x3)
% 147.41/147.67 Found (eq_ref0 x3) as proof of (((eq fofType) x3) x30)
% 147.41/147.67 Found ((eq_ref fofType) x3) as proof of (((eq fofType) x3) x30)
% 147.41/147.67 Found ((eq_ref fofType) x3) as proof of (((eq fofType) x3) x30)
% 147.41/147.67 Found ((eq_ref fofType) x3) as proof of (((eq fofType) x3) x30)
% 147.41/147.67 Found (eq_sym000 ((eq_ref fofType) x3)) as proof of (((eq fofType) x30) x3)
% 147.41/147.67 Found ((eq_sym00 x30) ((eq_ref fofType) x3)) as proof of (((eq fofType) x30) x3)
% 147.41/147.67 Found (((eq_sym0 x3) x30) ((eq_ref fofType) x3)) as proof of (((eq fofType) x30) x3)
% 147.41/147.67 Found ((((eq_sym fofType) x3) x30) ((eq_ref fofType) x3)) as proof of (((eq fofType) x30) x3)
% 147.41/147.67 Found (fun (x20:(forall (Xn:fofType), (not (((eq fofType) (cS Xn)) c0))))=> ((((eq_sym fofType) x3) x30) ((eq_ref fofType) x3))) as proof of (((eq fofType) x30) x3)
% 147.41/147.67 Found eq_ref00:=(eq_ref0 (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy)))))):(((eq Prop) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))
% 147.41/147.67 Found (eq_ref0 (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy)))))) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy)))))) b)
% 147.41/147.67 Found ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy)))))) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy)))))) b)
% 147.41/147.67 Found ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy)))))) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy)))))) b)
% 147.41/147.67 Found ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy)))))) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy)))))) b)
% 147.41/147.67 Found eq_ref000:=(eq_ref00 P):((P x5)->(P x5))
% 147.41/147.67 Found (eq_ref00 P) as proof of ((P x5)->(P Y))
% 147.41/147.67 Found ((eq_ref0 x5) P) as proof of ((P x5)->(P Y))
% 147.41/147.67 Found (((eq_ref fofType) x5) P) as proof of ((P x5)->(P Y))
% 147.41/147.67 Found (((eq_ref fofType) x5) P) as proof of ((P x5)->(P Y))
% 147.41/147.67 Found (fun (P:(fofType->Prop))=> (((eq_ref fofType) x5) P)) as proof of ((P x5)->(P Y))
% 147.41/147.67 Found (fun (P:(fofType->Prop))=> (((eq_ref fofType) x5) P)) as proof of (((eq fofType) x5) Y)
% 147.41/147.67 Found (fun (x00:((x0 Xx) Y)) (P:(fofType->Prop))=> (((eq_ref fofType) x5) P)) as proof of (((eq fofType) x5) Y)
% 147.41/147.67 Found x6:(P x5)
% 147.41/147.67 Instantiate: x5:=Y:fofType
% 147.41/147.67 Found (fun (x6:(P x5))=> x6) as proof of (P Y)
% 147.41/147.67 Found (fun (P:(fofType->Prop)) (x6:(P x5))=> x6) as proof of ((P x5)->(P Y))
% 147.41/147.67 Found (fun (P:(fofType->Prop)) (x6:(P x5))=> x6) as proof of (((eq fofType) x5) Y)
% 147.41/147.67 Found (fun (x00:((x0 Xx) Y)) (P:(fofType->Prop)) (x6:(P x5))=> x6) as proof of (((eq fofType) x5) Y)
% 147.41/147.67 Found eq_ref000:=(eq_ref00 P):((P x1)->(P x1))
% 147.41/147.67 Found (eq_ref00 P) as proof of ((P x1)->(P x'))
% 147.41/147.67 Found ((eq_ref0 x1) P) as proof of ((P x1)->(P x'))
% 147.41/147.67 Found (((eq_ref fofType) x1) P) as proof of ((P x1)->(P x'))
% 147.41/147.67 Found (((eq_ref fofType) x1) P) as proof of ((P x1)->(P x'))
% 147.41/147.67 Found (fun (P:(fofType->Prop))=> (((eq_ref fofType) x1) P)) as proof of ((P x1)->(P x'))
% 147.41/147.67 Found (fun (x00:((x0 Xx) x')) (P:(fofType->Prop))=> (((eq_ref fofType) x1) P)) as proof of (((eq fofType) x1) x')
% 147.41/147.67 Found x4:(P x1)
% 147.41/147.67 Instantiate: x1:=x':fofType
% 147.41/147.67 Found (fun (x4:(P x1))=> x4) as proof of (P x')
% 147.41/147.67 Found (fun (P:(fofType->Prop)) (x4:(P x1))=> x4) as proof of ((P x1)->(P x'))
% 147.41/147.67 Found (fun (x00:((x0 Xx) x')) (P:(fofType->Prop)) (x4:(P x1))=> x4) as proof of (((eq fofType) x1) x')
% 147.41/147.67 Found eq_ref000:=(eq_ref00 P):((P x5)->(P x5))
% 147.41/147.67 Found (eq_ref00 P) as proof of ((P x5)->(P Y))
% 147.41/147.67 Found ((eq_ref0 x5) P) as proof of ((P x5)->(P Y))
% 147.41/147.67 Found (((eq_ref fofType) x5) P) as proof of ((P x5)->(P Y))
% 147.41/147.67 Found (((eq_ref fofType) x5) P) as proof of ((P x5)->(P Y))
% 147.41/147.67 Found (fun (P:(fofType->Prop))=> (((eq_ref fofType) x5) P)) as proof of ((P x5)->(P Y))
% 147.41/147.67 Found (fun (P:(fofType->Prop))=> (((eq_ref fofType) x5) P)) as proof of (((eq fofType) x5) Y)
% 147.41/147.67 Found (fun (x00:((x2 Xx) Y)) (P:(fofType->Prop))=> (((eq_ref fofType) x5) P)) as proof of (((eq fofType) x5) Y)
% 147.72/147.92 Found x6:(P x5)
% 147.72/147.92 Instantiate: x5:=Y:fofType
% 147.72/147.92 Found (fun (x6:(P x5))=> x6) as proof of (P Y)
% 147.72/147.92 Found (fun (P:(fofType->Prop)) (x6:(P x5))=> x6) as proof of ((P x5)->(P Y))
% 147.72/147.92 Found (fun (P:(fofType->Prop)) (x6:(P x5))=> x6) as proof of (((eq fofType) x5) Y)
% 147.72/147.92 Found (fun (x00:((x2 Xx) Y)) (P:(fofType->Prop)) (x6:(P x5))=> x6) as proof of (((eq fofType) x5) Y)
% 147.72/147.92 Found eq_substitution0000:=(fun (a:fofType)=> ((eq_substitution000 a) (fun (x8:fofType)=> (cS Xx)))):(forall (a:fofType), ((((eq fofType) a) (cS Xy))->(((eq fofType) (cS Xx)) (cS Xx))))
% 147.72/147.92 Instantiate: x0:=(fun (x6:fofType) (x50:fofType)=> (forall (a:fofType), ((((eq fofType) a) (cS Xy))->(((eq fofType) x6) x6)))):(fofType->(fofType->Prop))
% 147.72/147.92 Found (fun (a:fofType)=> ((eq_substitution000 a) (fun (x8:fofType)=> (cS Xx)))) as proof of ((x0 (cS Xx)) (cS (cS Xy)))
% 147.72/147.92 Found (fun (a:fofType)=> (((fun (a:fofType)=> ((eq_substitution00 a) (cS Xy))) a) (fun (x8:fofType)=> (cS Xx)))) as proof of ((x0 (cS Xx)) (cS (cS Xy)))
% 147.72/147.92 Found (fun (a:fofType)=> (((fun (a:fofType)=> (((eq_substitution0 fofType) a) (cS Xy))) a) (fun (x8:fofType)=> (cS Xx)))) as proof of ((x0 (cS Xx)) (cS (cS Xy)))
% 147.72/147.92 Found (fun (a:fofType)=> (((fun (a:fofType)=> ((((eq_substitution fofType) fofType) a) (cS Xy))) a) (fun (x8:fofType)=> (cS Xx)))) as proof of ((x0 (cS Xx)) (cS (cS Xy)))
% 147.72/147.92 Found (fun (x4:(forall (Xx0:fofType) (Xy0:fofType), ((((eq fofType) (cS Xx0)) (cS Xy0))->(((eq fofType) Xx0) Xy0)))) (a:fofType)=> (((fun (a:fofType)=> ((((eq_substitution fofType) fofType) a) (cS Xy))) a) (fun (x8:fofType)=> (cS Xx)))) as proof of ((x0 (cS Xx)) (cS (cS Xy)))
% 147.72/147.92 Found (fun (x3:cIND) (x4:(forall (Xx0:fofType) (Xy0:fofType), ((((eq fofType) (cS Xx0)) (cS Xy0))->(((eq fofType) Xx0) Xy0)))) (a:fofType)=> (((fun (a:fofType)=> ((((eq_substitution fofType) fofType) a) (cS Xy))) a) (fun (x8:fofType)=> (cS Xx)))) as proof of ((forall (Xx0:fofType) (Xy0:fofType), ((((eq fofType) (cS Xx0)) (cS Xy0))->(((eq fofType) Xx0) Xy0)))->((x0 (cS Xx)) (cS (cS Xy))))
% 147.72/147.92 Found (fun (x3:cIND) (x4:(forall (Xx0:fofType) (Xy0:fofType), ((((eq fofType) (cS Xx0)) (cS Xy0))->(((eq fofType) Xx0) Xy0)))) (a:fofType)=> (((fun (a:fofType)=> ((((eq_substitution fofType) fofType) a) (cS Xy))) a) (fun (x8:fofType)=> (cS Xx)))) as proof of (cIND->((forall (Xx0:fofType) (Xy0:fofType), ((((eq fofType) (cS Xx0)) (cS Xy0))->(((eq fofType) Xx0) Xy0)))->((x0 (cS Xx)) (cS (cS Xy)))))
% 147.72/147.92 Found (and_rect10 (fun (x3:cIND) (x4:(forall (Xx0:fofType) (Xy0:fofType), ((((eq fofType) (cS Xx0)) (cS Xy0))->(((eq fofType) Xx0) Xy0)))) (a:fofType)=> (((fun (a:fofType)=> ((((eq_substitution fofType) fofType) a) (cS Xy))) a) (fun (x8:fofType)=> (cS Xx))))) as proof of ((x0 (cS Xx)) (cS (cS Xy)))
% 147.72/147.92 Found ((and_rect1 ((x0 (cS Xx)) (cS (cS Xy)))) (fun (x3:cIND) (x4:(forall (Xx0:fofType) (Xy0:fofType), ((((eq fofType) (cS Xx0)) (cS Xy0))->(((eq fofType) Xx0) Xy0)))) (a:fofType)=> (((fun (a:fofType)=> ((((eq_substitution fofType) fofType) a) (cS Xy))) a) (fun (x8:fofType)=> (cS Xx))))) as proof of ((x0 (cS Xx)) (cS (cS Xy)))
% 147.72/147.92 Found (((fun (P:Type) (x3:(cIND->((forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (cS Xx)) (cS Xy))->(((eq fofType) Xx) Xy)))->P)))=> (((((and_rect cIND) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (cS Xx)) (cS Xy))->(((eq fofType) Xx) Xy)))) P) x3) x1)) ((x0 (cS Xx)) (cS (cS Xy)))) (fun (x3:cIND) (x4:(forall (Xx0:fofType) (Xy0:fofType), ((((eq fofType) (cS Xx0)) (cS Xy0))->(((eq fofType) Xx0) Xy0)))) (a:fofType)=> (((fun (a:fofType)=> ((((eq_substitution fofType) fofType) a) (cS Xy))) a) (fun (x8:fofType)=> (cS Xx))))) as proof of ((x0 (cS Xx)) (cS (cS Xy)))
% 147.72/147.92 Found (fun (x00:((x0 Xx) Xy))=> (((fun (P:Type) (x3:(cIND->((forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (cS Xx)) (cS Xy))->(((eq fofType) Xx) Xy)))->P)))=> (((((and_rect cIND) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (cS Xx)) (cS Xy))->(((eq fofType) Xx) Xy)))) P) x3) x1)) ((x0 (cS Xx)) (cS (cS Xy)))) (fun (x3:cIND) (x4:(forall (Xx0:fofType) (Xy0:fofType), ((((eq fofType) (cS Xx0)) (cS Xy0))->(((eq fofType) Xx0) Xy0)))) (a:fofType)=> (((fun (a:fofType)=> ((((eq_substitution fofType) fofType) a) (cS Xy))) a) (fun (x8:fofType)=> (cS Xx)))))) as proof of ((x0 (cS Xx)) (cS (cS Xy)))
% 148.54/148.76 Found eq_substitution:=(fun (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)) (H:(((eq T) a) b))=> ((H (fun (x:T)=> (((eq U) (f a)) (f x)))) ((eq_ref U) (f a)))):(forall (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)), ((((eq T) a) b)->(((eq U) (f a)) (f b))))
% 148.54/148.76 Instantiate: b:=(forall (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)), ((((eq T) a) b)->(((eq U) (f a)) (f b)))):Prop
% 148.54/148.76 Found eq_substitution as proof of b
% 148.54/148.76 Found eq_ref00:=(eq_ref0 x30):(((eq fofType) x30) x30)
% 148.54/148.76 Found (eq_ref0 x30) as proof of (((eq fofType) x30) x3)
% 148.54/148.76 Found ((eq_ref fofType) x30) as proof of (((eq fofType) x30) x3)
% 148.54/148.76 Found ((eq_ref fofType) x30) as proof of (((eq fofType) x30) x3)
% 148.54/148.76 Found (fun (x5:(forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy))))=> ((eq_ref fofType) x30)) as proof of (((eq fofType) x30) x3)
% 148.54/148.76 Found (fun (x4:cIND) (x5:(forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy))))=> ((eq_ref fofType) x30)) as proof of ((forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy)))->(((eq fofType) x30) x3))
% 148.54/148.76 Found (fun (x4:cIND) (x5:(forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy))))=> ((eq_ref fofType) x30)) as proof of (cIND->((forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy)))->(((eq fofType) x30) x3)))
% 148.54/148.76 Found (and_rect10 (fun (x4:cIND) (x5:(forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy))))=> ((eq_ref fofType) x30))) as proof of (((eq fofType) x30) x3)
% 148.54/148.76 Found ((and_rect1 (((eq fofType) x30) x3)) (fun (x4:cIND) (x5:(forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy))))=> ((eq_ref fofType) x30))) as proof of (((eq fofType) x30) x3)
% 148.54/148.76 Found (((fun (P:Type) (x4:(cIND->((forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy)))->P)))=> (((((and_rect cIND) (forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy)))) P) x4) x10)) (((eq fofType) x30) x3)) (fun (x4:cIND) (x5:(forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy))))=> ((eq_ref fofType) x30))) as proof of (((eq fofType) x30) x3)
% 148.54/148.76 Found (fun (x20:(forall (Xn:fofType), (not (((eq fofType) (cS Xn)) c0))))=> (((fun (P:Type) (x4:(cIND->((forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy)))->P)))=> (((((and_rect cIND) (forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy)))) P) x4) x10)) (((eq fofType) x30) x3)) (fun (x4:cIND) (x5:(forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy))))=> ((eq_ref fofType) x30)))) as proof of (((eq fofType) x30) x3)
% 148.54/148.76 Found eq_ref00:=(eq_ref0 x1):(((eq fofType) x1) x1)
% 148.54/148.76 Found (eq_ref0 x1) as proof of (((eq fofType) x1) x')
% 148.54/148.76 Found ((eq_ref fofType) x1) as proof of (((eq fofType) x1) x')
% 148.54/148.76 Found ((eq_ref fofType) x1) as proof of (((eq fofType) x1) x')
% 148.54/148.76 Found ((eq_ref fofType) x1) as proof of (((eq fofType) x1) x')
% 148.54/148.76 Found (fun (x00:((x0 Xx) x'))=> ((eq_ref fofType) x1)) as proof of (((eq fofType) x1) x')
% 148.54/148.76 Found eq_ref00:=(eq_ref0 x30):(((eq fofType) x30) x30)
% 148.54/148.76 Found (eq_ref0 x30) as proof of (((eq fofType) x30) x3)
% 148.54/148.76 Found ((eq_ref fofType) x30) as proof of (((eq fofType) x30) x3)
% 148.54/148.76 Found ((eq_ref fofType) x30) as proof of (((eq fofType) x30) x3)
% 148.54/148.76 Found (fun (x5:(forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (cS Xx)) (cS Xy))->(((eq fofType) Xx) Xy))))=> ((eq_ref fofType) x30)) as proof of (((eq fofType) x30) x3)
% 148.54/148.76 Found (fun (x4:cIND) (x5:(forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (cS Xx)) (cS Xy))->(((eq fofType) Xx) Xy))))=> ((eq_ref fofType) x30)) as proof of ((forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (cS Xx)) (cS Xy))->(((eq fofType) Xx) Xy)))->(((eq fofType) x30) x3))
% 148.54/148.76 Found (fun (x4:cIND) (x5:(forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (cS Xx)) (cS Xy))->(((eq fofType) Xx) Xy))))=> ((eq_ref fofType) x30)) as proof of (cIND->((forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (cS Xx)) (cS Xy))->(((eq fofType) Xx) Xy)))->(((eq fofType) x30) x3)))
% 152.02/152.20 Found (and_rect10 (fun (x4:cIND) (x5:(forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (cS Xx)) (cS Xy))->(((eq fofType) Xx) Xy))))=> ((eq_ref fofType) x30))) as proof of (((eq fofType) x30) x3)
% 152.02/152.20 Found ((and_rect1 (((eq fofType) x30) x3)) (fun (x4:cIND) (x5:(forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (cS Xx)) (cS Xy))->(((eq fofType) Xx) Xy))))=> ((eq_ref fofType) x30))) as proof of (((eq fofType) x30) x3)
% 152.02/152.20 Found (((fun (P:Type) (x4:(cIND->((forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (cS Xx)) (cS Xy))->(((eq fofType) Xx) Xy)))->P)))=> (((((and_rect cIND) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (cS Xx)) (cS Xy))->(((eq fofType) Xx) Xy)))) P) x4) x0)) (((eq fofType) x30) x3)) (fun (x4:cIND) (x5:(forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (cS Xx)) (cS Xy))->(((eq fofType) Xx) Xy))))=> ((eq_ref fofType) x30))) as proof of (((eq fofType) x30) x3)
% 152.02/152.20 Found x00000:=(x0000 x3):(((eq fofType) x3) Y)
% 152.02/152.20 Found (x0000 x3) as proof of (((eq fofType) x3) Y)
% 152.02/152.20 Found ((fun (x30:fofType)=> ((x000 x30) x2)) x3) as proof of (((eq fofType) x3) Y)
% 152.02/152.20 Found ((fun (x30:fofType)=> (((x00 x1) x30) x2)) x3) as proof of (((eq fofType) x3) Y)
% 152.02/152.20 Found ((fun (x30:fofType)=> (((x00 x1) x30) x2)) x3) as proof of (((eq fofType) x3) Y)
% 152.02/152.20 Found (fun (x00:((x0 Xx) Y))=> ((fun (x30:fofType)=> (((x00 x1) x30) x2)) x3)) as proof of (((eq fofType) x3) Y)
% 152.02/152.20 Found (fun (Y:fofType) (x00:((x0 Xx) Y))=> ((fun (x30:fofType)=> (((x00 x1) x30) x2)) x3)) as proof of (((x0 Xx) Y)->(((eq fofType) x3) Y))
% 152.02/152.20 Found (fun (Y:fofType) (x00:((x0 Xx) Y))=> ((fun (x30:fofType)=> (((x00 x1) x30) x2)) x3)) as proof of (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) x3) Y)))
% 152.02/152.20 Found eq_ref00:=(eq_ref0 x30):(((eq fofType) x30) x30)
% 152.02/152.20 Found (eq_ref0 x30) as proof of (((eq fofType) x30) x3)
% 152.02/152.20 Found ((eq_ref fofType) x30) as proof of (((eq fofType) x30) x3)
% 152.02/152.20 Found ((eq_ref fofType) x30) as proof of (((eq fofType) x30) x3)
% 152.02/152.20 Found (fun (x20:(forall (Xn:fofType), (not (((eq fofType) (cS Xn)) c0))))=> ((eq_ref fofType) x30)) as proof of (((eq fofType) x30) x3)
% 152.02/152.20 Found x4:(P x30)
% 152.02/152.20 Instantiate: x3:=x30:fofType
% 152.02/152.20 Found (fun (x4:(P x30))=> x4) as proof of (P x3)
% 152.02/152.20 Found (fun (P:(fofType->Prop)) (x4:(P x30))=> x4) as proof of ((P x30)->(P x3))
% 152.02/152.20 Found (fun (x20:(forall (Xn:fofType), (not (((eq fofType) (cS Xn)) c0)))) (P:(fofType->Prop)) (x4:(P x30))=> x4) as proof of (((eq fofType) x30) x3)
% 152.02/152.20 Found eq_ref000:=(eq_ref00 P):((P x30)->(P x30))
% 152.02/152.20 Found (eq_ref00 P) as proof of ((P x30)->(P x3))
% 152.02/152.20 Found ((eq_ref0 x30) P) as proof of ((P x30)->(P x3))
% 152.02/152.20 Found (((eq_ref fofType) x30) P) as proof of ((P x30)->(P x3))
% 152.02/152.20 Found (((eq_ref fofType) x30) P) as proof of ((P x30)->(P x3))
% 152.02/152.20 Found (fun (P:(fofType->Prop))=> (((eq_ref fofType) x30) P)) as proof of ((P x30)->(P x3))
% 152.02/152.20 Found (fun (x20:(forall (Xn:fofType), (not (((eq fofType) (cS Xn)) c0)))) (P:(fofType->Prop))=> (((eq_ref fofType) x30) P)) as proof of (((eq fofType) x30) x3)
% 152.02/152.20 Found eq_sym000:=(eq_sym00 Y):((((eq fofType) x5) Y)->(((eq fofType) Y) x5))
% 152.02/152.20 Found (eq_sym00 Y) as proof of (((x0 Xx) Y)->(((eq fofType) x5) Y))
% 152.02/152.20 Found ((eq_sym0 x5) Y) as proof of (((x0 Xx) Y)->(((eq fofType) x5) Y))
% 152.02/152.20 Found (((eq_sym fofType) x5) Y) as proof of (((x0 Xx) Y)->(((eq fofType) x5) Y))
% 152.02/152.20 Found (((eq_sym fofType) x5) Y) as proof of (((x0 Xx) Y)->(((eq fofType) x5) Y))
% 152.02/152.20 Found (((eq_sym fofType) x5) Y) as proof of (((x0 Xx) Y)->(((eq fofType) x5) Y))
% 152.02/152.20 Found eq_sym000:=(eq_sym00 Y):((((eq fofType) x5) Y)->(((eq fofType) Y) x5))
% 152.02/152.20 Found (eq_sym00 Y) as proof of (((x2 Xx) Y)->(((eq fofType) x5) Y))
% 152.02/152.20 Found ((eq_sym0 x5) Y) as proof of (((x2 Xx) Y)->(((eq fofType) x5) Y))
% 152.02/152.20 Found (((eq_sym fofType) x5) Y) as proof of (((x2 Xx) Y)->(((eq fofType) x5) Y))
% 152.02/152.20 Found (((eq_sym fofType) x5) Y) as proof of (((x2 Xx) Y)->(((eq fofType) x5) Y))
% 152.02/152.20 Found (((eq_sym fofType) x5) Y) as proof of (((x2 Xx) Y)->(((eq fofType) x5) Y))
% 152.02/152.20 Found eq_ref00:=(eq_ref0 x5):(((eq fofType) x5) x5)
% 152.02/152.20 Found (eq_ref0 x5) as proof of (((eq fofType) x5) Y)
% 157.74/157.95 Found ((eq_ref fofType) x5) as proof of (((eq fofType) x5) Y)
% 157.74/157.95 Found ((eq_ref fofType) x5) as proof of (((eq fofType) x5) Y)
% 157.74/157.95 Found (fun (x00:((x0 Xx) Y))=> ((eq_ref fofType) x5)) as proof of (((eq fofType) x5) Y)
% 157.74/157.95 Found x00000:=(x0000 x5):(((eq fofType) x5) Y)
% 157.74/157.95 Found (x0000 x5) as proof of (((eq fofType) x5) Y)
% 157.74/157.95 Found ((x000 x4) x5) as proof of (((eq fofType) x5) Y)
% 157.74/157.95 Found (((x00 x3) x4) x5) as proof of (((eq fofType) x5) Y)
% 157.74/157.95 Found (((x00 x3) x4) x5) as proof of (((eq fofType) x5) Y)
% 157.74/157.95 Found (fun (x00:((x2 Xx) Y))=> (((x00 x3) x4) x5)) as proof of (((eq fofType) x5) Y)
% 157.74/157.95 Found (fun (Y:fofType) (x00:((x2 Xx) Y))=> (((x00 x3) x4) x5)) as proof of (((x2 Xx) Y)->(((eq fofType) x5) Y))
% 157.74/157.95 Found (fun (Y:fofType) (x00:((x2 Xx) Y))=> (((x00 x3) x4) x5)) as proof of (forall (Y:fofType), (((x2 Xx) Y)->(((eq fofType) x5) Y)))
% 157.74/157.95 Found eq_ref00:=(eq_ref0 x50):(((eq fofType) x50) x50)
% 157.74/157.95 Found (eq_ref0 x50) as proof of (((eq fofType) x50) x5)
% 157.74/157.95 Found ((eq_ref fofType) x50) as proof of (((eq fofType) x50) x5)
% 157.74/157.95 Found ((eq_ref fofType) x50) as proof of (((eq fofType) x50) x5)
% 157.74/157.95 Found x6:(P x50)
% 157.74/157.95 Instantiate: x5:=x50:fofType
% 157.74/157.95 Found (fun (x6:(P x50))=> x6) as proof of (P x5)
% 157.74/157.95 Found (fun (P:(fofType->Prop)) (x6:(P x50))=> x6) as proof of ((P x50)->(P x5))
% 157.74/157.95 Found eq_ref000:=(eq_ref00 P):((P x50)->(P x50))
% 157.74/157.95 Found (eq_ref00 P) as proof of ((P x50)->(P x5))
% 157.74/157.95 Found ((eq_ref0 x50) P) as proof of ((P x50)->(P x5))
% 157.74/157.95 Found (((eq_ref fofType) x50) P) as proof of ((P x50)->(P x5))
% 157.74/157.95 Found (((eq_ref fofType) x50) P) as proof of ((P x50)->(P x5))
% 157.74/157.95 Found (fun (P:(fofType->Prop))=> (((eq_ref fofType) x50) P)) as proof of ((P x50)->(P x5))
% 157.74/157.95 Found eq_ref00:=(eq_ref0 x5):(((eq fofType) x5) x5)
% 157.74/157.95 Found (eq_ref0 x5) as proof of (((eq fofType) x5) Y)
% 157.74/157.95 Found ((eq_ref fofType) x5) as proof of (((eq fofType) x5) Y)
% 157.74/157.95 Found ((eq_ref fofType) x5) as proof of (((eq fofType) x5) Y)
% 157.74/157.95 Found (fun (x00:((x2 Xx) Y))=> ((eq_ref fofType) x5)) as proof of (((eq fofType) x5) Y)
% 157.74/157.95 Found x4:(P x1)
% 157.74/157.95 Instantiate: x1:=Y:fofType
% 157.74/157.95 Found (fun (x4:(P x1))=> x4) as proof of (P Y)
% 157.74/157.95 Found (fun (P:(fofType->Prop)) (x4:(P x1))=> x4) as proof of ((P x1)->(P Y))
% 157.74/157.95 Found (fun (P:(fofType->Prop)) (x4:(P x1))=> x4) as proof of (((eq fofType) x1) Y)
% 157.74/157.95 Found (fun (x00:((x0 Xx) Y)) (P:(fofType->Prop)) (x4:(P x1))=> x4) as proof of (((eq fofType) x1) Y)
% 157.74/157.95 Found eq_ref000:=(eq_ref00 P):((P x1)->(P x1))
% 157.74/157.95 Found (eq_ref00 P) as proof of ((P x1)->(P Y))
% 157.74/157.95 Found ((eq_ref0 x1) P) as proof of ((P x1)->(P Y))
% 157.74/157.95 Found (((eq_ref fofType) x1) P) as proof of ((P x1)->(P Y))
% 157.74/157.95 Found (((eq_ref fofType) x1) P) as proof of ((P x1)->(P Y))
% 157.74/157.95 Found (fun (P:(fofType->Prop))=> (((eq_ref fofType) x1) P)) as proof of ((P x1)->(P Y))
% 157.74/157.95 Found (fun (P:(fofType->Prop))=> (((eq_ref fofType) x1) P)) as proof of (((eq fofType) x1) Y)
% 157.74/157.95 Found (fun (x00:((x0 Xx) Y)) (P:(fofType->Prop))=> (((eq_ref fofType) x1) P)) as proof of (((eq fofType) x1) Y)
% 157.74/157.95 Found eq_ref00:=(eq_ref0 (cS x50)):(((eq fofType) (cS x50)) (cS x50))
% 157.74/157.95 Found (eq_ref0 (cS x50)) as proof of (((eq fofType) (cS x50)) (cS x5))
% 157.74/157.95 Found ((eq_ref fofType) (cS x50)) as proof of (((eq fofType) (cS x50)) (cS x5))
% 157.74/157.95 Found ((eq_ref fofType) (cS x50)) as proof of (((eq fofType) (cS x50)) (cS x5))
% 157.74/157.95 Found ((eq_ref fofType) (cS x50)) as proof of (((eq fofType) (cS x50)) (cS x5))
% 157.74/157.95 Found (x410 ((eq_ref fofType) (cS x50))) as proof of (((eq fofType) x50) x5)
% 157.74/157.95 Found ((x41 x5) ((eq_ref fofType) (cS x50))) as proof of (((eq fofType) x50) x5)
% 157.74/157.95 Found (((x4 x50) x5) ((eq_ref fofType) (cS x50))) as proof of (((eq fofType) x50) x5)
% 157.74/157.95 Found eq_ref00:=(eq_ref0 (cS x50)):(((eq fofType) (cS x50)) (cS x50))
% 157.74/157.95 Found (eq_ref0 (cS x50)) as proof of (((eq fofType) (cS x50)) (cS x5))
% 157.74/157.95 Found ((eq_ref fofType) (cS x50)) as proof of (((eq fofType) (cS x50)) (cS x5))
% 157.74/157.95 Found ((eq_ref fofType) (cS x50)) as proof of (((eq fofType) (cS x50)) (cS x5))
% 157.74/157.95 Found ((eq_ref fofType) (cS x50)) as proof of (((eq fofType) (cS x50)) (cS x5))
% 157.74/157.95 Found (x4000 ((eq_ref fofType) (cS x50))) as proof of (((eq fofType) x50) x5)
% 157.74/157.95 Found ((x400 x5) ((eq_ref fofType) (cS x50))) as proof of (((eq fofType) x50) x5)
% 157.74/157.95 Found (((x40 x50) x5) ((eq_ref fofType) (cS x50))) as proof of (((eq fofType) x50) x5)
% 157.74/157.95 Found eq_substitution:=(fun (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)) (H:(((eq T) a) b))=> ((H (fun (x:T)=> (((eq U) (f a)) (f x)))) ((eq_ref U) (f a)))):(forall (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)), ((((eq T) a) b)->(((eq U) (f a)) (f b))))
% 158.52/158.73 Instantiate: b:=(forall (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)), ((((eq T) a) b)->(((eq U) (f a)) (f b)))):Prop
% 158.52/158.73 Found eq_substitution as proof of b
% 158.52/158.73 Found eq_substitution:=(fun (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)) (H:(((eq T) a) b))=> ((H (fun (x:T)=> (((eq U) (f a)) (f x)))) ((eq_ref U) (f a)))):(forall (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)), ((((eq T) a) b)->(((eq U) (f a)) (f b))))
% 158.52/158.73 Instantiate: b:=(forall (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)), ((((eq T) a) b)->(((eq U) (f a)) (f b)))):Prop
% 158.52/158.73 Found eq_substitution as proof of b
% 158.52/158.73 Found x00000:=(x0000 x1):(((eq fofType) x1) Y)
% 158.52/158.73 Found (x0000 x1) as proof of (((eq fofType) x1) Y)
% 158.52/158.73 Found ((fun (x10:fofType)=> ((x000 x10) x2)) x1) as proof of (((eq fofType) x1) Y)
% 158.52/158.73 Found ((fun (x10:fofType)=> (((fun (x10:fofType)=> ((x00 x10) x3)) x10) x2)) x1) as proof of (((eq fofType) x1) Y)
% 158.52/158.73 Found ((fun (x10:fofType)=> (((fun (x10:fofType)=> ((x00 x10) x3)) x10) x2)) x1) as proof of (((eq fofType) x1) Y)
% 158.52/158.73 Found (fun (x00:((x0 Xx) Y))=> ((fun (x10:fofType)=> (((fun (x10:fofType)=> ((x00 x10) x3)) x10) x2)) x1)) as proof of (((eq fofType) x1) Y)
% 158.52/158.73 Found (fun (Y:fofType) (x00:((x0 Xx) Y))=> ((fun (x10:fofType)=> (((fun (x10:fofType)=> ((x00 x10) x3)) x10) x2)) x1)) as proof of (((x0 Xx) Y)->(((eq fofType) x1) Y))
% 158.52/158.73 Found (fun (Y:fofType) (x00:((x0 Xx) Y))=> ((fun (x10:fofType)=> (((fun (x10:fofType)=> ((x00 x10) x3)) x10) x2)) x1)) as proof of (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) x1) Y)))
% 158.52/158.73 Found eq_ref00:=(eq_ref0 x10):(((eq fofType) x10) x10)
% 158.52/158.73 Found (eq_ref0 x10) as proof of (((eq fofType) x10) x1)
% 158.52/158.73 Found ((eq_ref fofType) x10) as proof of (((eq fofType) x10) x1)
% 158.52/158.73 Found ((eq_ref fofType) x10) as proof of (((eq fofType) x10) x1)
% 158.52/158.73 Found (fun (x20:((and cIND) (forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy)))))=> ((eq_ref fofType) x10)) as proof of (((eq fofType) x10) x1)
% 158.52/158.73 Found (fun (x30:(forall (Xn:fofType), (not (((eq fofType) (cS Xn)) c0)))) (x20:((and cIND) (forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy)))))=> ((eq_ref fofType) x10)) as proof of (((and cIND) (forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy))))->(((eq fofType) x10) x1))
% 158.52/158.73 Found x4:(P x10)
% 158.52/158.73 Instantiate: x1:=x10:fofType
% 158.52/158.73 Found (fun (x4:(P x10))=> x4) as proof of (P x1)
% 158.52/158.73 Found (fun (P:(fofType->Prop)) (x4:(P x10))=> x4) as proof of ((P x10)->(P x1))
% 158.52/158.73 Found (fun (x20:((and cIND) (forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy))))) (P:(fofType->Prop)) (x4:(P x10))=> x4) as proof of (((eq fofType) x10) x1)
% 158.52/158.73 Found (fun (x30:(forall (Xn:fofType), (not (((eq fofType) (cS Xn)) c0)))) (x20:((and cIND) (forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy))))) (P:(fofType->Prop)) (x4:(P x10))=> x4) as proof of (((and cIND) (forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy))))->(((eq fofType) x10) x1))
% 158.52/158.73 Found eq_ref000:=(eq_ref00 P):((P x10)->(P x10))
% 158.52/158.73 Found (eq_ref00 P) as proof of ((P x10)->(P x1))
% 158.52/158.73 Found ((eq_ref0 x10) P) as proof of ((P x10)->(P x1))
% 158.52/158.73 Found (((eq_ref fofType) x10) P) as proof of ((P x10)->(P x1))
% 158.52/158.73 Found (((eq_ref fofType) x10) P) as proof of ((P x10)->(P x1))
% 158.52/158.73 Found (fun (P:(fofType->Prop))=> (((eq_ref fofType) x10) P)) as proof of ((P x10)->(P x1))
% 158.52/158.73 Found (fun (x20:((and cIND) (forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy))))) (P:(fofType->Prop))=> (((eq_ref fofType) x10) P)) as proof of (((eq fofType) x10) x1)
% 158.52/158.73 Found (fun (x30:(forall (Xn:fofType), (not (((eq fofType) (cS Xn)) c0)))) (x20:((and cIND) (forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy))))) (P:(fofType->Prop))=> (((eq_ref fofType) x10) P)) as proof of (((and cIND) (forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy))))->(((eq fofType) x10) x1))
% 160.13/160.29 Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y)))))):(((eq (fofType->Prop)) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y)))))) (fun (x:fofType)=> ((and ((x0 Xx) x)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) x) Y))))))
% 160.13/160.29 Found (eta_expansion_dep00 (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y)))))) as proof of (((eq (fofType->Prop)) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y)))))) b)
% 160.13/160.29 Found ((eta_expansion_dep0 (fun (x2:fofType)=> Prop)) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y)))))) as proof of (((eq (fofType->Prop)) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y)))))) b)
% 160.13/160.29 Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y)))))) as proof of (((eq (fofType->Prop)) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y)))))) b)
% 160.13/160.29 Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y)))))) as proof of (((eq (fofType->Prop)) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y)))))) b)
% 160.13/160.29 Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y)))))) as proof of (((eq (fofType->Prop)) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y)))))) b)
% 160.13/160.29 Found eq_ref000:=(eq_ref00 P):((P x3)->(P x3))
% 160.13/160.29 Found (eq_ref00 P) as proof of ((P x3)->(P x'))
% 160.13/160.29 Found ((eq_ref0 x3) P) as proof of ((P x3)->(P x'))
% 160.13/160.29 Found (((eq_ref fofType) x3) P) as proof of ((P x3)->(P x'))
% 160.13/160.29 Found (((eq_ref fofType) x3) P) as proof of ((P x3)->(P x'))
% 160.13/160.29 Found (fun (P:(fofType->Prop))=> (((eq_ref fofType) x3) P)) as proof of ((P x3)->(P x'))
% 160.13/160.29 Found (fun (P:(fofType->Prop))=> (((eq_ref fofType) x3) P)) as proof of (((eq fofType) x3) x')
% 160.13/160.29 Found (fun (x00:((x0 Xx) x')) (P:(fofType->Prop))=> (((eq_ref fofType) x3) P)) as proof of (((eq fofType) x3) x')
% 160.13/160.29 Found x4:(P x3)
% 160.13/160.29 Instantiate: x3:=x':fofType
% 160.13/160.29 Found (fun (x4:(P x3))=> x4) as proof of (P x')
% 160.13/160.29 Found (fun (P:(fofType->Prop)) (x4:(P x3))=> x4) as proof of ((P x3)->(P x'))
% 160.13/160.29 Found (fun (P:(fofType->Prop)) (x4:(P x3))=> x4) as proof of (((eq fofType) x3) x')
% 160.13/160.29 Found (fun (x00:((x0 Xx) x')) (P:(fofType->Prop)) (x4:(P x3))=> x4) as proof of (((eq fofType) x3) x')
% 160.13/160.29 Found eq_ref00:=(eq_ref0 x30):(((eq fofType) x30) x30)
% 160.13/160.29 Found (eq_ref0 x30) as proof of (((eq fofType) x30) x3)
% 160.13/160.29 Found ((eq_ref fofType) x30) as proof of (((eq fofType) x30) x3)
% 160.13/160.29 Found ((eq_ref fofType) x30) as proof of (((eq fofType) x30) x3)
% 160.13/160.29 Found (fun (x20:(forall (Xn:fofType), (not (((eq fofType) (cS Xn)) c0))))=> ((eq_ref fofType) x30)) as proof of (((eq fofType) x30) x3)
% 160.13/160.29 Found eq_ref00:=(eq_ref0 (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy)))))):(((eq Prop) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))
% 160.13/160.29 Found (eq_ref0 (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy)))))) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy)))))) b)
% 160.13/160.29 Found ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy)))))) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy)))))) b)
% 160.13/160.29 Found ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy)))))) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy)))))) b)
% 164.34/164.56 Found ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy)))))) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy)))))) b)
% 164.34/164.56 Found eq_sym000:=(eq_sym00 Y):((((eq fofType) x3) Y)->(((eq fofType) Y) x3))
% 164.34/164.56 Found (eq_sym00 Y) as proof of (((x0 Xx) Y)->(((eq fofType) x3) Y))
% 164.34/164.56 Found ((eq_sym0 x3) Y) as proof of (((x0 Xx) Y)->(((eq fofType) x3) Y))
% 164.34/164.56 Found (((eq_sym fofType) x3) Y) as proof of (((x0 Xx) Y)->(((eq fofType) x3) Y))
% 164.34/164.56 Found (((eq_sym fofType) x3) Y) as proof of (((x0 Xx) Y)->(((eq fofType) x3) Y))
% 164.34/164.56 Found (((eq_sym fofType) x3) Y) as proof of (((x0 Xx) Y)->(((eq fofType) x3) Y))
% 164.34/164.56 Found eq_sym000:=(eq_sym00 Y):((((eq fofType) x3) Y)->(((eq fofType) Y) x3))
% 164.34/164.56 Found (eq_sym00 Y) as proof of (((x2 Xx) Y)->(((eq fofType) x3) Y))
% 164.34/164.56 Found ((eq_sym0 x3) Y) as proof of (((x2 Xx) Y)->(((eq fofType) x3) Y))
% 164.34/164.56 Found (((eq_sym fofType) x3) Y) as proof of (((x2 Xx) Y)->(((eq fofType) x3) Y))
% 164.34/164.56 Found (((eq_sym fofType) x3) Y) as proof of (((x2 Xx) Y)->(((eq fofType) x3) Y))
% 164.34/164.56 Found (((eq_sym fofType) x3) Y) as proof of (((x2 Xx) Y)->(((eq fofType) x3) Y))
% 164.34/164.56 Found eq_ref000:=(eq_ref00 P):((P x30)->(P x30))
% 164.34/164.56 Found (eq_ref00 P) as proof of ((P x30)->(P x3))
% 164.34/164.56 Found ((eq_ref0 x30) P) as proof of ((P x30)->(P x3))
% 164.34/164.56 Found (((eq_ref fofType) x30) P) as proof of ((P x30)->(P x3))
% 164.34/164.56 Found (((eq_ref fofType) x30) P) as proof of ((P x30)->(P x3))
% 164.34/164.56 Found (fun (P:(fofType->Prop))=> (((eq_ref fofType) x30) P)) as proof of ((P x30)->(P x3))
% 164.34/164.56 Found x6:(P x30)
% 164.34/164.56 Instantiate: x3:=x30:fofType
% 164.34/164.56 Found (fun (x6:(P x30))=> x6) as proof of (P x3)
% 164.34/164.56 Found (fun (P:(fofType->Prop)) (x6:(P x30))=> x6) as proof of ((P x30)->(P x3))
% 164.34/164.56 Found eq_ref00:=(eq_ref0 x3):(((eq fofType) x3) x3)
% 164.34/164.56 Found (eq_ref0 x3) as proof of (((eq fofType) x3) Y)
% 164.34/164.56 Found ((eq_ref fofType) x3) as proof of (((eq fofType) x3) Y)
% 164.34/164.56 Found ((eq_ref fofType) x3) as proof of (((eq fofType) x3) Y)
% 164.34/164.56 Found (fun (x00:((x0 Xx) Y))=> ((eq_ref fofType) x3)) as proof of (((eq fofType) x3) Y)
% 164.34/164.56 Found eq_ref00:=(eq_ref0 x3):(((eq fofType) x3) x3)
% 164.34/164.56 Found (eq_ref0 x3) as proof of (((eq fofType) x3) Y)
% 164.34/164.56 Found ((eq_ref fofType) x3) as proof of (((eq fofType) x3) Y)
% 164.34/164.56 Found ((eq_ref fofType) x3) as proof of (((eq fofType) x3) Y)
% 164.34/164.56 Found (fun (x00:((x2 Xx) Y))=> ((eq_ref fofType) x3)) as proof of (((eq fofType) x3) Y)
% 164.34/164.56 Found eq_substitution:=(fun (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)) (H:(((eq T) a) b))=> ((H (fun (x:T)=> (((eq U) (f a)) (f x)))) ((eq_ref U) (f a)))):(forall (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)), ((((eq T) a) b)->(((eq U) (f a)) (f b))))
% 164.34/164.56 Instantiate: b:=(forall (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)), ((((eq T) a) b)->(((eq U) (f a)) (f b)))):Prop
% 164.34/164.56 Found eq_substitution as proof of b
% 164.34/164.56 Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% 164.34/164.56 Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) b)
% 164.34/164.56 Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) b)
% 164.34/164.56 Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) b)
% 164.34/164.56 Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) b)
% 164.34/164.56 Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% 164.34/164.56 Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((and ((x2 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x2 Xx) Xy)->((x2 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x2 Xx) X)) (forall (Y:fofType), (((x2 Xx) Y)->(((eq fofType) X) Y)))))))))
% 164.34/164.56 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and ((x2 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x2 Xx) Xy)->((x2 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x2 Xx) X)) (forall (Y:fofType), (((x2 Xx) Y)->(((eq fofType) X) Y)))))))))
% 164.34/164.56 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and ((x2 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x2 Xx) Xy)->((x2 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x2 Xx) X)) (forall (Y:fofType), (((x2 Xx) Y)->(((eq fofType) X) Y)))))))))
% 164.34/164.56 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and ((x2 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x2 Xx) Xy)->((x2 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x2 Xx) X)) (forall (Y:fofType), (((x2 Xx) Y)->(((eq fofType) X) Y)))))))))
% 164.34/164.58 Found ((eq_trans0000 ((eq_ref Prop) (f x2))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f x2)) ((and ((and ((x2 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x2 Xx) Xy)->((x2 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x2 Xx) X)) (forall (Y:fofType), (((x2 Xx) Y)->(((eq fofType) X) Y)))))))))
% 164.34/164.58 Found (((eq_trans000 ((and ((and ((x2 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x2 Xx) Xy)->((x2 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x2 Xx) X)) (forall (Y:fofType), (((x2 Xx) Y)->(((eq fofType) X) Y))))))))) ((eq_ref Prop) (f x2))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f x2)) ((and ((and ((x2 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x2 Xx) Xy)->((x2 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x2 Xx) X)) (forall (Y:fofType), (((x2 Xx) Y)->(((eq fofType) X) Y)))))))))
% 164.34/164.58 Found ((((eq_trans00 ((and ((and ((x2 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x2 Xx) Xy)->((x2 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x2 Xx) X)) (forall (Y:fofType), (((x2 Xx) Y)->(((eq fofType) X) Y))))))))) ((and ((and ((x2 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x2 Xx) Xy)->((x2 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x2 Xx) X)) (forall (Y:fofType), (((x2 Xx) Y)->(((eq fofType) X) Y))))))))) ((eq_ref Prop) (f x2))) ((eq_ref Prop) ((and ((and ((x2 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x2 Xx) Xy)->((x2 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x2 Xx) X)) (forall (Y:fofType), (((x2 Xx) Y)->(((eq fofType) X) Y)))))))))) as proof of (((eq Prop) (f x2)) ((and ((and ((x2 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x2 Xx) Xy)->((x2 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x2 Xx) X)) (forall (Y:fofType), (((x2 Xx) Y)->(((eq fofType) X) Y)))))))))
