TSTP Solution File: SEU978^5 by cocATP---0.2.0
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- Process Solution
%------------------------------------------------------------------------------
% File : cocATP---0.2.0
% Problem : SEU978^5 : TPTP v6.1.0. Released v4.0.0.
% Transfm : none
% Format : tptp:raw
% Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% Computer : n091.star.cs.uiowa.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2609 0 2.40GHz
% Memory : 32286.75MB
% OS : Linux 2.6.32-431.20.3.el6.x86_64
% CPULimit : 300s
% DateTime : Thu Jul 17 13:33:27 EDT 2014
% Result : Timeout 300.11s
% Output : None
% Verified :
% SZS Type : None (Parsing solution fails)
% Syntax : Number of formulae : 0
% Comments :
%------------------------------------------------------------------------------
%----NO SOLUTION OUTPUT BY SYSTEM
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% % Problem : SEU978^5 : TPTP v6.1.0. Released v4.0.0.
% % Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n091.star.cs.uiowa.edu
% % Model : x86_64 x86_64
% % CPU : Intel(R) Xeon(R) CPU E5-2609 0 @ 2.40GHz
% % Memory : 32286.75MB
% % OS : Linux 2.6.32-431.20.3.el6.x86_64
% % CPULimit : 300
% % DateTime : Thu Jul 17 11:47:31 CDT 2014
% % CPUTime : 300.11
% Python 2.7.5
% Using paths ['/home/cristobal/cocATP/CASC/TPTP/', '/export/starexec/sandbox/benchmark/', '/export/starexec/sandbox/benchmark/']
% FOF formula (<kernel.Constant object at 0x23d6ab8>, <kernel.Type object at 0x23d6cf8>) of role type named a_type
% Using role type
% Declaring a:Type
% FOF formula (<kernel.Constant object at 0x25d4ab8>, <kernel.DependentProduct object at 0x23d6f80>) of role type named cR
% Using role type
% Declaring cR:(a->a)
% FOF formula (<kernel.Constant object at 0x23d6d40>, <kernel.DependentProduct object at 0x23d6950>) of role type named cP
% Using role type
% Declaring cP:(a->(a->a))
% FOF formula (<kernel.Constant object at 0x23d69e0>