% 164.34/164.58 Found (((((eq_trans0 (f x2)) ((and ((and ((x2 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x2 Xx) Xy)->((x2 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x2 Xx) X)) (forall (Y:fofType), (((x2 Xx) Y)->(((eq fofType) X) Y))))))))) ((and ((and ((x2 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x2 Xx) Xy)->((x2 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x2 Xx) X)) (forall (Y:fofType), (((x2 Xx) Y)->(((eq fofType) X) Y))))))))) ((eq_ref Prop) (f x2))) ((eq_ref Prop) ((and ((and ((x2 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x2 Xx) Xy)->((x2 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x2 Xx) X)) (forall (Y:fofType), (((x2 Xx) Y)->(((eq fofType) X) Y)))))))))) as proof of (((eq Prop) (f x2)) ((and ((and ((x2 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x2 Xx) Xy)->((x2 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x2 Xx) X)) (forall (Y:fofType), (((x2 Xx) Y)->(((eq fofType) X) Y)))))))))
% 164.34/164.58 Found ((((((eq_trans Prop) (f x2)) ((and ((and ((x2 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x2 Xx) Xy)->((x2 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x2 Xx) X)) (forall (Y:fofType), (((x2 Xx) Y)->(((eq fofType) X) Y))))))))) ((and ((and ((x2 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x2 Xx) Xy)->((x2 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x2 Xx) X)) (forall (Y:fofType), (((x2 Xx) Y)->(((eq fofType) X) Y))))))))) ((eq_ref Prop) (f x2))) ((eq_ref Prop) ((and ((and ((x2 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x2 Xx) Xy)->((x2 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x2 Xx) X)) (forall (Y:fofType), (((x2 Xx) Y)->(((eq fofType) X) Y)))))))))) as proof of (((eq Prop) (f x2)) ((and ((and ((x2 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x2 Xx) Xy)->((x2 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x2 Xx) X)) (forall (Y:fofType), (((x2 Xx) Y)->(((eq fofType) X) Y)))))))))
% 164.52/164.74 Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% 164.52/164.74 Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) b)
% 164.52/164.74 Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) b)
% 164.52/164.74 Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) b)
% 164.52/164.74 Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) b)
% 164.52/164.74 Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% 164.52/164.74 Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((and ((x2 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x2 Xx) Xy)->((x2 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x2 Xx) X)) (forall (Y:fofType), (((x2 Xx) Y)->(((eq fofType) X) Y)))))))))
% 164.52/164.74 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and ((x2 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x2 Xx) Xy)->((x2 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x2 Xx) X)) (forall (Y:fofType), (((x2 Xx) Y)->(((eq fofType) X) Y)))))))))
% 164.52/164.74 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and ((x2 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x2 Xx) Xy)->((x2 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x2 Xx) X)) (forall (Y:fofType), (((x2 Xx) Y)->(((eq fofType) X) Y)))))))))
% 164.52/164.74 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and ((x2 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x2 Xx) Xy)->((x2 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x2 Xx) X)) (forall (Y:fofType), (((x2 Xx) Y)->(((eq fofType) X) Y)))))))))
% 164.52/164.74 Found ((eq_trans0000 ((eq_ref Prop) (f x2))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f x2)) ((and ((and ((x2 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x2 Xx) Xy)->((x2 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x2 Xx) X)) (forall (Y:fofType), (((x2 Xx) Y)->(((eq fofType) X) Y)))))))))
% 164.52/164.74 Found (((eq_trans000 ((and ((and ((x2 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x2 Xx) Xy)->((x2 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x2 Xx) X)) (forall (Y:fofType), (((x2 Xx) Y)->(((eq fofType) X) Y))))))))) ((eq_ref Prop) (f x2))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f x2)) ((and ((and ((x2 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x2 Xx) Xy)->((x2 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x2 Xx) X)) (forall (Y:fofType), (((x2 Xx) Y)->(((eq fofType) X) Y)))))))))
% 164.52/164.74 Found ((((eq_trans00 ((and ((and ((x2 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x2 Xx) Xy)->((x2 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x2 Xx) X)) (forall (Y:fofType), (((x2 Xx) Y)->(((eq fofType) X) Y))))))))) ((and ((and ((x2 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x2 Xx) Xy)->((x2 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x2 Xx) X)) (forall (Y:fofType), (((x2 Xx) Y)->(((eq fofType) X) Y))))))))) ((eq_ref Prop) (f x2))) ((eq_ref Prop) ((and ((and ((x2 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x2 Xx) Xy)->((x2 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x2 Xx) X)) (forall (Y:fofType), (((x2 Xx) Y)->(((eq fofType) X) Y)))))))))) as proof of (((eq Prop) (f x2)) ((and ((and ((x2 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x2 Xx) Xy)->((x2 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x2 Xx) X)) (forall (Y:fofType), (((x2 Xx) Y)->(((eq fofType) X) Y)))))))))
% 164.52/164.74 Found (((((eq_trans0 (f x2)) ((and ((and ((x2 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x2 Xx) Xy)->((x2 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x2 Xx) X)) (forall (Y:fofType), (((x2 Xx) Y)->(((eq fofType) X) Y))))))))) ((and ((and ((x2 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x2 Xx) Xy)->((x2 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x2 Xx) X)) (forall (Y:fofType), (((x2 Xx) Y)->(((eq fofType) X) Y))))))))) ((eq_ref Prop) (f x2))) ((eq_ref Prop) ((and ((and ((x2 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x2 Xx) Xy)->((x2 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x2 Xx) X)) (forall (Y:fofType), (((x2 Xx) Y)->(((eq fofType) X) Y)))))))))) as proof of (((eq Prop) (f x2)) ((and ((and ((x2 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x2 Xx) Xy)->((x2 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x2 Xx) X)) (forall (Y:fofType), (((x2 Xx) Y)->(((eq fofType) X) Y)))))))))
% 164.66/164.88 Found ((((((eq_trans Prop) (f x2)) ((and ((and ((x2 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x2 Xx) Xy)->((x2 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x2 Xx) X)) (forall (Y:fofType), (((x2 Xx) Y)->(((eq fofType) X) Y))))))))) ((and ((and ((x2 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x2 Xx) Xy)->((x2 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x2 Xx) X)) (forall (Y:fofType), (((x2 Xx) Y)->(((eq fofType) X) Y))))))))) ((eq_ref Prop) (f x2))) ((eq_ref Prop) ((and ((and ((x2 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x2 Xx) Xy)->((x2 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x2 Xx) X)) (forall (Y:fofType), (((x2 Xx) Y)->(((eq fofType) X) Y)))))))))) as proof of (((eq Prop) (f x2)) ((and ((and ((x2 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x2 Xx) Xy)->((x2 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x2 Xx) X)) (forall (Y:fofType), (((x2 Xx) Y)->(((eq fofType) X) Y)))))))))
% 164.66/164.88 Found eq_substitution0000:=(fun (a:fofType)=> ((eq_substitution000 a) (fun (x8:fofType)=> (cS Xx)))):(forall (a:fofType), ((((eq fofType) a) (cS Xy))->(((eq fofType) (cS Xx)) (cS Xx))))
% 164.66/164.88 Instantiate: x0:=(fun (x6:fofType) (x50:fofType)=> (forall (a:fofType), ((((eq fofType) a) (cS Xy))->(((eq fofType) x6) x6)))):(fofType->(fofType->Prop))
% 164.66/164.88 Found (fun (a:fofType)=> ((eq_substitution000 a) (fun (x8:fofType)=> (cS Xx)))) as proof of ((x0 (cS Xx)) (cS (cS Xy)))
% 164.66/164.88 Found (fun (a:fofType)=> (((fun (a:fofType)=> ((eq_substitution00 a) (cS Xy))) a) (fun (x8:fofType)=> (cS Xx)))) as proof of ((x0 (cS Xx)) (cS (cS Xy)))
% 164.66/164.88 Found (fun (a:fofType)=> (((fun (a:fofType)=> (((eq_substitution0 fofType) a) (cS Xy))) a) (fun (x8:fofType)=> (cS Xx)))) as proof of ((x0 (cS Xx)) (cS (cS Xy)))
% 164.66/164.88 Found (fun (a:fofType)=> (((fun (a:fofType)=> ((((eq_substitution fofType) fofType) a) (cS Xy))) a) (fun (x8:fofType)=> (cS Xx)))) as proof of ((x0 (cS Xx)) (cS (cS Xy)))
% 164.66/164.88 Found (fun (x4:(forall (Xx0:fofType) (Xy0:fofType), ((((eq fofType) (cS Xx0)) (cS Xy0))->(((eq fofType) Xx0) Xy0)))) (a:fofType)=> (((fun (a:fofType)=> ((((eq_substitution fofType) fofType) a) (cS Xy))) a) (fun (x8:fofType)=> (cS Xx)))) as proof of ((x0 (cS Xx)) (cS (cS Xy)))
% 164.66/164.88 Found (fun (x3:cIND) (x4:(forall (Xx0:fofType) (Xy0:fofType), ((((eq fofType) (cS Xx0)) (cS Xy0))->(((eq fofType) Xx0) Xy0)))) (a:fofType)=> (((fun (a:fofType)=> ((((eq_substitution fofType) fofType) a) (cS Xy))) a) (fun (x8:fofType)=> (cS Xx)))) as proof of ((forall (Xx0:fofType) (Xy0:fofType), ((((eq fofType) (cS Xx0)) (cS Xy0))->(((eq fofType) Xx0) Xy0)))->((x0 (cS Xx)) (cS (cS Xy))))
% 164.66/164.88 Found (fun (x3:cIND) (x4:(forall (Xx0:fofType) (Xy0:fofType), ((((eq fofType) (cS Xx0)) (cS Xy0))->(((eq fofType) Xx0) Xy0)))) (a:fofType)=> (((fun (a:fofType)=> ((((eq_substitution fofType) fofType) a) (cS Xy))) a) (fun (x8:fofType)=> (cS Xx)))) as proof of (cIND->((forall (Xx0:fofType) (Xy0:fofType), ((((eq fofType) (cS Xx0)) (cS Xy0))->(((eq fofType) Xx0) Xy0)))->((x0 (cS Xx)) (cS (cS Xy)))))
% 164.66/164.88 Found (and_rect10 (fun (x3:cIND) (x4:(forall (Xx0:fofType) (Xy0:fofType), ((((eq fofType) (cS Xx0)) (cS Xy0))->(((eq fofType) Xx0) Xy0)))) (a:fofType)=> (((fun (a:fofType)=> ((((eq_substitution fofType) fofType) a) (cS Xy))) a) (fun (x8:fofType)=> (cS Xx))))) as proof of ((x0 (cS Xx)) (cS (cS Xy)))
% 165.71/165.90 Found ((and_rect1 ((x0 (cS Xx)) (cS (cS Xy)))) (fun (x3:cIND) (x4:(forall (Xx0:fofType) (Xy0:fofType), ((((eq fofType) (cS Xx0)) (cS Xy0))->(((eq fofType) Xx0) Xy0)))) (a:fofType)=> (((fun (a:fofType)=> ((((eq_substitution fofType) fofType) a) (cS Xy))) a) (fun (x8:fofType)=> (cS Xx))))) as proof of ((x0 (cS Xx)) (cS (cS Xy)))
% 165.71/165.90 Found (((fun (P:Type) (x3:(cIND->((forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (cS Xx)) (cS Xy))->(((eq fofType) Xx) Xy)))->P)))=> (((((and_rect cIND) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (cS Xx)) (cS Xy))->(((eq fofType) Xx) Xy)))) P) x3) x1)) ((x0 (cS Xx)) (cS (cS Xy)))) (fun (x3:cIND) (x4:(forall (Xx0:fofType) (Xy0:fofType), ((((eq fofType) (cS Xx0)) (cS Xy0))->(((eq fofType) Xx0) Xy0)))) (a:fofType)=> (((fun (a:fofType)=> ((((eq_substitution fofType) fofType) a) (cS Xy))) a) (fun (x8:fofType)=> (cS Xx))))) as proof of ((x0 (cS Xx)) (cS (cS Xy)))
% 165.71/165.90 Found (fun (x00:((x0 Xx) Xy))=> (((fun (P:Type) (x3:(cIND->((forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (cS Xx)) (cS Xy))->(((eq fofType) Xx) Xy)))->P)))=> (((((and_rect cIND) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (cS Xx)) (cS Xy))->(((eq fofType) Xx) Xy)))) P) x3) x1)) ((x0 (cS Xx)) (cS (cS Xy)))) (fun (x3:cIND) (x4:(forall (Xx0:fofType) (Xy0:fofType), ((((eq fofType) (cS Xx0)) (cS Xy0))->(((eq fofType) Xx0) Xy0)))) (a:fofType)=> (((fun (a:fofType)=> ((((eq_substitution fofType) fofType) a) (cS Xy))) a) (fun (x8:fofType)=> (cS Xx)))))) as proof of ((x0 (cS Xx)) (cS (cS Xy)))
% 165.71/165.90 Found eq_ref00:=(eq_ref0 x5):(((eq fofType) x5) x5)
% 165.71/165.90 Found (eq_ref0 x5) as proof of (((eq fofType) x5) x50)
% 165.71/165.90 Found ((eq_ref fofType) x5) as proof of (((eq fofType) x5) x50)
% 165.71/165.90 Found ((eq_ref fofType) x5) as proof of (((eq fofType) x5) x50)
% 165.71/165.90 Found ((eq_ref fofType) x5) as proof of (((eq fofType) x5) x50)
% 165.71/165.90 Found (eq_sym000 ((eq_ref fofType) x5)) as proof of (((eq fofType) x50) x5)
% 165.71/165.90 Found ((eq_sym00 x50) ((eq_ref fofType) x5)) as proof of (((eq fofType) x50) x5)
% 165.71/165.90 Found (((eq_sym0 x5) x50) ((eq_ref fofType) x5)) as proof of (((eq fofType) x50) x5)
% 165.71/165.90 Found ((((eq_sym fofType) x5) x50) ((eq_ref fofType) x5)) as proof of (((eq fofType) x50) x5)
% 165.71/165.90 Found x6:(P x5)
% 165.71/165.90 Instantiate: x5:=x':fofType
% 165.71/165.90 Found (fun (x6:(P x5))=> x6) as proof of (P x')
% 165.71/165.90 Found (fun (P:(fofType->Prop)) (x6:(P x5))=> x6) as proof of ((P x5)->(P x'))
% 165.71/165.90 Found (fun (P:(fofType->Prop)) (x6:(P x5))=> x6) as proof of (((eq fofType) x5) x')
% 165.71/165.90 Found (fun (x00:((x0 Xx) x')) (P:(fofType->Prop)) (x6:(P x5))=> x6) as proof of (((eq fofType) x5) x')
% 165.71/165.90 Found x00000:=(x0000 x3):(((eq fofType) x3) x')
% 165.71/165.90 Found (x0000 x3) as proof of (((eq fofType) x3) x')
% 165.71/165.90 Found ((fun (x30:fofType)=> ((x000 x30) x2)) x3) as proof of (((eq fofType) x3) x')
% 165.71/165.90 Found ((fun (x30:fofType)=> (((x00 x1) x30) x2)) x3) as proof of (((eq fofType) x3) x')
% 165.71/165.90 Found ((fun (x30:fofType)=> (((x00 x1) x30) x2)) x3) as proof of (((eq fofType) x3) x')
% 165.71/165.90 Found (fun (x00:((x0 Xx) x'))=> ((fun (x30:fofType)=> (((x00 x1) x30) x2)) x3)) as proof of (((eq fofType) x3) x')
% 165.71/165.90 Found (fun (x':fofType) (x00:((x0 Xx) x'))=> ((fun (x30:fofType)=> (((x00 x1) x30) x2)) x3)) as proof of (((x0 Xx) x')->(((eq fofType) x3) x'))
% 165.71/165.90 Found (fun (x':fofType) (x00:((x0 Xx) x'))=> ((fun (x30:fofType)=> (((x00 x1) x30) x2)) x3)) as proof of (forall (x':fofType), (((x0 Xx) x')->(((eq fofType) x3) x')))
% 165.71/165.90 Found eq_ref00:=(eq_ref0 x30):(((eq fofType) x30) x30)
% 165.71/165.90 Found (eq_ref0 x30) as proof of (((eq fofType) x30) x3)
% 165.71/165.90 Found ((eq_ref fofType) x30) as proof of (((eq fofType) x30) x3)
% 165.71/165.90 Found ((eq_ref fofType) x30) as proof of (((eq fofType) x30) x3)
% 165.71/165.90 Found (fun (x20:(forall (Xn:fofType), (not (((eq fofType) (cS Xn)) c0))))=> ((eq_ref fofType) x30)) as proof of (((eq fofType) x30) x3)
% 165.71/165.90 Found x4:(P x30)
% 165.71/165.90 Instantiate: x3:=x30:fofType
% 165.71/165.90 Found (fun (x4:(P x30))=> x4) as proof of (P x3)
% 165.71/165.90 Found (fun (P:(fofType->Prop)) (x4:(P x30))=> x4) as proof of ((P x30)->(P x3))
% 165.71/165.90 Found (fun (x20:(forall (Xn:fofType), (not (((eq fofType) (cS Xn)) c0)))) (P:(fofType->Prop)) (x4:(P x30))=> x4) as proof of (((eq fofType) x30) x3)
% 166.70/166.88 Found eq_ref000:=(eq_ref00 P):((P x30)->(P x30))
% 166.70/166.88 Found (eq_ref00 P) as proof of ((P x30)->(P x3))
% 166.70/166.88 Found ((eq_ref0 x30) P) as proof of ((P x30)->(P x3))
% 166.70/166.88 Found (((eq_ref fofType) x30) P) as proof of ((P x30)->(P x3))
% 166.70/166.88 Found (((eq_ref fofType) x30) P) as proof of ((P x30)->(P x3))
% 166.70/166.88 Found (fun (P:(fofType->Prop))=> (((eq_ref fofType) x30) P)) as proof of ((P x30)->(P x3))
% 166.70/166.88 Found (fun (x20:(forall (Xn:fofType), (not (((eq fofType) (cS Xn)) c0)))) (P:(fofType->Prop))=> (((eq_ref fofType) x30) P)) as proof of (((eq fofType) x30) x3)
% 166.70/166.88 Found eq_ref00:=(eq_ref0 x30):(((eq fofType) x30) x30)
% 166.70/166.88 Found (eq_ref0 x30) as proof of (((eq fofType) x30) x3)
% 166.70/166.88 Found ((eq_ref fofType) x30) as proof of (((eq fofType) x30) x3)
% 166.70/166.88 Found ((eq_ref fofType) x30) as proof of (((eq fofType) x30) x3)
% 166.70/166.88 Found (fun (x5:(forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (cS Xx)) (cS Xy))->(((eq fofType) Xx) Xy))))=> ((eq_ref fofType) x30)) as proof of (((eq fofType) x30) x3)
% 166.70/166.88 Found (fun (x4:cIND) (x5:(forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (cS Xx)) (cS Xy))->(((eq fofType) Xx) Xy))))=> ((eq_ref fofType) x30)) as proof of ((forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (cS Xx)) (cS Xy))->(((eq fofType) Xx) Xy)))->(((eq fofType) x30) x3))
% 166.70/166.88 Found (fun (x4:cIND) (x5:(forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (cS Xx)) (cS Xy))->(((eq fofType) Xx) Xy))))=> ((eq_ref fofType) x30)) as proof of (cIND->((forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (cS Xx)) (cS Xy))->(((eq fofType) Xx) Xy)))->(((eq fofType) x30) x3)))
% 166.70/166.88 Found (and_rect10 (fun (x4:cIND) (x5:(forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (cS Xx)) (cS Xy))->(((eq fofType) Xx) Xy))))=> ((eq_ref fofType) x30))) as proof of (((eq fofType) x30) x3)
% 166.70/166.88 Found ((and_rect1 (((eq fofType) x30) x3)) (fun (x4:cIND) (x5:(forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (cS Xx)) (cS Xy))->(((eq fofType) Xx) Xy))))=> ((eq_ref fofType) x30))) as proof of (((eq fofType) x30) x3)
% 166.70/166.88 Found (((fun (P:Type) (x4:(cIND->((forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (cS Xx)) (cS Xy))->(((eq fofType) Xx) Xy)))->P)))=> (((((and_rect cIND) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (cS Xx)) (cS Xy))->(((eq fofType) Xx) Xy)))) P) x4) x1)) (((eq fofType) x30) x3)) (fun (x4:cIND) (x5:(forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (cS Xx)) (cS Xy))->(((eq fofType) Xx) Xy))))=> ((eq_ref fofType) x30))) as proof of (((eq fofType) x30) x3)
% 166.70/166.88 Found (fun (x20:(forall (Xn:fofType), (not (((eq fofType) (cS Xn)) c0))))=> (((fun (P:Type) (x4:(cIND->((forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (cS Xx)) (cS Xy))->(((eq fofType) Xx) Xy)))->P)))=> (((((and_rect cIND) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (cS Xx)) (cS Xy))->(((eq fofType) Xx) Xy)))) P) x4) x1)) (((eq fofType) x30) x3)) (fun (x4:cIND) (x5:(forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (cS Xx)) (cS Xy))->(((eq fofType) Xx) Xy))))=> ((eq_ref fofType) x30)))) as proof of (((eq fofType) x30) x3)
% 166.70/166.88 Found x6:(P x5)
% 166.70/166.88 Instantiate: x5:=x':fofType
% 166.70/166.88 Found (fun (x6:(P x5))=> x6) as proof of (P x')
% 166.70/166.88 Found (fun (P:(fofType->Prop)) (x6:(P x5))=> x6) as proof of ((P x5)->(P x'))
% 166.70/166.88 Found (fun (P:(fofType->Prop)) (x6:(P x5))=> x6) as proof of (((eq fofType) x5) x')
% 166.70/166.88 Found (fun (x00:((x2 Xx) x')) (P:(fofType->Prop)) (x6:(P x5))=> x6) as proof of (((eq fofType) x5) x')
% 166.70/166.88 Found eq_ref000:=(eq_ref00 P):((P x5)->(P x5))
% 166.70/166.88 Found (eq_ref00 P) as proof of ((P x5)->(P x'))
% 166.70/166.88 Found ((eq_ref0 x5) P) as proof of ((P x5)->(P x'))
% 166.70/166.88 Found (((eq_ref fofType) x5) P) as proof of ((P x5)->(P x'))
% 166.70/166.88 Found (((eq_ref fofType) x5) P) as proof of ((P x5)->(P x'))
% 166.70/166.88 Found (fun (P:(fofType->Prop))=> (((eq_ref fofType) x5) P)) as proof of ((P x5)->(P x'))
% 166.70/166.88 Found (fun (P:(fofType->Prop))=> (((eq_ref fofType) x5) P)) as proof of (((eq fofType) x5) x')
% 166.70/166.88 Found (fun (x00:((x0 Xx) x')) (P:(fofType->Prop))=> (((eq_ref fofType) x5) P)) as proof of (((eq fofType) x5) x')
% 166.70/166.88 Found eq_ref000:=(eq_ref00 P):((P x5)->(P x5))
% 166.70/166.88 Found (eq_ref00 P) as proof of ((P x5)->(P x'))
% 166.70/166.88 Found ((eq_ref0 x5) P) as proof of ((P x5)->(P x'))
% 166.70/166.88 Found (((eq_ref fofType) x5) P) as proof of ((P x5)->(P x'))
% 166.70/166.88 Found (((eq_ref fofType) x5) P) as proof of ((P x5)->(P x'))
% 172.02/172.24 Found (fun (P:(fofType->Prop))=> (((eq_ref fofType) x5) P)) as proof of ((P x5)->(P x'))
% 172.02/172.24 Found (fun (P:(fofType->Prop))=> (((eq_ref fofType) x5) P)) as proof of (((eq fofType) x5) x')
% 172.02/172.24 Found (fun (x00:((x2 Xx) x')) (P:(fofType->Prop))=> (((eq_ref fofType) x5) P)) as proof of (((eq fofType) x5) x')
% 172.02/172.24 Found eq_ref000:=(eq_ref00 P0):((P0 (f x2))->(P0 (f x2)))
% 172.02/172.24 Found (eq_ref00 P0) as proof of (P1 (f x2))
% 172.02/172.24 Found ((eq_ref0 (f x2)) P0) as proof of (P1 (f x2))
% 172.02/172.24 Found (((eq_ref Prop) (f x2)) P0) as proof of (P1 (f x2))
% 172.02/172.24 Found (((eq_ref Prop) (f x2)) P0) as proof of (P1 (f x2))
% 172.02/172.24 Found eq_ref000:=(eq_ref00 P0):((P0 (f x2))->(P0 (f x2)))
% 172.02/172.24 Found (eq_ref00 P0) as proof of (P1 (f x2))
% 172.02/172.24 Found ((eq_ref0 (f x2)) P0) as proof of (P1 (f x2))
% 172.02/172.24 Found (((eq_ref Prop) (f x2)) P0) as proof of (P1 (f x2))
% 172.02/172.24 Found (((eq_ref Prop) (f x2)) P0) as proof of (P1 (f x2))
% 172.02/172.24 Found eq_sym000:=(eq_sym00 x'):((((eq fofType) x5) x')->(((eq fofType) x') x5))
% 172.02/172.24 Found (eq_sym00 x') as proof of (((x0 Xx) x')->(((eq fofType) x5) x'))
% 172.02/172.24 Found ((eq_sym0 x5) x') as proof of (((x0 Xx) x')->(((eq fofType) x5) x'))
% 172.02/172.24 Found (((eq_sym fofType) x5) x') as proof of (((x0 Xx) x')->(((eq fofType) x5) x'))
% 172.02/172.24 Found (((eq_sym fofType) x5) x') as proof of (((x0 Xx) x')->(((eq fofType) x5) x'))
% 172.02/172.24 Found (((eq_sym fofType) x5) x') as proof of (((x0 Xx) x')->(((eq fofType) x5) x'))
% 172.02/172.24 Found eq_sym000:=(eq_sym00 x'):((((eq fofType) x5) x')->(((eq fofType) x') x5))
% 172.02/172.24 Found (eq_sym00 x') as proof of (((x2 Xx) x')->(((eq fofType) x5) x'))
% 172.02/172.24 Found ((eq_sym0 x5) x') as proof of (((x2 Xx) x')->(((eq fofType) x5) x'))
% 172.02/172.24 Found (((eq_sym fofType) x5) x') as proof of (((x2 Xx) x')->(((eq fofType) x5) x'))
% 172.02/172.24 Found (((eq_sym fofType) x5) x') as proof of (((x2 Xx) x')->(((eq fofType) x5) x'))
% 172.02/172.24 Found (((eq_sym fofType) x5) x') as proof of (((x2 Xx) x')->(((eq fofType) x5) x'))
% 172.02/172.24 Found eq_ref00:=(eq_ref0 x10):(((eq fofType) x10) x10)
% 172.02/172.24 Found (eq_ref0 x10) as proof of (((eq fofType) x10) x1)
% 172.02/172.24 Found ((eq_ref fofType) x10) as proof of (((eq fofType) x10) x1)
% 172.02/172.24 Found ((eq_ref fofType) x10) as proof of (((eq fofType) x10) x1)
% 172.02/172.24 Found eq_ref00:=(eq_ref0 x5):(((eq fofType) x5) x5)
% 172.02/172.24 Found (eq_ref0 x5) as proof of (((eq fofType) x5) x')
% 172.02/172.24 Found ((eq_ref fofType) x5) as proof of (((eq fofType) x5) x')
% 172.02/172.24 Found ((eq_ref fofType) x5) as proof of (((eq fofType) x5) x')
% 172.02/172.24 Found (fun (x00:((x0 Xx) x'))=> ((eq_ref fofType) x5)) as proof of (((eq fofType) x5) x')
% 172.02/172.24 Found eq_ref00:=(eq_ref0 x1):(((eq fofType) x1) x1)
% 172.02/172.24 Found (eq_ref0 x1) as proof of (((eq fofType) x1) x10)
% 172.02/172.24 Found ((eq_ref fofType) x1) as proof of (((eq fofType) x1) x10)
% 172.02/172.24 Found ((eq_ref fofType) x1) as proof of (((eq fofType) x1) x10)
% 172.02/172.24 Found ((eq_ref fofType) x1) as proof of (((eq fofType) x1) x10)
% 172.02/172.24 Found (eq_sym000 ((eq_ref fofType) x1)) as proof of (((eq fofType) x10) x1)
% 172.02/172.24 Found ((eq_sym00 x10) ((eq_ref fofType) x1)) as proof of (((eq fofType) x10) x1)
% 172.02/172.24 Found (((eq_sym0 x1) x10) ((eq_ref fofType) x1)) as proof of (((eq fofType) x10) x1)
% 172.02/172.24 Found ((((eq_sym fofType) x1) x10) ((eq_ref fofType) x1)) as proof of (((eq fofType) x10) x1)
% 172.02/172.24 Found eq_ref00:=(eq_ref0 x5):(((eq fofType) x5) x5)
% 172.02/172.24 Found (eq_ref0 x5) as proof of (((eq fofType) x5) x')
% 172.02/172.24 Found ((eq_ref fofType) x5) as proof of (((eq fofType) x5) x')
% 172.02/172.24 Found ((eq_ref fofType) x5) as proof of (((eq fofType) x5) x')
% 172.02/172.24 Found (fun (x00:((x2 Xx) x'))=> ((eq_ref fofType) x5)) as proof of (((eq fofType) x5) x')
% 172.02/172.24 Found x4:(P x1)
% 172.02/172.24 Instantiate: x1:=x':fofType
% 172.02/172.24 Found (fun (x4:(P x1))=> x4) as proof of (P x')
% 172.02/172.24 Found (fun (P:(fofType->Prop)) (x4:(P x1))=> x4) as proof of ((P x1)->(P x'))
% 172.02/172.24 Found (fun (P:(fofType->Prop)) (x4:(P x1))=> x4) as proof of (((eq fofType) x1) x')
% 172.02/172.24 Found (fun (x00:((x0 Xx) x')) (P:(fofType->Prop)) (x4:(P x1))=> x4) as proof of (((eq fofType) x1) x')
% 172.02/172.24 Found eq_ref000:=(eq_ref00 P):((P x1)->(P x1))
% 172.02/172.24 Found (eq_ref00 P) as proof of ((P x1)->(P x'))
% 172.02/172.24 Found ((eq_ref0 x1) P) as proof of ((P x1)->(P x'))
% 172.02/172.24 Found (((eq_ref fofType) x1) P) as proof of ((P x1)->(P x'))
% 172.02/172.24 Found (((eq_ref fofType) x1) P) as proof of ((P x1)->(P x'))
% 172.02/172.24 Found (fun (P:(fofType->Prop))=> (((eq_ref fofType) x1) P)) as proof of ((P x1)->(P x'))
% 172.02/172.24 Found (fun (P:(fofType->Prop))=> (((eq_ref fofType) x1) P)) as proof of (((eq fofType) x1) x')
% 175.03/175.23 Found (fun (x00:((x0 Xx) x')) (P:(fofType->Prop))=> (((eq_ref fofType) x1) P)) as proof of (((eq fofType) x1) x')
% 175.03/175.23 Found x00000:=(x0000 x5):(((eq fofType) x5) x')
% 175.03/175.23 Found (x0000 x5) as proof of (((eq fofType) x5) x')
% 175.03/175.23 Found ((x000 x4) x5) as proof of (((eq fofType) x5) x')
% 175.03/175.23 Found (((x00 x3) x4) x5) as proof of (((eq fofType) x5) x')
% 175.03/175.23 Found (((x00 x3) x4) x5) as proof of (((eq fofType) x5) x')
% 175.03/175.23 Found (fun (x00:((x2 Xx) x'))=> (((x00 x3) x4) x5)) as proof of (((eq fofType) x5) x')
% 175.03/175.23 Found (fun (x':fofType) (x00:((x2 Xx) x'))=> (((x00 x3) x4) x5)) as proof of (((x2 Xx) x')->(((eq fofType) x5) x'))
% 175.03/175.23 Found (fun (x':fofType) (x00:((x2 Xx) x'))=> (((x00 x3) x4) x5)) as proof of (forall (x':fofType), (((x2 Xx) x')->(((eq fofType) x5) x')))
% 175.03/175.23 Found eq_ref00:=(eq_ref0 x50):(((eq fofType) x50) x50)
% 175.03/175.23 Found (eq_ref0 x50) as proof of (((eq fofType) x50) x5)
% 175.03/175.23 Found ((eq_ref fofType) x50) as proof of (((eq fofType) x50) x5)
% 175.03/175.23 Found ((eq_ref fofType) x50) as proof of (((eq fofType) x50) x5)
% 175.03/175.23 Found x6:(P x50)
% 175.03/175.23 Instantiate: x5:=x50:fofType
% 175.03/175.23 Found (fun (x6:(P x50))=> x6) as proof of (P x5)
% 175.03/175.23 Found (fun (P:(fofType->Prop)) (x6:(P x50))=> x6) as proof of ((P x50)->(P x5))
% 175.03/175.23 Found eq_ref000:=(eq_ref00 P):((P x50)->(P x50))
% 175.03/175.23 Found (eq_ref00 P) as proof of ((P x50)->(P x5))
% 175.03/175.23 Found ((eq_ref0 x50) P) as proof of ((P x50)->(P x5))
% 175.03/175.23 Found (((eq_ref fofType) x50) P) as proof of ((P x50)->(P x5))
% 175.03/175.23 Found (((eq_ref fofType) x50) P) as proof of ((P x50)->(P x5))
% 175.03/175.23 Found (fun (P:(fofType->Prop))=> (((eq_ref fofType) x50) P)) as proof of ((P x50)->(P x5))
% 175.03/175.23 Found eq_ref00:=(eq_ref0 x10):(((eq fofType) x10) x10)
% 175.03/175.23 Found (eq_ref0 x10) as proof of (((eq fofType) x10) x1)
% 175.03/175.23 Found ((eq_ref fofType) x10) as proof of (((eq fofType) x10) x1)
% 175.03/175.23 Found ((eq_ref fofType) x10) as proof of (((eq fofType) x10) x1)
% 175.03/175.23 Found (fun (x5:(forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy))))=> ((eq_ref fofType) x10)) as proof of (((eq fofType) x10) x1)
% 175.03/175.23 Found (fun (x4:cIND) (x5:(forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy))))=> ((eq_ref fofType) x10)) as proof of ((forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy)))->(((eq fofType) x10) x1))
% 175.03/175.23 Found (fun (x4:cIND) (x5:(forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy))))=> ((eq_ref fofType) x10)) as proof of (cIND->((forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy)))->(((eq fofType) x10) x1)))
% 175.03/175.23 Found (and_rect10 (fun (x4:cIND) (x5:(forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy))))=> ((eq_ref fofType) x10))) as proof of (((eq fofType) x10) x1)
% 175.03/175.23 Found ((and_rect1 (((eq fofType) x10) x1)) (fun (x4:cIND) (x5:(forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy))))=> ((eq_ref fofType) x10))) as proof of (((eq fofType) x10) x1)
% 175.03/175.23 Found (((fun (P:Type) (x4:(cIND->((forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy)))->P)))=> (((((and_rect cIND) (forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy)))) P) x4) x2)) (((eq fofType) x10) x1)) (fun (x4:cIND) (x5:(forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy))))=> ((eq_ref fofType) x10))) as proof of (((eq fofType) x10) x1)
% 175.03/175.23 Found eq_ref00:=(eq_ref0 ((unique fofType) (x0 Xx))):(((eq (fofType->Prop)) ((unique fofType) (x0 Xx))) ((unique fofType) (x0 Xx)))
% 175.03/175.23 Found (eq_ref0 ((unique fofType) (x0 Xx))) as proof of (((eq (fofType->Prop)) ((unique fofType) (x0 Xx))) b)
% 175.03/175.23 Found ((eq_ref (fofType->Prop)) ((unique fofType) (x0 Xx))) as proof of (((eq (fofType->Prop)) ((unique fofType) (x0 Xx))) b)
% 175.03/175.23 Found ((eq_ref (fofType->Prop)) ((unique fofType) (x0 Xx))) as proof of (((eq (fofType->Prop)) ((unique fofType) (x0 Xx))) b)
% 175.03/175.23 Found ((eq_ref (fofType->Prop)) ((unique fofType) (x0 Xx))) as proof of (((eq (fofType->Prop)) ((unique fofType) (x0 Xx))) b)
% 178.15/178.41 Found eq_substitution:=(fun (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)) (H:(((eq T) a) b))=> ((H (fun (x:T)=> (((eq U) (f a)) (f x)))) ((eq_ref U) (f a)))):(forall (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)), ((((eq T) a) b)->(((eq U) (f a)) (f b))))
% 178.15/178.41 Instantiate: b:=(forall (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)), ((((eq T) a) b)->(((eq U) (f a)) (f b)))):Prop
% 178.15/178.41 Found eq_substitution as proof of b
% 178.15/178.41 Found eq_ref000:=(eq_ref00 P):((P x5)->(P x5))
% 178.15/178.41 Found (eq_ref00 P) as proof of ((P x5)->(P Y))
% 178.15/178.41 Found ((eq_ref0 x5) P) as proof of ((P x5)->(P Y))
% 178.15/178.41 Found (((eq_ref fofType) x5) P) as proof of ((P x5)->(P Y))
% 178.15/178.41 Found (((eq_ref fofType) x5) P) as proof of ((P x5)->(P Y))
% 178.15/178.41 Found (fun (P:(fofType->Prop))=> (((eq_ref fofType) x5) P)) as proof of ((P x5)->(P Y))
% 178.15/178.41 Found (fun (x00:((x0 Xx) Y)) (P:(fofType->Prop))=> (((eq_ref fofType) x5) P)) as proof of (((eq fofType) x5) Y)
% 178.15/178.41 Found x6:(P x5)
% 178.15/178.41 Instantiate: x5:=Y:fofType
% 178.15/178.41 Found (fun (x6:(P x5))=> x6) as proof of (P Y)
% 178.15/178.41 Found (fun (P:(fofType->Prop)) (x6:(P x5))=> x6) as proof of ((P x5)->(P Y))
% 178.15/178.41 Found (fun (x00:((x0 Xx) Y)) (P:(fofType->Prop)) (x6:(P x5))=> x6) as proof of (((eq fofType) x5) Y)
% 178.15/178.41 Found eq_substitution:=(fun (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)) (H:(((eq T) a) b))=> ((H (fun (x:T)=> (((eq U) (f a)) (f x)))) ((eq_ref U) (f a)))):(forall (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)), ((((eq T) a) b)->(((eq U) (f a)) (f b))))
% 178.15/178.41 Instantiate: b:=(forall (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)), ((((eq T) a) b)->(((eq U) (f a)) (f b)))):Prop
% 178.15/178.41 Found eq_substitution as proof of b
% 178.15/178.41 Found eq_ref000:=(eq_ref00 P):((P x5)->(P x5))
% 178.15/178.41 Found (eq_ref00 P) as proof of ((P x5)->(P Y))
% 178.15/178.41 Found ((eq_ref0 x5) P) as proof of ((P x5)->(P Y))
% 178.15/178.41 Found (((eq_ref fofType) x5) P) as proof of ((P x5)->(P Y))
% 178.15/178.41 Found (((eq_ref fofType) x5) P) as proof of ((P x5)->(P Y))
% 178.15/178.41 Found (fun (P:(fofType->Prop))=> (((eq_ref fofType) x5) P)) as proof of ((P x5)->(P Y))
% 178.15/178.41 Found (fun (x00:((x2 Xx) Y)) (P:(fofType->Prop))=> (((eq_ref fofType) x5) P)) as proof of (((eq fofType) x5) Y)
% 178.15/178.41 Found x6:(P x5)
% 178.15/178.41 Instantiate: x5:=Y:fofType
% 178.15/178.41 Found (fun (x6:(P x5))=> x6) as proof of (P Y)
% 178.15/178.41 Found (fun (P:(fofType->Prop)) (x6:(P x5))=> x6) as proof of ((P x5)->(P Y))
% 178.15/178.41 Found (fun (x00:((x2 Xx) Y)) (P:(fofType->Prop)) (x6:(P x5))=> x6) as proof of (((eq fofType) x5) Y)
% 178.15/178.41 Found eq_ref00:=(eq_ref0 (cS x50)):(((eq fofType) (cS x50)) (cS x50))
% 178.15/178.41 Found (eq_ref0 (cS x50)) as proof of (((eq fofType) (cS x50)) (cS x5))
% 178.15/178.41 Found ((eq_ref fofType) (cS x50)) as proof of (((eq fofType) (cS x50)) (cS x5))
% 178.15/178.41 Found ((eq_ref fofType) (cS x50)) as proof of (((eq fofType) (cS x50)) (cS x5))
% 178.15/178.41 Found ((eq_ref fofType) (cS x50)) as proof of (((eq fofType) (cS x50)) (cS x5))
% 178.15/178.41 Found (x4000 ((eq_ref fofType) (cS x50))) as proof of (((eq fofType) x50) x5)
% 178.15/178.41 Found ((x400 x5) ((eq_ref fofType) (cS x50))) as proof of (((eq fofType) x50) x5)
% 178.15/178.41 Found (((x40 x50) x5) ((eq_ref fofType) (cS x50))) as proof of (((eq fofType) x50) x5)
% 178.15/178.41 Found eq_ref00:=(eq_ref0 (cS x50)):(((eq fofType) (cS x50)) (cS x50))
% 178.15/178.41 Found (eq_ref0 (cS x50)) as proof of (((eq fofType) (cS x50)) (cS x5))
% 178.15/178.41 Found ((eq_ref fofType) (cS x50)) as proof of (((eq fofType) (cS x50)) (cS x5))
% 178.15/178.41 Found ((eq_ref fofType) (cS x50)) as proof of (((eq fofType) (cS x50)) (cS x5))
% 178.15/178.41 Found ((eq_ref fofType) (cS x50)) as proof of (((eq fofType) (cS x50)) (cS x5))
% 178.15/178.41 Found (x410 ((eq_ref fofType) (cS x50))) as proof of (((eq fofType) x50) x5)
% 178.15/178.41 Found ((x41 x5) ((eq_ref fofType) (cS x50))) as proof of (((eq fofType) x50) x5)
% 178.15/178.41 Found (((x4 x50) x5) ((eq_ref fofType) (cS x50))) as proof of (((eq fofType) x50) x5)
% 178.15/178.41 Found eq_substitution:=(fun (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)) (H:(((eq T) a) b))=> ((H (fun (x:T)=> (((eq U) (f a)) (f x)))) ((eq_ref U) (f a)))):(forall (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)), ((((eq T) a) b)->(((eq U) (f a)) (f b))))
% 178.15/178.41 Instantiate: b:=(forall (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)), ((((eq T) a) b)->(((eq U) (f a)) (f b)))):Prop
% 178.15/178.41 Found eq_substitution as proof of b
% 178.15/178.41 Found x00000:=(x0000 x1):(((eq fofType) x1) x')
% 178.15/178.41 Found (x0000 x1) as proof of (((eq fofType) x1) x')
% 178.15/178.41 Found ((fun (x10:fofType)=> ((x000 x10) x2)) x1) as proof of (((eq fofType) x1) x')
% 180.24/180.45 Found ((fun (x10:fofType)=> (((fun (x10:fofType)=> ((x00 x10) x3)) x10) x2)) x1) as proof of (((eq fofType) x1) x')
% 180.24/180.45 Found ((fun (x10:fofType)=> (((fun (x10:fofType)=> ((x00 x10) x3)) x10) x2)) x1) as proof of (((eq fofType) x1) x')
% 180.24/180.45 Found (fun (x00:((x0 Xx) x'))=> ((fun (x10:fofType)=> (((fun (x10:fofType)=> ((x00 x10) x3)) x10) x2)) x1)) as proof of (((eq fofType) x1) x')
% 180.24/180.45 Found (fun (x':fofType) (x00:((x0 Xx) x'))=> ((fun (x10:fofType)=> (((fun (x10:fofType)=> ((x00 x10) x3)) x10) x2)) x1)) as proof of (((x0 Xx) x')->(((eq fofType) x1) x'))
% 180.24/180.45 Found (fun (x':fofType) (x00:((x0 Xx) x'))=> ((fun (x10:fofType)=> (((fun (x10:fofType)=> ((x00 x10) x3)) x10) x2)) x1)) as proof of (forall (x':fofType), (((x0 Xx) x')->(((eq fofType) x1) x')))
% 180.24/180.45 Found eq_ref00:=(eq_ref0 x10):(((eq fofType) x10) x10)
% 180.24/180.45 Found (eq_ref0 x10) as proof of (((eq fofType) x10) x1)
% 180.24/180.45 Found ((eq_ref fofType) x10) as proof of (((eq fofType) x10) x1)
% 180.24/180.45 Found ((eq_ref fofType) x10) as proof of (((eq fofType) x10) x1)
% 180.24/180.45 Found (fun (x20:((and cIND) (forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy)))))=> ((eq_ref fofType) x10)) as proof of (((eq fofType) x10) x1)
% 180.24/180.45 Found (fun (x30:(forall (Xn:fofType), (not (((eq fofType) (cS Xn)) c0)))) (x20:((and cIND) (forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy)))))=> ((eq_ref fofType) x10)) as proof of (((and cIND) (forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy))))->(((eq fofType) x10) x1))
% 180.24/180.45 Found x4:(P x10)
% 180.24/180.45 Instantiate: x1:=x10:fofType
% 180.24/180.45 Found (fun (x4:(P x10))=> x4) as proof of (P x1)
% 180.24/180.45 Found (fun (P:(fofType->Prop)) (x4:(P x10))=> x4) as proof of ((P x10)->(P x1))
% 180.24/180.45 Found (fun (x20:((and cIND) (forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy))))) (P:(fofType->Prop)) (x4:(P x10))=> x4) as proof of (((eq fofType) x10) x1)
% 180.24/180.45 Found (fun (x30:(forall (Xn:fofType), (not (((eq fofType) (cS Xn)) c0)))) (x20:((and cIND) (forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy))))) (P:(fofType->Prop)) (x4:(P x10))=> x4) as proof of (((and cIND) (forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy))))->(((eq fofType) x10) x1))
% 180.24/180.45 Found eq_ref000:=(eq_ref00 P):((P x10)->(P x10))
% 180.24/180.45 Found (eq_ref00 P) as proof of ((P x10)->(P x1))
% 180.24/180.45 Found ((eq_ref0 x10) P) as proof of ((P x10)->(P x1))
% 180.24/180.45 Found (((eq_ref fofType) x10) P) as proof of ((P x10)->(P x1))
% 180.24/180.45 Found (((eq_ref fofType) x10) P) as proof of ((P x10)->(P x1))
% 180.24/180.45 Found (fun (P:(fofType->Prop))=> (((eq_ref fofType) x10) P)) as proof of ((P x10)->(P x1))
% 180.24/180.45 Found (fun (x20:((and cIND) (forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy))))) (P:(fofType->Prop))=> (((eq_ref fofType) x10) P)) as proof of (((eq fofType) x10) x1)
% 180.24/180.45 Found (fun (x30:(forall (Xn:fofType), (not (((eq fofType) (cS Xn)) c0)))) (x20:((and cIND) (forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy))))) (P:(fofType->Prop))=> (((eq_ref fofType) x10) P)) as proof of (((and cIND) (forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy))))->(((eq fofType) x10) x1))
% 180.24/180.45 Found eq_ref00:=(eq_ref0 x5):(((eq fofType) x5) x5)
% 180.24/180.45 Found (eq_ref0 x5) as proof of (((eq fofType) x5) Y)
% 180.24/180.45 Found ((eq_ref fofType) x5) as proof of (((eq fofType) x5) Y)
% 180.24/180.45 Found ((eq_ref fofType) x5) as proof of (((eq fofType) x5) Y)
% 180.24/180.45 Found ((eq_ref fofType) x5) as proof of (((eq fofType) x5) Y)
% 180.24/180.45 Found (fun (x00:((x0 Xx) Y))=> ((eq_ref fofType) x5)) as proof of (((eq fofType) x5) Y)
% 180.24/180.45 Found eq_substitution:=(fun (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)) (H:(((eq T) a) b))=> ((H (fun (x:T)=> (((eq U) (f a)) (f x)))) ((eq_ref U) (f a)))):(forall (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)), ((((eq T) a) b)->(((eq U) (f a)) (f b))))
% 180.24/180.45 Instantiate: b:=(forall (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)), ((((eq T) a) b)->(((eq U) (f a)) (f b)))):Prop
% 180.24/180.45 Found eq_substitution as proof of b
% 180.24/180.45 Found eq_ref00:=(eq_ref0 x5):(((eq fofType) x5) x5)
% 182.62/182.82 Found (eq_ref0 x5) as proof of (((eq fofType) x5) Y)
% 182.62/182.82 Found ((eq_ref fofType) x5) as proof of (((eq fofType) x5) Y)
% 182.62/182.82 Found ((eq_ref fofType) x5) as proof of (((eq fofType) x5) Y)
% 182.62/182.82 Found ((eq_ref fofType) x5) as proof of (((eq fofType) x5) Y)
% 182.62/182.82 Found (fun (x00:((x2 Xx) Y))=> ((eq_ref fofType) x5)) as proof of (((eq fofType) x5) Y)
% 182.62/182.82 Found eq_ref000:=(eq_ref00 P):((P x30)->(P x30))
% 182.62/182.82 Found (eq_ref00 P) as proof of ((P x30)->(P x3))
% 182.62/182.82 Found ((eq_ref0 x30) P) as proof of ((P x30)->(P x3))
% 182.62/182.82 Found (((eq_ref fofType) x30) P) as proof of ((P x30)->(P x3))
% 182.62/182.82 Found (((eq_ref fofType) x30) P) as proof of ((P x30)->(P x3))
% 182.62/182.82 Found (fun (P:(fofType->Prop))=> (((eq_ref fofType) x30) P)) as proof of ((P x30)->(P x3))
% 182.62/182.82 Found (fun (x20:(forall (Xn:fofType), (not (((eq fofType) (cS Xn)) c0)))) (P:(fofType->Prop))=> (((eq_ref fofType) x30) P)) as proof of (((eq fofType) x30) x3)
% 182.62/182.82 Found eq_substitution:=(fun (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)) (H:(((eq T) a) b))=> ((H (fun (x:T)=> (((eq U) (f a)) (f x)))) ((eq_ref U) (f a)))):(forall (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)), ((((eq T) a) b)->(((eq U) (f a)) (f b))))
% 182.62/182.82 Instantiate: b:=(forall (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)), ((((eq T) a) b)->(((eq U) (f a)) (f b)))):Prop
% 182.62/182.82 Found eq_substitution as proof of b
% 182.62/182.82 Found x6:(P x30)
% 182.62/182.82 Instantiate: x3:=x30:fofType
% 182.62/182.82 Found (fun (x6:(P x30))=> x6) as proof of (P x3)
% 182.62/182.82 Found (fun (P:(fofType->Prop)) (x6:(P x30))=> x6) as proof of ((P x30)->(P x3))
% 182.62/182.82 Found (fun (x20:(forall (Xn:fofType), (not (((eq fofType) (cS Xn)) c0)))) (P:(fofType->Prop)) (x6:(P x30))=> x6) as proof of (((eq fofType) x30) x3)
% 182.62/182.82 Found eq_ref00:=(eq_ref0 x30):(((eq fofType) x30) x30)
% 182.62/182.82 Found (eq_ref0 x30) as proof of (((eq fofType) x30) x3)
% 182.62/182.82 Found ((eq_ref fofType) x30) as proof of (((eq fofType) x30) x3)
% 182.62/182.82 Found ((eq_ref fofType) x30) as proof of (((eq fofType) x30) x3)
% 182.62/182.82 Found (fun (x20:(forall (Xn:fofType), (not (((eq fofType) (cS Xn)) c0))))=> ((eq_ref fofType) x30)) as proof of (((eq fofType) x30) x3)
% 182.62/182.82 Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% 182.62/182.82 Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) b)
% 182.62/182.82 Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) b)
% 182.62/182.82 Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) b)
% 182.62/182.82 Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) b)
% 182.62/182.82 Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% 182.62/182.82 Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((and ((x2 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x2 Xx) Xy)->((x2 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x2 Xx))))))
% 182.62/182.82 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and ((x2 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x2 Xx) Xy)->((x2 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x2 Xx))))))
% 182.62/182.82 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and ((x2 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x2 Xx) Xy)->((x2 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x2 Xx))))))
% 182.62/182.82 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and ((x2 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x2 Xx) Xy)->((x2 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x2 Xx))))))
% 182.62/182.82 Found ((eq_trans0000 ((eq_ref Prop) (f x2))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f x2)) ((and ((and ((x2 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x2 Xx) Xy)->((x2 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x2 Xx))))))
% 182.62/182.82 Found (((eq_trans000 ((and ((and ((x2 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x2 Xx) Xy)->((x2 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x2 Xx)))))) ((eq_ref Prop) (f x2))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f x2)) ((and ((and ((x2 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x2 Xx) Xy)->((x2 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x2 Xx))))))
% 182.62/182.82 Found ((((eq_trans00 ((and ((and ((x2 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x2 Xx) Xy)->((x2 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x2 Xx)))))) ((and ((and ((x2 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x2 Xx) Xy)->((x2 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x2 Xx)))))) ((eq_ref Prop) (f x2))) ((eq_ref Prop) ((and ((and ((x2 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x2 Xx) Xy)->((x2 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x2 Xx))))))) as proof of (((eq Prop) (f x2)) ((and ((and ((x2 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x2 Xx) Xy)->((x2 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x2 Xx))))))
% 182.74/182.97 Found (((((eq_trans0 (f x2)) ((and ((and ((x2 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x2 Xx) Xy)->((x2 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x2 Xx)))))) ((and ((and ((x2 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x2 Xx) Xy)->((x2 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x2 Xx)))))) ((eq_ref Prop) (f x2))) ((eq_ref Prop) ((and ((and ((x2 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x2 Xx) Xy)->((x2 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x2 Xx))))))) as proof of (((eq Prop) (f x2)) ((and ((and ((x2 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x2 Xx) Xy)->((x2 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x2 Xx))))))
% 182.74/182.97 Found ((((((eq_trans Prop) (f x2)) ((and ((and ((x2 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x2 Xx) Xy)->((x2 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x2 Xx)))))) ((and ((and ((x2 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x2 Xx) Xy)->((x2 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x2 Xx)))))) ((eq_ref Prop) (f x2))) ((eq_ref Prop) ((and ((and ((x2 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x2 Xx) Xy)->((x2 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x2 Xx))))))) as proof of (((eq Prop) (f x2)) ((and ((and ((x2 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x2 Xx) Xy)->((x2 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x2 Xx))))))
% 182.74/182.97 Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% 182.74/182.97 Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) b)
% 182.74/182.97 Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) b)
% 182.74/182.97 Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) b)
% 182.74/182.97 Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) b)
% 182.74/182.97 Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% 182.74/182.97 Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((and ((x2 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x2 Xx) Xy)->((x2 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x2 Xx))))))
% 182.74/182.97 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and ((x2 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x2 Xx) Xy)->((x2 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x2 Xx))))))
% 182.74/182.97 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and ((x2 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x2 Xx) Xy)->((x2 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x2 Xx))))))
% 182.74/182.97 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and ((x2 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x2 Xx) Xy)->((x2 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x2 Xx))))))
% 182.74/182.97 Found ((eq_trans0000 ((eq_ref Prop) (f x2))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f x2)) ((and ((and ((x2 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x2 Xx) Xy)->((x2 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x2 Xx))))))
% 182.74/182.97 Found (((eq_trans000 ((and ((and ((x2 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x2 Xx) Xy)->((x2 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x2 Xx)))))) ((eq_ref Prop) (f x2))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f x2)) ((and ((and ((x2 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x2 Xx) Xy)->((x2 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x2 Xx))))))
% 185.06/185.30 Found ((((eq_trans00 ((and ((and ((x2 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x2 Xx) Xy)->((x2 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x2 Xx)))))) ((and ((and ((x2 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x2 Xx) Xy)->((x2 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x2 Xx)))))) ((eq_ref Prop) (f x2))) ((eq_ref Prop) ((and ((and ((x2 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x2 Xx) Xy)->((x2 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x2 Xx))))))) as proof of (((eq Prop) (f x2)) ((and ((and ((x2 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x2 Xx) Xy)->((x2 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x2 Xx))))))
% 185.06/185.30 Found (((((eq_trans0 (f x2)) ((and ((and ((x2 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x2 Xx) Xy)->((x2 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x2 Xx)))))) ((and ((and ((x2 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x2 Xx) Xy)->((x2 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x2 Xx)))))) ((eq_ref Prop) (f x2))) ((eq_ref Prop) ((and ((and ((x2 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x2 Xx) Xy)->((x2 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x2 Xx))))))) as proof of (((eq Prop) (f x2)) ((and ((and ((x2 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x2 Xx) Xy)->((x2 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x2 Xx))))))
% 185.06/185.30 Found ((((((eq_trans Prop) (f x2)) ((and ((and ((x2 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x2 Xx) Xy)->((x2 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x2 Xx)))))) ((and ((and ((x2 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x2 Xx) Xy)->((x2 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x2 Xx)))))) ((eq_ref Prop) (f x2))) ((eq_ref Prop) ((and ((and ((x2 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x2 Xx) Xy)->((x2 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x2 Xx))))))) as proof of (((eq Prop) (f x2)) ((and ((and ((x2 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x2 Xx) Xy)->((x2 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x2 Xx))))))
% 185.06/185.30 Found eq_sym000:=(eq_sym00 x'):((((eq fofType) x3) x')->(((eq fofType) x') x3))
% 185.06/185.30 Found (eq_sym00 x') as proof of (((x0 Xx) x')->(((eq fofType) x3) x'))
% 185.06/185.30 Found ((eq_sym0 x3) x') as proof of (((x0 Xx) x')->(((eq fofType) x3) x'))
% 185.06/185.30 Found (((eq_sym fofType) x3) x') as proof of (((x0 Xx) x')->(((eq fofType) x3) x'))
% 185.06/185.30 Found (((eq_sym fofType) x3) x') as proof of (((x0 Xx) x')->(((eq fofType) x3) x'))
% 185.06/185.30 Found (((eq_sym fofType) x3) x') as proof of (((x0 Xx) x')->(((eq fofType) x3) x'))
% 185.06/185.30 Found eq_sym000:=(eq_sym00 x'):((((eq fofType) x3) x')->(((eq fofType) x') x3))
% 185.06/185.30 Found (eq_sym00 x') as proof of (((x2 Xx) x')->(((eq fofType) x3) x'))
% 185.06/185.30 Found ((eq_sym0 x3) x') as proof of (((x2 Xx) x')->(((eq fofType) x3) x'))
% 185.06/185.30 Found (((eq_sym fofType) x3) x') as proof of (((x2 Xx) x')->(((eq fofType) x3) x'))
% 185.06/185.30 Found (((eq_sym fofType) x3) x') as proof of (((x2 Xx) x')->(((eq fofType) x3) x'))
% 185.06/185.30 Found (((eq_sym fofType) x3) x') as proof of (((x2 Xx) x')->(((eq fofType) x3) x'))
% 185.06/185.30 Found eq_ref00:=(eq_ref0 x3):(((eq fofType) x3) x3)
% 185.06/185.30 Found (eq_ref0 x3) as proof of (((eq fofType) x3) x')
% 185.06/185.30 Found ((eq_ref fofType) x3) as proof of (((eq fofType) x3) x')
% 185.06/185.30 Found ((eq_ref fofType) x3) as proof of (((eq fofType) x3) x')
% 185.06/185.30 Found (fun (x00:((x0 Xx) x'))=> ((eq_ref fofType) x3)) as proof of (((eq fofType) x3) x')
% 185.06/185.30 Found eq_ref00:=(eq_ref0 x3):(((eq fofType) x3) x3)
% 185.06/185.30 Found (eq_ref0 x3) as proof of (((eq fofType) x3) x30)
% 185.06/185.30 Found ((eq_ref fofType) x3) as proof of (((eq fofType) x3) x30)
% 185.06/185.30 Found ((eq_ref fofType) x3) as proof of (((eq fofType) x3) x30)
% 185.06/185.30 Found ((eq_ref fofType) x3) as proof of (((eq fofType) x3) x30)
% 185.06/185.30 Found (eq_sym000 ((eq_ref fofType) x3)) as proof of (((eq fofType) x30) x3)
% 188.64/188.82 Found ((eq_sym00 x30) ((eq_ref fofType) x3)) as proof of (((eq fofType) x30) x3)
% 188.64/188.82 Found (((eq_sym0 x3) x30) ((eq_ref fofType) x3)) as proof of (((eq fofType) x30) x3)
% 188.64/188.82 Found ((((eq_sym fofType) x3) x30) ((eq_ref fofType) x3)) as proof of (((eq fofType) x30) x3)
% 188.64/188.82 Found (fun (x20:(forall (Xn:fofType), (not (((eq fofType) (cS Xn)) c0))))=> ((((eq_sym fofType) x3) x30) ((eq_ref fofType) x3))) as proof of (((eq fofType) x30) x3)
% 188.64/188.82 Found eq_ref00:=(eq_ref0 x3):(((eq fofType) x3) x3)
% 188.64/188.82 Found (eq_ref0 x3) as proof of (((eq fofType) x3) x')
% 188.64/188.82 Found ((eq_ref fofType) x3) as proof of (((eq fofType) x3) x')
% 188.64/188.82 Found ((eq_ref fofType) x3) as proof of (((eq fofType) x3) x')
% 188.64/188.82 Found (fun (x00:((x2 Xx) x'))=> ((eq_ref fofType) x3)) as proof of (((eq fofType) x3) x')
% 188.64/188.82 Found eq_ref00:=(eq_ref0 c0):(((eq fofType) c0) c0)
% 188.64/188.82 Found (eq_ref0 c0) as proof of (((eq fofType) c0) x5)
% 188.64/188.82 Found ((eq_ref fofType) c0) as proof of (((eq fofType) c0) x5)
% 188.64/188.82 Found ((eq_ref fofType) c0) as proof of (((eq fofType) c0) x5)
% 188.64/188.82 Found ((eq_ref fofType) c0) as proof of (((eq fofType) c0) x5)
% 188.64/188.82 Found eq_ref00:=(eq_ref0 x30):(((eq fofType) x30) x30)
% 188.64/188.82 Found (eq_ref0 x30) as proof of (((eq fofType) x30) x3)
% 188.64/188.82 Found ((eq_ref fofType) x30) as proof of (((eq fofType) x30) x3)
% 188.64/188.82 Found ((eq_ref fofType) x30) as proof of (((eq fofType) x30) x3)
% 188.64/188.82 Found (fun (x5:(forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy))))=> ((eq_ref fofType) x30)) as proof of (((eq fofType) x30) x3)
% 188.64/188.82 Found (fun (x4:cIND) (x5:(forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy))))=> ((eq_ref fofType) x30)) as proof of ((forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy)))->(((eq fofType) x30) x3))
% 188.64/188.82 Found (fun (x4:cIND) (x5:(forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy))))=> ((eq_ref fofType) x30)) as proof of (cIND->((forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy)))->(((eq fofType) x30) x3)))
% 188.64/188.82 Found (and_rect10 (fun (x4:cIND) (x5:(forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy))))=> ((eq_ref fofType) x30))) as proof of (((eq fofType) x30) x3)
% 188.64/188.82 Found ((and_rect1 (((eq fofType) x30) x3)) (fun (x4:cIND) (x5:(forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy))))=> ((eq_ref fofType) x30))) as proof of (((eq fofType) x30) x3)
% 188.64/188.82 Found (((fun (P:Type) (x4:(cIND->((forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy)))->P)))=> (((((and_rect cIND) (forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy)))) P) x4) x10)) (((eq fofType) x30) x3)) (fun (x4:cIND) (x5:(forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy))))=> ((eq_ref fofType) x30))) as proof of (((eq fofType) x30) x3)
% 188.64/188.82 Found (fun (x20:(forall (Xn:fofType), (not (((eq fofType) (cS Xn)) c0))))=> (((fun (P:Type) (x4:(cIND->((forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy)))->P)))=> (((((and_rect cIND) (forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy)))) P) x4) x10)) (((eq fofType) x30) x3)) (fun (x4:cIND) (x5:(forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy))))=> ((eq_ref fofType) x30)))) as proof of (((eq fofType) x30) x3)
% 188.64/188.82 Found eq_ref000:=(eq_ref00 P0):((P0 (f x2))->(P0 (f x2)))
% 188.64/188.82 Found (eq_ref00 P0) as proof of (P1 (f x2))
% 188.64/188.82 Found ((eq_ref0 (f x2)) P0) as proof of (P1 (f x2))
% 188.64/188.82 Found (((eq_ref Prop) (f x2)) P0) as proof of (P1 (f x2))
% 188.64/188.82 Found (((eq_ref Prop) (f x2)) P0) as proof of (P1 (f x2))
% 188.64/188.82 Found eq_ref000:=(eq_ref00 P0):((P0 (f x2))->(P0 (f x2)))
% 188.64/188.82 Found (eq_ref00 P0) as proof of (P1 (f x2))
% 188.64/188.82 Found ((eq_ref0 (f x2)) P0) as proof of (P1 (f x2))
% 188.64/188.82 Found (((eq_ref Prop) (f x2)) P0) as proof of (P1 (f x2))
% 188.64/188.82 Found (((eq_ref Prop) (f x2)) P0) as proof of (P1 (f x2))
% 191.55/191.71 Found eq_ref00:=(eq_ref0 x30):(((eq fofType) x30) x30)
% 191.55/191.71 Found (eq_ref0 x30) as proof of (((eq fofType) x30) x3)
% 191.55/191.71 Found ((eq_ref fofType) x30) as proof of (((eq fofType) x30) x3)
% 191.55/191.71 Found ((eq_ref fofType) x30) as proof of (((eq fofType) x30) x3)
% 191.55/191.71 Found (fun (x5:(forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (cS Xx)) (cS Xy))->(((eq fofType) Xx) Xy))))=> ((eq_ref fofType) x30)) as proof of (((eq fofType) x30) x3)
% 191.55/191.71 Found (fun (x4:cIND) (x5:(forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (cS Xx)) (cS Xy))->(((eq fofType) Xx) Xy))))=> ((eq_ref fofType) x30)) as proof of ((forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (cS Xx)) (cS Xy))->(((eq fofType) Xx) Xy)))->(((eq fofType) x30) x3))
% 191.55/191.71 Found (fun (x4:cIND) (x5:(forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (cS Xx)) (cS Xy))->(((eq fofType) Xx) Xy))))=> ((eq_ref fofType) x30)) as proof of (cIND->((forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (cS Xx)) (cS Xy))->(((eq fofType) Xx) Xy)))->(((eq fofType) x30) x3)))
% 191.55/191.71 Found (and_rect10 (fun (x4:cIND) (x5:(forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (cS Xx)) (cS Xy))->(((eq fofType) Xx) Xy))))=> ((eq_ref fofType) x30))) as proof of (((eq fofType) x30) x3)
% 191.55/191.71 Found ((and_rect1 (((eq fofType) x30) x3)) (fun (x4:cIND) (x5:(forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (cS Xx)) (cS Xy))->(((eq fofType) Xx) Xy))))=> ((eq_ref fofType) x30))) as proof of (((eq fofType) x30) x3)
% 191.55/191.71 Found (((fun (P:Type) (x4:(cIND->((forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (cS Xx)) (cS Xy))->(((eq fofType) Xx) Xy)))->P)))=> (((((and_rect cIND) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (cS Xx)) (cS Xy))->(((eq fofType) Xx) Xy)))) P) x4) x1)) (((eq fofType) x30) x3)) (fun (x4:cIND) (x5:(forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (cS Xx)) (cS Xy))->(((eq fofType) Xx) Xy))))=> ((eq_ref fofType) x30))) as proof of (((eq fofType) x30) x3)
% 191.55/191.71 Found (fun (x20:(forall (Xn:fofType), (not (((eq fofType) (cS Xn)) c0))))=> (((fun (P:Type) (x4:(cIND->((forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (cS Xx)) (cS Xy))->(((eq fofType) Xx) Xy)))->P)))=> (((((and_rect cIND) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (cS Xx)) (cS Xy))->(((eq fofType) Xx) Xy)))) P) x4) x1)) (((eq fofType) x30) x3)) (fun (x4:cIND) (x5:(forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (cS Xx)) (cS Xy))->(((eq fofType) Xx) Xy))))=> ((eq_ref fofType) x30)))) as proof of (((eq fofType) x30) x3)
% 191.55/191.71 Found eq_ref00:=(eq_ref0 (forall (Xx:fofType) (Xy:fofType), (((x2 Xx) Xy)->((x2 (cS Xx)) (cS (cS Xy)))))):(((eq Prop) (forall (Xx:fofType) (Xy:fofType), (((x2 Xx) Xy)->((x2 (cS Xx)) (cS (cS Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((x2 Xx) Xy)->((x2 (cS Xx)) (cS (cS Xy))))))
% 191.55/191.71 Found (eq_ref0 (forall (Xx:fofType) (Xy:fofType), (((x2 Xx) Xy)->((x2 (cS Xx)) (cS (cS Xy)))))) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), (((x2 Xx) Xy)->((x2 (cS Xx)) (cS (cS Xy)))))) b)
% 191.55/191.71 Found ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), (((x2 Xx) Xy)->((x2 (cS Xx)) (cS (cS Xy)))))) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), (((x2 Xx) Xy)->((x2 (cS Xx)) (cS (cS Xy)))))) b)
% 191.55/191.71 Found ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), (((x2 Xx) Xy)->((x2 (cS Xx)) (cS (cS Xy)))))) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), (((x2 Xx) Xy)->((x2 (cS Xx)) (cS (cS Xy)))))) b)
% 191.55/191.71 Found ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), (((x2 Xx) Xy)->((x2 (cS Xx)) (cS (cS Xy)))))) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), (((x2 Xx) Xy)->((x2 (cS Xx)) (cS (cS Xy)))))) b)
% 191.55/191.71 Found x6:(P x10)
% 191.55/191.71 Instantiate: x1:=x10:fofType
% 191.55/191.71 Found (fun (x6:(P x10))=> x6) as proof of (P x1)
% 191.55/191.71 Found (fun (P:(fofType->Prop)) (x6:(P x10))=> x6) as proof of ((P x10)->(P x1))
% 191.55/191.71 Found eq_ref000:=(eq_ref00 P):((P x10)->(P x10))
% 191.55/191.71 Found (eq_ref00 P) as proof of ((P x10)->(P x1))
% 191.55/191.71 Found ((eq_ref0 x10) P) as proof of ((P x10)->(P x1))
% 191.55/191.71 Found (((eq_ref fofType) x10) P) as proof of ((P x10)->(P x1))
% 191.55/191.71 Found (((eq_ref fofType) x10) P) as proof of ((P x10)->(P x1))
% 191.55/191.71 Found (fun (P:(fofType->Prop))=> (((eq_ref fofType) x10) P)) as proof of ((P x10)->(P x1))
% 191.55/191.71 Found x6:(P x3)
% 191.55/191.71 Instantiate: x3:=Y:fofType
% 196.11/196.34 Found (fun (x6:(P x3))=> x6) as proof of (P Y)
% 196.11/196.34 Found (fun (P:(fofType->Prop)) (x6:(P x3))=> x6) as proof of ((P x3)->(P Y))
% 196.11/196.34 Found (fun (x00:((x0 Xx) Y)) (P:(fofType->Prop)) (x6:(P x3))=> x6) as proof of (((eq fofType) x3) Y)
% 196.11/196.34 Found eq_ref000:=(eq_ref00 P):((P x3)->(P x3))
% 196.11/196.34 Found (eq_ref00 P) as proof of ((P x3)->(P Y))
% 196.11/196.34 Found ((eq_ref0 x3) P) as proof of ((P x3)->(P Y))
% 196.11/196.34 Found (((eq_ref fofType) x3) P) as proof of ((P x3)->(P Y))
% 196.11/196.34 Found (((eq_ref fofType) x3) P) as proof of ((P x3)->(P Y))
% 196.11/196.34 Found (fun (P:(fofType->Prop))=> (((eq_ref fofType) x3) P)) as proof of ((P x3)->(P Y))
% 196.11/196.34 Found (fun (x00:((x0 Xx) Y)) (P:(fofType->Prop))=> (((eq_ref fofType) x3) P)) as proof of (((eq fofType) x3) Y)
% 196.11/196.34 Found eq_ref000:=(eq_ref00 P):((P x3)->(P x3))
% 196.11/196.34 Found (eq_ref00 P) as proof of ((P x3)->(P Y))
% 196.11/196.34 Found ((eq_ref0 x3) P) as proof of ((P x3)->(P Y))
% 196.11/196.34 Found (((eq_ref fofType) x3) P) as proof of ((P x3)->(P Y))
% 196.11/196.34 Found (((eq_ref fofType) x3) P) as proof of ((P x3)->(P Y))
% 196.11/196.34 Found (fun (P:(fofType->Prop))=> (((eq_ref fofType) x3) P)) as proof of ((P x3)->(P Y))
% 196.11/196.34 Found (fun (x00:((x2 Xx) Y)) (P:(fofType->Prop))=> (((eq_ref fofType) x3) P)) as proof of (((eq fofType) x3) Y)
% 196.11/196.34 Found x6:(P x3)
% 196.11/196.34 Instantiate: x3:=Y:fofType
% 196.11/196.34 Found (fun (x6:(P x3))=> x6) as proof of (P Y)
% 196.11/196.34 Found (fun (P:(fofType->Prop)) (x6:(P x3))=> x6) as proof of ((P x3)->(P Y))
% 196.11/196.34 Found (fun (x00:((x2 Xx) Y)) (P:(fofType->Prop)) (x6:(P x3))=> x6) as proof of (((eq fofType) x3) Y)
% 196.11/196.34 Found eq_ref00:=(eq_ref0 x10):(((eq fofType) x10) x10)
% 196.11/196.34 Found (eq_ref0 x10) as proof of (((eq fofType) x10) x1)
% 196.11/196.34 Found ((eq_ref fofType) x10) as proof of (((eq fofType) x10) x1)
% 196.11/196.34 Found ((eq_ref fofType) x10) as proof of (((eq fofType) x10) x1)
% 196.11/196.34 Found eq_ref00:=(eq_ref0 x3):(((eq fofType) x3) x3)
% 196.11/196.34 Found (eq_ref0 x3) as proof of (((eq fofType) x3) Y)
% 196.11/196.34 Found ((eq_ref fofType) x3) as proof of (((eq fofType) x3) Y)
% 196.11/196.34 Found ((eq_ref fofType) x3) as proof of (((eq fofType) x3) Y)
% 196.11/196.34 Found ((eq_ref fofType) x3) as proof of (((eq fofType) x3) Y)
% 196.11/196.34 Found (fun (x00:((x0 Xx) Y))=> ((eq_ref fofType) x3)) as proof of (((eq fofType) x3) Y)
% 196.11/196.34 Found eq_ref00:=(eq_ref0 x3):(((eq fofType) x3) x3)
% 196.11/196.34 Found (eq_ref0 x3) as proof of (((eq fofType) x3) Y)
% 196.11/196.34 Found ((eq_ref fofType) x3) as proof of (((eq fofType) x3) Y)
% 196.11/196.34 Found ((eq_ref fofType) x3) as proof of (((eq fofType) x3) Y)
% 196.11/196.34 Found ((eq_ref fofType) x3) as proof of (((eq fofType) x3) Y)
% 196.11/196.34 Found (fun (x00:((x2 Xx) Y))=> ((eq_ref fofType) x3)) as proof of (((eq fofType) x3) Y)
% 196.11/196.34 Found eq_ref00:=(eq_ref0 x10):(((eq fofType) x10) x10)
% 196.11/196.34 Found (eq_ref0 x10) as proof of (forall (P:(fofType->Prop)), ((P x10)->(P x1)))
% 196.11/196.34 Found ((eq_ref fofType) x10) as proof of (forall (P:(fofType->Prop)), ((P x10)->(P x1)))
% 196.11/196.34 Found ((eq_ref fofType) x10) as proof of (forall (P:(fofType->Prop)), ((P x10)->(P x1)))
% 196.11/196.34 Found ((eq_ref fofType) x10) as proof of (forall (P:(fofType->Prop)), ((P x10)->(P x1)))
% 196.11/196.34 Found eq_ref00:=(eq_ref0 x1):(((eq fofType) x1) x1)
% 196.11/196.34 Found (eq_ref0 x1) as proof of (((eq fofType) x1) x10)
% 196.11/196.34 Found ((eq_ref fofType) x1) as proof of (((eq fofType) x1) x10)
% 196.11/196.34 Found ((eq_ref fofType) x1) as proof of (((eq fofType) x1) x10)
% 196.11/196.34 Found ((eq_ref fofType) x1) as proof of (((eq fofType) x1) x10)
% 196.11/196.34 Found (eq_sym000 ((eq_ref fofType) x1)) as proof of (((eq fofType) x10) x1)
% 196.11/196.34 Found ((eq_sym00 x10) ((eq_ref fofType) x1)) as proof of (((eq fofType) x10) x1)
% 196.11/196.34 Found (((eq_sym0 x1) x10) ((eq_ref fofType) x1)) as proof of (((eq fofType) x10) x1)
% 196.11/196.34 Found ((((eq_sym fofType) x1) x10) ((eq_ref fofType) x1)) as proof of (((eq fofType) x10) x1)
% 196.11/196.34 Found eq_ref00:=(eq_ref0 x10):(((eq fofType) x10) x10)
% 196.11/196.34 Found (eq_ref0 x10) as proof of (((eq fofType) x10) x1)
% 196.11/196.34 Found ((eq_ref fofType) x10) as proof of (((eq fofType) x10) x1)
% 196.11/196.34 Found ((eq_ref fofType) x10) as proof of (((eq fofType) x10) x1)
% 196.11/196.34 Found (fun (x5:(forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy))))=> ((eq_ref fofType) x10)) as proof of (((eq fofType) x10) x1)
% 196.11/196.34 Found (fun (x4:cIND) (x5:(forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy))))=> ((eq_ref fofType) x10)) as proof of ((forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy)))->(((eq fofType) x10) x1))
% 196.53/196.70 Found (fun (x4:cIND) (x5:(forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy))))=> ((eq_ref fofType) x10)) as proof of (cIND->((forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy)))->(((eq fofType) x10) x1)))
% 196.53/196.70 Found (and_rect10 (fun (x4:cIND) (x5:(forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy))))=> ((eq_ref fofType) x10))) as proof of (((eq fofType) x10) x1)
% 196.53/196.70 Found ((and_rect1 (((eq fofType) x10) x1)) (fun (x4:cIND) (x5:(forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy))))=> ((eq_ref fofType) x10))) as proof of (((eq fofType) x10) x1)
% 196.53/196.70 Found (((fun (P:Type) (x4:(cIND->((forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy)))->P)))=> (((((and_rect cIND) (forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy)))) P) x4) x20)) (((eq fofType) x10) x1)) (fun (x4:cIND) (x5:(forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy))))=> ((eq_ref fofType) x10))) as proof of (((eq fofType) x10) x1)
% 196.53/196.70 Found (fun (x20:((and cIND) (forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy)))))=> (((fun (P:Type) (x4:(cIND->((forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy)))->P)))=> (((((and_rect cIND) (forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy)))) P) x4) x20)) (((eq fofType) x10) x1)) (fun (x4:cIND) (x5:(forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy))))=> ((eq_ref fofType) x10)))) as proof of (((eq fofType) x10) x1)
% 196.53/196.70 Found (fun (x30:(forall (Xn:fofType), (not (((eq fofType) (cS Xn)) c0)))) (x20:((and cIND) (forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy)))))=> (((fun (P:Type) (x4:(cIND->((forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy)))->P)))=> (((((and_rect cIND) (forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy)))) P) x4) x20)) (((eq fofType) x10) x1)) (fun (x4:cIND) (x5:(forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy))))=> ((eq_ref fofType) x10)))) as proof of (((and cIND) (forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy))))->(((eq fofType) x10) x1))
% 196.53/196.70 Found eq_ref00:=(eq_ref0 x10):(((eq fofType) x10) x10)
% 196.53/196.70 Found (eq_ref0 x10) as proof of (((eq fofType) x10) x1)
% 196.53/196.70 Found ((eq_ref fofType) x10) as proof of (((eq fofType) x10) x1)
% 196.53/196.70 Found ((eq_ref fofType) x10) as proof of (((eq fofType) x10) x1)
% 196.53/196.70 Found (fun (x5:(forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy))))=> ((eq_ref fofType) x10)) as proof of (((eq fofType) x10) x1)
% 196.53/196.70 Found (fun (x4:cIND) (x5:(forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy))))=> ((eq_ref fofType) x10)) as proof of ((forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy)))->(((eq fofType) x10) x1))
% 196.53/196.70 Found (fun (x4:cIND) (x5:(forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy))))=> ((eq_ref fofType) x10)) as proof of (cIND->((forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy)))->(((eq fofType) x10) x1)))
% 196.53/196.70 Found (and_rect10 (fun (x4:cIND) (x5:(forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy))))=> ((eq_ref fofType) x10))) as proof of (((eq fofType) x10) x1)
% 196.53/196.70 Found ((and_rect1 (((eq fofType) x10) x1)) (fun (x4:cIND) (x5:(forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy))))=> ((eq_ref fofType) x10))) as proof of (((eq fofType) x10) x1)
% 196.53/196.70 Found (((fun (P:Type) (x4:(cIND->((forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy)))->P)))=> (((((and_rect cIND) (forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy)))) P) x4) x2)) (((eq fofType) x10) x1)) (fun (x4:cIND) (x5:(forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy))))=> ((eq_ref fofType) x10))) as proof of (((eq fofType) x10) x1)
% 198.64/198.81 Found eq_ref00:=(eq_ref0 x5):(((eq fofType) x5) x5)
% 198.64/198.81 Found (eq_ref0 x5) as proof of (((eq fofType) x5) x50)
% 198.64/198.81 Found ((eq_ref fofType) x5) as proof of (((eq fofType) x5) x50)
% 198.64/198.81 Found ((eq_ref fofType) x5) as proof of (((eq fofType) x5) x50)
% 198.64/198.81 Found ((eq_ref fofType) x5) as proof of (((eq fofType) x5) x50)
% 198.64/198.81 Found (eq_sym000 ((eq_ref fofType) x5)) as proof of (((eq fofType) x50) x5)
% 198.64/198.81 Found ((eq_sym00 x50) ((eq_ref fofType) x5)) as proof of (((eq fofType) x50) x5)
% 198.64/198.81 Found (((eq_sym0 x5) x50) ((eq_ref fofType) x5)) as proof of (((eq fofType) x50) x5)
% 198.64/198.81 Found ((((eq_sym fofType) x5) x50) ((eq_ref fofType) x5)) as proof of (((eq fofType) x50) x5)
% 198.64/198.81 Found eq_ref00:=(eq_ref0 x10):(((eq fofType) x10) x10)
% 198.64/198.81 Found (eq_ref0 x10) as proof of (((eq fofType) x10) x1)
% 198.64/198.81 Found ((eq_ref fofType) x10) as proof of (((eq fofType) x10) x1)
% 198.64/198.81 Found ((eq_ref fofType) x10) as proof of (((eq fofType) x10) x1)
% 198.64/198.81 Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% 198.64/198.81 Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) ((and ((x0 Xx) x1)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) x1) Y)))))
% 198.64/198.81 Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and ((x0 Xx) x1)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) x1) Y)))))
% 198.64/198.81 Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and ((x0 Xx) x1)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) x1) Y)))))
% 198.64/198.81 Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) ((and ((x0 Xx) x1)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) x1) Y)))))
% 198.64/198.81 Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:fofType), (((eq Prop) (f x)) ((and ((x0 Xx) x)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) x) Y))))))
% 198.64/198.81 Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% 198.64/198.81 Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) ((and ((x0 Xx) x1)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) x1) Y)))))
% 198.64/198.81 Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and ((x0 Xx) x1)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) x1) Y)))))
% 198.64/198.81 Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and ((x0 Xx) x1)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) x1) Y)))))
% 198.64/198.81 Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) ((and ((x0 Xx) x1)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) x1) Y)))))
% 198.64/198.81 Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:fofType), (((eq Prop) (f x)) ((and ((x0 Xx) x)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) x) Y))))))
% 198.64/198.81 Found eq_ref00:=(eq_ref0 x1):(((eq fofType) x1) x1)
% 198.64/198.81 Found (eq_ref0 x1) as proof of (((eq fofType) x1) x10)
% 198.64/198.81 Found ((eq_ref fofType) x1) as proof of (((eq fofType) x1) x10)
% 198.64/198.81 Found ((eq_ref fofType) x1) as proof of (((eq fofType) x1) x10)
% 198.64/198.81 Found ((eq_ref fofType) x1) as proof of (((eq fofType) x1) x10)
% 198.64/198.81 Found (eq_sym000 ((eq_ref fofType) x1)) as proof of (((eq fofType) x10) x1)
% 198.64/198.81 Found ((eq_sym00 x10) ((eq_ref fofType) x1)) as proof of (((eq fofType) x10) x1)
% 198.64/198.81 Found (((eq_sym0 x1) x10) ((eq_ref fofType) x1)) as proof of (((eq fofType) x10) x1)
% 198.64/198.81 Found ((((eq_sym fofType) x1) x10) ((eq_ref fofType) x1)) as proof of (((eq fofType) x10) x1)
% 198.64/198.81 Found (fun (x20:((and cIND) (forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy)))))=> ((((eq_sym fofType) x1) x10) ((eq_ref fofType) x1))) as proof of (((eq fofType) x10) x1)
% 198.64/198.81 Found (fun (x30:(forall (Xn:fofType), (not (((eq fofType) (cS Xn)) c0)))) (x20:((and cIND) (forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy)))))=> ((((eq_sym fofType) x1) x10) ((eq_ref fofType) x1))) as proof of (((and cIND) (forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy))))->(((eq fofType) x10) x1))
% 201.22/201.40 Found eq_substitution:=(fun (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)) (H:(((eq T) a) b))=> ((H (fun (x:T)=> (((eq U) (f a)) (f x)))) ((eq_ref U) (f a)))):(forall (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)), ((((eq T) a) b)->(((eq U) (f a)) (f b))))
% 201.22/201.40 Instantiate: b:=(forall (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)), ((((eq T) a) b)->(((eq U) (f a)) (f b)))):Prop
% 201.22/201.40 Found eq_substitution as proof of b
% 201.22/201.40 Found eq_sym000:=(eq_sym00 Y):((((eq fofType) x5) Y)->(((eq fofType) Y) x5))
% 201.22/201.40 Found (eq_sym00 Y) as proof of (((x0 Xx) Y)->(((eq fofType) x5) Y))
% 201.22/201.40 Found ((eq_sym0 x5) Y) as proof of (((x0 Xx) Y)->(((eq fofType) x5) Y))
% 201.22/201.40 Found (((eq_sym fofType) x5) Y) as proof of (((x0 Xx) Y)->(((eq fofType) x5) Y))
% 201.22/201.40 Found (((eq_sym fofType) x5) Y) as proof of (((x0 Xx) Y)->(((eq fofType) x5) Y))
% 201.22/201.40 Found (((eq_sym fofType) x5) Y) as proof of (((x0 Xx) Y)->(((eq fofType) x5) Y))
% 201.22/201.40 Found eq_substitution:=(fun (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)) (H:(((eq T) a) b))=> ((H (fun (x:T)=> (((eq U) (f a)) (f x)))) ((eq_ref U) (f a)))):(forall (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)), ((((eq T) a) b)->(((eq U) (f a)) (f b))))
% 201.22/201.40 Instantiate: b:=(forall (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)), ((((eq T) a) b)->(((eq U) (f a)) (f b)))):Prop
% 201.22/201.40 Found eq_substitution as proof of b
% 201.22/201.40 Found eq_substitution:=(fun (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)) (H:(((eq T) a) b))=> ((H (fun (x:T)=> (((eq U) (f a)) (f x)))) ((eq_ref U) (f a)))):(forall (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)), ((((eq T) a) b)->(((eq U) (f a)) (f b))))
% 201.22/201.40 Instantiate: b:=(forall (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)), ((((eq T) a) b)->(((eq U) (f a)) (f b)))):Prop
% 201.22/201.40 Found eq_substitution as proof of b
% 201.22/201.40 Found eq_ref000:=(eq_ref00 P):((P x5)->(P x5))
% 201.22/201.40 Found (eq_ref00 P) as proof of ((P x5)->(P x'))
% 201.22/201.40 Found ((eq_ref0 x5) P) as proof of ((P x5)->(P x'))
% 201.22/201.40 Found (((eq_ref fofType) x5) P) as proof of ((P x5)->(P x'))
% 201.22/201.40 Found (((eq_ref fofType) x5) P) as proof of ((P x5)->(P x'))
% 201.22/201.40 Found (fun (P:(fofType->Prop))=> (((eq_ref fofType) x5) P)) as proof of ((P x5)->(P x'))
% 201.22/201.40 Found (fun (x00:((x0 Xx) x')) (P:(fofType->Prop))=> (((eq_ref fofType) x5) P)) as proof of (((eq fofType) x5) x')
% 201.22/201.40 Found x6:(P x5)
% 201.22/201.40 Instantiate: x5:=x':fofType
% 201.22/201.40 Found (fun (x6:(P x5))=> x6) as proof of (P x')
% 201.22/201.40 Found (fun (P:(fofType->Prop)) (x6:(P x5))=> x6) as proof of ((P x5)->(P x'))
% 201.22/201.40 Found (fun (x00:((x0 Xx) x')) (P:(fofType->Prop)) (x6:(P x5))=> x6) as proof of (((eq fofType) x5) x')
% 201.22/201.40 Found eq_ref00:=(eq_ref0 x5):(((eq fofType) x5) x5)
% 201.22/201.40 Found (eq_ref0 x5) as proof of (((eq fofType) x5) Y)
% 201.22/201.40 Found ((eq_ref fofType) x5) as proof of (((eq fofType) x5) Y)
% 201.22/201.40 Found ((eq_ref fofType) x5) as proof of (((eq fofType) x5) Y)
% 201.22/201.40 Found (fun (x00:((x0 Xx) Y))=> ((eq_ref fofType) x5)) as proof of (((eq fofType) x5) Y)
% 201.22/201.40 Found x6:(P x5)
% 201.22/201.40 Instantiate: x5:=x':fofType
% 201.22/201.40 Found (fun (x6:(P x5))=> x6) as proof of (P x')
% 201.22/201.40 Found (fun (P:(fofType->Prop)) (x6:(P x5))=> x6) as proof of ((P x5)->(P x'))
% 201.22/201.40 Found (fun (x00:((x2 Xx) x')) (P:(fofType->Prop)) (x6:(P x5))=> x6) as proof of (((eq fofType) x5) x')
% 201.22/201.40 Found eq_ref000:=(eq_ref00 P):((P x5)->(P x5))
% 201.22/201.40 Found (eq_ref00 P) as proof of ((P x5)->(P x'))
% 201.22/201.40 Found ((eq_ref0 x5) P) as proof of ((P x5)->(P x'))
% 201.22/201.40 Found (((eq_ref fofType) x5) P) as proof of ((P x5)->(P x'))
% 201.22/201.40 Found (((eq_ref fofType) x5) P) as proof of ((P x5)->(P x'))
% 201.22/201.40 Found (fun (P:(fofType->Prop))=> (((eq_ref fofType) x5) P)) as proof of ((P x5)->(P x'))
% 201.22/201.40 Found (fun (x00:((x2 Xx) x')) (P:(fofType->Prop))=> (((eq_ref fofType) x5) P)) as proof of (((eq fofType) x5) x')
% 201.22/201.40 Found eta_expansion:=(fun (A:Type) (B:Type)=> ((eta_expansion_dep A) (fun (x1:A)=> B))):(forall (A:Type) (B:Type) (f:(A->B)), (((eq (A->B)) f) (fun (x:A)=> (f x))))
% 201.22/201.40 Instantiate: b:=(forall (A:Type) (B:Type) (f:(A->B)), (((eq (A->B)) f) (fun (x:A)=> (f x)))):Prop
% 201.22/201.40 Found eta_expansion as proof of b
% 201.22/201.40 Found eq_ref00:=(eq_ref0 x5):(((eq fofType) x5) x5)
% 201.22/201.40 Found (eq_ref0 x5) as proof of (((eq fofType) x5) x')
% 201.22/201.40 Found ((eq_ref fofType) x5) as proof of (((eq fofType) x5) x')
% 201.22/201.40 Found ((eq_ref fofType) x5) as proof of (((eq fofType) x5) x')
% 205.82/205.99 Found ((eq_ref fofType) x5) as proof of (((eq fofType) x5) x')
% 205.82/205.99 Found (fun (x00:((x0 Xx) x'))=> ((eq_ref fofType) x5)) as proof of (((eq fofType) x5) x')
% 205.82/205.99 Found eq_ref00:=(eq_ref0 (cS Xx0)):(((eq fofType) (cS Xx0)) (cS Xx0))
% 205.82/205.99 Found (eq_ref0 (cS Xx0)) as proof of (((eq fofType) (cS Xx0)) x5)
% 205.82/205.99 Found ((eq_ref fofType) (cS Xx0)) as proof of (((eq fofType) (cS Xx0)) x5)
% 205.82/205.99 Found ((eq_ref fofType) (cS Xx0)) as proof of (((eq fofType) (cS Xx0)) x5)
% 205.82/205.99 Found (fun (x6:(((eq fofType) Xx0) x5))=> ((eq_ref fofType) (cS Xx0))) as proof of (((eq fofType) (cS Xx0)) x5)
% 205.82/205.99 Found eq_ref00:=(eq_ref0 x5):(((eq fofType) x5) x5)
% 205.82/205.99 Found (eq_ref0 x5) as proof of (((eq fofType) x5) x')
% 205.82/205.99 Found ((eq_ref fofType) x5) as proof of (((eq fofType) x5) x')
% 205.82/205.99 Found ((eq_ref fofType) x5) as proof of (((eq fofType) x5) x')
% 205.82/205.99 Found ((eq_ref fofType) x5) as proof of (((eq fofType) x5) x')
% 205.82/205.99 Found (fun (x00:((x2 Xx) x'))=> ((eq_ref fofType) x5)) as proof of (((eq fofType) x5) x')
% 205.82/205.99 Found eq_ref000:=(eq_ref00 P):((P x30)->(P x30))
% 205.82/205.99 Found (eq_ref00 P) as proof of ((P x30)->(P x3))
% 205.82/205.99 Found ((eq_ref0 x30) P) as proof of ((P x30)->(P x3))
% 205.82/205.99 Found (((eq_ref fofType) x30) P) as proof of ((P x30)->(P x3))
% 205.82/205.99 Found (((eq_ref fofType) x30) P) as proof of ((P x30)->(P x3))
% 205.82/205.99 Found (fun (P:(fofType->Prop))=> (((eq_ref fofType) x30) P)) as proof of ((P x30)->(P x3))
% 205.82/205.99 Found (fun (x20:(forall (Xn:fofType), (not (((eq fofType) (cS Xn)) c0)))) (P:(fofType->Prop))=> (((eq_ref fofType) x30) P)) as proof of (((eq fofType) x30) x3)
% 205.82/205.99 Found x6:(P x30)
% 205.82/205.99 Instantiate: x3:=x30:fofType
% 205.82/205.99 Found (fun (x6:(P x30))=> x6) as proof of (P x3)
% 205.82/205.99 Found (fun (P:(fofType->Prop)) (x6:(P x30))=> x6) as proof of ((P x30)->(P x3))
% 205.82/205.99 Found (fun (x20:(forall (Xn:fofType), (not (((eq fofType) (cS Xn)) c0)))) (P:(fofType->Prop)) (x6:(P x30))=> x6) as proof of (((eq fofType) x30) x3)
% 205.82/205.99 Found eq_ref00:=(eq_ref0 (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy)))))):(((eq Prop) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))
% 205.82/205.99 Found (eq_ref0 (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy)))))) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy)))))) b)
% 205.82/205.99 Found ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy)))))) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy)))))) b)
% 205.82/205.99 Found ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy)))))) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy)))))) b)
% 205.82/205.99 Found ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy)))))) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy)))))) b)
% 205.82/205.99 Found eq_ref00:=(eq_ref0 (cS x5)):(((eq fofType) (cS x5)) (cS x5))
% 205.82/205.99 Found (eq_ref0 (cS x5)) as proof of (((eq fofType) (cS x5)) (cS Y))
% 205.82/205.99 Found ((eq_ref fofType) (cS x5)) as proof of (((eq fofType) (cS x5)) (cS Y))
% 205.82/205.99 Found ((eq_ref fofType) (cS x5)) as proof of (((eq fofType) (cS x5)) (cS Y))
% 205.82/205.99 Found ((eq_ref fofType) (cS x5)) as proof of (((eq fofType) (cS x5)) (cS Y))
% 205.82/205.99 Found (x400 ((eq_ref fofType) (cS x5))) as proof of (((eq fofType) x5) Y)
% 205.82/205.99 Found ((x40 Y) ((eq_ref fofType) (cS x5))) as proof of (((eq fofType) x5) Y)
% 205.82/205.99 Found (((x4 x5) Y) ((eq_ref fofType) (cS x5))) as proof of (((eq fofType) x5) Y)
% 205.82/205.99 Found (fun (x00:((x0 Xx) Y))=> (((x4 x5) Y) ((eq_ref fofType) (cS x5)))) as proof of (((eq fofType) x5) Y)
% 205.82/205.99 Found eq_ref00:=(eq_ref0 (cS x30)):(((eq fofType) (cS x30)) (cS x30))
% 205.82/205.99 Found (eq_ref0 (cS x30)) as proof of (((eq fofType) (cS x30)) (cS x3))
% 205.82/205.99 Found ((eq_ref fofType) (cS x30)) as proof of (((eq fofType) (cS x30)) (cS x3))
% 205.82/205.99 Found ((eq_ref fofType) (cS x30)) as proof of (((eq fofType) (cS x30)) (cS x3))
% 205.82/205.99 Found ((eq_ref fofType) (cS x30)) as proof of (((eq fofType) (cS x30)) (cS x3))
% 205.82/205.99 Found (x500 ((eq_ref fofType) (cS x30))) as proof of (((eq fofType) x30) x3)
% 208.64/208.86 Found ((x50 x3) ((eq_ref fofType) (cS x30))) as proof of (((eq fofType) x30) x3)
% 208.64/208.86 Found (((x5 x30) x3) ((eq_ref fofType) (cS x30))) as proof of (((eq fofType) x30) x3)
% 208.64/208.86 Found eq_ref00:=(eq_ref0 (fun (X:fofType)=> ((and ((x2 Xx) X)) (forall (Y:fofType), (((x2 Xx) Y)->(((eq fofType) X) Y)))))):(((eq (fofType->Prop)) (fun (X:fofType)=> ((and ((x2 Xx) X)) (forall (Y:fofType), (((x2 Xx) Y)->(((eq fofType) X) Y)))))) (fun (X:fofType)=> ((and ((x2 Xx) X)) (forall (Y:fofType), (((x2 Xx) Y)->(((eq fofType) X) Y))))))
% 208.64/208.86 Found (eq_ref0 (fun (X:fofType)=> ((and ((x2 Xx) X)) (forall (Y:fofType), (((x2 Xx) Y)->(((eq fofType) X) Y)))))) as proof of (((eq (fofType->Prop)) (fun (X:fofType)=> ((and ((x2 Xx) X)) (forall (Y:fofType), (((x2 Xx) Y)->(((eq fofType) X) Y)))))) b)
% 208.64/208.86 Found ((eq_ref (fofType->Prop)) (fun (X:fofType)=> ((and ((x2 Xx) X)) (forall (Y:fofType), (((x2 Xx) Y)->(((eq fofType) X) Y)))))) as proof of (((eq (fofType->Prop)) (fun (X:fofType)=> ((and ((x2 Xx) X)) (forall (Y:fofType), (((x2 Xx) Y)->(((eq fofType) X) Y)))))) b)
% 208.64/208.86 Found ((eq_ref (fofType->Prop)) (fun (X:fofType)=> ((and ((x2 Xx) X)) (forall (Y:fofType), (((x2 Xx) Y)->(((eq fofType) X) Y)))))) as proof of (((eq (fofType->Prop)) (fun (X:fofType)=> ((and ((x2 Xx) X)) (forall (Y:fofType), (((x2 Xx) Y)->(((eq fofType) X) Y)))))) b)
% 208.64/208.86 Found ((eq_ref (fofType->Prop)) (fun (X:fofType)=> ((and ((x2 Xx) X)) (forall (Y:fofType), (((x2 Xx) Y)->(((eq fofType) X) Y)))))) as proof of (((eq (fofType->Prop)) (fun (X:fofType)=> ((and ((x2 Xx) X)) (forall (Y:fofType), (((x2 Xx) Y)->(((eq fofType) X) Y)))))) b)
% 208.64/208.86 Found eq_ref00:=(eq_ref0 x3):(((eq fofType) x3) x3)
% 208.64/208.86 Found (eq_ref0 x3) as proof of (((eq fofType) x3) x30)
% 208.64/208.86 Found ((eq_ref fofType) x3) as proof of (((eq fofType) x3) x30)
% 208.64/208.86 Found ((eq_ref fofType) x3) as proof of (((eq fofType) x3) x30)
% 208.64/208.86 Found ((eq_ref fofType) x3) as proof of (((eq fofType) x3) x30)
% 208.64/208.86 Found (eq_sym000 ((eq_ref fofType) x3)) as proof of (((eq fofType) x30) x3)
% 208.64/208.86 Found ((eq_sym00 x30) ((eq_ref fofType) x3)) as proof of (((eq fofType) x30) x3)
% 208.64/208.86 Found (((eq_sym0 x3) x30) ((eq_ref fofType) x3)) as proof of (((eq fofType) x30) x3)
% 208.64/208.86 Found ((((eq_sym fofType) x3) x30) ((eq_ref fofType) x3)) as proof of (((eq fofType) x30) x3)
% 208.64/208.86 Found (fun (x20:(forall (Xn:fofType), (not (((eq fofType) (cS Xn)) c0))))=> ((((eq_sym fofType) x3) x30) ((eq_ref fofType) x3))) as proof of (((eq fofType) x30) x3)
% 208.64/208.86 Found eq_ref00:=(eq_ref0 (forall (Xx:fofType) (Xy:fofType), (((x2 Xx) Xy)->((x2 (cS Xx)) (cS (cS Xy)))))):(((eq Prop) (forall (Xx:fofType) (Xy:fofType), (((x2 Xx) Xy)->((x2 (cS Xx)) (cS (cS Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((x2 Xx) Xy)->((x2 (cS Xx)) (cS (cS Xy))))))
% 208.64/208.86 Found (eq_ref0 (forall (Xx:fofType) (Xy:fofType), (((x2 Xx) Xy)->((x2 (cS Xx)) (cS (cS Xy)))))) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), (((x2 Xx) Xy)->((x2 (cS Xx)) (cS (cS Xy)))))) b)
% 208.64/208.86 Found ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), (((x2 Xx) Xy)->((x2 (cS Xx)) (cS (cS Xy)))))) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), (((x2 Xx) Xy)->((x2 (cS Xx)) (cS (cS Xy)))))) b)
% 208.64/208.86 Found ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), (((x2 Xx) Xy)->((x2 (cS Xx)) (cS (cS Xy)))))) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), (((x2 Xx) Xy)->((x2 (cS Xx)) (cS (cS Xy)))))) b)
% 208.64/208.86 Found ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), (((x2 Xx) Xy)->((x2 (cS Xx)) (cS (cS Xy)))))) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), (((x2 Xx) Xy)->((x2 (cS Xx)) (cS (cS Xy)))))) b)
% 208.64/208.86 Found x6:(P x5)
% 208.64/208.86 Instantiate: x5:=Y:fofType
% 208.64/208.86 Found (fun (x6:(P x5))=> x6) as proof of (P Y)
% 208.64/208.86 Found (fun (P:(fofType->Prop)) (x6:(P x5))=> x6) as proof of ((P x5)->(P Y))
% 208.64/208.86 Found (fun (P:(fofType->Prop)) (x6:(P x5))=> x6) as proof of (((eq fofType) x5) Y)
% 208.64/208.86 Found (fun (x00:((x0 Xx) Y)) (P:(fofType->Prop)) (x6:(P x5))=> x6) as proof of (((eq fofType) x5) Y)
% 208.64/208.86 Found eq_ref000:=(eq_ref00 P):((P x5)->(P x5))
% 208.64/208.86 Found (eq_ref00 P) as proof of ((P x5)->(P Y))
% 208.64/208.86 Found ((eq_ref0 x5) P) as proof of ((P x5)->(P Y))
% 208.64/208.86 Found (((eq_ref fofType) x5) P) as proof of ((P x5)->(P Y))
% 208.64/208.86 Found (((eq_ref fofType) x5) P) as proof of ((P x5)->(P Y))
% 210.21/210.43 Found (fun (P:(fofType->Prop))=> (((eq_ref fofType) x5) P)) as proof of ((P x5)->(P Y))
% 210.21/210.43 Found (fun (P:(fofType->Prop))=> (((eq_ref fofType) x5) P)) as proof of (((eq fofType) x5) Y)
% 210.21/210.43 Found (fun (x00:((x0 Xx) Y)) (P:(fofType->Prop))=> (((eq_ref fofType) x5) P)) as proof of (((eq fofType) x5) Y)
% 210.21/210.43 Found x6:(P x5)
% 210.21/210.43 Instantiate: x5:=Y:fofType
% 210.21/210.43 Found (fun (x6:(P x5))=> x6) as proof of (P Y)
% 210.21/210.43 Found (fun (P:(fofType->Prop)) (x6:(P x5))=> x6) as proof of ((P x5)->(P Y))
% 210.21/210.43 Found (fun (P:(fofType->Prop)) (x6:(P x5))=> x6) as proof of (((eq fofType) x5) Y)
% 210.21/210.43 Found (fun (x00:((x2 Xx) Y)) (P:(fofType->Prop)) (x6:(P x5))=> x6) as proof of (((eq fofType) x5) Y)
% 210.21/210.43 Found eq_ref000:=(eq_ref00 P):((P x5)->(P x5))
% 210.21/210.43 Found (eq_ref00 P) as proof of ((P x5)->(P Y))
% 210.21/210.43 Found ((eq_ref0 x5) P) as proof of ((P x5)->(P Y))
% 210.21/210.43 Found (((eq_ref fofType) x5) P) as proof of ((P x5)->(P Y))
% 210.21/210.43 Found (((eq_ref fofType) x5) P) as proof of ((P x5)->(P Y))
% 210.21/210.43 Found (fun (P:(fofType->Prop))=> (((eq_ref fofType) x5) P)) as proof of ((P x5)->(P Y))
% 210.21/210.43 Found (fun (P:(fofType->Prop))=> (((eq_ref fofType) x5) P)) as proof of (((eq fofType) x5) Y)
% 210.21/210.43 Found (fun (x00:((x2 Xx) Y)) (P:(fofType->Prop))=> (((eq_ref fofType) x5) P)) as proof of (((eq fofType) x5) Y)
% 210.21/210.43 Found eq_ref000:=(eq_ref00 P):((P x10)->(P x10))
% 210.21/210.43 Found (eq_ref00 P) as proof of ((P x10)->(P x1))
% 210.21/210.43 Found ((eq_ref0 x10) P) as proof of ((P x10)->(P x1))
% 210.21/210.43 Found (((eq_ref fofType) x10) P) as proof of ((P x10)->(P x1))
% 210.21/210.43 Found (((eq_ref fofType) x10) P) as proof of ((P x10)->(P x1))
% 210.21/210.43 Found (fun (P:(fofType->Prop))=> (((eq_ref fofType) x10) P)) as proof of ((P x10)->(P x1))
% 210.21/210.43 Found (fun (P:(fofType->Prop))=> (((eq_ref fofType) x10) P)) as proof of (forall (P:(fofType->Prop)), ((P x10)->(P x1)))
% 210.21/210.43 Found eq_ref00:=(eq_ref0 x30):(((eq fofType) x30) x30)
% 210.21/210.43 Found (eq_ref0 x30) as proof of (((eq fofType) x30) x3)
% 210.21/210.43 Found ((eq_ref fofType) x30) as proof of (((eq fofType) x30) x3)
% 210.21/210.43 Found ((eq_ref fofType) x30) as proof of (((eq fofType) x30) x3)
% 210.21/210.43 Found (fun (x5:(forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy))))=> ((eq_ref fofType) x30)) as proof of (((eq fofType) x30) x3)
% 210.21/210.43 Found (fun (x4:cIND) (x5:(forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy))))=> ((eq_ref fofType) x30)) as proof of ((forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy)))->(((eq fofType) x30) x3))
% 210.21/210.43 Found (fun (x4:cIND) (x5:(forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy))))=> ((eq_ref fofType) x30)) as proof of (cIND->((forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy)))->(((eq fofType) x30) x3)))
% 210.21/210.43 Found (and_rect10 (fun (x4:cIND) (x5:(forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy))))=> ((eq_ref fofType) x30))) as proof of (((eq fofType) x30) x3)
% 210.21/210.43 Found ((and_rect1 (((eq fofType) x30) x3)) (fun (x4:cIND) (x5:(forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy))))=> ((eq_ref fofType) x30))) as proof of (((eq fofType) x30) x3)
% 210.21/210.43 Found (((fun (P:Type) (x4:(cIND->((forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy)))->P)))=> (((((and_rect cIND) (forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy)))) P) x4) x10)) (((eq fofType) x30) x3)) (fun (x4:cIND) (x5:(forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy))))=> ((eq_ref fofType) x30))) as proof of (((eq fofType) x30) x3)
% 210.21/210.43 Found (fun (x20:(forall (Xn:fofType), (not (((eq fofType) (cS Xn)) c0))))=> (((fun (P:Type) (x4:(cIND->((forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy)))->P)))=> (((((and_rect cIND) (forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy)))) P) x4) x10)) (((eq fofType) x30) x3)) (fun (x4:cIND) (x5:(forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy))))=> ((eq_ref fofType) x30)))) as proof of (((eq fofType) x30) x3)
% 214.65/214.83 Found x2:(P x10)
% 214.65/214.83 Instantiate: x1:=x10:fofType
% 214.65/214.83 Found (fun (x2:(P x10))=> x2) as proof of (P x1)
% 214.65/214.83 Found (fun (P:(fofType->Prop)) (x2:(P x10))=> x2) as proof of ((P x10)->(P x1))
% 214.65/214.83 Found (fun (P:(fofType->Prop)) (x2:(P x10))=> x2) as proof of (forall (P:(fofType->Prop)), ((P x10)->(P x1)))
% 214.65/214.83 Found eq_ref00:=(eq_ref0 x10):(((eq fofType) x10) x10)
% 214.65/214.83 Found (eq_ref0 x10) as proof of (((eq fofType) x10) x1)
% 214.65/214.83 Found ((eq_ref fofType) x10) as proof of (((eq fofType) x10) x1)
% 214.65/214.83 Found ((eq_ref fofType) x10) as proof of (((eq fofType) x10) x1)
% 214.65/214.83 Found (fun (x20:((and cIND) (forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy)))))=> ((eq_ref fofType) x10)) as proof of (((eq fofType) x10) x1)
% 214.65/214.83 Found (fun (x30:(forall (Xn:fofType), (not (((eq fofType) (cS Xn)) c0)))) (x20:((and cIND) (forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy)))))=> ((eq_ref fofType) x10)) as proof of (((and cIND) (forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy))))->(((eq fofType) x10) x1))
% 214.65/214.83 Found eq_ref00:=(eq_ref0 x30):(((eq fofType) x30) x30)
% 214.65/214.83 Found (eq_ref0 x30) as proof of (((eq fofType) x30) x3)
% 214.65/214.83 Found ((eq_ref fofType) x30) as proof of (((eq fofType) x30) x3)
% 214.65/214.83 Found ((eq_ref fofType) x30) as proof of (((eq fofType) x30) x3)
% 214.65/214.83 Found (fun (x20:(forall (Xn:fofType), (not (((eq fofType) (cS Xn)) c0))))=> ((eq_ref fofType) x30)) as proof of (((eq fofType) x30) x3)
% 214.65/214.83 Found eq_ref00:=(eq_ref0 c0):(((eq fofType) c0) c0)
% 214.65/214.83 Found (eq_ref0 c0) as proof of (((eq fofType) c0) x5)
% 214.65/214.83 Found ((eq_ref fofType) c0) as proof of (((eq fofType) c0) x5)
% 214.65/214.83 Found ((eq_ref fofType) c0) as proof of (((eq fofType) c0) x5)
% 214.65/214.83 Found ((eq_ref fofType) c0) as proof of (((eq fofType) c0) x5)
% 214.65/214.83 Found eq_sym000:=(eq_sym00 Y):((((eq fofType) x1) Y)->(((eq fofType) Y) x1))
% 214.65/214.83 Found (eq_sym00 Y) as proof of (((x0 Xx) Y)->(((eq fofType) x1) Y))
% 214.65/214.83 Found ((eq_sym0 x1) Y) as proof of (((x0 Xx) Y)->(((eq fofType) x1) Y))
% 214.65/214.83 Found (((eq_sym fofType) x1) Y) as proof of (((x0 Xx) Y)->(((eq fofType) x1) Y))
% 214.65/214.83 Found (((eq_sym fofType) x1) Y) as proof of (((x0 Xx) Y)->(((eq fofType) x1) Y))
% 214.65/214.83 Found (((eq_sym fofType) x1) Y) as proof of (((x0 Xx) Y)->(((eq fofType) x1) Y))
% 214.65/214.83 Found eq_ref00:=(eq_ref0 x10):(((eq fofType) x10) x10)
% 214.65/214.83 Found (eq_ref0 x10) as proof of (forall (P:(fofType->Prop)), ((P x10)->(P x1)))
% 214.65/214.83 Found ((eq_ref fofType) x10) as proof of (forall (P:(fofType->Prop)), ((P x10)->(P x1)))
% 214.65/214.83 Found ((eq_ref fofType) x10) as proof of (forall (P:(fofType->Prop)), ((P x10)->(P x1)))
% 214.65/214.83 Found ((eq_ref fofType) x10) as proof of (forall (P:(fofType->Prop)), ((P x10)->(P x1)))
% 214.65/214.83 Found x2:(P x10)
% 214.65/214.83 Instantiate: x1:=x10:fofType
% 214.65/214.83 Found (fun (x2:(P x10))=> x2) as proof of (P x1)
% 214.65/214.83 Found (fun (P:(fofType->Prop)) (x2:(P x10))=> x2) as proof of ((P x10)->(P x1))
% 214.65/214.83 Found eq_ref000:=(eq_ref00 P):((P x10)->(P x10))
% 214.65/214.83 Found (eq_ref00 P) as proof of ((P x10)->(P x1))
% 214.65/214.83 Found ((eq_ref0 x10) P) as proof of ((P x10)->(P x1))
% 214.65/214.83 Found (((eq_ref fofType) x10) P) as proof of ((P x10)->(P x1))
% 214.65/214.83 Found (((eq_ref fofType) x10) P) as proof of ((P x10)->(P x1))
% 214.65/214.83 Found (fun (P:(fofType->Prop))=> (((eq_ref fofType) x10) P)) as proof of ((P x10)->(P x1))
% 214.65/214.83 Found eq_sym000:=(eq_sym00 Y):((((eq fofType) x3) Y)->(((eq fofType) Y) x3))
% 214.65/214.83 Found (eq_sym00 Y) as proof of (((x0 Xx) Y)->(((eq fofType) x3) Y))
% 214.65/214.83 Found ((eq_sym0 x3) Y) as proof of (((x0 Xx) Y)->(((eq fofType) x3) Y))
% 214.65/214.83 Found (((eq_sym fofType) x3) Y) as proof of (((x0 Xx) Y)->(((eq fofType) x3) Y))
% 214.65/214.83 Found (((eq_sym fofType) x3) Y) as proof of (((x0 Xx) Y)->(((eq fofType) x3) Y))
% 214.65/214.83 Found (((eq_sym fofType) x3) Y) as proof of (((x0 Xx) Y)->(((eq fofType) x3) Y))
% 214.65/214.83 Found x6:(P x10)
% 214.65/214.83 Instantiate: x1:=x10:fofType
% 214.65/214.83 Found (fun (x6:(P x10))=> x6) as proof of (P x1)
% 214.65/214.83 Found (fun (P:(fofType->Prop)) (x6:(P x10))=> x6) as proof of ((P x10)->(P x1))
% 214.65/214.83 Found eq_ref000:=(eq_ref00 P):((P x10)->(P x10))
% 214.65/214.83 Found (eq_ref00 P) as proof of ((P x10)->(P x1))
% 214.65/214.83 Found ((eq_ref0 x10) P) as proof of ((P x10)->(P x1))
% 214.65/214.83 Found (((eq_ref fofType) x10) P) as proof of ((P x10)->(P x1))
% 214.65/214.83 Found (((eq_ref fofType) x10) P) as proof of ((P x10)->(P x1))
% 217.26/217.42 Found (fun (P:(fofType->Prop))=> (((eq_ref fofType) x10) P)) as proof of ((P x10)->(P x1))
% 217.26/217.42 Found x0000000:=(x000000 x5):(((eq fofType) x5) Y)
% 217.26/217.42 Found (x000000 x5) as proof of (((eq fofType) x5) Y)
% 217.26/217.42 Found ((x00000 x4) x5) as proof of (((eq fofType) x5) Y)
% 217.26/217.42 Found (((x0000 x3) x4) x5) as proof of (((eq fofType) x5) Y)
% 217.26/217.42 Found ((((x000 x2) x3) x4) x5) as proof of (((eq fofType) x5) Y)
% 217.26/217.42 Found (((((x00 x1) x2) x3) x4) x5) as proof of (((eq fofType) x5) Y)
% 217.26/217.42 Found (((((x00 x1) x2) x3) x4) x5) as proof of (((eq fofType) x5) Y)
% 217.26/217.42 Found (fun (x00:((x0 Xx) Y))=> (((((x00 x1) x2) x3) x4) x5)) as proof of (((eq fofType) x5) Y)
% 217.26/217.42 Found (fun (Y:fofType) (x00:((x0 Xx) Y))=> (((((x00 x1) x2) x3) x4) x5)) as proof of (((x0 Xx) Y)->(((eq fofType) x5) Y))
% 217.26/217.42 Found (fun (Y:fofType) (x00:((x0 Xx) Y))=> (((((x00 x1) x2) x3) x4) x5)) as proof of (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) x5) Y)))
% 217.26/217.42 Found eq_ref00:=(eq_ref0 x50):(((eq fofType) x50) x50)
% 217.26/217.42 Found (eq_ref0 x50) as proof of (((eq fofType) x50) x5)
% 217.26/217.42 Found ((eq_ref fofType) x50) as proof of (((eq fofType) x50) x5)
% 217.26/217.42 Found ((eq_ref fofType) x50) as proof of (((eq fofType) x50) x5)
% 217.26/217.42 Found x6:(P x50)
% 217.26/217.42 Instantiate: x5:=x50:fofType
% 217.26/217.42 Found (fun (x6:(P x50))=> x6) as proof of (P x5)
% 217.26/217.42 Found (fun (P:(fofType->Prop)) (x6:(P x50))=> x6) as proof of ((P x50)->(P x5))
% 217.26/217.42 Found eq_ref000:=(eq_ref00 P):((P x50)->(P x50))
% 217.26/217.42 Found (eq_ref00 P) as proof of ((P x50)->(P x5))
% 217.26/217.42 Found ((eq_ref0 x50) P) as proof of ((P x50)->(P x5))
% 217.26/217.42 Found (((eq_ref fofType) x50) P) as proof of ((P x50)->(P x5))
% 217.26/217.42 Found (((eq_ref fofType) x50) P) as proof of ((P x50)->(P x5))
% 217.26/217.42 Found (fun (P:(fofType->Prop))=> (((eq_ref fofType) x50) P)) as proof of ((P x50)->(P x5))
% 217.26/217.42 Found eq_ref00:=(eq_ref0 (cS x50)):(((eq fofType) (cS x50)) (cS x50))
% 217.26/217.42 Found (eq_ref0 (cS x50)) as proof of (((eq fofType) (cS x50)) (cS x5))
% 217.26/217.42 Found ((eq_ref fofType) (cS x50)) as proof of (((eq fofType) (cS x50)) (cS x5))
% 217.26/217.42 Found ((eq_ref fofType) (cS x50)) as proof of (((eq fofType) (cS x50)) (cS x5))
% 217.26/217.42 Found ((eq_ref fofType) (cS x50)) as proof of (((eq fofType) (cS x50)) (cS x5))
% 217.26/217.42 Found (x410 ((eq_ref fofType) (cS x50))) as proof of (((eq fofType) x50) x5)
% 217.26/217.42 Found ((x41 x5) ((eq_ref fofType) (cS x50))) as proof of (((eq fofType) x50) x5)
% 217.26/217.42 Found (((x4 x50) x5) ((eq_ref fofType) (cS x50))) as proof of (((eq fofType) x50) x5)
% 217.26/217.42 Found eq_ref00:=(eq_ref0 (cS x50)):(((eq fofType) (cS x50)) (cS x50))
% 217.26/217.42 Found (eq_ref0 (cS x50)) as proof of (((eq fofType) (cS x50)) (cS x5))
% 217.26/217.42 Found ((eq_ref fofType) (cS x50)) as proof of (((eq fofType) (cS x50)) (cS x5))
% 217.26/217.42 Found ((eq_ref fofType) (cS x50)) as proof of (((eq fofType) (cS x50)) (cS x5))
% 217.26/217.42 Found ((eq_ref fofType) (cS x50)) as proof of (((eq fofType) (cS x50)) (cS x5))
% 217.26/217.42 Found (x4000 ((eq_ref fofType) (cS x50))) as proof of (((eq fofType) x50) x5)
% 217.26/217.42 Found ((x400 x5) ((eq_ref fofType) (cS x50))) as proof of (((eq fofType) x50) x5)
% 217.26/217.42 Found (((x40 x50) x5) ((eq_ref fofType) (cS x50))) as proof of (((eq fofType) x50) x5)
% 217.26/217.42 Found eq_ref00:=(eq_ref0 x1):(((eq fofType) x1) x1)
% 217.26/217.42 Found (eq_ref0 x1) as proof of (((eq fofType) x1) Y)
% 217.26/217.42 Found ((eq_ref fofType) x1) as proof of (((eq fofType) x1) Y)
% 217.26/217.42 Found ((eq_ref fofType) x1) as proof of (((eq fofType) x1) Y)
% 217.26/217.42 Found (fun (x00:((x0 Xx) Y))=> ((eq_ref fofType) x1)) as proof of (((eq fofType) x1) Y)
% 217.26/217.42 Found x6:(P x3)
% 217.26/217.42 Instantiate: x3:=x':fofType
% 217.26/217.42 Found (fun (x6:(P x3))=> x6) as proof of (P x')
% 217.26/217.42 Found (fun (P:(fofType->Prop)) (x6:(P x3))=> x6) as proof of ((P x3)->(P x'))
% 217.26/217.42 Found (fun (x00:((x0 Xx) x')) (P:(fofType->Prop)) (x6:(P x3))=> x6) as proof of (((eq fofType) x3) x')
% 217.26/217.42 Found eq_ref000:=(eq_ref00 P):((P x3)->(P x3))
% 217.26/217.42 Found (eq_ref00 P) as proof of ((P x3)->(P x'))
% 217.26/217.42 Found ((eq_ref0 x3) P) as proof of ((P x3)->(P x'))
% 217.26/217.42 Found (((eq_ref fofType) x3) P) as proof of ((P x3)->(P x'))
% 217.26/217.42 Found (((eq_ref fofType) x3) P) as proof of ((P x3)->(P x'))
% 217.26/217.42 Found (fun (P:(fofType->Prop))=> (((eq_ref fofType) x3) P)) as proof of ((P x3)->(P x'))
% 217.26/217.42 Found (fun (x00:((x0 Xx) x')) (P:(fofType->Prop))=> (((eq_ref fofType) x3) P)) as proof of (((eq fofType) x3) x')
% 217.26/217.42 Found eq_ref00:=(eq_ref0 x10):(((eq fofType) x10) x10)
% 217.26/217.42 Found (eq_ref0 x10) as proof of (((eq fofType) x10) x1)
% 218.46/218.64 Found ((eq_ref fofType) x10) as proof of (((eq fofType) x10) x1)
% 218.46/218.64 Found ((eq_ref fofType) x10) as proof of (((eq fofType) x10) x1)
% 218.46/218.64 Found eq_ref00:=(eq_ref0 x3):(((eq fofType) x3) x3)
% 218.46/218.64 Found (eq_ref0 x3) as proof of (((eq fofType) x3) Y)
% 218.46/218.64 Found ((eq_ref fofType) x3) as proof of (((eq fofType) x3) Y)
% 218.46/218.64 Found ((eq_ref fofType) x3) as proof of (((eq fofType) x3) Y)
% 218.46/218.64 Found (fun (x00:((x0 Xx) Y))=> ((eq_ref fofType) x3)) as proof of (((eq fofType) x3) Y)
% 218.46/218.64 Found x6:(P x3)
% 218.46/218.64 Instantiate: x3:=x':fofType
% 218.46/218.64 Found (fun (x6:(P x3))=> x6) as proof of (P x')
% 218.46/218.64 Found (fun (P:(fofType->Prop)) (x6:(P x3))=> x6) as proof of ((P x3)->(P x'))
% 218.46/218.64 Found (fun (x00:((x2 Xx) x')) (P:(fofType->Prop)) (x6:(P x3))=> x6) as proof of (((eq fofType) x3) x')
% 218.46/218.64 Found eq_ref000:=(eq_ref00 P):((P x3)->(P x3))
% 218.46/218.64 Found (eq_ref00 P) as proof of ((P x3)->(P x'))
% 218.46/218.64 Found ((eq_ref0 x3) P) as proof of ((P x3)->(P x'))
% 218.46/218.64 Found (((eq_ref fofType) x3) P) as proof of ((P x3)->(P x'))
% 218.46/218.64 Found (((eq_ref fofType) x3) P) as proof of ((P x3)->(P x'))
% 218.46/218.64 Found (fun (P:(fofType->Prop))=> (((eq_ref fofType) x3) P)) as proof of ((P x3)->(P x'))
% 218.46/218.64 Found (fun (x00:((x2 Xx) x')) (P:(fofType->Prop))=> (((eq_ref fofType) x3) P)) as proof of (((eq fofType) x3) x')
% 218.46/218.64 Found eq_ref00:=(eq_ref0 x3):(((eq fofType) x3) x3)
% 218.46/218.64 Found (eq_ref0 x3) as proof of (((eq fofType) x3) x')
% 218.46/218.64 Found ((eq_ref fofType) x3) as proof of (((eq fofType) x3) x')
% 218.46/218.64 Found ((eq_ref fofType) x3) as proof of (((eq fofType) x3) x')
% 218.46/218.64 Found ((eq_ref fofType) x3) as proof of (((eq fofType) x3) x')
% 218.46/218.64 Found (fun (x00:((x0 Xx) x'))=> ((eq_ref fofType) x3)) as proof of (((eq fofType) x3) x')
% 218.46/218.64 Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% 218.46/218.64 Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) (((unique fofType) (x0 Xx)) x1))
% 218.46/218.64 Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (((unique fofType) (x0 Xx)) x1))
% 218.46/218.64 Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (((unique fofType) (x0 Xx)) x1))
% 218.46/218.64 Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) (((unique fofType) (x0 Xx)) x1))
% 218.46/218.64 Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (((unique fofType) (x0 Xx)) x)))
% 218.46/218.64 Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% 218.46/218.64 Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) (((unique fofType) (x0 Xx)) x1))
% 218.46/218.64 Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (((unique fofType) (x0 Xx)) x1))
% 218.46/218.64 Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (((unique fofType) (x0 Xx)) x1))
% 218.46/218.64 Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) (((unique fofType) (x0 Xx)) x1))
% 218.46/218.64 Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (((unique fofType) (x0 Xx)) x)))
% 218.46/218.64 Found eq_ref00:=(eq_ref0 x3):(((eq fofType) x3) x3)
% 218.46/218.64 Found (eq_ref0 x3) as proof of (((eq fofType) x3) x')
% 218.46/218.64 Found ((eq_ref fofType) x3) as proof of (((eq fofType) x3) x')
% 218.46/218.64 Found ((eq_ref fofType) x3) as proof of (((eq fofType) x3) x')
% 218.46/218.64 Found ((eq_ref fofType) x3) as proof of (((eq fofType) x3) x')
% 218.46/218.64 Found (fun (x00:((x2 Xx) x'))=> ((eq_ref fofType) x3)) as proof of (((eq fofType) x3) x')
% 218.46/218.64 Found x00000:=(x0000 x5):(((eq fofType) x5) Y)
% 218.46/218.64 Found (x0000 x5) as proof of (((eq fofType) x5) Y)
% 218.46/218.64 Found ((x000 x4) x5) as proof of (((eq fofType) x5) Y)
% 218.46/218.64 Found (((x00 x3) x4) x5) as proof of (((eq fofType) x5) Y)
% 218.46/218.64 Found (((x00 x3) x4) x5) as proof of (((eq fofType) x5) Y)
% 218.46/218.64 Found (fun (x00:((x2 Xx) Y))=> (((x00 x3) x4) x5)) as proof of (((eq fofType) x5) Y)
% 218.46/218.64 Found (fun (Y:fofType) (x00:((x2 Xx) Y))=> (((x00 x3) x4) x5)) as proof of (((x2 Xx) Y)->(((eq fofType) x5) Y))
% 218.46/218.64 Found (fun (Y:fofType) (x00:((x2 Xx) Y))=> (((x00 x3) x4) x5)) as proof of (forall (Y:fofType), (((x2 Xx) Y)->(((eq fofType) x5) Y)))
% 218.46/218.64 Found eq_ref00:=(eq_ref0 x50):(((eq fofType) x50) x50)
% 218.46/218.64 Found (eq_ref0 x50) as proof of (((eq fofType) x50) x5)
% 218.46/218.64 Found ((eq_ref fofType) x50) as proof of (((eq fofType) x50) x5)
% 218.46/218.64 Found ((eq_ref fofType) x50) as proof of (((eq fofType) x50) x5)
% 218.46/218.64 Found x6:(P x50)
% 218.46/218.64 Instantiate: x5:=x50:fofType
% 218.46/218.64 Found (fun (x6:(P x50))=> x6) as proof of (P x5)
% 220.63/220.85 Found (fun (P:(fofType->Prop)) (x6:(P x50))=> x6) as proof of ((P x50)->(P x5))
% 220.63/220.85 Found eq_ref000:=(eq_ref00 P):((P x50)->(P x50))
% 220.63/220.85 Found (eq_ref00 P) as proof of ((P x50)->(P x5))
% 220.63/220.85 Found ((eq_ref0 x50) P) as proof of ((P x50)->(P x5))
% 220.63/220.85 Found (((eq_ref fofType) x50) P) as proof of ((P x50)->(P x5))
% 220.63/220.85 Found (((eq_ref fofType) x50) P) as proof of ((P x50)->(P x5))
% 220.63/220.85 Found (fun (P:(fofType->Prop))=> (((eq_ref fofType) x50) P)) as proof of ((P x50)->(P x5))
% 220.63/220.85 Found eq_ref00:=(eq_ref0 x10):(((eq fofType) x10) x10)
% 220.63/220.85 Found (eq_ref0 x10) as proof of (((eq fofType) x10) x1)
% 220.63/220.85 Found ((eq_ref fofType) x10) as proof of (((eq fofType) x10) x1)
% 220.63/220.85 Found ((eq_ref fofType) x10) as proof of (((eq fofType) x10) x1)
% 220.63/220.85 Found (fun (x5:(forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy))))=> ((eq_ref fofType) x10)) as proof of (((eq fofType) x10) x1)
% 220.63/220.85 Found (fun (x4:cIND) (x5:(forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy))))=> ((eq_ref fofType) x10)) as proof of ((forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy)))->(((eq fofType) x10) x1))
% 220.63/220.85 Found (fun (x4:cIND) (x5:(forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy))))=> ((eq_ref fofType) x10)) as proof of (cIND->((forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy)))->(((eq fofType) x10) x1)))
% 220.63/220.85 Found (and_rect10 (fun (x4:cIND) (x5:(forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy))))=> ((eq_ref fofType) x10))) as proof of (((eq fofType) x10) x1)
% 220.63/220.85 Found ((and_rect1 (((eq fofType) x10) x1)) (fun (x4:cIND) (x5:(forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy))))=> ((eq_ref fofType) x10))) as proof of (((eq fofType) x10) x1)
% 220.63/220.85 Found (((fun (P:Type) (x4:(cIND->((forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy)))->P)))=> (((((and_rect cIND) (forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy)))) P) x4) x2)) (((eq fofType) x10) x1)) (fun (x4:cIND) (x5:(forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy))))=> ((eq_ref fofType) x10))) as proof of (((eq fofType) x10) x1)
% 220.63/220.85 Found (fun (x20:((and cIND) (forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy)))))=> (((fun (P:Type) (x4:(cIND->((forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy)))->P)))=> (((((and_rect cIND) (forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy)))) P) x4) x2)) (((eq fofType) x10) x1)) (fun (x4:cIND) (x5:(forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy))))=> ((eq_ref fofType) x10)))) as proof of (((eq fofType) x10) x1)
% 220.63/220.85 Found (fun (x30:(forall (Xn:fofType), (not (((eq fofType) (cS Xn)) c0)))) (x20:((and cIND) (forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy)))))=> (((fun (P:Type) (x4:(cIND->((forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy)))->P)))=> (((((and_rect cIND) (forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy)))) P) x4) x2)) (((eq fofType) x10) x1)) (fun (x4:cIND) (x5:(forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy))))=> ((eq_ref fofType) x10)))) as proof of (((and cIND) (forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy))))->(((eq fofType) x10) x1))
% 220.63/220.85 Found eq_ref00:=(eq_ref0 x10):(((eq fofType) x10) x10)
% 220.63/220.85 Found (eq_ref0 x10) as proof of (((eq fofType) x10) x1)
% 220.63/220.85 Found ((eq_ref fofType) x10) as proof of (((eq fofType) x10) x1)
% 220.63/220.85 Found ((eq_ref fofType) x10) as proof of (((eq fofType) x10) x1)
% 220.63/220.85 Found (fun (x5:(forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy))))=> ((eq_ref fofType) x10)) as proof of (((eq fofType) x10) x1)
% 220.63/220.87 Found (fun (x4:cIND) (x5:(forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy))))=> ((eq_ref fofType) x10)) as proof of ((forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy)))->(((eq fofType) x10) x1))
% 220.63/220.87 Found (fun (x4:cIND) (x5:(forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy))))=> ((eq_ref fofType) x10)) as proof of (cIND->((forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy)))->(((eq fofType) x10) x1)))
% 220.63/220.87 Found (and_rect10 (fun (x4:cIND) (x5:(forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy))))=> ((eq_ref fofType) x10))) as proof of (((eq fofType) x10) x1)
% 220.63/220.87 Found ((and_rect1 (((eq fofType) x10) x1)) (fun (x4:cIND) (x5:(forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy))))=> ((eq_ref fofType) x10))) as proof of (((eq fofType) x10) x1)
% 220.63/220.87 Found (((fun (P:Type) (x4:(cIND->((forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy)))->P)))=> (((((and_rect cIND) (forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy)))) P) x4) x20)) (((eq fofType) x10) x1)) (fun (x4:cIND) (x5:(forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy))))=> ((eq_ref fofType) x10))) as proof of (((eq fofType) x10) x1)
% 220.63/220.87 Found (fun (x20:((and cIND) (forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy)))))=> (((fun (P:Type) (x4:(cIND->((forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy)))->P)))=> (((((and_rect cIND) (forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy)))) P) x4) x20)) (((eq fofType) x10) x1)) (fun (x4:cIND) (x5:(forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy))))=> ((eq_ref fofType) x10)))) as proof of (((eq fofType) x10) x1)
% 220.63/220.87 Found (fun (x30:(forall (Xn:fofType), (not (((eq fofType) (cS Xn)) c0)))) (x20:((and cIND) (forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy)))))=> (((fun (P:Type) (x4:(cIND->((forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy)))->P)))=> (((((and_rect cIND) (forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy)))) P) x4) x20)) (((eq fofType) x10) x1)) (fun (x4:cIND) (x5:(forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy))))=> ((eq_ref fofType) x10)))) as proof of (((and cIND) (forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy))))->(((eq fofType) x10) x1))
% 220.63/220.87 Found eq_ref00:=(eq_ref0 x30):(((eq fofType) x30) x30)
% 220.63/220.87 Found (eq_ref0 x30) as proof of (((eq fofType) x30) x3)
% 220.63/220.87 Found ((eq_ref fofType) x30) as proof of (((eq fofType) x30) x3)
% 220.63/220.87 Found ((eq_ref fofType) x30) as proof of (((eq fofType) x30) x3)
% 220.63/220.87 Found (fun (x5:(forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy))))=> ((eq_ref fofType) x30)) as proof of (((eq fofType) x30) x3)
% 220.63/220.87 Found (fun (x4:cIND) (x5:(forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy))))=> ((eq_ref fofType) x30)) as proof of ((forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy)))->(((eq fofType) x30) x3))
% 220.63/220.87 Found (fun (x4:cIND) (x5:(forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy))))=> ((eq_ref fofType) x30)) as proof of (cIND->((forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy)))->(((eq fofType) x30) x3)))
% 220.63/220.87 Found (and_rect10 (fun (x4:cIND) (x5:(forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy))))=> ((eq_ref fofType) x30))) as proof of (((eq fofType) x30) x3)
% 222.74/222.96 Found ((and_rect1 (((eq fofType) x30) x3)) (fun (x4:cIND) (x5:(forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy))))=> ((eq_ref fofType) x30))) as proof of (((eq fofType) x30) x3)
% 222.74/222.96 Found (((fun (P:Type) (x4:(cIND->((forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy)))->P)))=> (((((and_rect cIND) (forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy)))) P) x4) x1)) (((eq fofType) x30) x3)) (fun (x4:cIND) (x5:(forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy))))=> ((eq_ref fofType) x30))) as proof of (((eq fofType) x30) x3)
% 222.74/222.96 Found (fun (x20:(forall (Xn:fofType), (not (((eq fofType) (cS Xn)) c0))))=> (((fun (P:Type) (x4:(cIND->((forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy)))->P)))=> (((((and_rect cIND) (forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy)))) P) x4) x1)) (((eq fofType) x30) x3)) (fun (x4:cIND) (x5:(forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy))))=> ((eq_ref fofType) x30)))) as proof of (((eq fofType) x30) x3)
% 222.74/222.96 Found eta_expansion:=(fun (A:Type) (B:Type)=> ((eta_expansion_dep A) (fun (x1:A)=> B))):(forall (A:Type) (B:Type) (f:(A->B)), (((eq (A->B)) f) (fun (x:A)=> (f x))))
% 222.74/222.96 Instantiate: b:=(forall (A:Type) (B:Type) (f:(A->B)), (((eq (A->B)) f) (fun (x:A)=> (f x)))):Prop
% 222.74/222.96 Found eta_expansion as proof of b
% 222.74/222.96 Found x0:((and cIND) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (cS Xx)) (cS Xy))->(((eq fofType) Xx) Xy))))
% 222.74/222.96 Instantiate: b:=(forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (cS Xx)) (cS Xy))->(((eq fofType) Xx) Xy))):Prop;x4:=(fun (x6:fofType) (x50:fofType)=> cIND):(fofType->(fofType->Prop))
% 222.74/222.96 Found x0 as proof of (P b)
% 222.74/222.96 Found eq_ref00:=(eq_ref0 x5):(((eq fofType) x5) x5)
% 222.74/222.96 Found (eq_ref0 x5) as proof of (((eq fofType) x5) x50)
% 222.74/222.96 Found ((eq_ref fofType) x5) as proof of (((eq fofType) x5) x50)
% 222.74/222.96 Found ((eq_ref fofType) x5) as proof of (((eq fofType) x5) x50)
% 222.74/222.96 Found ((eq_ref fofType) x5) as proof of (((eq fofType) x5) x50)
% 222.74/222.96 Found (eq_sym000 ((eq_ref fofType) x5)) as proof of (((eq fofType) x50) x5)
% 222.74/222.96 Found ((eq_sym00 x50) ((eq_ref fofType) x5)) as proof of (((eq fofType) x50) x5)
% 222.74/222.96 Found (((eq_sym0 x5) x50) ((eq_ref fofType) x5)) as proof of (((eq fofType) x50) x5)
% 222.74/222.96 Found ((((eq_sym fofType) x5) x50) ((eq_ref fofType) x5)) as proof of (((eq fofType) x50) x5)
% 222.74/222.96 Found eq_ref00:=(eq_ref0 x1):(((eq fofType) x1) x1)
% 222.74/222.96 Found (eq_ref0 x1) as proof of (((eq fofType) x1) x10)
% 222.74/222.96 Found ((eq_ref fofType) x1) as proof of (((eq fofType) x1) x10)
% 222.74/222.96 Found ((eq_ref fofType) x1) as proof of (((eq fofType) x1) x10)
% 222.74/222.96 Found ((eq_ref fofType) x1) as proof of (((eq fofType) x1) x10)
% 222.74/222.96 Found (eq_sym000 ((eq_ref fofType) x1)) as proof of (((eq fofType) x10) x1)
% 222.74/222.96 Found ((eq_sym00 x10) ((eq_ref fofType) x1)) as proof of (((eq fofType) x10) x1)
% 222.74/222.96 Found (((eq_sym0 x1) x10) ((eq_ref fofType) x1)) as proof of (((eq fofType) x10) x1)
% 222.74/222.96 Found ((((eq_sym fofType) x1) x10) ((eq_ref fofType) x1)) as proof of (((eq fofType) x10) x1)
% 222.74/222.96 Found (fun (x20:((and cIND) (forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy)))))=> ((((eq_sym fofType) x1) x10) ((eq_ref fofType) x1))) as proof of (((eq fofType) x10) x1)
% 222.74/222.96 Found (fun (x30:(forall (Xn:fofType), (not (((eq fofType) (cS Xn)) c0)))) (x20:((and cIND) (forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy)))))=> ((((eq_sym fofType) x1) x10) ((eq_ref fofType) x1))) as proof of (((and cIND) (forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy))))->(((eq fofType) x10) x1))
% 222.74/222.96 Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y)))))):(((eq (fofType->Prop)) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y)))))) (fun (x:fofType)=> ((and ((x0 Xx) x)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) x) Y))))))
% 223.94/224.19 Found (eta_expansion_dep00 (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y)))))) as proof of (((eq (fofType->Prop)) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y)))))) b)
% 223.94/224.19 Found ((eta_expansion_dep0 (fun (x4:fofType)=> Prop)) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y)))))) as proof of (((eq (fofType->Prop)) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y)))))) b)
% 223.94/224.19 Found (((eta_expansion_dep fofType) (fun (x4:fofType)=> Prop)) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y)))))) as proof of (((eq (fofType->Prop)) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y)))))) b)
% 223.94/224.19 Found (((eta_expansion_dep fofType) (fun (x4:fofType)=> Prop)) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y)))))) as proof of (((eq (fofType->Prop)) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y)))))) b)
% 223.94/224.19 Found (((eta_expansion_dep fofType) (fun (x4:fofType)=> Prop)) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y)))))) as proof of (((eq (fofType->Prop)) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y)))))) b)
% 223.94/224.19 Found eq_ref00:=(eq_ref0 (cS x50)):(((eq fofType) (cS x50)) (cS x50))
% 223.94/224.19 Found (eq_ref0 (cS x50)) as proof of (((eq fofType) (cS x50)) (cS x5))
% 223.94/224.19 Found ((eq_ref fofType) (cS x50)) as proof of (((eq fofType) (cS x50)) (cS x5))
% 223.94/224.19 Found ((eq_ref fofType) (cS x50)) as proof of (((eq fofType) (cS x50)) (cS x5))
% 223.94/224.19 Found ((eq_ref fofType) (cS x50)) as proof of (((eq fofType) (cS x50)) (cS x5))
% 223.94/224.19 Found (x4000 ((eq_ref fofType) (cS x50))) as proof of (((eq fofType) x50) x5)
% 223.94/224.19 Found ((x400 x5) ((eq_ref fofType) (cS x50))) as proof of (((eq fofType) x50) x5)
% 223.94/224.19 Found (((x40 x50) x5) ((eq_ref fofType) (cS x50))) as proof of (((eq fofType) x50) x5)
% 223.94/224.19 Found eq_ref00:=(eq_ref0 (cS x50)):(((eq fofType) (cS x50)) (cS x50))
% 223.94/224.19 Found (eq_ref0 (cS x50)) as proof of (((eq fofType) (cS x50)) (cS x5))
% 223.94/224.19 Found ((eq_ref fofType) (cS x50)) as proof of (((eq fofType) (cS x50)) (cS x5))
% 223.94/224.19 Found ((eq_ref fofType) (cS x50)) as proof of (((eq fofType) (cS x50)) (cS x5))
% 223.94/224.19 Found ((eq_ref fofType) (cS x50)) as proof of (((eq fofType) (cS x50)) (cS x5))
% 223.94/224.19 Found (x410 ((eq_ref fofType) (cS x50))) as proof of (((eq fofType) x50) x5)
% 223.94/224.19 Found ((x41 x5) ((eq_ref fofType) (cS x50))) as proof of (((eq fofType) x50) x5)
% 223.94/224.19 Found (((x4 x50) x5) ((eq_ref fofType) (cS x50))) as proof of (((eq fofType) x50) x5)
% 223.94/224.19 Found eq_sym000:=(eq_sym00 x'):((((eq fofType) x5) x')->(((eq fofType) x') x5))
% 223.94/224.19 Found (eq_sym00 x') as proof of (((x0 Xx) x')->(((eq fofType) x5) x'))
% 223.94/224.19 Found ((eq_sym0 x5) x') as proof of (((x0 Xx) x')->(((eq fofType) x5) x'))
% 223.94/224.19 Found (((eq_sym fofType) x5) x') as proof of (((x0 Xx) x')->(((eq fofType) x5) x'))
% 223.94/224.19 Found (((eq_sym fofType) x5) x') as proof of (((x0 Xx) x')->(((eq fofType) x5) x'))
% 223.94/224.19 Found (((eq_sym fofType) x5) x') as proof of (((x0 Xx) x')->(((eq fofType) x5) x'))
% 223.94/224.19 Found eq_ref00:=(eq_ref0 (f x4)):(((eq Prop) (f x4)) (f x4))
% 223.94/224.19 Found (eq_ref0 (f x4)) as proof of (((eq Prop) (f x4)) b)
% 223.94/224.19 Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) b)
% 223.94/224.19 Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) b)
% 223.94/224.19 Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) b)
% 223.94/224.19 Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% 223.94/224.19 Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((and ((x4 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x4 Xx) Xy)->((x4 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x4 Xx) X)) (forall (Y:fofType), (((x4 Xx) Y)->(((eq fofType) X) Y)))))))))
% 223.94/224.19 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and ((x4 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x4 Xx) Xy)->((x4 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x4 Xx) X)) (forall (Y:fofType), (((x4 Xx) Y)->(((eq fofType) X) Y)))))))))
% 224.03/224.20 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and ((x4 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x4 Xx) Xy)->((x4 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x4 Xx) X)) (forall (Y:fofType), (((x4 Xx) Y)->(((eq fofType) X) Y)))))))))
% 224.03/224.20 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and ((x4 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x4 Xx) Xy)->((x4 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x4 Xx) X)) (forall (Y:fofType), (((x4 Xx) Y)->(((eq fofType) X) Y)))))))))
% 224.03/224.20 Found ((eq_trans0000 ((eq_ref Prop) (f x4))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f x4)) ((and ((and ((x4 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x4 Xx) Xy)->((x4 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x4 Xx) X)) (forall (Y:fofType), (((x4 Xx) Y)->(((eq fofType) X) Y)))))))))
% 224.03/224.20 Found (((eq_trans000 ((and ((and ((x4 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x4 Xx) Xy)->((x4 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x4 Xx) X)) (forall (Y:fofType), (((x4 Xx) Y)->(((eq fofType) X) Y))))))))) ((eq_ref Prop) (f x4))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f x4)) ((and ((and ((x4 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x4 Xx) Xy)->((x4 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x4 Xx) X)) (forall (Y:fofType), (((x4 Xx) Y)->(((eq fofType) X) Y)))))))))
% 224.03/224.20 Found ((((eq_trans00 ((and ((and ((x4 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x4 Xx) Xy)->((x4 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x4 Xx) X)) (forall (Y:fofType), (((x4 Xx) Y)->(((eq fofType) X) Y))))))))) ((and ((and ((x4 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x4 Xx) Xy)->((x4 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x4 Xx) X)) (forall (Y:fofType), (((x4 Xx) Y)->(((eq fofType) X) Y))))))))) ((eq_ref Prop) (f x4))) ((eq_ref Prop) ((and ((and ((x4 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x4 Xx) Xy)->((x4 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x4 Xx) X)) (forall (Y:fofType), (((x4 Xx) Y)->(((eq fofType) X) Y)))))))))) as proof of (((eq Prop) (f x4)) ((and ((and ((x4 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x4 Xx) Xy)->((x4 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x4 Xx) X)) (forall (Y:fofType), (((x4 Xx) Y)->(((eq fofType) X) Y)))))))))
% 224.03/224.20 Found (((((eq_trans0 (f x4)) ((and ((and ((x4 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x4 Xx) Xy)->((x4 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x4 Xx) X)) (forall (Y:fofType), (((x4 Xx) Y)->(((eq fofType) X) Y))))))))) ((and ((and ((x4 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x4 Xx) Xy)->((x4 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x4 Xx) X)) (forall (Y:fofType), (((x4 Xx) Y)->(((eq fofType) X) Y))))))))) ((eq_ref Prop) (f x4))) ((eq_ref Prop) ((and ((and ((x4 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x4 Xx) Xy)->((x4 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x4 Xx) X)) (forall (Y:fofType), (((x4 Xx) Y)->(((eq fofType) X) Y)))))))))) as proof of (((eq Prop) (f x4)) ((and ((and ((x4 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x4 Xx) Xy)->((x4 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x4 Xx) X)) (forall (Y:fofType), (((x4 Xx) Y)->(((eq fofType) X) Y)))))))))
% 224.03/224.20 Found ((((((eq_trans Prop) (f x4)) ((and ((and ((x4 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x4 Xx) Xy)->((x4 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x4 Xx) X)) (forall (Y:fofType), (((x4 Xx) Y)->(((eq fofType) X) Y))))))))) ((and ((and ((x4 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x4 Xx) Xy)->((x4 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x4 Xx) X)) (forall (Y:fofType), (((x4 Xx) Y)->(((eq fofType) X) Y))))))))) ((eq_ref Prop) (f x4))) ((eq_ref Prop) ((and ((and ((x4 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x4 Xx) Xy)->((x4 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x4 Xx) X)) (forall (Y:fofType), (((x4 Xx) Y)->(((eq fofType) X) Y)))))))))) as proof of (((eq Prop) (f x4)) ((and ((and ((x4 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x4 Xx) Xy)->((x4 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x4 Xx) X)) (forall (Y:fofType), (((x4 Xx) Y)->(((eq fofType) X) Y)))))))))
% 224.15/224.35 Found eq_ref00:=(eq_ref0 (f x4)):(((eq Prop) (f x4)) (f x4))
% 224.15/224.35 Found (eq_ref0 (f x4)) as proof of (((eq Prop) (f x4)) b)
% 224.15/224.35 Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) b)
% 224.15/224.35 Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) b)
% 224.15/224.35 Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) b)
% 224.15/224.35 Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% 224.15/224.35 Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((and ((x4 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x4 Xx) Xy)->((x4 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x4 Xx) X)) (forall (Y:fofType), (((x4 Xx) Y)->(((eq fofType) X) Y)))))))))
% 224.15/224.35 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and ((x4 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x4 Xx) Xy)->((x4 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x4 Xx) X)) (forall (Y:fofType), (((x4 Xx) Y)->(((eq fofType) X) Y)))))))))
% 224.15/224.35 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and ((x4 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x4 Xx) Xy)->((x4 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x4 Xx) X)) (forall (Y:fofType), (((x4 Xx) Y)->(((eq fofType) X) Y)))))))))
% 224.15/224.35 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and ((x4 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x4 Xx) Xy)->((x4 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x4 Xx) X)) (forall (Y:fofType), (((x4 Xx) Y)->(((eq fofType) X) Y)))))))))
% 224.15/224.35 Found ((eq_trans0000 ((eq_ref Prop) (f x4))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f x4)) ((and ((and ((x4 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x4 Xx) Xy)->((x4 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x4 Xx) X)) (forall (Y:fofType), (((x4 Xx) Y)->(((eq fofType) X) Y)))))))))
% 224.15/224.35 Found (((eq_trans000 ((and ((and ((x4 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x4 Xx) Xy)->((x4 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x4 Xx) X)) (forall (Y:fofType), (((x4 Xx) Y)->(((eq fofType) X) Y))))))))) ((eq_ref Prop) (f x4))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f x4)) ((and ((and ((x4 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x4 Xx) Xy)->((x4 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x4 Xx) X)) (forall (Y:fofType), (((x4 Xx) Y)->(((eq fofType) X) Y)))))))))
% 224.15/224.35 Found ((((eq_trans00 ((and ((and ((x4 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x4 Xx) Xy)->((x4 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x4 Xx) X)) (forall (Y:fofType), (((x4 Xx) Y)->(((eq fofType) X) Y))))))))) ((and ((and ((x4 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x4 Xx) Xy)->((x4 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x4 Xx) X)) (forall (Y:fofType), (((x4 Xx) Y)->(((eq fofType) X) Y))))))))) ((eq_ref Prop) (f x4))) ((eq_ref Prop) ((and ((and ((x4 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x4 Xx) Xy)->((x4 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x4 Xx) X)) (forall (Y:fofType), (((x4 Xx) Y)->(((eq fofType) X) Y)))))))))) as proof of (((eq Prop) (f x4)) ((and ((and ((x4 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x4 Xx) Xy)->((x4 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x4 Xx) X)) (forall (Y:fofType), (((x4 Xx) Y)->(((eq fofType) X) Y)))))))))
% 225.21/225.44 Found (((((eq_trans0 (f x4)) ((and ((and ((x4 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x4 Xx) Xy)->((x4 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x4 Xx) X)) (forall (Y:fofType), (((x4 Xx) Y)->(((eq fofType) X) Y))))))))) ((and ((and ((x4 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x4 Xx) Xy)->((x4 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x4 Xx) X)) (forall (Y:fofType), (((x4 Xx) Y)->(((eq fofType) X) Y))))))))) ((eq_ref Prop) (f x4))) ((eq_ref Prop) ((and ((and ((x4 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x4 Xx) Xy)->((x4 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x4 Xx) X)) (forall (Y:fofType), (((x4 Xx) Y)->(((eq fofType) X) Y)))))))))) as proof of (((eq Prop) (f x4)) ((and ((and ((x4 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x4 Xx) Xy)->((x4 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x4 Xx) X)) (forall (Y:fofType), (((x4 Xx) Y)->(((eq fofType) X) Y)))))))))
% 225.21/225.44 Found ((((((eq_trans Prop) (f x4)) ((and ((and ((x4 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x4 Xx) Xy)->((x4 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x4 Xx) X)) (forall (Y:fofType), (((x4 Xx) Y)->(((eq fofType) X) Y))))))))) ((and ((and ((x4 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x4 Xx) Xy)->((x4 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x4 Xx) X)) (forall (Y:fofType), (((x4 Xx) Y)->(((eq fofType) X) Y))))))))) ((eq_ref Prop) (f x4))) ((eq_ref Prop) ((and ((and ((x4 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x4 Xx) Xy)->((x4 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x4 Xx) X)) (forall (Y:fofType), (((x4 Xx) Y)->(((eq fofType) X) Y)))))))))) as proof of (((eq Prop) (f x4)) ((and ((and ((x4 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x4 Xx) Xy)->((x4 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x4 Xx) X)) (forall (Y:fofType), (((x4 Xx) Y)->(((eq fofType) X) Y)))))))))
% 225.21/225.44 Found eq_ref00:=(eq_ref0 (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy)))))):(((eq Prop) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))
% 225.21/225.44 Found (eq_ref0 (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy)))))) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy)))))) b)
% 225.21/225.44 Found ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy)))))) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy)))))) b)
% 225.21/225.44 Found ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy)))))) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy)))))) b)
% 225.21/225.44 Found ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy)))))) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy)))))) b)
% 225.21/225.44 Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% 225.21/225.44 Found (eq_ref0 b) as proof of (((eq Prop) b) (f x0))
% 225.21/225.44 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% 225.21/225.44 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% 225.21/225.44 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% 225.21/225.44 Found eq_ref00:=(eq_ref0 ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y))))))))):(((eq Prop) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y))))))))) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y)))))))))
% 225.62/225.80 Found (eq_ref0 ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y))))))))) as proof of (((eq Prop) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y))))))))) b)
% 225.62/225.80 Found ((eq_ref Prop) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y))))))))) as proof of (((eq Prop) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y))))))))) b)
% 225.62/225.80 Found ((eq_ref Prop) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y))))))))) as proof of (((eq Prop) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y))))))))) b)
% 225.62/225.80 Found ((eq_ref Prop) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y))))))))) as proof of (((eq Prop) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y))))))))) b)
% 225.62/225.80 Found eq_ref00:=(eq_ref0 ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y))))))))):(((eq Prop) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y))))))))) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y)))))))))
% 225.62/225.80 Found (eq_ref0 ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y))))))))) as proof of (((eq Prop) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y))))))))) b)
% 225.62/225.80 Found ((eq_ref Prop) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y))))))))) as proof of (((eq Prop) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y))))))))) b)
% 226.33/226.55 Found ((eq_ref Prop) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y))))))))) as proof of (((eq Prop) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y))))))))) b)
% 226.33/226.55 Found ((eq_ref Prop) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y))))))))) as proof of (((eq Prop) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y))))))))) b)
% 226.33/226.55 Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% 226.33/226.55 Found (eq_ref0 b) as proof of (((eq Prop) b) (f x0))
% 226.33/226.55 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% 226.33/226.55 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% 226.33/226.55 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% 226.33/226.55 Found x6:(P x3)
% 226.33/226.55 Instantiate: x3:=Y:fofType
% 226.33/226.55 Found (fun (x6:(P x3))=> x6) as proof of (P Y)
% 226.33/226.55 Found (fun (P:(fofType->Prop)) (x6:(P x3))=> x6) as proof of ((P x3)->(P Y))
% 226.33/226.55 Found (fun (P:(fofType->Prop)) (x6:(P x3))=> x6) as proof of (((eq fofType) x3) Y)
% 226.33/226.55 Found (fun (x00:((x0 Xx) Y)) (P:(fofType->Prop)) (x6:(P x3))=> x6) as proof of (((eq fofType) x3) Y)
% 226.33/226.55 Found eq_ref000:=(eq_ref00 P):((P x3)->(P x3))
% 226.33/226.55 Found (eq_ref00 P) as proof of ((P x3)->(P Y))
% 226.33/226.55 Found ((eq_ref0 x3) P) as proof of ((P x3)->(P Y))
% 226.33/226.55 Found (((eq_ref fofType) x3) P) as proof of ((P x3)->(P Y))
% 226.33/226.55 Found (((eq_ref fofType) x3) P) as proof of ((P x3)->(P Y))
% 226.33/226.55 Found (fun (P:(fofType->Prop))=> (((eq_ref fofType) x3) P)) as proof of ((P x3)->(P Y))
% 226.33/226.55 Found (fun (P:(fofType->Prop))=> (((eq_ref fofType) x3) P)) as proof of (((eq fofType) x3) Y)
% 226.33/226.55 Found (fun (x00:((x0 Xx) Y)) (P:(fofType->Prop))=> (((eq_ref fofType) x3) P)) as proof of (((eq fofType) x3) Y)
% 226.33/226.55 Found eq_ref00:=(eq_ref0 (forall (Xx:fofType) (Xy:fofType), (((x4 Xx) Xy)->((x4 (cS Xx)) (cS (cS Xy)))))):(((eq Prop) (forall (Xx:fofType) (Xy:fofType), (((x4 Xx) Xy)->((x4 (cS Xx)) (cS (cS Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((x4 Xx) Xy)->((x4 (cS Xx)) (cS (cS Xy))))))
% 226.33/226.55 Found (eq_ref0 (forall (Xx:fofType) (Xy:fofType), (((x4 Xx) Xy)->((x4 (cS Xx)) (cS (cS Xy)))))) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), (((x4 Xx) Xy)->((x4 (cS Xx)) (cS (cS Xy)))))) b)
% 226.33/226.55 Found ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), (((x4 Xx) Xy)->((x4 (cS Xx)) (cS (cS Xy)))))) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), (((x4 Xx) Xy)->((x4 (cS Xx)) (cS (cS Xy)))))) b)
% 226.33/226.55 Found ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), (((x4 Xx) Xy)->((x4 (cS Xx)) (cS (cS Xy)))))) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), (((x4 Xx) Xy)->((x4 (cS Xx)) (cS (cS Xy)))))) b)
% 226.33/226.55 Found ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), (((x4 Xx) Xy)->((x4 (cS Xx)) (cS (cS Xy)))))) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), (((x4 Xx) Xy)->((x4 (cS Xx)) (cS (cS Xy)))))) b)
% 226.33/226.55 Found x6:(P x3)
% 226.33/226.55 Instantiate: x3:=Y:fofType
% 226.33/226.55 Found (fun (x6:(P x3))=> x6) as proof of (P Y)
% 226.33/226.55 Found (fun (P:(fofType->Prop)) (x6:(P x3))=> x6) as proof of ((P x3)->(P Y))
% 226.33/226.55 Found (fun (P:(fofType->Prop)) (x6:(P x3))=> x6) as proof of (((eq fofType) x3) Y)
% 226.33/226.55 Found (fun (x00:((x2 Xx) Y)) (P:(fofType->Prop)) (x6:(P x3))=> x6) as proof of (((eq fofType) x3) Y)
% 233.92/234.11 Found eq_ref000:=(eq_ref00 P):((P x3)->(P x3))
% 233.92/234.11 Found (eq_ref00 P) as proof of ((P x3)->(P Y))
% 233.92/234.11 Found ((eq_ref0 x3) P) as proof of ((P x3)->(P Y))
% 233.92/234.11 Found (((eq_ref fofType) x3) P) as proof of ((P x3)->(P Y))
% 233.92/234.11 Found (((eq_ref fofType) x3) P) as proof of ((P x3)->(P Y))
% 233.92/234.11 Found (fun (P:(fofType->Prop))=> (((eq_ref fofType) x3) P)) as proof of ((P x3)->(P Y))
% 233.92/234.11 Found (fun (P:(fofType->Prop))=> (((eq_ref fofType) x3) P)) as proof of (((eq fofType) x3) Y)
% 233.92/234.11 Found (fun (x00:((x2 Xx) Y)) (P:(fofType->Prop))=> (((eq_ref fofType) x3) P)) as proof of (((eq fofType) x3) Y)
% 233.92/234.11 Found eq_ref00:=(eq_ref0 x5):(((eq fofType) x5) x5)
% 233.92/234.11 Found (eq_ref0 x5) as proof of (((eq fofType) x5) x')
% 233.92/234.11 Found ((eq_ref fofType) x5) as proof of (((eq fofType) x5) x')
% 233.92/234.11 Found ((eq_ref fofType) x5) as proof of (((eq fofType) x5) x')
% 233.92/234.11 Found (fun (x00:((x0 Xx) x'))=> ((eq_ref fofType) x5)) as proof of (((eq fofType) x5) x')
% 233.92/234.11 Found x7:(P (cS Xx0))
% 233.92/234.11 Instantiate: x5:=(cS Xx0):fofType
% 233.92/234.11 Found (fun (x7:(P (cS Xx0)))=> x7) as proof of (P x5)
% 233.92/234.11 Found (fun (P:(fofType->Prop)) (x7:(P (cS Xx0)))=> x7) as proof of ((P (cS Xx0))->(P x5))
% 233.92/234.11 Found (fun (x6:(((eq fofType) Xx0) x5)) (P:(fofType->Prop)) (x7:(P (cS Xx0)))=> x7) as proof of (((eq fofType) (cS Xx0)) x5)
% 233.92/234.11 Found eq_ref000:=(eq_ref00 P):((P (cS Xx0))->(P (cS Xx0)))
% 233.92/234.11 Found (eq_ref00 P) as proof of ((P (cS Xx0))->(P x5))
% 233.92/234.11 Found ((eq_ref0 (cS Xx0)) P) as proof of ((P (cS Xx0))->(P x5))
% 233.92/234.11 Found (((eq_ref fofType) (cS Xx0)) P) as proof of ((P (cS Xx0))->(P x5))
% 233.92/234.11 Found (((eq_ref fofType) (cS Xx0)) P) as proof of ((P (cS Xx0))->(P x5))
% 233.92/234.11 Found (fun (P:(fofType->Prop))=> (((eq_ref fofType) (cS Xx0)) P)) as proof of ((P (cS Xx0))->(P x5))
% 233.92/234.11 Found (fun (x6:(((eq fofType) Xx0) x5)) (P:(fofType->Prop))=> (((eq_ref fofType) (cS Xx0)) P)) as proof of (((eq fofType) (cS Xx0)) x5)
% 233.92/234.11 Found eq_ref000:=(eq_ref00 (ex fofType)):(((ex fofType) ((unique fofType) (x4 Xx)))->((ex fofType) ((unique fofType) (x4 Xx))))
% 233.92/234.11 Found (eq_ref00 (ex fofType)) as proof of (P ((unique fofType) (x4 Xx)))
% 233.92/234.11 Found ((eq_ref0 ((unique fofType) (x4 Xx))) (ex fofType)) as proof of (P ((unique fofType) (x4 Xx)))
% 233.92/234.11 Found (((eq_ref (fofType->Prop)) ((unique fofType) (x4 Xx))) (ex fofType)) as proof of (P ((unique fofType) (x4 Xx)))
% 233.92/234.11 Found (((eq_ref (fofType->Prop)) ((unique fofType) (x4 Xx))) (ex fofType)) as proof of (P ((unique fofType) (x4 Xx)))
% 233.92/234.11 Found eq_ref00:=(eq_ref0 (cS Xx0)):(((eq fofType) (cS Xx0)) (cS Xx0))
% 233.92/234.11 Found (eq_ref0 (cS Xx0)) as proof of (((eq fofType) (cS Xx0)) x5)
% 233.92/234.11 Found ((eq_ref fofType) (cS Xx0)) as proof of (((eq fofType) (cS Xx0)) x5)
% 233.92/234.11 Found ((eq_ref fofType) (cS Xx0)) as proof of (((eq fofType) (cS Xx0)) x5)
% 233.92/234.11 Found (fun (x6:(((eq fofType) Xx0) x5))=> ((eq_ref fofType) (cS Xx0))) as proof of (((eq fofType) (cS Xx0)) x5)
% 233.92/234.11 Found eta_expansion000:=(eta_expansion00 ((unique fofType) (x2 Xx))):(((eq (fofType->Prop)) ((unique fofType) (x2 Xx))) (fun (x:fofType)=> (((unique fofType) (x2 Xx)) x)))
% 233.92/234.11 Found (eta_expansion00 ((unique fofType) (x2 Xx))) as proof of (((eq (fofType->Prop)) ((unique fofType) (x2 Xx))) b)
% 233.92/234.11 Found ((eta_expansion0 Prop) ((unique fofType) (x2 Xx))) as proof of (((eq (fofType->Prop)) ((unique fofType) (x2 Xx))) b)
% 233.92/234.11 Found (((eta_expansion fofType) Prop) ((unique fofType) (x2 Xx))) as proof of (((eq (fofType->Prop)) ((unique fofType) (x2 Xx))) b)
% 233.92/234.11 Found (((eta_expansion fofType) Prop) ((unique fofType) (x2 Xx))) as proof of (((eq (fofType->Prop)) ((unique fofType) (x2 Xx))) b)
% 233.92/234.11 Found (((eta_expansion fofType) Prop) ((unique fofType) (x2 Xx))) as proof of (((eq (fofType->Prop)) ((unique fofType) (x2 Xx))) b)
% 233.92/234.11 Found eq_ref00:=(eq_ref0 (cS x30)):(((eq fofType) (cS x30)) (cS x30))
% 233.92/234.11 Found (eq_ref0 (cS x30)) as proof of (((eq fofType) (cS x30)) (cS x3))
% 233.92/234.11 Found ((eq_ref fofType) (cS x30)) as proof of (((eq fofType) (cS x30)) (cS x3))
% 233.92/234.11 Found ((eq_ref fofType) (cS x30)) as proof of (((eq fofType) (cS x30)) (cS x3))
% 233.92/234.11 Found ((eq_ref fofType) (cS x30)) as proof of (((eq fofType) (cS x30)) (cS x3))
% 233.92/234.11 Found (x500 ((eq_ref fofType) (cS x30))) as proof of (((eq fofType) x30) x3)
% 233.92/234.11 Found ((x50 x3) ((eq_ref fofType) (cS x30))) as proof of (((eq fofType) x30) x3)
% 233.92/234.11 Found (((x5 x30) x3) ((eq_ref fofType) (cS x30))) as proof of (((eq fofType) x30) x3)
% 234.82/235.00 Found eq_ref00:=(eq_ref0 (cS x5)):(((eq fofType) (cS x5)) (cS x5))
% 234.82/235.00 Found (eq_ref0 (cS x5)) as proof of (((eq fofType) (cS x5)) (cS x'))
% 234.82/235.00 Found ((eq_ref fofType) (cS x5)) as proof of (((eq fofType) (cS x5)) (cS x'))
% 234.82/235.00 Found ((eq_ref fofType) (cS x5)) as proof of (((eq fofType) (cS x5)) (cS x'))
% 234.82/235.00 Found ((eq_ref fofType) (cS x5)) as proof of (((eq fofType) (cS x5)) (cS x'))
% 234.82/235.00 Found (x400 ((eq_ref fofType) (cS x5))) as proof of (((eq fofType) x5) x')
% 234.82/235.00 Found ((x40 x') ((eq_ref fofType) (cS x5))) as proof of (((eq fofType) x5) x')
% 234.82/235.00 Found (((x4 x5) x') ((eq_ref fofType) (cS x5))) as proof of (((eq fofType) x5) x')
% 234.82/235.00 Found (fun (x00:((x0 Xx) x'))=> (((x4 x5) x') ((eq_ref fofType) (cS x5)))) as proof of (((eq fofType) x5) x')
% 234.82/235.00 Found eq_ref000:=(eq_ref00 P0):((P0 (f x4))->(P0 (f x4)))
% 234.82/235.00 Found (eq_ref00 P0) as proof of (P1 (f x4))
% 234.82/235.00 Found ((eq_ref0 (f x4)) P0) as proof of (P1 (f x4))
% 234.82/235.00 Found (((eq_ref Prop) (f x4)) P0) as proof of (P1 (f x4))
% 234.82/235.00 Found (((eq_ref Prop) (f x4)) P0) as proof of (P1 (f x4))
% 234.82/235.00 Found eq_ref000:=(eq_ref00 P0):((P0 (f x4))->(P0 (f x4)))
% 234.82/235.00 Found (eq_ref00 P0) as proof of (P1 (f x4))
% 234.82/235.00 Found ((eq_ref0 (f x4)) P0) as proof of (P1 (f x4))
% 234.82/235.00 Found (((eq_ref Prop) (f x4)) P0) as proof of (P1 (f x4))
% 234.82/235.00 Found (((eq_ref Prop) (f x4)) P0) as proof of (P1 (f x4))
% 234.82/235.00 Found eq_ref000:=(eq_ref00 P):((P x10)->(P x10))
% 234.82/235.00 Found (eq_ref00 P) as proof of ((P x10)->(P x1))
% 234.82/235.00 Found ((eq_ref0 x10) P) as proof of ((P x10)->(P x1))
% 234.82/235.00 Found (((eq_ref fofType) x10) P) as proof of ((P x10)->(P x1))
% 234.82/235.00 Found (((eq_ref fofType) x10) P) as proof of ((P x10)->(P x1))
% 234.82/235.00 Found (fun (P:(fofType->Prop))=> (((eq_ref fofType) x10) P)) as proof of ((P x10)->(P x1))
% 234.82/235.00 Found (fun (P:(fofType->Prop))=> (((eq_ref fofType) x10) P)) as proof of (forall (P:(fofType->Prop)), ((P x10)->(P x1)))
% 234.82/235.00 Found x2:(P x10)
% 234.82/235.00 Instantiate: x1:=x10:fofType
% 234.82/235.00 Found (fun (x2:(P x10))=> x2) as proof of (P x1)
% 234.82/235.00 Found (fun (P:(fofType->Prop)) (x2:(P x10))=> x2) as proof of ((P x10)->(P x1))
% 234.82/235.00 Found (fun (P:(fofType->Prop)) (x2:(P x10))=> x2) as proof of (forall (P:(fofType->Prop)), ((P x10)->(P x1)))
% 234.82/235.00 Found x00000:=(x0000 x3):(((eq fofType) x3) Y)
% 234.82/235.00 Found (x0000 x3) as proof of (((eq fofType) x3) Y)
% 234.82/235.00 Found ((fun (x30:fofType)=> ((x000 x30) x5)) x3) as proof of (((eq fofType) x3) Y)
% 234.82/235.00 Found ((fun (x30:fofType)=> (((fun (x30:fofType)=> ((x00 x30) x4)) x30) x5)) x3) as proof of (((eq fofType) x3) Y)
% 234.82/235.00 Found ((fun (x30:fofType)=> (((fun (x30:fofType)=> ((x00 x30) x4)) x30) x5)) x3) as proof of (((eq fofType) x3) Y)
% 234.82/235.00 Found (fun (x00:((x2 Xx) Y))=> ((fun (x30:fofType)=> (((fun (x30:fofType)=> ((x00 x30) x4)) x30) x5)) x3)) as proof of (((eq fofType) x3) Y)
% 234.82/235.00 Found (fun (Y:fofType) (x00:((x2 Xx) Y))=> ((fun (x30:fofType)=> (((fun (x30:fofType)=> ((x00 x30) x4)) x30) x5)) x3)) as proof of (((x2 Xx) Y)->(((eq fofType) x3) Y))
% 234.82/235.00 Found (fun (Y:fofType) (x00:((x2 Xx) Y))=> ((fun (x30:fofType)=> (((fun (x30:fofType)=> ((x00 x30) x4)) x30) x5)) x3)) as proof of (forall (Y:fofType), (((x2 Xx) Y)->(((eq fofType) x3) Y)))
% 234.82/235.00 Found eq_ref00:=(eq_ref0 x30):(((eq fofType) x30) x30)
% 234.82/235.00 Found (eq_ref0 x30) as proof of (((eq fofType) x30) x3)
% 234.82/235.00 Found ((eq_ref fofType) x30) as proof of (((eq fofType) x30) x3)
% 234.82/235.00 Found ((eq_ref fofType) x30) as proof of (((eq fofType) x30) x3)
% 234.82/235.00 Found (fun (x50:(forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy))))=> ((eq_ref fofType) x30)) as proof of (((eq fofType) x30) x3)
% 234.82/235.00 Found (fun (x40:cIND) (x50:(forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy))))=> ((eq_ref fofType) x30)) as proof of ((forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy)))->(((eq fofType) x30) x3))
% 234.82/235.00 Found x6:(P x30)
% 234.82/235.00 Instantiate: x3:=x30:fofType
% 234.82/235.00 Found (fun (x6:(P x30))=> x6) as proof of (P x3)
% 234.82/235.00 Found (fun (P:(fofType->Prop)) (x6:(P x30))=> x6) as proof of ((P x30)->(P x3))
% 234.82/235.00 Found (fun (x50:(forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy)))) (P:(fofType->Prop)) (x6:(P x30))=> x6) as proof of (((eq fofType) x30) x3)
% 234.82/235.00 Found (fun (x40:cIND) (x50:(forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy)))) (P:(fofType->Prop)) (x6:(P x30))=> x6) as proof of ((forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy)))->(((eq fofType) x30) x3))
% 238.27/238.50 Found eq_ref000:=(eq_ref00 P):((P x30)->(P x30))
% 238.27/238.50 Found (eq_ref00 P) as proof of ((P x30)->(P x3))
% 238.27/238.50 Found ((eq_ref0 x30) P) as proof of ((P x30)->(P x3))
% 238.27/238.50 Found (((eq_ref fofType) x30) P) as proof of ((P x30)->(P x3))
% 238.27/238.50 Found (((eq_ref fofType) x30) P) as proof of ((P x30)->(P x3))
% 238.27/238.50 Found (fun (P:(fofType->Prop))=> (((eq_ref fofType) x30) P)) as proof of ((P x30)->(P x3))
% 238.27/238.50 Found (fun (x50:(forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy)))) (P:(fofType->Prop))=> (((eq_ref fofType) x30) P)) as proof of (((eq fofType) x30) x3)
% 238.27/238.50 Found (fun (x40:cIND) (x50:(forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy)))) (P:(fofType->Prop))=> (((eq_ref fofType) x30) P)) as proof of ((forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy)))->(((eq fofType) x30) x3))
% 238.27/238.50 Found eq_ref000:=(eq_ref00 P):((P x5)->(P x5))
% 238.27/238.50 Found (eq_ref00 P) as proof of ((P x5)->(P Y))
% 238.27/238.50 Found ((eq_ref0 x5) P) as proof of ((P x5)->(P Y))
% 238.27/238.50 Found (((eq_ref fofType) x5) P) as proof of ((P x5)->(P Y))
% 238.27/238.50 Found (((eq_ref fofType) x5) P) as proof of ((P x5)->(P Y))
% 238.27/238.50 Found (fun (P:(fofType->Prop))=> (((eq_ref fofType) x5) P)) as proof of ((P x5)->(P Y))
% 238.27/238.50 Found (fun (x00:((x0 Xx) Y)) (P:(fofType->Prop))=> (((eq_ref fofType) x5) P)) as proof of (((eq fofType) x5) Y)
% 238.27/238.50 Found x6:(P x5)
% 238.27/238.50 Instantiate: x5:=Y:fofType
% 238.27/238.50 Found (fun (x6:(P x5))=> x6) as proof of (P Y)
% 238.27/238.50 Found (fun (P:(fofType->Prop)) (x6:(P x5))=> x6) as proof of ((P x5)->(P Y))
% 238.27/238.50 Found (fun (x00:((x0 Xx) Y)) (P:(fofType->Prop)) (x6:(P x5))=> x6) as proof of (((eq fofType) x5) Y)
% 238.27/238.50 Found eq_ref00:=(eq_ref0 x5):(((eq fofType) x5) x5)
% 238.27/238.50 Found (eq_ref0 x5) as proof of (((eq fofType) x5) Y)
% 238.27/238.50 Found ((eq_ref fofType) x5) as proof of (((eq fofType) x5) Y)
% 238.27/238.50 Found ((eq_ref fofType) x5) as proof of (((eq fofType) x5) Y)
% 238.27/238.50 Found ((eq_ref fofType) x5) as proof of (((eq fofType) x5) Y)
% 238.27/238.50 Found (fun (x00:((x0 Xx) Y))=> ((eq_ref fofType) x5)) as proof of (((eq fofType) x5) Y)
% 238.27/238.50 Found x2:(P x10)
% 238.27/238.50 Instantiate: x1:=x10:fofType
% 238.27/238.50 Found (fun (x2:(P x10))=> x2) as proof of (P x1)
% 238.27/238.50 Found (fun (P:(fofType->Prop)) (x2:(P x10))=> x2) as proof of ((P x10)->(P x1))
% 238.27/238.50 Found x6:(P x5)
% 238.27/238.50 Instantiate: x5:=x':fofType
% 238.27/238.50 Found (fun (x6:(P x5))=> x6) as proof of (P x')
% 238.27/238.50 Found (fun (P:(fofType->Prop)) (x6:(P x5))=> x6) as proof of ((P x5)->(P x'))
% 238.27/238.50 Found (fun (P:(fofType->Prop)) (x6:(P x5))=> x6) as proof of (((eq fofType) x5) x')
% 238.27/238.50 Found (fun (x00:((x0 Xx) x')) (P:(fofType->Prop)) (x6:(P x5))=> x6) as proof of (((eq fofType) x5) x')
% 238.27/238.50 Found eq_ref000:=(eq_ref00 P):((P x10)->(P x10))
% 238.27/238.50 Found (eq_ref00 P) as proof of ((P x10)->(P x1))
% 238.27/238.50 Found ((eq_ref0 x10) P) as proof of ((P x10)->(P x1))
% 238.27/238.50 Found (((eq_ref fofType) x10) P) as proof of ((P x10)->(P x1))
% 238.27/238.50 Found (((eq_ref fofType) x10) P) as proof of ((P x10)->(P x1))
% 238.27/238.50 Found (fun (P:(fofType->Prop))=> (((eq_ref fofType) x10) P)) as proof of ((P x10)->(P x1))
% 238.27/238.50 Found eq_ref000:=(eq_ref00 P):((P x30)->(P x30))
% 238.27/238.50 Found (eq_ref00 P) as proof of ((P x30)->(P x3))
% 238.27/238.50 Found ((eq_ref0 x30) P) as proof of ((P x30)->(P x3))
% 238.27/238.50 Found (((eq_ref fofType) x30) P) as proof of ((P x30)->(P x3))
% 238.27/238.50 Found (((eq_ref fofType) x30) P) as proof of ((P x30)->(P x3))
% 238.27/238.50 Found (fun (P:(fofType->Prop))=> (((eq_ref fofType) x30) P)) as proof of ((P x30)->(P x3))
% 238.27/238.50 Found (fun (x20:(forall (Xn:fofType), (not (((eq fofType) (cS Xn)) c0)))) (P:(fofType->Prop))=> (((eq_ref fofType) x30) P)) as proof of (((eq fofType) x30) x3)
% 238.27/238.50 Found eq_ref000:=(eq_ref00 P):((P x10)->(P x10))
% 238.27/238.50 Found (eq_ref00 P) as proof of ((P x10)->(P x1))
% 238.27/238.50 Found ((eq_ref0 x10) P) as proof of ((P x10)->(P x1))
% 238.27/238.50 Found (((eq_ref fofType) x10) P) as proof of ((P x10)->(P x1))
% 238.27/238.50 Found (((eq_ref fofType) x10) P) as proof of ((P x10)->(P x1))
% 238.27/238.50 Found (fun (P:(fofType->Prop))=> (((eq_ref fofType) x10) P)) as proof of ((P x10)->(P x1))
% 238.27/238.50 Found (fun (x20:((and cIND) (forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy))))) (P:(fofType->Prop))=> (((eq_ref fofType) x10) P)) as proof of (((eq fofType) x10) x1)
% 240.22/240.41 Found (fun (x30:(forall (Xn:fofType), (not (((eq fofType) (cS Xn)) c0)))) (x20:((and cIND) (forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy))))) (P:(fofType->Prop))=> (((eq_ref fofType) x10) P)) as proof of (((and cIND) (forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy))))->(((eq fofType) x10) x1))
% 240.22/240.41 Found x6:(P x30)
% 240.22/240.41 Instantiate: x3:=x30:fofType
% 240.22/240.41 Found (fun (x6:(P x30))=> x6) as proof of (P x3)
% 240.22/240.41 Found (fun (P:(fofType->Prop)) (x6:(P x30))=> x6) as proof of ((P x30)->(P x3))
% 240.22/240.41 Found (fun (x20:(forall (Xn:fofType), (not (((eq fofType) (cS Xn)) c0)))) (P:(fofType->Prop)) (x6:(P x30))=> x6) as proof of (((eq fofType) x30) x3)
% 240.22/240.41 Found x6:(P x10)
% 240.22/240.41 Instantiate: x1:=x10:fofType
% 240.22/240.41 Found (fun (x6:(P x10))=> x6) as proof of (P x1)
% 240.22/240.41 Found (fun (P:(fofType->Prop)) (x6:(P x10))=> x6) as proof of ((P x10)->(P x1))
% 240.22/240.41 Found (fun (x20:((and cIND) (forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy))))) (P:(fofType->Prop)) (x6:(P x10))=> x6) as proof of (((eq fofType) x10) x1)
% 240.22/240.41 Found (fun (x30:(forall (Xn:fofType), (not (((eq fofType) (cS Xn)) c0)))) (x20:((and cIND) (forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy))))) (P:(fofType->Prop)) (x6:(P x10))=> x6) as proof of (((and cIND) (forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy))))->(((eq fofType) x10) x1))
% 240.22/240.41 Found x6:(P x5)
% 240.22/240.41 Instantiate: x5:=x':fofType
% 240.22/240.41 Found (fun (x6:(P x5))=> x6) as proof of (P x')
% 240.22/240.41 Found (fun (P:(fofType->Prop)) (x6:(P x5))=> x6) as proof of ((P x5)->(P x'))
% 240.22/240.41 Found (fun (P:(fofType->Prop)) (x6:(P x5))=> x6) as proof of (((eq fofType) x5) x')
% 240.22/240.41 Found (fun (x00:((x2 Xx) x')) (P:(fofType->Prop)) (x6:(P x5))=> x6) as proof of (((eq fofType) x5) x')
% 240.22/240.41 Found eq_ref000:=(eq_ref00 P):((P x5)->(P x5))
% 240.22/240.41 Found (eq_ref00 P) as proof of ((P x5)->(P x'))
% 240.22/240.41 Found ((eq_ref0 x5) P) as proof of ((P x5)->(P x'))
% 240.22/240.41 Found (((eq_ref fofType) x5) P) as proof of ((P x5)->(P x'))
% 240.22/240.41 Found (((eq_ref fofType) x5) P) as proof of ((P x5)->(P x'))
% 240.22/240.41 Found (fun (P:(fofType->Prop))=> (((eq_ref fofType) x5) P)) as proof of ((P x5)->(P x'))
% 240.22/240.41 Found (fun (P:(fofType->Prop))=> (((eq_ref fofType) x5) P)) as proof of (((eq fofType) x5) x')
% 240.22/240.41 Found (fun (x00:((x0 Xx) x')) (P:(fofType->Prop))=> (((eq_ref fofType) x5) P)) as proof of (((eq fofType) x5) x')
% 240.22/240.41 Found eq_ref000:=(eq_ref00 P):((P x5)->(P x5))
% 240.22/240.41 Found (eq_ref00 P) as proof of ((P x5)->(P x'))
% 240.22/240.41 Found ((eq_ref0 x5) P) as proof of ((P x5)->(P x'))
% 240.22/240.41 Found (((eq_ref fofType) x5) P) as proof of ((P x5)->(P x'))
% 240.22/240.41 Found (((eq_ref fofType) x5) P) as proof of ((P x5)->(P x'))
% 240.22/240.41 Found (fun (P:(fofType->Prop))=> (((eq_ref fofType) x5) P)) as proof of ((P x5)->(P x'))
% 240.22/240.41 Found (fun (P:(fofType->Prop))=> (((eq_ref fofType) x5) P)) as proof of (((eq fofType) x5) x')
% 240.22/240.41 Found (fun (x00:((x2 Xx) x')) (P:(fofType->Prop))=> (((eq_ref fofType) x5) P)) as proof of (((eq fofType) x5) x')
% 240.22/240.41 Found eq_ref00:=(eq_ref0 x10):(((eq fofType) x10) x10)
% 240.22/240.41 Found (eq_ref0 x10) as proof of (((eq fofType) x10) x1)
% 240.22/240.41 Found ((eq_ref fofType) x10) as proof of (((eq fofType) x10) x1)
% 240.22/240.41 Found ((eq_ref fofType) x10) as proof of (((eq fofType) x10) x1)
% 240.22/240.41 Found (fun (x20:((and cIND) (forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy)))))=> ((eq_ref fofType) x10)) as proof of (((eq fofType) x10) x1)
% 240.22/240.41 Found (fun (x30:(forall (Xn:fofType), (not (((eq fofType) (cS Xn)) c0)))) (x20:((and cIND) (forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy)))))=> ((eq_ref fofType) x10)) as proof of (((and cIND) (forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy))))->(((eq fofType) x10) x1))
% 240.22/240.41 Found eq_ref00:=(eq_ref0 x30):(((eq fofType) x30) x30)
% 240.22/240.41 Found (eq_ref0 x30) as proof of (((eq fofType) x30) x3)
% 240.22/240.41 Found ((eq_ref fofType) x30) as proof of (((eq fofType) x30) x3)
% 243.35/243.54 Found ((eq_ref fofType) x30) as proof of (((eq fofType) x30) x3)
% 243.35/243.54 Found (fun (x20:(forall (Xn:fofType), (not (((eq fofType) (cS Xn)) c0))))=> ((eq_ref fofType) x30)) as proof of (((eq fofType) x30) x3)
% 243.35/243.54 Found eq_ref00:=(eq_ref0 (cS x30)):(((eq fofType) (cS x30)) (cS x30))
% 243.35/243.54 Found (eq_ref0 (cS x30)) as proof of (((eq fofType) (cS x30)) (cS x3))
% 243.35/243.54 Found ((eq_ref fofType) (cS x30)) as proof of (((eq fofType) (cS x30)) (cS x3))
% 243.35/243.54 Found ((eq_ref fofType) (cS x30)) as proof of (((eq fofType) (cS x30)) (cS x3))
% 243.35/243.54 Found ((eq_ref fofType) (cS x30)) as proof of (((eq fofType) (cS x30)) (cS x3))
% 243.35/243.54 Found (x5000 ((eq_ref fofType) (cS x30))) as proof of (((eq fofType) x30) x3)
% 243.35/243.54 Found ((x500 x3) ((eq_ref fofType) (cS x30))) as proof of (((eq fofType) x30) x3)
% 243.35/243.54 Found (((x50 x30) x3) ((eq_ref fofType) (cS x30))) as proof of (((eq fofType) x30) x3)
% 243.35/243.54 Found (fun (x50:(forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy))))=> (((x50 x30) x3) ((eq_ref fofType) (cS x30)))) as proof of (((eq fofType) x30) x3)
% 243.35/243.54 Found (fun (x40:cIND) (x50:(forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy))))=> (((x50 x30) x3) ((eq_ref fofType) (cS x30)))) as proof of ((forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy)))->(((eq fofType) x30) x3))
% 243.35/243.54 Found eq_ref00:=(eq_ref0 (cS x30)):(((eq fofType) (cS x30)) (cS x30))
% 243.35/243.54 Found (eq_ref0 (cS x30)) as proof of (((eq fofType) (cS x30)) (cS x3))
% 243.35/243.54 Found ((eq_ref fofType) (cS x30)) as proof of (((eq fofType) (cS x30)) (cS x3))
% 243.35/243.54 Found ((eq_ref fofType) (cS x30)) as proof of (((eq fofType) (cS x30)) (cS x3))
% 243.35/243.54 Found ((eq_ref fofType) (cS x30)) as proof of (((eq fofType) (cS x30)) (cS x3))
% 243.35/243.54 Found (x510 ((eq_ref fofType) (cS x30))) as proof of (((eq fofType) x30) x3)
% 243.35/243.54 Found ((x51 x3) ((eq_ref fofType) (cS x30))) as proof of (((eq fofType) x30) x3)
% 243.35/243.54 Found (((x5 x30) x3) ((eq_ref fofType) (cS x30))) as proof of (((eq fofType) x30) x3)
% 243.35/243.54 Found (fun (x50:(forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy))))=> (((x5 x30) x3) ((eq_ref fofType) (cS x30)))) as proof of (((eq fofType) x30) x3)
% 243.35/243.54 Found (fun (x40:cIND) (x50:(forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy))))=> (((x5 x30) x3) ((eq_ref fofType) (cS x30)))) as proof of ((forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy)))->(((eq fofType) x30) x3))
% 243.35/243.54 Found eq_sym000:=(eq_sym00 x'):((((eq fofType) x1) x')->(((eq fofType) x') x1))
% 243.35/243.54 Found (eq_sym00 x') as proof of (((x0 Xx) x')->(((eq fofType) x1) x'))
% 243.35/243.54 Found ((eq_sym0 x1) x') as proof of (((x0 Xx) x')->(((eq fofType) x1) x'))
% 243.35/243.54 Found (((eq_sym fofType) x1) x') as proof of (((x0 Xx) x')->(((eq fofType) x1) x'))
% 243.35/243.54 Found (((eq_sym fofType) x1) x') as proof of (((x0 Xx) x')->(((eq fofType) x1) x'))
% 243.35/243.54 Found (((eq_sym fofType) x1) x') as proof of (((x0 Xx) x')->(((eq fofType) x1) x'))
% 243.35/243.54 Found eq_sym000:=(eq_sym00 x'):((((eq fofType) x3) x')->(((eq fofType) x') x3))
% 243.35/243.54 Found (eq_sym00 x') as proof of (((x0 Xx) x')->(((eq fofType) x3) x'))
% 243.35/243.54 Found ((eq_sym0 x3) x') as proof of (((x0 Xx) x')->(((eq fofType) x3) x'))
% 243.35/243.54 Found (((eq_sym fofType) x3) x') as proof of (((x0 Xx) x')->(((eq fofType) x3) x'))
% 243.35/243.54 Found (((eq_sym fofType) x3) x') as proof of (((x0 Xx) x')->(((eq fofType) x3) x'))
% 243.35/243.54 Found (((eq_sym fofType) x3) x') as proof of (((x0 Xx) x')->(((eq fofType) x3) x'))
% 243.35/243.54 Found eq_ref00:=(eq_ref0 x1):(((eq fofType) x1) x1)
% 243.35/243.54 Found (eq_ref0 x1) as proof of (((eq fofType) x1) x')
% 243.35/243.54 Found ((eq_ref fofType) x1) as proof of (((eq fofType) x1) x')
% 243.35/243.54 Found ((eq_ref fofType) x1) as proof of (((eq fofType) x1) x')
% 243.35/243.54 Found (fun (x00:((x0 Xx) x'))=> ((eq_ref fofType) x1)) as proof of (((eq fofType) x1) x')
% 243.35/243.54 Found eq_ref00:=(eq_ref0 x5):(((eq fofType) x5) x5)
% 243.35/243.54 Found (eq_ref0 x5) as proof of (((eq fofType) x5) x50)
% 243.35/243.54 Found ((eq_ref fofType) x5) as proof of (((eq fofType) x5) x50)
% 243.35/243.54 Found ((eq_ref fofType) x5) as proof of (((eq fofType) x5) x50)
% 243.35/243.54 Found ((eq_ref fofType) x5) as proof of (((eq fofType) x5) x50)
% 243.35/243.54 Found (eq_sym000 ((eq_ref fofType) x5)) as proof of (((eq fofType) x50) x5)
% 245.72/245.90 Found ((eq_sym00 x50) ((eq_ref fofType) x5)) as proof of (((eq fofType) x50) x5)
% 245.72/245.90 Found (((eq_sym0 x5) x50) ((eq_ref fofType) x5)) as proof of (((eq fofType) x50) x5)
% 245.72/245.90 Found ((((eq_sym fofType) x5) x50) ((eq_ref fofType) x5)) as proof of (((eq fofType) x50) x5)
% 245.72/245.90 Found eq_ref00:=(eq_ref0 x3):(((eq fofType) x3) x3)
% 245.72/245.90 Found (eq_ref0 x3) as proof of (((eq fofType) x3) x')
% 245.72/245.90 Found ((eq_ref fofType) x3) as proof of (((eq fofType) x3) x')
% 245.72/245.90 Found ((eq_ref fofType) x3) as proof of (((eq fofType) x3) x')
% 245.72/245.90 Found (fun (x00:((x0 Xx) x'))=> ((eq_ref fofType) x3)) as proof of (((eq fofType) x3) x')
% 245.72/245.90 Found x0000000:=(x000000 x5):(((eq fofType) x5) x')
% 245.72/245.90 Found (x000000 x5) as proof of (((eq fofType) x5) x')
% 245.72/245.90 Found ((x00000 x4) x5) as proof of (((eq fofType) x5) x')
% 245.72/245.90 Found (((x0000 x3) x4) x5) as proof of (((eq fofType) x5) x')
% 245.72/245.90 Found ((((x000 x2) x3) x4) x5) as proof of (((eq fofType) x5) x')
% 245.72/245.90 Found (((((x00 x1) x2) x3) x4) x5) as proof of (((eq fofType) x5) x')
% 245.72/245.90 Found (((((x00 x1) x2) x3) x4) x5) as proof of (((eq fofType) x5) x')
% 245.72/245.92 Found (fun (x00:((x0 Xx) x'))=> (((((x00 x1) x2) x3) x4) x5)) as proof of (((eq fofType) x5) x')
% 245.72/245.92 Found (fun (x':fofType) (x00:((x0 Xx) x'))=> (((((x00 x1) x2) x3) x4) x5)) as proof of (((x0 Xx) x')->(((eq fofType) x5) x'))
% 245.72/245.92 Found (fun (x':fofType) (x00:((x0 Xx) x'))=> (((((x00 x1) x2) x3) x4) x5)) as proof of (forall (x':fofType), (((x0 Xx) x')->(((eq fofType) x5) x')))
% 245.72/245.92 Found eq_ref00:=(eq_ref0 x50):(((eq fofType) x50) x50)
% 245.72/245.92 Found (eq_ref0 x50) as proof of (((eq fofType) x50) x5)
% 245.72/245.92 Found ((eq_ref fofType) x50) as proof of (((eq fofType) x50) x5)
% 245.72/245.92 Found ((eq_ref fofType) x50) as proof of (((eq fofType) x50) x5)
% 245.72/245.92 Found x6:(P x50)
% 245.72/245.92 Instantiate: x5:=x50:fofType
% 245.72/245.92 Found (fun (x6:(P x50))=> x6) as proof of (P x5)
% 245.72/245.92 Found (fun (P:(fofType->Prop)) (x6:(P x50))=> x6) as proof of ((P x50)->(P x5))
% 245.72/245.92 Found eq_ref000:=(eq_ref00 P):((P x50)->(P x50))
% 245.72/245.92 Found (eq_ref00 P) as proof of ((P x50)->(P x5))
% 245.72/245.92 Found ((eq_ref0 x50) P) as proof of ((P x50)->(P x5))
% 245.72/245.92 Found (((eq_ref fofType) x50) P) as proof of ((P x50)->(P x5))
% 245.72/245.92 Found (((eq_ref fofType) x50) P) as proof of ((P x50)->(P x5))
% 245.72/245.92 Found (fun (P:(fofType->Prop))=> (((eq_ref fofType) x50) P)) as proof of ((P x50)->(P x5))
% 245.72/245.92 Found eq_ref00:=(eq_ref0 (cS x50)):(((eq fofType) (cS x50)) (cS x50))
% 245.72/245.92 Found (eq_ref0 (cS x50)) as proof of (((eq fofType) (cS x50)) (cS x5))
% 245.72/245.92 Found ((eq_ref fofType) (cS x50)) as proof of (((eq fofType) (cS x50)) (cS x5))
% 245.72/245.92 Found ((eq_ref fofType) (cS x50)) as proof of (((eq fofType) (cS x50)) (cS x5))
% 245.72/245.92 Found ((eq_ref fofType) (cS x50)) as proof of (((eq fofType) (cS x50)) (cS x5))
% 245.72/245.92 Found (x4000 ((eq_ref fofType) (cS x50))) as proof of (((eq fofType) x50) x5)
% 245.72/245.92 Found ((x400 x5) ((eq_ref fofType) (cS x50))) as proof of (((eq fofType) x50) x5)
% 245.72/245.92 Found (((x40 x50) x5) ((eq_ref fofType) (cS x50))) as proof of (((eq fofType) x50) x5)
% 245.72/245.92 Found eq_ref00:=(eq_ref0 (cS x50)):(((eq fofType) (cS x50)) (cS x50))
% 245.72/245.92 Found (eq_ref0 (cS x50)) as proof of (((eq fofType) (cS x50)) (cS x5))
% 245.72/245.92 Found ((eq_ref fofType) (cS x50)) as proof of (((eq fofType) (cS x50)) (cS x5))
% 245.72/245.92 Found ((eq_ref fofType) (cS x50)) as proof of (((eq fofType) (cS x50)) (cS x5))
% 245.72/245.92 Found ((eq_ref fofType) (cS x50)) as proof of (((eq fofType) (cS x50)) (cS x5))
% 245.72/245.92 Found (x410 ((eq_ref fofType) (cS x50))) as proof of (((eq fofType) x50) x5)
% 245.72/245.92 Found ((x41 x5) ((eq_ref fofType) (cS x50))) as proof of (((eq fofType) x50) x5)
% 245.72/245.92 Found (((x4 x50) x5) ((eq_ref fofType) (cS x50))) as proof of (((eq fofType) x50) x5)
% 245.72/245.92 Found eq_ref000:=(eq_ref00 (ex fofType)):(((ex fofType) ((unique fofType) (x0 Xx)))->((ex fofType) ((unique fofType) (x0 Xx))))
% 245.72/245.92 Found (eq_ref00 (ex fofType)) as proof of (P ((unique fofType) (x0 Xx)))
% 245.72/245.92 Found ((eq_ref0 ((unique fofType) (x0 Xx))) (ex fofType)) as proof of (P ((unique fofType) (x0 Xx)))
% 245.72/245.92 Found (((eq_ref (fofType->Prop)) ((unique fofType) (x0 Xx))) (ex fofType)) as proof of (P ((unique fofType) (x0 Xx)))
% 245.72/245.92 Found (((eq_ref (fofType->Prop)) ((unique fofType) (x0 Xx))) (ex fofType)) as proof of (P ((unique fofType) (x0 Xx)))
% 245.72/245.92 Found x00000:=(x0000 x5):(((eq fofType) x5) x')
% 245.72/245.92 Found (x0000 x5) as proof of (((eq fofType) x5) x')
% 248.02/248.25 Found ((x000 x4) x5) as proof of (((eq fofType) x5) x')
% 248.02/248.25 Found (((x00 x3) x4) x5) as proof of (((eq fofType) x5) x')
% 248.02/248.25 Found (((x00 x3) x4) x5) as proof of (((eq fofType) x5) x')
% 248.02/248.25 Found (fun (x00:((x2 Xx) x'))=> (((x00 x3) x4) x5)) as proof of (((eq fofType) x5) x')
% 248.02/248.25 Found (fun (x':fofType) (x00:((x2 Xx) x'))=> (((x00 x3) x4) x5)) as proof of (((x2 Xx) x')->(((eq fofType) x5) x'))
% 248.02/248.25 Found (fun (x':fofType) (x00:((x2 Xx) x'))=> (((x00 x3) x4) x5)) as proof of (forall (x':fofType), (((x2 Xx) x')->(((eq fofType) x5) x')))
% 248.02/248.25 Found eq_ref00:=(eq_ref0 x50):(((eq fofType) x50) x50)
% 248.02/248.25 Found (eq_ref0 x50) as proof of (((eq fofType) x50) x5)
% 248.02/248.25 Found ((eq_ref fofType) x50) as proof of (((eq fofType) x50) x5)
% 248.02/248.25 Found ((eq_ref fofType) x50) as proof of (((eq fofType) x50) x5)
% 248.02/248.25 Found x6:(P x50)
% 248.02/248.25 Instantiate: x5:=x50:fofType
% 248.02/248.25 Found (fun (x6:(P x50))=> x6) as proof of (P x5)
% 248.02/248.25 Found (fun (P:(fofType->Prop)) (x6:(P x50))=> x6) as proof of ((P x50)->(P x5))
% 248.02/248.25 Found eq_ref000:=(eq_ref00 P):((P x50)->(P x50))
% 248.02/248.25 Found (eq_ref00 P) as proof of ((P x50)->(P x5))
% 248.02/248.26 Found ((eq_ref0 x50) P) as proof of ((P x50)->(P x5))
% 248.02/248.26 Found (((eq_ref fofType) x50) P) as proof of ((P x50)->(P x5))
% 248.02/248.26 Found (((eq_ref fofType) x50) P) as proof of ((P x50)->(P x5))
% 248.02/248.26 Found (fun (P:(fofType->Prop))=> (((eq_ref fofType) x50) P)) as proof of ((P x50)->(P x5))
% 248.02/248.26 Found eq_ref000:=(eq_ref00 (ex fofType)):(((ex fofType) ((unique fofType) (x2 Xx)))->((ex fofType) ((unique fofType) (x2 Xx))))
% 248.02/248.26 Found (eq_ref00 (ex fofType)) as proof of (P ((unique fofType) (x2 Xx)))
% 248.02/248.26 Found ((eq_ref0 ((unique fofType) (x2 Xx))) (ex fofType)) as proof of (P ((unique fofType) (x2 Xx)))
% 248.02/248.26 Found (((eq_ref (fofType->Prop)) ((unique fofType) (x2 Xx))) (ex fofType)) as proof of (P ((unique fofType) (x2 Xx)))
% 248.02/248.26 Found (((eq_ref (fofType->Prop)) ((unique fofType) (x2 Xx))) (ex fofType)) as proof of (P ((unique fofType) (x2 Xx)))
% 248.02/248.26 Found eq_ref00:=(eq_ref0 ((unique fofType) (x0 Xx))):(((eq (fofType->Prop)) ((unique fofType) (x0 Xx))) ((unique fofType) (x0 Xx)))
% 248.02/248.26 Found (eq_ref0 ((unique fofType) (x0 Xx))) as proof of (((eq (fofType->Prop)) ((unique fofType) (x0 Xx))) b)
% 248.02/248.26 Found ((eq_ref (fofType->Prop)) ((unique fofType) (x0 Xx))) as proof of (((eq (fofType->Prop)) ((unique fofType) (x0 Xx))) b)
% 248.02/248.26 Found ((eq_ref (fofType->Prop)) ((unique fofType) (x0 Xx))) as proof of (((eq (fofType->Prop)) ((unique fofType) (x0 Xx))) b)
% 248.02/248.26 Found ((eq_ref (fofType->Prop)) ((unique fofType) (x0 Xx))) as proof of (((eq (fofType->Prop)) ((unique fofType) (x0 Xx))) b)
% 248.02/248.26 Found x0:((and cIND) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (cS Xx)) (cS Xy))->(((eq fofType) Xx) Xy))))
% 248.02/248.26 Instantiate: b:=(forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (cS Xx)) (cS Xy))->(((eq fofType) Xx) Xy))):Prop;x4:=(fun (x6:fofType) (x50:fofType)=> cIND):(fofType->(fofType->Prop))
% 248.02/248.26 Found x0 as proof of (P b)
% 248.02/248.26 Found eq_ref00:=(eq_ref0 ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx)))))):(((eq Prop) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx)))))) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx))))))
% 248.02/248.26 Found (eq_ref0 ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx)))))) as proof of (((eq Prop) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx)))))) b)
% 248.02/248.26 Found ((eq_ref Prop) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx)))))) as proof of (((eq Prop) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx)))))) b)
% 249.21/249.41 Found ((eq_ref Prop) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx)))))) as proof of (((eq Prop) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx)))))) b)
% 249.21/249.41 Found ((eq_ref Prop) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx)))))) as proof of (((eq Prop) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx)))))) b)
% 249.21/249.41 Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% 249.21/249.41 Found (eq_ref0 b) as proof of (((eq Prop) b) (f x0))
% 249.21/249.41 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% 249.21/249.41 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% 249.21/249.41 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% 249.21/249.41 Found eq_ref00:=(eq_ref0 ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx)))))):(((eq Prop) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx)))))) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx))))))
% 249.21/249.41 Found (eq_ref0 ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx)))))) as proof of (((eq Prop) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx)))))) b)
% 249.21/249.41 Found ((eq_ref Prop) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx)))))) as proof of (((eq Prop) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx)))))) b)
% 249.21/249.41 Found ((eq_ref Prop) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx)))))) as proof of (((eq Prop) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx)))))) b)
% 249.21/249.41 Found ((eq_ref Prop) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx)))))) as proof of (((eq Prop) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx)))))) b)
% 249.21/249.41 Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% 249.21/249.41 Found (eq_ref0 b) as proof of (((eq Prop) b) (f x0))
% 249.21/249.41 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% 249.21/249.41 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% 249.21/249.41 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% 249.21/249.41 Found eq_ref00:=(eq_ref0 x10):(((eq fofType) x10) x10)
% 249.21/249.41 Found (eq_ref0 x10) as proof of (((eq fofType) x10) x1)
% 249.21/249.41 Found ((eq_ref fofType) x10) as proof of (((eq fofType) x10) x1)
% 249.21/249.41 Found ((eq_ref fofType) x10) as proof of (((eq fofType) x10) x1)
% 249.21/249.41 Found (fun (x5:(forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy))))=> ((eq_ref fofType) x10)) as proof of (((eq fofType) x10) x1)
% 249.21/249.41 Found (fun (x4:cIND) (x5:(forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy))))=> ((eq_ref fofType) x10)) as proof of ((forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy)))->(((eq fofType) x10) x1))
% 249.35/249.59 Found (fun (x4:cIND) (x5:(forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy))))=> ((eq_ref fofType) x10)) as proof of (cIND->((forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy)))->(((eq fofType) x10) x1)))
% 249.35/249.59 Found (and_rect10 (fun (x4:cIND) (x5:(forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy))))=> ((eq_ref fofType) x10))) as proof of (((eq fofType) x10) x1)
% 249.35/249.59 Found ((and_rect1 (((eq fofType) x10) x1)) (fun (x4:cIND) (x5:(forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy))))=> ((eq_ref fofType) x10))) as proof of (((eq fofType) x10) x1)
% 249.35/249.59 Found (((fun (P:Type) (x4:(cIND->((forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy)))->P)))=> (((((and_rect cIND) (forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy)))) P) x4) x2)) (((eq fofType) x10) x1)) (fun (x4:cIND) (x5:(forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy))))=> ((eq_ref fofType) x10))) as proof of (((eq fofType) x10) x1)
% 249.35/249.59 Found (fun (x20:((and cIND) (forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy)))))=> (((fun (P:Type) (x4:(cIND->((forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy)))->P)))=> (((((and_rect cIND) (forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy)))) P) x4) x2)) (((eq fofType) x10) x1)) (fun (x4:cIND) (x5:(forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy))))=> ((eq_ref fofType) x10)))) as proof of (((eq fofType) x10) x1)
% 249.35/249.59 Found (fun (x30:(forall (Xn:fofType), (not (((eq fofType) (cS Xn)) c0)))) (x20:((and cIND) (forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy)))))=> (((fun (P:Type) (x4:(cIND->((forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy)))->P)))=> (((((and_rect cIND) (forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy)))) P) x4) x2)) (((eq fofType) x10) x1)) (fun (x4:cIND) (x5:(forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy))))=> ((eq_ref fofType) x10)))) as proof of (((and cIND) (forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy))))->(((eq fofType) x10) x1))
% 249.35/249.59 Found eq_ref00:=(eq_ref0 x30):(((eq fofType) x30) x30)
% 249.35/249.59 Found (eq_ref0 x30) as proof of (((eq fofType) x30) x3)
% 249.35/249.59 Found ((eq_ref fofType) x30) as proof of (((eq fofType) x30) x3)
% 249.35/249.59 Found ((eq_ref fofType) x30) as proof of (((eq fofType) x30) x3)
% 249.35/249.59 Found (fun (x5:(forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy))))=> ((eq_ref fofType) x30)) as proof of (((eq fofType) x30) x3)
% 249.35/249.59 Found (fun (x4:cIND) (x5:(forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy))))=> ((eq_ref fofType) x30)) as proof of ((forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy)))->(((eq fofType) x30) x3))
% 249.35/249.59 Found (fun (x4:cIND) (x5:(forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy))))=> ((eq_ref fofType) x30)) as proof of (cIND->((forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy)))->(((eq fofType) x30) x3)))
% 249.35/249.59 Found (and_rect10 (fun (x4:cIND) (x5:(forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy))))=> ((eq_ref fofType) x30))) as proof of (((eq fofType) x30) x3)
% 249.35/249.59 Found ((and_rect1 (((eq fofType) x30) x3)) (fun (x4:cIND) (x5:(forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy))))=> ((eq_ref fofType) x30))) as proof of (((eq fofType) x30) x3)
% 250.25/250.46 Found (((fun (P:Type) (x4:(cIND->((forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy)))->P)))=> (((((and_rect cIND) (forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy)))) P) x4) x1)) (((eq fofType) x30) x3)) (fun (x4:cIND) (x5:(forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy))))=> ((eq_ref fofType) x30))) as proof of (((eq fofType) x30) x3)
% 250.25/250.46 Found (fun (x20:(forall (Xn:fofType), (not (((eq fofType) (cS Xn)) c0))))=> (((fun (P:Type) (x4:(cIND->((forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy)))->P)))=> (((((and_rect cIND) (forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy)))) P) x4) x1)) (((eq fofType) x30) x3)) (fun (x4:cIND) (x5:(forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy))))=> ((eq_ref fofType) x30)))) as proof of (((eq fofType) x30) x3)
% 250.25/250.46 Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% 250.25/250.46 Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) b)
% 250.25/250.46 Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% 250.25/250.46 Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% 250.25/250.46 Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% 250.25/250.46 Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% 250.25/250.46 Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y)))))))))
% 250.25/250.46 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y)))))))))
% 250.25/250.46 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y)))))))))
% 250.25/250.46 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y)))))))))
% 250.25/250.46 Found ((eq_trans0000 ((eq_ref Prop) (f x0))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f x0)) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y)))))))))
% 250.25/250.46 Found (((eq_trans000 ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y))))))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f x0)) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y)))))))))
% 250.25/250.46 Found ((((eq_trans00 ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y))))))))) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y))))))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y)))))))))) as proof of (((eq Prop) (f x0)) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y)))))))))
% 250.25/250.47 Found (((((eq_trans0 (f x0)) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y))))))))) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y))))))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y)))))))))) as proof of (((eq Prop) (f x0)) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y)))))))))
% 250.25/250.47 Found ((((((eq_trans Prop) (f x0)) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y))))))))) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y))))))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y)))))))))) as proof of (((eq Prop) (f x0)) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y)))))))))
% 250.25/250.47 Found (fun (x2:(forall (Xn:fofType), (not (((eq fofType) (cS Xn)) c0))))=> ((((((eq_trans Prop) (f x0)) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y))))))))) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y))))))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y))))))))))) as proof of (((eq Prop) (f x0)) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y)))))))))
% 250.25/250.47 Found (fun (x1:((and cIND) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (cS Xx)) (cS Xy))->(((eq fofType) Xx) Xy))))) (x2:(forall (Xn:fofType), (not (((eq fofType) (cS Xn)) c0))))=> ((((((eq_trans Prop) (f x0)) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y))))))))) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y))))))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y))))))))))) as proof of ((forall (Xn:fofType), (not (((eq fofType) (cS Xn)) c0)))->(((eq Prop) (f x0)) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y))))))))))
% 250.25/250.49 Found (fun (x1:((and cIND) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (cS Xx)) (cS Xy))->(((eq fofType) Xx) Xy))))) (x2:(forall (Xn:fofType), (not (((eq fofType) (cS Xn)) c0))))=> ((((((eq_trans Prop) (f x0)) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y))))))))) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y))))))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y))))))))))) as proof of (((and cIND) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (cS Xx)) (cS Xy))->(((eq fofType) Xx) Xy))))->((forall (Xn:fofType), (not (((eq fofType) (cS Xn)) c0)))->(((eq Prop) (f x0)) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y)))))))))))
% 250.25/250.49 Found (and_rect00 (fun (x1:((and cIND) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (cS Xx)) (cS Xy))->(((eq fofType) Xx) Xy))))) (x2:(forall (Xn:fofType), (not (((eq fofType) (cS Xn)) c0))))=> ((((((eq_trans Prop) (f x0)) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y))))))))) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y))))))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y)))))))))))) as proof of (((eq Prop) (f x0)) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y)))))))))
% 250.25/250.49 Found ((and_rect0 (((eq Prop) (f x0)) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y)))))))))) (fun (x1:((and cIND) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (cS Xx)) (cS Xy))->(((eq fofType) Xx) Xy))))) (x2:(forall (Xn:fofType), (not (((eq fofType) (cS Xn)) c0))))=> ((((((eq_trans Prop) (f x0)) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y))))))))) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y))))))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y)))))))))))) as proof of (((eq Prop) (f x0)) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y)))))))))
% 250.42/250.63 Found (((fun (P0:Type) (x1:(((and cIND) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (cS Xx)) (cS Xy))->(((eq fofType) Xx) Xy))))->((forall (Xn:fofType), (not (((eq fofType) (cS Xn)) c0)))->P0)))=> (((((and_rect ((and cIND) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (cS Xx)) (cS Xy))->(((eq fofType) Xx) Xy))))) (forall (Xn:fofType), (not (((eq fofType) (cS Xn)) c0)))) P0) x1) x)) (((eq Prop) (f x0)) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y)))))))))) (fun (x1:((and cIND) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (cS Xx)) (cS Xy))->(((eq fofType) Xx) Xy))))) (x2:(forall (Xn:fofType), (not (((eq fofType) (cS Xn)) c0))))=> ((((((eq_trans Prop) (f x0)) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y))))))))) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y))))))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y)))))))))))) as proof of (((eq Prop) (f x0)) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y)))))))))
% 250.42/250.63 Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% 250.42/250.63 Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) b)
% 250.42/250.63 Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% 250.42/250.63 Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% 250.42/250.63 Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% 250.42/250.63 Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% 250.42/250.63 Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y)))))))))
% 250.42/250.63 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y)))))))))
% 250.42/250.65 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y)))))))))
% 250.42/250.65 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y)))))))))
% 250.42/250.65 Found ((eq_trans0000 ((eq_ref Prop) (f x0))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f x0)) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y)))))))))
% 250.42/250.65 Found (((eq_trans000 ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y))))))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f x0)) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y)))))))))
% 250.42/250.65 Found ((((eq_trans00 ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y))))))))) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y))))))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y)))))))))) as proof of (((eq Prop) (f x0)) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y)))))))))
% 250.42/250.65 Found (((((eq_trans0 (f x0)) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y))))))))) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y))))))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y)))))))))) as proof of (((eq Prop) (f x0)) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y)))))))))
% 250.42/250.65 Found ((((((eq_trans Prop) (f x0)) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y))))))))) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y))))))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y)))))))))) as proof of (((eq Prop) (f x0)) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y)))))))))
% 250.42/250.66 Found (fun (x2:(forall (Xn:fofType), (not (((eq fofType) (cS Xn)) c0))))=> ((((((eq_trans Prop) (f x0)) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y))))))))) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y))))))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y))))))))))) as proof of (((eq Prop) (f x0)) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y)))))))))
% 250.42/250.66 Found (fun (x1:((and cIND) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (cS Xx)) (cS Xy))->(((eq fofType) Xx) Xy))))) (x2:(forall (Xn:fofType), (not (((eq fofType) (cS Xn)) c0))))=> ((((((eq_trans Prop) (f x0)) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y))))))))) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y))))))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y))))))))))) as proof of ((forall (Xn:fofType), (not (((eq fofType) (cS Xn)) c0)))->(((eq Prop) (f x0)) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y))))))))))
% 250.42/250.66 Found (fun (x1:((and cIND) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (cS Xx)) (cS Xy))->(((eq fofType) Xx) Xy))))) (x2:(forall (Xn:fofType), (not (((eq fofType) (cS Xn)) c0))))=> ((((((eq_trans Prop) (f x0)) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y))))))))) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y))))))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y))))))))))) as proof of (((and cIND) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (cS Xx)) (cS Xy))->(((eq fofType) Xx) Xy))))->((forall (Xn:fofType), (not (((eq fofType) (cS Xn)) c0)))->(((eq Prop) (f x0)) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y)))))))))))
% 250.52/250.69 Found (and_rect00 (fun (x1:((and cIND) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (cS Xx)) (cS Xy))->(((eq fofType) Xx) Xy))))) (x2:(forall (Xn:fofType), (not (((eq fofType) (cS Xn)) c0))))=> ((((((eq_trans Prop) (f x0)) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y))))))))) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y))))))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y)))))))))))) as proof of (((eq Prop) (f x0)) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y)))))))))
% 250.52/250.69 Found ((and_rect0 (((eq Prop) (f x0)) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y)))))))))) (fun (x1:((and cIND) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (cS Xx)) (cS Xy))->(((eq fofType) Xx) Xy))))) (x2:(forall (Xn:fofType), (not (((eq fofType) (cS Xn)) c0))))=> ((((((eq_trans Prop) (f x0)) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y))))))))) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y))))))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y)))))))))))) as proof of (((eq Prop) (f x0)) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y)))))))))
% 250.52/250.69 Found (((fun (P0:Type) (x1:(((and cIND) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (cS Xx)) (cS Xy))->(((eq fofType) Xx) Xy))))->((forall (Xn:fofType), (not (((eq fofType) (cS Xn)) c0)))->P0)))=> (((((and_rect ((and cIND) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (cS Xx)) (cS Xy))->(((eq fofType) Xx) Xy))))) (forall (Xn:fofType), (not (((eq fofType) (cS Xn)) c0)))) P0) x1) x)) (((eq Prop) (f x0)) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y)))))))))) (fun (x1:((and cIND) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (cS Xx)) (cS Xy))->(((eq fofType) Xx) Xy))))) (x2:(forall (Xn:fofType), (not (((eq fofType) (cS Xn)) c0))))=> ((((((eq_trans Prop) (f x0)) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y))))))))) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y))))))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y)))))))))))) as proof of (((eq Prop) (f x0)) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x0 Xx) X)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) X) Y)))))))))
% 250.64/250.89 Found eq_ref00:=(eq_ref0 (f x4)):(((eq Prop) (f x4)) (f x4))
% 250.64/250.89 Found (eq_ref0 (f x4)) as proof of (((eq Prop) (f x4)) b)
% 250.64/250.89 Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) b)
% 250.64/250.89 Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) b)
% 250.64/250.89 Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) b)
% 250.64/250.89 Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% 250.64/250.89 Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((and ((x4 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x4 Xx) Xy)->((x4 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x4 Xx))))))
% 250.64/250.89 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and ((x4 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x4 Xx) Xy)->((x4 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x4 Xx))))))
% 250.64/250.89 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and ((x4 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x4 Xx) Xy)->((x4 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x4 Xx))))))
% 250.64/250.89 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and ((x4 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x4 Xx) Xy)->((x4 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x4 Xx))))))
% 250.64/250.89 Found ((eq_trans0000 ((eq_ref Prop) (f x4))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f x4)) ((and ((and ((x4 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x4 Xx) Xy)->((x4 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x4 Xx))))))
% 250.64/250.89 Found (((eq_trans000 ((and ((and ((x4 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x4 Xx) Xy)->((x4 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x4 Xx)))))) ((eq_ref Prop) (f x4))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f x4)) ((and ((and ((x4 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x4 Xx) Xy)->((x4 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x4 Xx))))))
% 250.64/250.89 Found ((((eq_trans00 ((and ((and ((x4 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x4 Xx) Xy)->((x4 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x4 Xx)))))) ((and ((and ((x4 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x4 Xx) Xy)->((x4 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x4 Xx)))))) ((eq_ref Prop) (f x4))) ((eq_ref Prop) ((and ((and ((x4 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x4 Xx) Xy)->((x4 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x4 Xx))))))) as proof of (((eq Prop) (f x4)) ((and ((and ((x4 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x4 Xx) Xy)->((x4 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x4 Xx))))))
% 250.64/250.89 Found (((((eq_trans0 (f x4)) ((and ((and ((x4 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x4 Xx) Xy)->((x4 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x4 Xx)))))) ((and ((and ((x4 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x4 Xx) Xy)->((x4 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x4 Xx)))))) ((eq_ref Prop) (f x4))) ((eq_ref Prop) ((and ((and ((x4 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x4 Xx) Xy)->((x4 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x4 Xx))))))) as proof of (((eq Prop) (f x4)) ((and ((and ((x4 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x4 Xx) Xy)->((x4 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x4 Xx))))))
% 251.12/251.30 Found ((((((eq_trans Prop) (f x4)) ((and ((and ((x4 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x4 Xx) Xy)->((x4 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x4 Xx)))))) ((and ((and ((x4 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x4 Xx) Xy)->((x4 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x4 Xx)))))) ((eq_ref Prop) (f x4))) ((eq_ref Prop) ((and ((and ((x4 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x4 Xx) Xy)->((x4 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x4 Xx))))))) as proof of (((eq Prop) (f x4)) ((and ((and ((x4 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x4 Xx) Xy)->((x4 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x4 Xx))))))
% 251.12/251.30 Found eq_ref00:=(eq_ref0 (f x4)):(((eq Prop) (f x4)) (f x4))
% 251.12/251.30 Found (eq_ref0 (f x4)) as proof of (((eq Prop) (f x4)) b)
% 251.12/251.30 Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) b)
% 251.12/251.30 Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) b)
% 251.12/251.30 Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) b)
% 251.12/251.30 Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% 251.12/251.30 Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((and ((x4 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x4 Xx) Xy)->((x4 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x4 Xx))))))
% 251.12/251.30 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and ((x4 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x4 Xx) Xy)->((x4 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x4 Xx))))))
% 251.12/251.30 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and ((x4 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x4 Xx) Xy)->((x4 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x4 Xx))))))
% 251.12/251.30 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and ((x4 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x4 Xx) Xy)->((x4 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x4 Xx))))))
% 251.12/251.30 Found ((eq_trans0000 ((eq_ref Prop) (f x4))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f x4)) ((and ((and ((x4 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x4 Xx) Xy)->((x4 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x4 Xx))))))
% 251.12/251.30 Found (((eq_trans000 ((and ((and ((x4 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x4 Xx) Xy)->((x4 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x4 Xx)))))) ((eq_ref Prop) (f x4))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f x4)) ((and ((and ((x4 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x4 Xx) Xy)->((x4 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x4 Xx))))))
% 251.12/251.30 Found ((((eq_trans00 ((and ((and ((x4 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x4 Xx) Xy)->((x4 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x4 Xx)))))) ((and ((and ((x4 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x4 Xx) Xy)->((x4 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x4 Xx)))))) ((eq_ref Prop) (f x4))) ((eq_ref Prop) ((and ((and ((x4 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x4 Xx) Xy)->((x4 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x4 Xx))))))) as proof of (((eq Prop) (f x4)) ((and ((and ((x4 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x4 Xx) Xy)->((x4 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x4 Xx))))))
% 252.73/252.94 Found (((((eq_trans0 (f x4)) ((and ((and ((x4 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x4 Xx) Xy)->((x4 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x4 Xx)))))) ((and ((and ((x4 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x4 Xx) Xy)->((x4 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x4 Xx)))))) ((eq_ref Prop) (f x4))) ((eq_ref Prop) ((and ((and ((x4 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x4 Xx) Xy)->((x4 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x4 Xx))))))) as proof of (((eq Prop) (f x4)) ((and ((and ((x4 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x4 Xx) Xy)->((x4 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x4 Xx))))))
% 252.73/252.94 Found ((((((eq_trans Prop) (f x4)) ((and ((and ((x4 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x4 Xx) Xy)->((x4 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x4 Xx)))))) ((and ((and ((x4 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x4 Xx) Xy)->((x4 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x4 Xx)))))) ((eq_ref Prop) (f x4))) ((eq_ref Prop) ((and ((and ((x4 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x4 Xx) Xy)->((x4 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x4 Xx))))))) as proof of (((eq Prop) (f x4)) ((and ((and ((x4 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x4 Xx) Xy)->((x4 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x4 Xx))))))
% 252.73/252.94 Found x6:(P x1)
% 252.73/252.94 Instantiate: x1:=Y:fofType
% 252.73/252.94 Found (fun (x6:(P x1))=> x6) as proof of (P Y)
% 252.73/252.94 Found (fun (P:(fofType->Prop)) (x6:(P x1))=> x6) as proof of ((P x1)->(P Y))
% 252.73/252.94 Found (fun (x00:((x0 Xx) Y)) (P:(fofType->Prop)) (x6:(P x1))=> x6) as proof of (((eq fofType) x1) Y)
% 252.73/252.94 Found eq_ref000:=(eq_ref00 P):((P x1)->(P x1))
% 252.73/252.94 Found (eq_ref00 P) as proof of ((P x1)->(P Y))
% 252.73/252.94 Found ((eq_ref0 x1) P) as proof of ((P x1)->(P Y))
% 252.73/252.94 Found (((eq_ref fofType) x1) P) as proof of ((P x1)->(P Y))
% 252.73/252.94 Found (((eq_ref fofType) x1) P) as proof of ((P x1)->(P Y))
% 252.73/252.94 Found (fun (P:(fofType->Prop))=> (((eq_ref fofType) x1) P)) as proof of ((P x1)->(P Y))
% 252.73/252.94 Found (fun (x00:((x0 Xx) Y)) (P:(fofType->Prop))=> (((eq_ref fofType) x1) P)) as proof of (((eq fofType) x1) Y)
% 252.73/252.94 Found eq_ref00:=(eq_ref0 (forall (Xx:fofType) (Xy:fofType), (((x4 Xx) Xy)->((x4 (cS Xx)) (cS (cS Xy)))))):(((eq Prop) (forall (Xx:fofType) (Xy:fofType), (((x4 Xx) Xy)->((x4 (cS Xx)) (cS (cS Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((x4 Xx) Xy)->((x4 (cS Xx)) (cS (cS Xy))))))
% 252.73/252.94 Found (eq_ref0 (forall (Xx:fofType) (Xy:fofType), (((x4 Xx) Xy)->((x4 (cS Xx)) (cS (cS Xy)))))) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), (((x4 Xx) Xy)->((x4 (cS Xx)) (cS (cS Xy)))))) b)
% 252.73/252.94 Found ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), (((x4 Xx) Xy)->((x4 (cS Xx)) (cS (cS Xy)))))) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), (((x4 Xx) Xy)->((x4 (cS Xx)) (cS (cS Xy)))))) b)
% 252.73/252.94 Found ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), (((x4 Xx) Xy)->((x4 (cS Xx)) (cS (cS Xy)))))) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), (((x4 Xx) Xy)->((x4 (cS Xx)) (cS (cS Xy)))))) b)
% 252.73/252.94 Found ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), (((x4 Xx) Xy)->((x4 (cS Xx)) (cS (cS Xy)))))) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), (((x4 Xx) Xy)->((x4 (cS Xx)) (cS (cS Xy)))))) b)
% 252.73/252.94 Found eq_ref000:=(eq_ref00 P):((P x3)->(P x3))
% 252.73/252.94 Found (eq_ref00 P) as proof of ((P x3)->(P Y))
% 252.73/252.94 Found ((eq_ref0 x3) P) as proof of ((P x3)->(P Y))
% 252.73/252.94 Found (((eq_ref fofType) x3) P) as proof of ((P x3)->(P Y))
% 252.73/252.94 Found (((eq_ref fofType) x3) P) as proof of ((P x3)->(P Y))
% 252.73/252.94 Found (fun (P:(fofType->Prop))=> (((eq_ref fofType) x3) P)) as proof of ((P x3)->(P Y))
% 252.73/252.94 Found (fun (x00:((x0 Xx) Y)) (P:(fofType->Prop))=> (((eq_ref fofType) x3) P)) as proof of (((eq fofType) x3) Y)
% 256.84/257.02 Found x6:(P x3)
% 256.84/257.02 Instantiate: x3:=Y:fofType
% 256.84/257.02 Found (fun (x6:(P x3))=> x6) as proof of (P Y)
% 256.84/257.02 Found (fun (P:(fofType->Prop)) (x6:(P x3))=> x6) as proof of ((P x3)->(P Y))
% 256.84/257.02 Found (fun (x00:((x0 Xx) Y)) (P:(fofType->Prop)) (x6:(P x3))=> x6) as proof of (((eq fofType) x3) Y)
% 256.84/257.02 Found eq_ref00:=(eq_ref0 x1):(((eq fofType) x1) x1)
% 256.84/257.02 Found (eq_ref0 x1) as proof of (((eq fofType) x1) Y)
% 256.84/257.02 Found ((eq_ref fofType) x1) as proof of (((eq fofType) x1) Y)
% 256.84/257.02 Found ((eq_ref fofType) x1) as proof of (((eq fofType) x1) Y)
% 256.84/257.02 Found ((eq_ref fofType) x1) as proof of (((eq fofType) x1) Y)
% 256.84/257.02 Found (fun (x00:((x0 Xx) Y))=> ((eq_ref fofType) x1)) as proof of (((eq fofType) x1) Y)
% 256.84/257.02 Found eq_ref00:=(eq_ref0 c0):(((eq fofType) c0) c0)
% 256.84/257.02 Found (eq_ref0 c0) as proof of (((eq fofType) c0) x5)
% 256.84/257.02 Found ((eq_ref fofType) c0) as proof of (((eq fofType) c0) x5)
% 256.84/257.02 Found ((eq_ref fofType) c0) as proof of (((eq fofType) c0) x5)
% 256.84/257.02 Found ((eq_ref fofType) c0) as proof of (((eq fofType) c0) x5)
% 256.84/257.02 Found eq_ref00:=(eq_ref0 c0):(((eq fofType) c0) c0)
% 256.84/257.02 Found (eq_ref0 c0) as proof of (((eq fofType) c0) x5)
% 256.84/257.02 Found ((eq_ref fofType) c0) as proof of (((eq fofType) c0) x5)
% 256.84/257.02 Found ((eq_ref fofType) c0) as proof of (((eq fofType) c0) x5)
% 256.84/257.02 Found ((eq_ref fofType) c0) as proof of (((eq fofType) c0) x5)
% 256.84/257.02 Found eq_ref00:=(eq_ref0 x3):(((eq fofType) x3) x3)
% 256.84/257.02 Found (eq_ref0 x3) as proof of (((eq fofType) x3) Y)
% 256.84/257.02 Found ((eq_ref fofType) x3) as proof of (((eq fofType) x3) Y)
% 256.84/257.02 Found ((eq_ref fofType) x3) as proof of (((eq fofType) x3) Y)
% 256.84/257.02 Found ((eq_ref fofType) x3) as proof of (((eq fofType) x3) Y)
% 256.84/257.02 Found (fun (x00:((x0 Xx) Y))=> ((eq_ref fofType) x3)) as proof of (((eq fofType) x3) Y)
% 256.84/257.02 Found eq_ref00:=(eq_ref0 (cS x50)):(((eq fofType) (cS x50)) (cS x50))
% 256.84/257.02 Found (eq_ref0 (cS x50)) as proof of (((eq fofType) (cS x50)) (cS x5))
% 256.84/257.02 Found ((eq_ref fofType) (cS x50)) as proof of (((eq fofType) (cS x50)) (cS x5))
% 256.84/257.02 Found ((eq_ref fofType) (cS x50)) as proof of (((eq fofType) (cS x50)) (cS x5))
% 256.84/257.02 Found ((eq_ref fofType) (cS x50)) as proof of (((eq fofType) (cS x50)) (cS x5))
% 256.84/257.02 Found (x4000 ((eq_ref fofType) (cS x50))) as proof of (((eq fofType) x50) x5)
% 256.84/257.02 Found ((x400 x5) ((eq_ref fofType) (cS x50))) as proof of (((eq fofType) x50) x5)
% 256.84/257.02 Found (((x40 x50) x5) ((eq_ref fofType) (cS x50))) as proof of (((eq fofType) x50) x5)
% 256.84/257.02 Found eq_ref00:=(eq_ref0 (cS x50)):(((eq fofType) (cS x50)) (cS x50))
% 256.84/257.02 Found (eq_ref0 (cS x50)) as proof of (((eq fofType) (cS x50)) (cS x5))
% 256.84/257.02 Found ((eq_ref fofType) (cS x50)) as proof of (((eq fofType) (cS x50)) (cS x5))
% 256.84/257.02 Found ((eq_ref fofType) (cS x50)) as proof of (((eq fofType) (cS x50)) (cS x5))
% 256.84/257.02 Found ((eq_ref fofType) (cS x50)) as proof of (((eq fofType) (cS x50)) (cS x5))
% 256.84/257.02 Found (x410 ((eq_ref fofType) (cS x50))) as proof of (((eq fofType) x50) x5)
% 256.84/257.02 Found ((x41 x5) ((eq_ref fofType) (cS x50))) as proof of (((eq fofType) x50) x5)
% 256.84/257.02 Found (((x4 x50) x5) ((eq_ref fofType) (cS x50))) as proof of (((eq fofType) x50) x5)
% 256.84/257.02 Found x6:(P x3)
% 256.84/257.02 Instantiate: x3:=x':fofType
% 256.84/257.02 Found (fun (x6:(P x3))=> x6) as proof of (P x')
% 256.84/257.02 Found (fun (P:(fofType->Prop)) (x6:(P x3))=> x6) as proof of ((P x3)->(P x'))
% 256.84/257.02 Found (fun (P:(fofType->Prop)) (x6:(P x3))=> x6) as proof of (((eq fofType) x3) x')
% 256.84/257.02 Found (fun (x00:((x0 Xx) x')) (P:(fofType->Prop)) (x6:(P x3))=> x6) as proof of (((eq fofType) x3) x')
% 256.84/257.02 Found x6:(P x3)
% 256.84/257.02 Instantiate: x3:=x':fofType
% 256.84/257.02 Found (fun (x6:(P x3))=> x6) as proof of (P x')
% 256.84/257.02 Found (fun (P:(fofType->Prop)) (x6:(P x3))=> x6) as proof of ((P x3)->(P x'))
% 256.84/257.02 Found (fun (P:(fofType->Prop)) (x6:(P x3))=> x6) as proof of (((eq fofType) x3) x')
% 256.84/257.02 Found (fun (x00:((x2 Xx) x')) (P:(fofType->Prop)) (x6:(P x3))=> x6) as proof of (((eq fofType) x3) x')
% 256.84/257.02 Found eq_ref000:=(eq_ref00 P):((P x3)->(P x3))
% 256.84/257.02 Found (eq_ref00 P) as proof of ((P x3)->(P x'))
% 256.84/257.02 Found ((eq_ref0 x3) P) as proof of ((P x3)->(P x'))
% 256.84/257.02 Found (((eq_ref fofType) x3) P) as proof of ((P x3)->(P x'))
% 256.84/257.02 Found (((eq_ref fofType) x3) P) as proof of ((P x3)->(P x'))
% 256.84/257.02 Found (fun (P:(fofType->Prop))=> (((eq_ref fofType) x3) P)) as proof of ((P x3)->(P x'))
% 256.84/257.02 Found (fun (P:(fofType->Prop))=> (((eq_ref fofType) x3) P)) as proof of (((eq fofType) x3) x')
% 256.84/257.02 Found (fun (x00:((x0 Xx) x')) (P:(fofType->Prop))=> (((eq_ref fofType) x3) P)) as proof of (((eq fofType) x3) x')
% 261.50/261.72 Found eq_ref000:=(eq_ref00 P):((P x3)->(P x3))
% 261.50/261.72 Found (eq_ref00 P) as proof of ((P x3)->(P x'))
% 261.50/261.72 Found ((eq_ref0 x3) P) as proof of ((P x3)->(P x'))
% 261.50/261.72 Found (((eq_ref fofType) x3) P) as proof of ((P x3)->(P x'))
% 261.50/261.72 Found (((eq_ref fofType) x3) P) as proof of ((P x3)->(P x'))
% 261.50/261.72 Found (fun (P:(fofType->Prop))=> (((eq_ref fofType) x3) P)) as proof of ((P x3)->(P x'))
% 261.50/261.72 Found (fun (P:(fofType->Prop))=> (((eq_ref fofType) x3) P)) as proof of (((eq fofType) x3) x')
% 261.50/261.72 Found (fun (x00:((x2 Xx) x')) (P:(fofType->Prop))=> (((eq_ref fofType) x3) P)) as proof of (((eq fofType) x3) x')
% 261.50/261.72 Found eq_ref00:=(eq_ref0 x3):(((eq fofType) x3) x3)
% 261.50/261.72 Found (eq_ref0 x3) as proof of (((eq fofType) x3) x30)
% 261.50/261.72 Found ((eq_ref fofType) x3) as proof of (((eq fofType) x3) x30)
% 261.50/261.72 Found ((eq_ref fofType) x3) as proof of (((eq fofType) x3) x30)
% 261.50/261.72 Found ((eq_ref fofType) x3) as proof of (((eq fofType) x3) x30)
% 261.50/261.72 Found (eq_sym000 ((eq_ref fofType) x3)) as proof of (((eq fofType) x30) x3)
% 261.50/261.72 Found ((eq_sym00 x30) ((eq_ref fofType) x3)) as proof of (((eq fofType) x30) x3)
% 261.50/261.72 Found (((eq_sym0 x3) x30) ((eq_ref fofType) x3)) as proof of (((eq fofType) x30) x3)
% 261.50/261.72 Found ((((eq_sym fofType) x3) x30) ((eq_ref fofType) x3)) as proof of (((eq fofType) x30) x3)
% 261.50/261.72 Found eq_ref00:=(eq_ref0 (cS x30)):(((eq fofType) (cS x30)) (cS x30))
% 261.50/261.72 Found (eq_ref0 (cS x30)) as proof of (((eq fofType) (cS x30)) (cS x3))
% 261.50/261.72 Found ((eq_ref fofType) (cS x30)) as proof of (((eq fofType) (cS x30)) (cS x3))
% 261.50/261.72 Found ((eq_ref fofType) (cS x30)) as proof of (((eq fofType) (cS x30)) (cS x3))
% 261.50/261.72 Found ((eq_ref fofType) (cS x30)) as proof of (((eq fofType) (cS x30)) (cS x3))
% 261.50/261.72 Found (x500 ((eq_ref fofType) (cS x30))) as proof of (((eq fofType) x30) x3)
% 261.50/261.72 Found ((x50 x3) ((eq_ref fofType) (cS x30))) as proof of (((eq fofType) x30) x3)
% 261.50/261.72 Found (((x5 x30) x3) ((eq_ref fofType) (cS x30))) as proof of (((eq fofType) x30) x3)
% 261.50/261.72 Found (fun (x20:(forall (Xn:fofType), (not (((eq fofType) (cS Xn)) c0))))=> (((x5 x30) x3) ((eq_ref fofType) (cS x30)))) as proof of (((eq fofType) x30) x3)
% 261.50/261.72 Found eq_ref000:=(eq_ref00 P):((P (cS Xx0))->(P (cS Xx0)))
% 261.50/261.72 Found (eq_ref00 P) as proof of ((P (cS Xx0))->(P x5))
% 261.50/261.72 Found ((eq_ref0 (cS Xx0)) P) as proof of ((P (cS Xx0))->(P x5))
% 261.50/261.72 Found (((eq_ref fofType) (cS Xx0)) P) as proof of ((P (cS Xx0))->(P x5))
% 261.50/261.72 Found (((eq_ref fofType) (cS Xx0)) P) as proof of ((P (cS Xx0))->(P x5))
% 261.50/261.72 Found (fun (P:(fofType->Prop))=> (((eq_ref fofType) (cS Xx0)) P)) as proof of ((P (cS Xx0))->(P x5))
% 261.50/261.72 Found (fun (x6:(((eq fofType) Xx0) x5)) (P:(fofType->Prop))=> (((eq_ref fofType) (cS Xx0)) P)) as proof of (((eq fofType) (cS Xx0)) x5)
% 261.50/261.72 Found x7:(P (cS Xx0))
% 261.50/261.72 Instantiate: x5:=(cS Xx0):fofType
% 261.50/261.72 Found (fun (x7:(P (cS Xx0)))=> x7) as proof of (P x5)
% 261.50/261.72 Found (fun (P:(fofType->Prop)) (x7:(P (cS Xx0)))=> x7) as proof of ((P (cS Xx0))->(P x5))
% 261.50/261.72 Found (fun (x6:(((eq fofType) Xx0) x5)) (P:(fofType->Prop)) (x7:(P (cS Xx0)))=> x7) as proof of (((eq fofType) (cS Xx0)) x5)
% 261.50/261.72 Found eq_ref00:=(eq_ref0 x30):(((eq fofType) x30) x30)
% 261.50/261.72 Found (eq_ref0 x30) as proof of (forall (P:(fofType->Prop)), ((P x30)->(P x3)))
% 261.50/261.72 Found ((eq_ref fofType) x30) as proof of (forall (P:(fofType->Prop)), ((P x30)->(P x3)))
% 261.50/261.72 Found ((eq_ref fofType) x30) as proof of (forall (P:(fofType->Prop)), ((P x30)->(P x3)))
% 261.50/261.72 Found ((eq_ref fofType) x30) as proof of (forall (P:(fofType->Prop)), ((P x30)->(P x3)))
% 261.50/261.72 Found eq_ref00:=(eq_ref0 (f x3)):(((eq Prop) (f x3)) (f x3))
% 261.50/261.72 Found (eq_ref0 (f x3)) as proof of (((eq Prop) (f x3)) ((and ((x2 Xx) x3)) (forall (Y:fofType), (((x2 Xx) Y)->(((eq fofType) x3) Y)))))
% 261.50/261.72 Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((and ((x2 Xx) x3)) (forall (Y:fofType), (((x2 Xx) Y)->(((eq fofType) x3) Y)))))
% 261.50/261.72 Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((and ((x2 Xx) x3)) (forall (Y:fofType), (((x2 Xx) Y)->(((eq fofType) x3) Y)))))
% 261.50/261.72 Found (fun (x3:fofType)=> ((eq_ref Prop) (f x3))) as proof of (((eq Prop) (f x3)) ((and ((x2 Xx) x3)) (forall (Y:fofType), (((x2 Xx) Y)->(((eq fofType) x3) Y)))))
% 261.50/261.72 Found (fun (x3:fofType)=> ((eq_ref Prop) (f x3))) as proof of (forall (x:fofType), (((eq Prop) (f x)) ((and ((x2 Xx) x)) (forall (Y:fofType), (((x2 Xx) Y)->(((eq fofType) x) Y))))))
% 266.15/266.35 Found eq_ref000:=(eq_ref00 P0):((P0 (f x4))->(P0 (f x4)))
% 266.15/266.35 Found (eq_ref00 P0) as proof of (P1 (f x4))
% 266.15/266.35 Found ((eq_ref0 (f x4)) P0) as proof of (P1 (f x4))
% 266.15/266.35 Found (((eq_ref Prop) (f x4)) P0) as proof of (P1 (f x4))
% 266.15/266.35 Found (((eq_ref Prop) (f x4)) P0) as proof of (P1 (f x4))
% 266.15/266.35 Found eq_ref00:=(eq_ref0 (f x3)):(((eq Prop) (f x3)) (f x3))
% 266.15/266.35 Found (eq_ref0 (f x3)) as proof of (((eq Prop) (f x3)) ((and ((x2 Xx) x3)) (forall (Y:fofType), (((x2 Xx) Y)->(((eq fofType) x3) Y)))))
% 266.15/266.35 Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((and ((x2 Xx) x3)) (forall (Y:fofType), (((x2 Xx) Y)->(((eq fofType) x3) Y)))))
% 266.15/266.35 Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((and ((x2 Xx) x3)) (forall (Y:fofType), (((x2 Xx) Y)->(((eq fofType) x3) Y)))))
% 266.15/266.35 Found (fun (x3:fofType)=> ((eq_ref Prop) (f x3))) as proof of (((eq Prop) (f x3)) ((and ((x2 Xx) x3)) (forall (Y:fofType), (((x2 Xx) Y)->(((eq fofType) x3) Y)))))
% 266.15/266.35 Found (fun (x3:fofType)=> ((eq_ref Prop) (f x3))) as proof of (forall (x:fofType), (((eq Prop) (f x)) ((and ((x2 Xx) x)) (forall (Y:fofType), (((x2 Xx) Y)->(((eq fofType) x) Y))))))
% 266.15/266.35 Found eq_ref000:=(eq_ref00 P0):((P0 (f x4))->(P0 (f x4)))
% 266.15/266.35 Found (eq_ref00 P0) as proof of (P1 (f x4))
% 266.15/266.35 Found ((eq_ref0 (f x4)) P0) as proof of (P1 (f x4))
% 266.15/266.35 Found (((eq_ref Prop) (f x4)) P0) as proof of (P1 (f x4))
% 266.15/266.35 Found (((eq_ref Prop) (f x4)) P0) as proof of (P1 (f x4))
% 266.15/266.35 Found eq_ref00:=(eq_ref0 x30):(((eq fofType) x30) x30)
% 266.15/266.35 Found (eq_ref0 x30) as proof of (((eq fofType) x30) x3)
% 266.15/266.35 Found ((eq_ref fofType) x30) as proof of (((eq fofType) x30) x3)
% 266.15/266.35 Found ((eq_ref fofType) x30) as proof of (((eq fofType) x30) x3)
% 266.15/266.35 Found x6:(P x5)
% 266.15/266.35 Instantiate: x5:=x':fofType
% 266.15/266.35 Found (fun (x6:(P x5))=> x6) as proof of (P x')
% 266.15/266.35 Found (fun (P:(fofType->Prop)) (x6:(P x5))=> x6) as proof of ((P x5)->(P x'))
% 266.15/266.35 Found (fun (x00:((x0 Xx) x')) (P:(fofType->Prop)) (x6:(P x5))=> x6) as proof of (((eq fofType) x5) x')
% 266.15/266.35 Found x00000:=(x0000 x3):(((eq fofType) x3) x')
% 266.15/266.35 Found (x0000 x3) as proof of (((eq fofType) x3) x')
% 266.15/266.35 Found ((fun (x30:fofType)=> ((x000 x30) x5)) x3) as proof of (((eq fofType) x3) x')
% 266.15/266.35 Found ((fun (x30:fofType)=> (((fun (x30:fofType)=> ((x00 x30) x4)) x30) x5)) x3) as proof of (((eq fofType) x3) x')
% 266.15/266.35 Found ((fun (x30:fofType)=> (((fun (x30:fofType)=> ((x00 x30) x4)) x30) x5)) x3) as proof of (((eq fofType) x3) x')
% 266.15/266.35 Found (fun (x00:((x2 Xx) x'))=> ((fun (x30:fofType)=> (((fun (x30:fofType)=> ((x00 x30) x4)) x30) x5)) x3)) as proof of (((eq fofType) x3) x')
% 266.15/266.35 Found (fun (x':fofType) (x00:((x2 Xx) x'))=> ((fun (x30:fofType)=> (((fun (x30:fofType)=> ((x00 x30) x4)) x30) x5)) x3)) as proof of (((x2 Xx) x')->(((eq fofType) x3) x'))
% 266.15/266.35 Found (fun (x':fofType) (x00:((x2 Xx) x'))=> ((fun (x30:fofType)=> (((fun (x30:fofType)=> ((x00 x30) x4)) x30) x5)) x3)) as proof of (forall (x':fofType), (((x2 Xx) x')->(((eq fofType) x3) x')))
% 266.15/266.35 Found eq_ref00:=(eq_ref0 x30):(((eq fofType) x30) x30)
% 266.15/266.35 Found (eq_ref0 x30) as proof of (((eq fofType) x30) x3)
% 266.15/266.35 Found ((eq_ref fofType) x30) as proof of (((eq fofType) x30) x3)
% 266.15/266.35 Found ((eq_ref fofType) x30) as proof of (((eq fofType) x30) x3)
% 266.15/266.35 Found (fun (x50:(forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy))))=> ((eq_ref fofType) x30)) as proof of (((eq fofType) x30) x3)
% 266.15/266.35 Found (fun (x40:cIND) (x50:(forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy))))=> ((eq_ref fofType) x30)) as proof of ((forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy)))->(((eq fofType) x30) x3))
% 266.15/266.35 Found x6:(P x30)
% 266.15/266.35 Instantiate: x3:=x30:fofType
% 266.15/266.35 Found (fun (x6:(P x30))=> x6) as proof of (P x3)
% 266.15/266.35 Found (fun (P:(fofType->Prop)) (x6:(P x30))=> x6) as proof of ((P x30)->(P x3))
% 266.15/266.35 Found (fun (x50:(forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy)))) (P:(fofType->Prop)) (x6:(P x30))=> x6) as proof of (((eq fofType) x30) x3)
% 266.15/266.35 Found (fun (x40:cIND) (x50:(forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy)))) (P:(fofType->Prop)) (x6:(P x30))=> x6) as proof of ((forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy)))->(((eq fofType) x30) x3))
% 269.15/269.39 Found eq_ref000:=(eq_ref00 P):((P x30)->(P x30))
% 269.15/269.39 Found (eq_ref00 P) as proof of ((P x30)->(P x3))
% 269.15/269.39 Found ((eq_ref0 x30) P) as proof of ((P x30)->(P x3))
% 269.15/269.39 Found (((eq_ref fofType) x30) P) as proof of ((P x30)->(P x3))
% 269.15/269.39 Found (((eq_ref fofType) x30) P) as proof of ((P x30)->(P x3))
% 269.15/269.39 Found (fun (P:(fofType->Prop))=> (((eq_ref fofType) x30) P)) as proof of ((P x30)->(P x3))
% 269.15/269.39 Found (fun (x50:(forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy)))) (P:(fofType->Prop))=> (((eq_ref fofType) x30) P)) as proof of (((eq fofType) x30) x3)
% 269.15/269.39 Found (fun (x40:cIND) (x50:(forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy)))) (P:(fofType->Prop))=> (((eq_ref fofType) x30) P)) as proof of ((forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy)))->(((eq fofType) x30) x3))
% 269.15/269.39 Found eq_ref000:=(eq_ref00 P):((P x5)->(P x5))
% 269.15/269.39 Found (eq_ref00 P) as proof of ((P x5)->(P x'))
% 269.15/269.39 Found ((eq_ref0 x5) P) as proof of ((P x5)->(P x'))
% 269.15/269.39 Found (((eq_ref fofType) x5) P) as proof of ((P x5)->(P x'))
% 269.15/269.39 Found (((eq_ref fofType) x5) P) as proof of ((P x5)->(P x'))
% 269.15/269.39 Found (fun (P:(fofType->Prop))=> (((eq_ref fofType) x5) P)) as proof of ((P x5)->(P x'))
% 269.15/269.39 Found (fun (x00:((x0 Xx) x')) (P:(fofType->Prop))=> (((eq_ref fofType) x5) P)) as proof of (((eq fofType) x5) x')
% 269.15/269.39 Found eq_ref00:=(eq_ref0 x5):(((eq fofType) x5) x5)
% 269.15/269.39 Found (eq_ref0 x5) as proof of (((eq fofType) x5) x')
% 269.15/269.39 Found ((eq_ref fofType) x5) as proof of (((eq fofType) x5) x')
% 269.15/269.39 Found ((eq_ref fofType) x5) as proof of (((eq fofType) x5) x')
% 269.15/269.39 Found ((eq_ref fofType) x5) as proof of (((eq fofType) x5) x')
% 269.15/269.39 Found (fun (x00:((x0 Xx) x'))=> ((eq_ref fofType) x5)) as proof of (((eq fofType) x5) x')
% 269.15/269.39 Found eq_sym0:=(eq_sym Prop):(forall (a:Prop) (b:Prop), ((((eq Prop) a) b)->(((eq Prop) b) a)))
% 269.15/269.39 Instantiate: b:=(forall (a:Prop) (b:Prop), ((((eq Prop) a) b)->(((eq Prop) b) a))):Prop
% 269.15/269.39 Found eq_sym0 as proof of b
% 269.15/269.39 Found eq_ref000:=(eq_ref00 P):((P x30)->(P x30))
% 269.15/269.39 Found (eq_ref00 P) as proof of ((P x30)->(P x3))
% 269.15/269.39 Found ((eq_ref0 x30) P) as proof of ((P x30)->(P x3))
% 269.15/269.39 Found (((eq_ref fofType) x30) P) as proof of ((P x30)->(P x3))
% 269.15/269.39 Found (((eq_ref fofType) x30) P) as proof of ((P x30)->(P x3))
% 269.15/269.39 Found (fun (P:(fofType->Prop))=> (((eq_ref fofType) x30) P)) as proof of ((P x30)->(P x3))
% 269.15/269.39 Found (fun (x20:(forall (Xn:fofType), (not (((eq fofType) (cS Xn)) c0)))) (P:(fofType->Prop))=> (((eq_ref fofType) x30) P)) as proof of (((eq fofType) x30) x3)
% 269.15/269.39 Found x6:(P x30)
% 269.15/269.39 Instantiate: x3:=x30:fofType
% 269.15/269.39 Found (fun (x6:(P x30))=> x6) as proof of (P x3)
% 269.15/269.39 Found (fun (P:(fofType->Prop)) (x6:(P x30))=> x6) as proof of ((P x30)->(P x3))
% 269.15/269.39 Found (fun (x20:(forall (Xn:fofType), (not (((eq fofType) (cS Xn)) c0)))) (P:(fofType->Prop)) (x6:(P x30))=> x6) as proof of (((eq fofType) x30) x3)
% 269.15/269.39 Found eq_ref000:=(eq_ref00 P):((P x10)->(P x10))
% 269.15/269.39 Found (eq_ref00 P) as proof of ((P x10)->(P x1))
% 269.15/269.39 Found ((eq_ref0 x10) P) as proof of ((P x10)->(P x1))
% 269.15/269.39 Found (((eq_ref fofType) x10) P) as proof of ((P x10)->(P x1))
% 269.15/269.39 Found (((eq_ref fofType) x10) P) as proof of ((P x10)->(P x1))
% 269.15/269.39 Found (fun (P:(fofType->Prop))=> (((eq_ref fofType) x10) P)) as proof of ((P x10)->(P x1))
% 269.15/269.39 Found (fun (x20:((and cIND) (forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy))))) (P:(fofType->Prop))=> (((eq_ref fofType) x10) P)) as proof of (((eq fofType) x10) x1)
% 269.15/269.39 Found (fun (x30:(forall (Xn:fofType), (not (((eq fofType) (cS Xn)) c0)))) (x20:((and cIND) (forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy))))) (P:(fofType->Prop))=> (((eq_ref fofType) x10) P)) as proof of (((and cIND) (forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy))))->(((eq fofType) x10) x1))
% 269.15/269.39 Found x6:(P x10)
% 269.15/269.39 Instantiate: x1:=x10:fofType
% 269.15/269.39 Found (fun (x6:(P x10))=> x6) as proof of (P x1)
% 269.15/269.39 Found (fun (P:(fofType->Prop)) (x6:(P x10))=> x6) as proof of ((P x10)->(P x1))
% 272.60/272.84 Found (fun (x20:((and cIND) (forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy))))) (P:(fofType->Prop)) (x6:(P x10))=> x6) as proof of (((eq fofType) x10) x1)
% 272.60/272.84 Found (fun (x30:(forall (Xn:fofType), (not (((eq fofType) (cS Xn)) c0)))) (x20:((and cIND) (forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy))))) (P:(fofType->Prop)) (x6:(P x10))=> x6) as proof of (((and cIND) (forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy))))->(((eq fofType) x10) x1))
% 272.60/272.84 Found eq_ref00:=(eq_ref0 (cS x10)):(((eq fofType) (cS x10)) (cS x10))
% 272.60/272.84 Found (eq_ref0 (cS x10)) as proof of (((eq fofType) (cS x10)) (cS x1))
% 272.60/272.84 Found ((eq_ref fofType) (cS x10)) as proof of (((eq fofType) (cS x10)) (cS x1))
% 272.60/272.84 Found ((eq_ref fofType) (cS x10)) as proof of (((eq fofType) (cS x10)) (cS x1))
% 272.60/272.84 Found ((eq_ref fofType) (cS x10)) as proof of (((eq fofType) (cS x10)) (cS x1))
% 272.60/272.84 Found (x500 ((eq_ref fofType) (cS x10))) as proof of (((eq fofType) x10) x1)
% 272.60/272.84 Found ((x50 x1) ((eq_ref fofType) (cS x10))) as proof of (((eq fofType) x10) x1)
% 272.60/272.84 Found (((x5 x10) x1) ((eq_ref fofType) (cS x10))) as proof of (((eq fofType) x10) x1)
% 272.60/272.84 Found eq_ref000:=(eq_ref00 (ex fofType)):(((ex fofType) ((unique fofType) (x4 Xx)))->((ex fofType) ((unique fofType) (x4 Xx))))
% 272.60/272.84 Found (eq_ref00 (ex fofType)) as proof of (P ((unique fofType) (x4 Xx)))
% 272.60/272.84 Found ((eq_ref0 ((unique fofType) (x4 Xx))) (ex fofType)) as proof of (P ((unique fofType) (x4 Xx)))
% 272.60/272.84 Found (((eq_ref (fofType->Prop)) ((unique fofType) (x4 Xx))) (ex fofType)) as proof of (P ((unique fofType) (x4 Xx)))
% 272.60/272.84 Found (((eq_ref (fofType->Prop)) ((unique fofType) (x4 Xx))) (ex fofType)) as proof of (P ((unique fofType) (x4 Xx)))
% 272.60/272.84 Found eq_ref000:=(eq_ref00 (ex fofType)):(((ex fofType) ((unique fofType) (x4 Xx)))->((ex fofType) ((unique fofType) (x4 Xx))))
% 272.60/272.84 Found (eq_ref00 (ex fofType)) as proof of (P ((unique fofType) (x4 Xx)))
% 272.60/272.84 Found ((eq_ref0 ((unique fofType) (x4 Xx))) (ex fofType)) as proof of (P ((unique fofType) (x4 Xx)))
% 272.60/272.84 Found (((eq_ref (fofType->Prop)) ((unique fofType) (x4 Xx))) (ex fofType)) as proof of (P ((unique fofType) (x4 Xx)))
% 272.60/272.84 Found (((eq_ref (fofType->Prop)) ((unique fofType) (x4 Xx))) (ex fofType)) as proof of (P ((unique fofType) (x4 Xx)))
% 272.60/272.84 Found eq_ref00:=(eq_ref0 (cS x30)):(((eq fofType) (cS x30)) (cS x30))
% 272.60/272.84 Found (eq_ref0 (cS x30)) as proof of (((eq fofType) (cS x30)) (cS x3))
% 272.60/272.84 Found ((eq_ref fofType) (cS x30)) as proof of (((eq fofType) (cS x30)) (cS x3))
% 272.60/272.84 Found ((eq_ref fofType) (cS x30)) as proof of (((eq fofType) (cS x30)) (cS x3))
% 272.60/272.84 Found ((eq_ref fofType) (cS x30)) as proof of (((eq fofType) (cS x30)) (cS x3))
% 272.60/272.84 Found (x5000 ((eq_ref fofType) (cS x30))) as proof of (((eq fofType) x30) x3)
% 272.60/272.84 Found ((x500 x3) ((eq_ref fofType) (cS x30))) as proof of (((eq fofType) x30) x3)
% 272.60/272.84 Found (((x50 x30) x3) ((eq_ref fofType) (cS x30))) as proof of (((eq fofType) x30) x3)
% 272.60/272.84 Found (fun (x50:(forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy))))=> (((x50 x30) x3) ((eq_ref fofType) (cS x30)))) as proof of (((eq fofType) x30) x3)
% 272.60/272.84 Found (fun (x40:cIND) (x50:(forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy))))=> (((x50 x30) x3) ((eq_ref fofType) (cS x30)))) as proof of ((forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy)))->(((eq fofType) x30) x3))
% 272.60/272.84 Found eq_ref00:=(eq_ref0 (cS x30)):(((eq fofType) (cS x30)) (cS x30))
% 272.60/272.84 Found (eq_ref0 (cS x30)) as proof of (((eq fofType) (cS x30)) (cS x3))
% 272.60/272.84 Found ((eq_ref fofType) (cS x30)) as proof of (((eq fofType) (cS x30)) (cS x3))
% 272.60/272.84 Found ((eq_ref fofType) (cS x30)) as proof of (((eq fofType) (cS x30)) (cS x3))
% 272.60/272.84 Found ((eq_ref fofType) (cS x30)) as proof of (((eq fofType) (cS x30)) (cS x3))
% 272.60/272.84 Found (x510 ((eq_ref fofType) (cS x30))) as proof of (((eq fofType) x30) x3)
% 272.60/272.84 Found ((x51 x3) ((eq_ref fofType) (cS x30))) as proof of (((eq fofType) x30) x3)
% 272.60/272.84 Found (((x5 x30) x3) ((eq_ref fofType) (cS x30))) as proof of (((eq fofType) x30) x3)
% 275.52/275.72 Found (fun (x50:(forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy))))=> (((x5 x30) x3) ((eq_ref fofType) (cS x30)))) as proof of (((eq fofType) x30) x3)
% 275.52/275.72 Found (fun (x40:cIND) (x50:(forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy))))=> (((x5 x30) x3) ((eq_ref fofType) (cS x30)))) as proof of ((forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy)))->(((eq fofType) x30) x3))
% 275.52/275.72 Found eq_ref00:=(eq_ref0 (cS Xx0)):(((eq fofType) (cS Xx0)) (cS Xx0))
% 275.52/275.72 Found (eq_ref0 (cS Xx0)) as proof of (((eq fofType) (cS Xx0)) x5)
% 275.52/275.72 Found ((eq_ref fofType) (cS Xx0)) as proof of (((eq fofType) (cS Xx0)) x5)
% 275.52/275.72 Found ((eq_ref fofType) (cS Xx0)) as proof of (((eq fofType) (cS Xx0)) x5)
% 275.52/275.72 Found (fun (x6:(((eq fofType) Xx0) x5))=> ((eq_ref fofType) (cS Xx0))) as proof of (((eq fofType) (cS Xx0)) x5)
% 275.52/275.72 Found eq_ref00:=(eq_ref0 (cS Xx0)):(((eq fofType) (cS Xx0)) (cS Xx0))
% 275.52/275.72 Found (eq_ref0 (cS Xx0)) as proof of (((eq fofType) (cS Xx0)) x5)
% 275.52/275.72 Found ((eq_ref fofType) (cS Xx0)) as proof of (((eq fofType) (cS Xx0)) x5)
% 275.52/275.72 Found ((eq_ref fofType) (cS Xx0)) as proof of (((eq fofType) (cS Xx0)) x5)
% 275.52/275.72 Found (fun (x6:(((eq fofType) Xx0) x5))=> ((eq_ref fofType) (cS Xx0))) as proof of (((eq fofType) (cS Xx0)) x5)
% 275.52/275.72 Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% 275.52/275.72 Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) b)
% 275.52/275.72 Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% 275.52/275.72 Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% 275.52/275.72 Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% 275.52/275.72 Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% 275.52/275.72 Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx))))))
% 275.52/275.72 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx))))))
% 275.52/275.72 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx))))))
% 275.52/275.72 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx))))))
% 275.52/275.72 Found ((eq_trans0000 ((eq_ref Prop) (f x0))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f x0)) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx))))))
% 275.52/275.72 Found (((eq_trans000 ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx)))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f x0)) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx))))))
% 275.52/275.72 Found ((((eq_trans00 ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx)))))) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx)))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx))))))) as proof of (((eq Prop) (f x0)) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx))))))
% 275.52/275.72 Found (((((eq_trans0 (f x0)) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx)))))) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx)))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx))))))) as proof of (((eq Prop) (f x0)) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx))))))
% 275.52/275.72 Found ((((((eq_trans Prop) (f x0)) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx)))))) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx)))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx))))))) as proof of (((eq Prop) (f x0)) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx))))))
% 275.52/275.72 Found (fun (x2:(forall (Xn:fofType), (not (((eq fofType) (cS Xn)) c0))))=> ((((((eq_trans Prop) (f x0)) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx)))))) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx)))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx)))))))) as proof of (((eq Prop) (f x0)) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx))))))
% 275.52/275.72 Found (fun (x1:((and cIND) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (cS Xx)) (cS Xy))->(((eq fofType) Xx) Xy))))) (x2:(forall (Xn:fofType), (not (((eq fofType) (cS Xn)) c0))))=> ((((((eq_trans Prop) (f x0)) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx)))))) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx)))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx)))))))) as proof of ((forall (Xn:fofType), (not (((eq fofType) (cS Xn)) c0)))->(((eq Prop) (f x0)) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx)))))))
% 275.52/275.72 Found (fun (x1:((and cIND) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (cS Xx)) (cS Xy))->(((eq fofType) Xx) Xy))))) (x2:(forall (Xn:fofType), (not (((eq fofType) (cS Xn)) c0))))=> ((((((eq_trans Prop) (f x0)) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx)))))) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx)))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx)))))))) as proof of (((and cIND) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (cS Xx)) (cS Xy))->(((eq fofType) Xx) Xy))))->((forall (Xn:fofType), (not (((eq fofType) (cS Xn)) c0)))->(((eq Prop) (f x0)) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx))))))))
% 275.52/275.75 Found (and_rect00 (fun (x1:((and cIND) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (cS Xx)) (cS Xy))->(((eq fofType) Xx) Xy))))) (x2:(forall (Xn:fofType), (not (((eq fofType) (cS Xn)) c0))))=> ((((((eq_trans Prop) (f x0)) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx)))))) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx)))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx))))))))) as proof of (((eq Prop) (f x0)) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx))))))
% 275.52/275.75 Found ((and_rect0 (((eq Prop) (f x0)) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx))))))) (fun (x1:((and cIND) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (cS Xx)) (cS Xy))->(((eq fofType) Xx) Xy))))) (x2:(forall (Xn:fofType), (not (((eq fofType) (cS Xn)) c0))))=> ((((((eq_trans Prop) (f x0)) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx)))))) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx)))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx))))))))) as proof of (((eq Prop) (f x0)) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx))))))
% 275.52/275.75 Found (((fun (P0:Type) (x1:(((and cIND) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (cS Xx)) (cS Xy))->(((eq fofType) Xx) Xy))))->((forall (Xn:fofType), (not (((eq fofType) (cS Xn)) c0)))->P0)))=> (((((and_rect ((and cIND) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (cS Xx)) (cS Xy))->(((eq fofType) Xx) Xy))))) (forall (Xn:fofType), (not (((eq fofType) (cS Xn)) c0)))) P0) x1) x)) (((eq Prop) (f x0)) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx))))))) (fun (x1:((and cIND) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (cS Xx)) (cS Xy))->(((eq fofType) Xx) Xy))))) (x2:(forall (Xn:fofType), (not (((eq fofType) (cS Xn)) c0))))=> ((((((eq_trans Prop) (f x0)) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx)))))) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx)))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx))))))))) as proof of (((eq Prop) (f x0)) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx))))))
% 275.74/275.96 Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% 275.74/275.96 Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) b)
% 275.74/275.96 Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% 275.74/275.96 Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% 275.74/275.96 Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% 275.74/275.96 Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% 275.74/275.96 Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx))))))
% 275.74/275.96 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx))))))
% 275.74/275.96 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx))))))
% 275.74/275.96 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx))))))
% 275.74/275.96 Found ((eq_trans0000 ((eq_ref Prop) (f x0))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f x0)) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx))))))
% 275.74/275.96 Found (((eq_trans000 ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx)))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f x0)) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx))))))
% 275.74/275.96 Found ((((eq_trans00 ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx)))))) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx)))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx))))))) as proof of (((eq Prop) (f x0)) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx))))))
% 275.74/275.96 Found (((((eq_trans0 (f x0)) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx)))))) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx)))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx))))))) as proof of (((eq Prop) (f x0)) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx))))))
% 275.74/275.96 Found ((((((eq_trans Prop) (f x0)) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx)))))) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx)))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx))))))) as proof of (((eq Prop) (f x0)) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx))))))
% 275.74/275.99 Found (fun (x2:(forall (Xn:fofType), (not (((eq fofType) (cS Xn)) c0))))=> ((((((eq_trans Prop) (f x0)) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx)))))) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx)))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx)))))))) as proof of (((eq Prop) (f x0)) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx))))))
% 275.74/275.99 Found (fun (x1:((and cIND) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (cS Xx)) (cS Xy))->(((eq fofType) Xx) Xy))))) (x2:(forall (Xn:fofType), (not (((eq fofType) (cS Xn)) c0))))=> ((((((eq_trans Prop) (f x0)) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx)))))) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx)))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx)))))))) as proof of ((forall (Xn:fofType), (not (((eq fofType) (cS Xn)) c0)))->(((eq Prop) (f x0)) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx)))))))
% 275.74/275.99 Found (fun (x1:((and cIND) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (cS Xx)) (cS Xy))->(((eq fofType) Xx) Xy))))) (x2:(forall (Xn:fofType), (not (((eq fofType) (cS Xn)) c0))))=> ((((((eq_trans Prop) (f x0)) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx)))))) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx)))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx)))))))) as proof of (((and cIND) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (cS Xx)) (cS Xy))->(((eq fofType) Xx) Xy))))->((forall (Xn:fofType), (not (((eq fofType) (cS Xn)) c0)))->(((eq Prop) (f x0)) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx))))))))
% 275.74/275.99 Found (and_rect00 (fun (x1:((and cIND) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (cS Xx)) (cS Xy))->(((eq fofType) Xx) Xy))))) (x2:(forall (Xn:fofType), (not (((eq fofType) (cS Xn)) c0))))=> ((((((eq_trans Prop) (f x0)) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx)))))) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx)))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx))))))))) as proof of (((eq Prop) (f x0)) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx))))))
% 275.98/276.19 Found ((and_rect0 (((eq Prop) (f x0)) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx))))))) (fun (x1:((and cIND) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (cS Xx)) (cS Xy))->(((eq fofType) Xx) Xy))))) (x2:(forall (Xn:fofType), (not (((eq fofType) (cS Xn)) c0))))=> ((((((eq_trans Prop) (f x0)) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx)))))) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx)))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx))))))))) as proof of (((eq Prop) (f x0)) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx))))))
% 275.98/276.19 Found (((fun (P0:Type) (x1:(((and cIND) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (cS Xx)) (cS Xy))->(((eq fofType) Xx) Xy))))->((forall (Xn:fofType), (not (((eq fofType) (cS Xn)) c0)))->P0)))=> (((((and_rect ((and cIND) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (cS Xx)) (cS Xy))->(((eq fofType) Xx) Xy))))) (forall (Xn:fofType), (not (((eq fofType) (cS Xn)) c0)))) P0) x1) x)) (((eq Prop) (f x0)) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx))))))) (fun (x1:((and cIND) (forall (Xx:fofType) (Xy:fofType), ((((eq fofType) (cS Xx)) (cS Xy))->(((eq fofType) Xx) Xy))))) (x2:(forall (Xn:fofType), (not (((eq fofType) (cS Xn)) c0))))=> ((((((eq_trans Prop) (f x0)) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx)))))) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx)))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx))))))))) as proof of (((eq Prop) (f x0)) ((and ((and ((x0 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) ((unique fofType) (x0 Xx))))))
% 275.98/276.19 Found x6:(P x5)
% 275.98/276.19 Instantiate: x5:=Y:fofType
% 275.98/276.19 Found (fun (x6:(P x5))=> x6) as proof of (P Y)
% 275.98/276.19 Found (fun (P:(fofType->Prop)) (x6:(P x5))=> x6) as proof of ((P x5)->(P Y))
% 275.98/276.19 Found (fun (P:(fofType->Prop)) (x6:(P x5))=> x6) as proof of (((eq fofType) x5) Y)
% 275.98/276.19 Found (fun (x00:((x0 Xx) Y)) (P:(fofType->Prop)) (x6:(P x5))=> x6) as proof of (((eq fofType) x5) Y)
% 275.98/276.19 Found eq_ref000:=(eq_ref00 P):((P x5)->(P x5))
% 275.98/276.19 Found (eq_ref00 P) as proof of ((P x5)->(P Y))
% 275.98/276.19 Found ((eq_ref0 x5) P) as proof of ((P x5)->(P Y))
% 275.98/276.19 Found (((eq_ref fofType) x5) P) as proof of ((P x5)->(P Y))
% 275.98/276.19 Found (((eq_ref fofType) x5) P) as proof of ((P x5)->(P Y))
% 275.98/276.19 Found (fun (P:(fofType->Prop))=> (((eq_ref fofType) x5) P)) as proof of ((P x5)->(P Y))
% 275.98/276.19 Found (fun (P:(fofType->Prop))=> (((eq_ref fofType) x5) P)) as proof of (((eq fofType) x5) Y)
% 279.34/279.52 Found (fun (x00:((x0 Xx) Y)) (P:(fofType->Prop))=> (((eq_ref fofType) x5) P)) as proof of (((eq fofType) x5) Y)
% 279.34/279.52 Found eq_ref00:=(eq_ref0 (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy)))))):(((eq Prop) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))
% 279.34/279.52 Found (eq_ref0 (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy)))))) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy)))))) b)
% 279.34/279.52 Found ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy)))))) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy)))))) b)
% 279.34/279.52 Found ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy)))))) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy)))))) b)
% 279.34/279.52 Found ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy)))))) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy)))))) b)
% 279.34/279.52 Found eq_ref00:=(eq_ref0 x5):(((eq fofType) x5) x5)
% 279.34/279.52 Found (eq_ref0 x5) as proof of (((eq fofType) x5) x50)
% 279.34/279.52 Found ((eq_ref fofType) x5) as proof of (((eq fofType) x5) x50)
% 279.34/279.52 Found ((eq_ref fofType) x5) as proof of (((eq fofType) x5) x50)
% 279.34/279.52 Found ((eq_ref fofType) x5) as proof of (((eq fofType) x5) x50)
% 279.34/279.52 Found (eq_sym000 ((eq_ref fofType) x5)) as proof of (((eq fofType) x50) x5)
% 279.34/279.52 Found ((eq_sym00 x50) ((eq_ref fofType) x5)) as proof of (((eq fofType) x50) x5)
% 279.34/279.52 Found (((eq_sym0 x5) x50) ((eq_ref fofType) x5)) as proof of (((eq fofType) x50) x5)
% 279.34/279.52 Found ((((eq_sym fofType) x5) x50) ((eq_ref fofType) x5)) as proof of (((eq fofType) x50) x5)
% 279.34/279.52 Found eq_ref00:=(eq_ref0 x5):(((eq fofType) x5) x5)
% 279.34/279.52 Found (eq_ref0 x5) as proof of (((eq fofType) x5) x50)
% 279.34/279.52 Found ((eq_ref fofType) x5) as proof of (((eq fofType) x5) x50)
% 279.34/279.52 Found ((eq_ref fofType) x5) as proof of (((eq fofType) x5) x50)
% 279.34/279.52 Found ((eq_ref fofType) x5) as proof of (((eq fofType) x5) x50)
% 279.34/279.52 Found (eq_sym000 ((eq_ref fofType) x5)) as proof of (((eq fofType) x50) x5)
% 279.34/279.52 Found ((eq_sym00 x50) ((eq_ref fofType) x5)) as proof of (((eq fofType) x50) x5)
% 279.34/279.52 Found (((eq_sym0 x5) x50) ((eq_ref fofType) x5)) as proof of (((eq fofType) x50) x5)
% 279.34/279.52 Found ((((eq_sym fofType) x5) x50) ((eq_ref fofType) x5)) as proof of (((eq fofType) x50) x5)
% 279.34/279.52 Found eq_ref00:=(eq_ref0 (forall (Xx:fofType) (Xy:fofType), (((x2 Xx) Xy)->((x2 (cS Xx)) (cS (cS Xy)))))):(((eq Prop) (forall (Xx:fofType) (Xy:fofType), (((x2 Xx) Xy)->((x2 (cS Xx)) (cS (cS Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((x2 Xx) Xy)->((x2 (cS Xx)) (cS (cS Xy))))))
% 279.34/279.52 Found (eq_ref0 (forall (Xx:fofType) (Xy:fofType), (((x2 Xx) Xy)->((x2 (cS Xx)) (cS (cS Xy)))))) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), (((x2 Xx) Xy)->((x2 (cS Xx)) (cS (cS Xy)))))) b)
% 279.34/279.52 Found ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), (((x2 Xx) Xy)->((x2 (cS Xx)) (cS (cS Xy)))))) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), (((x2 Xx) Xy)->((x2 (cS Xx)) (cS (cS Xy)))))) b)
% 279.34/279.52 Found ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), (((x2 Xx) Xy)->((x2 (cS Xx)) (cS (cS Xy)))))) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), (((x2 Xx) Xy)->((x2 (cS Xx)) (cS (cS Xy)))))) b)
% 279.34/279.52 Found ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), (((x2 Xx) Xy)->((x2 (cS Xx)) (cS (cS Xy)))))) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), (((x2 Xx) Xy)->((x2 (cS Xx)) (cS (cS Xy)))))) b)
% 279.34/279.52 Found eq_ref00:=(eq_ref0 (f x3)):(((eq Prop) (f x3)) (f x3))
% 279.34/279.52 Found (eq_ref0 (f x3)) as proof of (((eq Prop) (f x3)) ((and ((x0 Xx) x3)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) x3) Y)))))
% 279.34/279.52 Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((and ((x0 Xx) x3)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) x3) Y)))))
% 279.34/279.52 Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((and ((x0 Xx) x3)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) x3) Y)))))
% 283.84/284.06 Found (fun (x3:fofType)=> ((eq_ref Prop) (f x3))) as proof of (((eq Prop) (f x3)) ((and ((x0 Xx) x3)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) x3) Y)))))
% 283.84/284.06 Found (fun (x3:fofType)=> ((eq_ref Prop) (f x3))) as proof of (forall (x:fofType), (((eq Prop) (f x)) ((and ((x0 Xx) x)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) x) Y))))))
% 283.84/284.06 Found eq_ref00:=(eq_ref0 (f x3)):(((eq Prop) (f x3)) (f x3))
% 283.84/284.06 Found (eq_ref0 (f x3)) as proof of (((eq Prop) (f x3)) ((and ((x0 Xx) x3)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) x3) Y)))))
% 283.84/284.06 Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((and ((x0 Xx) x3)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) x3) Y)))))
% 283.84/284.06 Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((and ((x0 Xx) x3)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) x3) Y)))))
% 283.84/284.06 Found (fun (x3:fofType)=> ((eq_ref Prop) (f x3))) as proof of (((eq Prop) (f x3)) ((and ((x0 Xx) x3)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) x3) Y)))))
% 283.84/284.06 Found (fun (x3:fofType)=> ((eq_ref Prop) (f x3))) as proof of (forall (x:fofType), (((eq Prop) (f x)) ((and ((x0 Xx) x)) (forall (Y:fofType), (((x0 Xx) Y)->(((eq fofType) x) Y))))))
% 283.84/284.06 Found x6:(P x1)
% 283.84/284.06 Instantiate: x1:=x':fofType
% 283.84/284.06 Found (fun (x6:(P x1))=> x6) as proof of (P x')
% 283.84/284.06 Found (fun (P:(fofType->Prop)) (x6:(P x1))=> x6) as proof of ((P x1)->(P x'))
% 283.84/284.06 Found (fun (x00:((x0 Xx) x')) (P:(fofType->Prop)) (x6:(P x1))=> x6) as proof of (((eq fofType) x1) x')
% 283.84/284.06 Found eq_ref00:=(eq_ref0 (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy)))))):(((eq Prop) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy)))))) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy))))))
% 283.84/284.06 Found (eq_ref0 (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy)))))) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy)))))) b)
% 283.84/284.06 Found ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy)))))) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy)))))) b)
% 283.84/284.06 Found ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy)))))) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy)))))) b)
% 283.84/284.06 Found ((eq_ref Prop) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy)))))) as proof of (((eq Prop) (forall (Xx:fofType) (Xy:fofType), (((x0 Xx) Xy)->((x0 (cS Xx)) (cS (cS Xy)))))) b)
% 283.84/284.06 Found eq_ref000:=(eq_ref00 P):((P x1)->(P x1))
% 283.84/284.06 Found (eq_ref00 P) as proof of ((P x1)->(P x'))
% 283.84/284.06 Found ((eq_ref0 x1) P) as proof of ((P x1)->(P x'))
% 283.84/284.06 Found (((eq_ref fofType) x1) P) as proof of ((P x1)->(P x'))
% 283.84/284.06 Found (((eq_ref fofType) x1) P) as proof of ((P x1)->(P x'))
% 283.84/284.06 Found (fun (P:(fofType->Prop))=> (((eq_ref fofType) x1) P)) as proof of ((P x1)->(P x'))
% 283.84/284.06 Found (fun (x00:((x0 Xx) x')) (P:(fofType->Prop))=> (((eq_ref fofType) x1) P)) as proof of (((eq fofType) x1) x')
% 283.84/284.06 Found x6:(P x3)
% 283.84/284.06 Instantiate: x3:=x':fofType
% 283.84/284.06 Found (fun (x6:(P x3))=> x6) as proof of (P x')
% 283.84/284.06 Found (fun (P:(fofType->Prop)) (x6:(P x3))=> x6) as proof of ((P x3)->(P x'))
% 283.84/284.06 Found (fun (x00:((x0 Xx) x')) (P:(fofType->Prop)) (x6:(P x3))=> x6) as proof of (((eq fofType) x3) x')
% 283.84/284.06 Found eq_ref000:=(eq_ref00 P):((P x3)->(P x3))
% 283.84/284.06 Found (eq_ref00 P) as proof of ((P x3)->(P x'))
% 283.84/284.06 Found ((eq_ref0 x3) P) as proof of ((P x3)->(P x'))
% 283.84/284.06 Found (((eq_ref fofType) x3) P) as proof of ((P x3)->(P x'))
% 283.84/284.06 Found (((eq_ref fofType) x3) P) as proof of ((P x3)->(P x'))
% 283.84/284.06 Found (fun (P:(fofType->Prop))=> (((eq_ref fofType) x3) P)) as proof of ((P x3)->(P x'))
% 283.84/284.06 Found (fun (x00:((x0 Xx) x')) (P:(fofType->Prop))=> (((eq_ref fofType) x3) P)) as proof of (((eq fofType) x3) x')
% 283.84/284.06 Found eq_ref00:=(eq_ref0 x1):(((eq fofType) x1) x1)
% 283.84/284.06 Found (eq_ref0 x1) as proof of (((eq fofType) x1) x')
% 283.84/284.06 Found ((eq_ref fofType) x1) as proof of (((eq fofType) x1) x')
% 283.84/284.06 Found ((eq_ref fofType) x1) as proof of (((eq fofType) x1) x')
% 283.84/284.06 Found ((eq_ref fofType) x1) as proof of (((eq fofType) x1) x')
% 283.84/284.06 Found (fun (x00:((x0 Xx) x'))=> ((eq_ref fofType) x1)) as proof of (((eq fofType) x1) x')
% 286.71/286.94 Found eq_ref00:=(eq_ref0 x3):(((eq fofType) x3) x3)
% 286.71/286.94 Found (eq_ref0 x3) as proof of (((eq fofType) x3) x')
% 286.71/286.94 Found ((eq_ref fofType) x3) as proof of (((eq fofType) x3) x')
% 286.71/286.94 Found ((eq_ref fofType) x3) as proof of (((eq fofType) x3) x')
% 286.71/286.94 Found ((eq_ref fofType) x3) as proof of (((eq fofType) x3) x')
% 286.71/286.94 Found (fun (x00:((x0 Xx) x'))=> ((eq_ref fofType) x3)) as proof of (((eq fofType) x3) x')
% 286.71/286.94 Found eq_sym0:=(eq_sym Prop):(forall (a:Prop) (b:Prop), ((((eq Prop) a) b)->(((eq Prop) b) a)))
% 286.71/286.94 Instantiate: b:=(forall (a:Prop) (b:Prop), ((((eq Prop) a) b)->(((eq Prop) b) a))):Prop
% 286.71/286.94 Found eq_sym0 as proof of b
% 286.71/286.94 Found x4:(P x30)
% 286.71/286.94 Instantiate: x3:=x30:fofType
% 286.71/286.94 Found (fun (x4:(P x30))=> x4) as proof of (P x3)
% 286.71/286.94 Found (fun (P:(fofType->Prop)) (x4:(P x30))=> x4) as proof of ((P x30)->(P x3))
% 286.71/286.94 Found (fun (P:(fofType->Prop)) (x4:(P x30))=> x4) as proof of (forall (P:(fofType->Prop)), ((P x30)->(P x3)))
% 286.71/286.94 Found eq_ref000:=(eq_ref00 P):((P x30)->(P x30))
% 286.71/286.94 Found (eq_ref00 P) as proof of ((P x30)->(P x3))
% 286.71/286.94 Found ((eq_ref0 x30) P) as proof of ((P x30)->(P x3))
% 286.71/286.94 Found (((eq_ref fofType) x30) P) as proof of ((P x30)->(P x3))
% 286.71/286.94 Found (((eq_ref fofType) x30) P) as proof of ((P x30)->(P x3))
% 286.71/286.94 Found (fun (P:(fofType->Prop))=> (((eq_ref fofType) x30) P)) as proof of ((P x30)->(P x3))
% 286.71/286.94 Found (fun (P:(fofType->Prop))=> (((eq_ref fofType) x30) P)) as proof of (forall (P:(fofType->Prop)), ((P x30)->(P x3)))
% 286.71/286.94 Found eta_expansion000:=(eta_expansion00 (fun (X:fofType)=> ((and ((x4 Xx) X)) (forall (Y:fofType), (((x4 Xx) Y)->(((eq fofType) X) Y)))))):(((eq (fofType->Prop)) (fun (X:fofType)=> ((and ((x4 Xx) X)) (forall (Y:fofType), (((x4 Xx) Y)->(((eq fofType) X) Y)))))) (fun (x:fofType)=> ((and ((x4 Xx) x)) (forall (Y:fofType), (((x4 Xx) Y)->(((eq fofType) x) Y))))))
% 286.71/286.94 Found (eta_expansion00 (fun (X:fofType)=> ((and ((x4 Xx) X)) (forall (Y:fofType), (((x4 Xx) Y)->(((eq fofType) X) Y)))))) as proof of (((eq (fofType->Prop)) (fun (X:fofType)=> ((and ((x4 Xx) X)) (forall (Y:fofType), (((x4 Xx) Y)->(((eq fofType) X) Y)))))) b)
% 286.71/286.94 Found ((eta_expansion0 Prop) (fun (X:fofType)=> ((and ((x4 Xx) X)) (forall (Y:fofType), (((x4 Xx) Y)->(((eq fofType) X) Y)))))) as proof of (((eq (fofType->Prop)) (fun (X:fofType)=> ((and ((x4 Xx) X)) (forall (Y:fofType), (((x4 Xx) Y)->(((eq fofType) X) Y)))))) b)
% 286.71/286.94 Found (((eta_expansion fofType) Prop) (fun (X:fofType)=> ((and ((x4 Xx) X)) (forall (Y:fofType), (((x4 Xx) Y)->(((eq fofType) X) Y)))))) as proof of (((eq (fofType->Prop)) (fun (X:fofType)=> ((and ((x4 Xx) X)) (forall (Y:fofType), (((x4 Xx) Y)->(((eq fofType) X) Y)))))) b)
% 286.71/286.94 Found (((eta_expansion fofType) Prop) (fun (X:fofType)=> ((and ((x4 Xx) X)) (forall (Y:fofType), (((x4 Xx) Y)->(((eq fofType) X) Y)))))) as proof of (((eq (fofType->Prop)) (fun (X:fofType)=> ((and ((x4 Xx) X)) (forall (Y:fofType), (((x4 Xx) Y)->(((eq fofType) X) Y)))))) b)
% 286.71/286.94 Found (((eta_expansion fofType) Prop) (fun (X:fofType)=> ((and ((x4 Xx) X)) (forall (Y:fofType), (((x4 Xx) Y)->(((eq fofType) X) Y)))))) as proof of (((eq (fofType->Prop)) (fun (X:fofType)=> ((and ((x4 Xx) X)) (forall (Y:fofType), (((x4 Xx) Y)->(((eq fofType) X) Y)))))) b)
% 286.71/286.94 Found eq_ref00:=(eq_ref0 c0):(((eq fofType) c0) c0)
% 286.71/286.94 Found (eq_ref0 c0) as proof of (((eq fofType) c0) x5)
% 286.71/286.94 Found ((eq_ref fofType) c0) as proof of (((eq fofType) c0) x5)
% 286.71/286.94 Found ((eq_ref fofType) c0) as proof of (((eq fofType) c0) x5)
% 286.71/286.94 Found ((eq_ref fofType) c0) as proof of (((eq fofType) c0) x5)
% 286.71/286.94 Found eq_ref00:=(eq_ref0 c0):(((eq fofType) c0) c0)
% 286.71/286.94 Found (eq_ref0 c0) as proof of (((eq fofType) c0) x5)
% 286.71/286.94 Found ((eq_ref fofType) c0) as proof of (((eq fofType) c0) x5)
% 286.71/286.94 Found ((eq_ref fofType) c0) as proof of (((eq fofType) c0) x5)
% 286.71/286.94 Found ((eq_ref fofType) c0) as proof of (((eq fofType) c0) x5)
% 286.71/286.94 Found eq_ref00:=(eq_ref0 (f x3)):(((eq Prop) (f x3)) (f x3))
% 286.71/286.94 Found (eq_ref0 (f x3)) as proof of (((eq Prop) (f x3)) (((unique fofType) (x2 Xx)) x3))
% 286.71/286.94 Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) (((unique fofType) (x2 Xx)) x3))
% 286.71/286.94 Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) (((unique fofType) (x2 Xx)) x3))
% 286.71/286.94 Found (fun (x3:fofType)=> ((eq_ref Prop) (f x3))) as proof of (((eq Prop) (f x3)) (((unique fofType) (x2 Xx)) x3))
% 289.98/290.24 Found (fun (x3:fofType)=> ((eq_ref Prop) (f x3))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (((unique fofType) (x2 Xx)) x)))
% 289.98/290.24 Found eq_ref00:=(eq_ref0 (f x3)):(((eq Prop) (f x3)) (f x3))
% 289.98/290.24 Found (eq_ref0 (f x3)) as proof of (((eq Prop) (f x3)) (((unique fofType) (x2 Xx)) x3))
% 289.98/290.24 Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) (((unique fofType) (x2 Xx)) x3))
% 289.98/290.24 Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) (((unique fofType) (x2 Xx)) x3))
% 289.98/290.24 Found (fun (x3:fofType)=> ((eq_ref Prop) (f x3))) as proof of (((eq Prop) (f x3)) (((unique fofType) (x2 Xx)) x3))
% 289.98/290.24 Found (fun (x3:fofType)=> ((eq_ref Prop) (f x3))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (((unique fofType) (x2 Xx)) x)))
% 289.98/290.24 Found eq_ref000:=(eq_ref00 P):((P x30)->(P x30))
% 289.98/290.24 Found (eq_ref00 P) as proof of ((P x30)->(P x3))
% 289.98/290.24 Found ((eq_ref0 x30) P) as proof of ((P x30)->(P x3))
% 289.98/290.24 Found (((eq_ref fofType) x30) P) as proof of ((P x30)->(P x3))
% 289.98/290.24 Found (((eq_ref fofType) x30) P) as proof of ((P x30)->(P x3))
% 289.98/290.24 Found (fun (P:(fofType->Prop))=> (((eq_ref fofType) x30) P)) as proof of ((P x30)->(P x3))
% 289.98/290.24 Found x4:(P x30)
% 289.98/290.24 Instantiate: x3:=x30:fofType
% 289.98/290.24 Found (fun (x4:(P x30))=> x4) as proof of (P x3)
% 289.98/290.24 Found (fun (P:(fofType->Prop)) (x4:(P x30))=> x4) as proof of ((P x30)->(P x3))
% 289.98/290.24 Found eq_ref00:=(eq_ref0 x30):(((eq fofType) x30) x30)
% 289.98/290.24 Found (eq_ref0 x30) as proof of (forall (P:(fofType->Prop)), ((P x30)->(P x3)))
% 289.98/290.24 Found ((eq_ref fofType) x30) as proof of (forall (P:(fofType->Prop)), ((P x30)->(P x3)))
% 289.98/290.24 Found ((eq_ref fofType) x30) as proof of (forall (P:(fofType->Prop)), ((P x30)->(P x3)))
% 289.98/290.24 Found ((eq_ref fofType) x30) as proof of (forall (P:(fofType->Prop)), ((P x30)->(P x3)))
% 289.98/290.24 Found eq_ref00:=(eq_ref0 x3):(((eq fofType) x3) x3)
% 289.98/290.24 Found (eq_ref0 x3) as proof of (((eq fofType) x3) x30)
% 289.98/290.24 Found ((eq_ref fofType) x3) as proof of (((eq fofType) x3) x30)
% 289.98/290.24 Found ((eq_ref fofType) x3) as proof of (((eq fofType) x3) x30)
% 289.98/290.24 Found ((eq_ref fofType) x3) as proof of (((eq fofType) x3) x30)
% 289.98/290.24 Found (eq_sym000 ((eq_ref fofType) x3)) as proof of (((eq fofType) x30) x3)
% 289.98/290.24 Found ((eq_sym00 x30) ((eq_ref fofType) x3)) as proof of (((eq fofType) x30) x3)
% 289.98/290.24 Found (((eq_sym0 x3) x30) ((eq_ref fofType) x3)) as proof of (((eq fofType) x30) x3)
% 289.98/290.24 Found ((((eq_sym fofType) x3) x30) ((eq_ref fofType) x3)) as proof of (((eq fofType) x30) x3)
% 289.98/290.24 Found eq_ref000:=(eq_ref00 (ex fofType)):(((ex fofType) ((unique fofType) (x0 Xx)))->((ex fofType) ((unique fofType) (x0 Xx))))
% 289.98/290.24 Found (eq_ref00 (ex fofType)) as proof of (P ((unique fofType) (x0 Xx)))
% 289.98/290.24 Found ((eq_ref0 ((unique fofType) (x0 Xx))) (ex fofType)) as proof of (P ((unique fofType) (x0 Xx)))
% 289.98/290.24 Found (((eq_ref (fofType->Prop)) ((unique fofType) (x0 Xx))) (ex fofType)) as proof of (P ((unique fofType) (x0 Xx)))
% 289.98/290.24 Found (((eq_ref (fofType->Prop)) ((unique fofType) (x0 Xx))) (ex fofType)) as proof of (P ((unique fofType) (x0 Xx)))
% 289.98/290.24 Found eq_ref000:=(eq_ref00 (ex fofType)):(((ex fofType) ((unique fofType) (x0 Xx)))->((ex fofType) ((unique fofType) (x0 Xx))))
% 289.98/290.24 Found (eq_ref00 (ex fofType)) as proof of (P ((unique fofType) (x0 Xx)))
% 289.98/290.24 Found ((eq_ref0 ((unique fofType) (x0 Xx))) (ex fofType)) as proof of (P ((unique fofType) (x0 Xx)))
% 289.98/290.24 Found (((eq_ref (fofType->Prop)) ((unique fofType) (x0 Xx))) (ex fofType)) as proof of (P ((unique fofType) (x0 Xx)))
% 289.98/290.24 Found (((eq_ref (fofType->Prop)) ((unique fofType) (x0 Xx))) (ex fofType)) as proof of (P ((unique fofType) (x0 Xx)))
% 289.98/290.24 Found eq_ref00:=(eq_ref0 (cS x5)):(((eq fofType) (cS x5)) (cS x5))
% 289.98/290.24 Found (eq_ref0 (cS x5)) as proof of (((eq fofType) (cS x5)) (cS Y))
% 289.98/290.24 Found ((eq_ref fofType) (cS x5)) as proof of (((eq fofType) (cS x5)) (cS Y))
% 289.98/290.24 Found ((eq_ref fofType) (cS x5)) as proof of (((eq fofType) (cS x5)) (cS Y))
% 289.98/290.24 Found ((eq_ref fofType) (cS x5)) as proof of (((eq fofType) (cS x5)) (cS Y))
% 289.98/290.24 Found (x400 ((eq_ref fofType) (cS x5))) as proof of (((eq fofType) x5) Y)
% 289.98/290.24 Found ((x40 Y) ((eq_ref fofType) (cS x5))) as proof of (((eq fofType) x5) Y)
% 289.98/290.24 Found (((x4 x5) Y) ((eq_ref fofType) (cS x5))) as proof of (((eq fofType) x5) Y)
% 289.98/290.24 Found (fun (x00:((x0 Xx) Y))=> (((x4 x5) Y) ((eq_ref fofType) (cS x5)))) as proof of (((eq fofType) x5) Y)
% 293.99/294.20 Found eq_ref000:=(eq_ref00 (ex fofType)):(((ex fofType) ((unique fofType) (x2 Xx)))->((ex fofType) ((unique fofType) (x2 Xx))))
% 293.99/294.20 Found (eq_ref00 (ex fofType)) as proof of (P ((unique fofType) (x2 Xx)))
% 293.99/294.20 Found ((eq_ref0 ((unique fofType) (x2 Xx))) (ex fofType)) as proof of (P ((unique fofType) (x2 Xx)))
% 293.99/294.20 Found (((eq_ref (fofType->Prop)) ((unique fofType) (x2 Xx))) (ex fofType)) as proof of (P ((unique fofType) (x2 Xx)))
% 293.99/294.20 Found (((eq_ref (fofType->Prop)) ((unique fofType) (x2 Xx))) (ex fofType)) as proof of (P ((unique fofType) (x2 Xx)))
% 293.99/294.20 Found eq_ref000:=(eq_ref00 (ex fofType)):(((ex fofType) ((unique fofType) (x2 Xx)))->((ex fofType) ((unique fofType) (x2 Xx))))
% 293.99/294.20 Found (eq_ref00 (ex fofType)) as proof of (P ((unique fofType) (x2 Xx)))
% 293.99/294.20 Found ((eq_ref0 ((unique fofType) (x2 Xx))) (ex fofType)) as proof of (P ((unique fofType) (x2 Xx)))
% 293.99/294.20 Found (((eq_ref (fofType->Prop)) ((unique fofType) (x2 Xx))) (ex fofType)) as proof of (P ((unique fofType) (x2 Xx)))
% 293.99/294.20 Found (((eq_ref (fofType->Prop)) ((unique fofType) (x2 Xx))) (ex fofType)) as proof of (P ((unique fofType) (x2 Xx)))
% 293.99/294.20 Found eq_ref00:=(eq_ref0 (cS x30)):(((eq fofType) (cS x30)) (cS x30))
% 293.99/294.20 Found (eq_ref0 (cS x30)) as proof of (((eq fofType) (cS x30)) (cS x3))
% 293.99/294.20 Found ((eq_ref fofType) (cS x30)) as proof of (((eq fofType) (cS x30)) (cS x3))
% 293.99/294.20 Found ((eq_ref fofType) (cS x30)) as proof of (((eq fofType) (cS x30)) (cS x3))
% 293.99/294.20 Found ((eq_ref fofType) (cS x30)) as proof of (((eq fofType) (cS x30)) (cS x3))
% 293.99/294.20 Found (x500 ((eq_ref fofType) (cS x30))) as proof of (((eq fofType) x30) x3)
% 293.99/294.20 Found ((x50 x3) ((eq_ref fofType) (cS x30))) as proof of (((eq fofType) x30) x3)
% 293.99/294.20 Found (((x5 x30) x3) ((eq_ref fofType) (cS x30))) as proof of (((eq fofType) x30) x3)
% 293.99/294.20 Found (fun (x20:(forall (Xn:fofType), (not (((eq fofType) (cS Xn)) c0))))=> (((x5 x30) x3) ((eq_ref fofType) (cS x30)))) as proof of (((eq fofType) x30) x3)
% 293.99/294.20 Found eq_ref00:=(eq_ref0 x30):(((eq fofType) x30) x30)
% 293.99/294.20 Found (eq_ref0 x30) as proof of (((eq fofType) x30) x3)
% 293.99/294.20 Found ((eq_ref fofType) x30) as proof of (((eq fofType) x30) x3)
% 293.99/294.20 Found ((eq_ref fofType) x30) as proof of (((eq fofType) x30) x3)
% 293.99/294.20 Found eq_sym0:=(eq_sym Prop):(forall (a:Prop) (b:Prop), ((((eq Prop) a) b)->(((eq Prop) b) a)))
% 293.99/294.20 Instantiate: b:=(forall (a:Prop) (b:Prop), ((((eq Prop) a) b)->(((eq Prop) b) a))):Prop
% 293.99/294.20 Found eq_sym0 as proof of b
% 293.99/294.20 Found x6:(P x1)
% 293.99/294.20 Instantiate: x1:=Y:fofType
% 293.99/294.20 Found (fun (x6:(P x1))=> x6) as proof of (P Y)
% 293.99/294.20 Found (fun (P:(fofType->Prop)) (x6:(P x1))=> x6) as proof of ((P x1)->(P Y))
% 293.99/294.20 Found (fun (P:(fofType->Prop)) (x6:(P x1))=> x6) as proof of (((eq fofType) x1) Y)
% 293.99/294.20 Found (fun (x00:((x0 Xx) Y)) (P:(fofType->Prop)) (x6:(P x1))=> x6) as proof of (((eq fofType) x1) Y)
% 293.99/294.20 Found eq_ref000:=(eq_ref00 P):((P x1)->(P x1))
% 293.99/294.20 Found (eq_ref00 P) as proof of ((P x1)->(P Y))
% 293.99/294.20 Found ((eq_ref0 x1) P) as proof of ((P x1)->(P Y))
% 293.99/294.20 Found (((eq_ref fofType) x1) P) as proof of ((P x1)->(P Y))
% 293.99/294.20 Found (((eq_ref fofType) x1) P) as proof of ((P x1)->(P Y))
% 293.99/294.20 Found (fun (P:(fofType->Prop))=> (((eq_ref fofType) x1) P)) as proof of ((P x1)->(P Y))
% 293.99/294.20 Found (fun (P:(fofType->Prop))=> (((eq_ref fofType) x1) P)) as proof of (((eq fofType) x1) Y)
% 293.99/294.20 Found (fun (x00:((x0 Xx) Y)) (P:(fofType->Prop))=> (((eq_ref fofType) x1) P)) as proof of (((eq fofType) x1) Y)
% 293.99/294.20 Found x6:(P x3)
% 293.99/294.20 Instantiate: x3:=Y:fofType
% 293.99/294.20 Found (fun (x6:(P x3))=> x6) as proof of (P Y)
% 293.99/294.20 Found (fun (P:(fofType->Prop)) (x6:(P x3))=> x6) as proof of ((P x3)->(P Y))
% 293.99/294.20 Found (fun (P:(fofType->Prop)) (x6:(P x3))=> x6) as proof of (((eq fofType) x3) Y)
% 293.99/294.20 Found (fun (x00:((x0 Xx) Y)) (P:(fofType->Prop)) (x6:(P x3))=> x6) as proof of (((eq fofType) x3) Y)
% 293.99/294.20 Found eq_ref000:=(eq_ref00 P):((P x3)->(P x3))
% 293.99/294.20 Found (eq_ref00 P) as proof of ((P x3)->(P Y))
% 293.99/294.20 Found ((eq_ref0 x3) P) as proof of ((P x3)->(P Y))
% 293.99/294.20 Found (((eq_ref fofType) x3) P) as proof of ((P x3)->(P Y))
% 293.99/294.20 Found (((eq_ref fofType) x3) P) as proof of ((P x3)->(P Y))
% 293.99/294.20 Found (fun (P:(fofType->Prop))=> (((eq_ref fofType) x3) P)) as proof of ((P x3)->(P Y))
% 293.99/294.20 Found (fun (P:(fofType->Prop))=> (((eq_ref fofType) x3) P)) as proof of (((eq fofType) x3) Y)
% 295.85/296.05 Found (fun (x00:((x0 Xx) Y)) (P:(fofType->Prop))=> (((eq_ref fofType) x3) P)) as proof of (((eq fofType) x3) Y)
% 295.85/296.05 Found eq_ref00:=(eq_ref0 x3):(((eq fofType) x3) x3)
% 295.85/296.05 Found (eq_ref0 x3) as proof of (((eq fofType) x3) x30)
% 295.85/296.05 Found ((eq_ref fofType) x3) as proof of (((eq fofType) x3) x30)
% 295.85/296.05 Found ((eq_ref fofType) x3) as proof of (((eq fofType) x3) x30)
% 295.85/296.05 Found ((eq_ref fofType) x3) as proof of (((eq fofType) x3) x30)
% 295.85/296.05 Found (eq_sym000 ((eq_ref fofType) x3)) as proof of (((eq fofType) x30) x3)
% 295.85/296.05 Found ((eq_sym00 x30) ((eq_ref fofType) x3)) as proof of (((eq fofType) x30) x3)
% 295.85/296.05 Found (((eq_sym0 x3) x30) ((eq_ref fofType) x3)) as proof of (((eq fofType) x30) x3)
% 295.85/296.05 Found ((((eq_sym fofType) x3) x30) ((eq_ref fofType) x3)) as proof of (((eq fofType) x30) x3)
% 295.85/296.05 Found (fun (x50:(forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy))))=> ((((eq_sym fofType) x3) x30) ((eq_ref fofType) x3))) as proof of (((eq fofType) x30) x3)
% 295.85/296.05 Found (fun (x40:cIND) (x50:(forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy))))=> ((((eq_sym fofType) x3) x30) ((eq_ref fofType) x3))) as proof of ((forall (Xx0:fofType) (Xy:fofType), ((((eq fofType) (cS Xx0)) (cS Xy))->(((eq fofType) Xx0) Xy)))->(((eq fofType) x30) x3))
% 295.85/296.05 Found eq_ref00:=(eq_ref0 ((and ((and ((x2 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x2 Xx) Xy)->((x2 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x2 Xx) X)) (forall (Y:fofType), (((x2 Xx) Y)->(((eq fofType) X) Y))))))))):(((eq Prop) ((and ((and ((x2 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x2 Xx) Xy)->((x2 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x2 Xx) X)) (forall (Y:fofType), (((x2 Xx) Y)->(((eq fofType) X) Y))))))))) ((and ((and ((x2 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x2 Xx) Xy)->((x2 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x2 Xx) X)) (forall (Y:fofType), (((x2 Xx) Y)->(((eq fofType) X) Y)))))))))
% 295.85/296.05 Found (eq_ref0 ((and ((and ((x2 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x2 Xx) Xy)->((x2 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x2 Xx) X)) (forall (Y:fofType), (((x2 Xx) Y)->(((eq fofType) X) Y))))))))) as proof of (((eq Prop) ((and ((and ((x2 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x2 Xx) Xy)->((x2 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x2 Xx) X)) (forall (Y:fofType), (((x2 Xx) Y)->(((eq fofType) X) Y))))))))) b)
% 295.85/296.05 Found ((eq_ref Prop) ((and ((and ((x2 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x2 Xx) Xy)->((x2 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x2 Xx) X)) (forall (Y:fofType), (((x2 Xx) Y)->(((eq fofType) X) Y))))))))) as proof of (((eq Prop) ((and ((and ((x2 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x2 Xx) Xy)->((x2 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x2 Xx) X)) (forall (Y:fofType), (((x2 Xx) Y)->(((eq fofType) X) Y))))))))) b)
% 295.85/296.05 Found ((eq_ref Prop) ((and ((and ((x2 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x2 Xx) Xy)->((x2 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x2 Xx) X)) (forall (Y:fofType), (((x2 Xx) Y)->(((eq fofType) X) Y))))))))) as proof of (((eq Prop) ((and ((and ((x2 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x2 Xx) Xy)->((x2 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x2 Xx) X)) (forall (Y:fofType), (((x2 Xx) Y)->(((eq fofType) X) Y))))))))) b)
% 295.85/296.05 Found ((eq_ref Prop) ((and ((and ((x2 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x2 Xx) Xy)->((x2 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fofType) (fun (X:fofType)=> ((and ((x2 Xx) X)) (forall (Y:fofType), (((x2 Xx) Y)->(((eq fofType) X) Y))))))))) as proof of (((eq Prop) ((and ((and ((x2 c0) c0)) (forall (Xx:fofType) (Xy:fofType), (((x2 Xx) Xy)->((x2 (cS Xx)) (cS (cS Xy))))))) (forall (Xx:fofType), ((ex fo
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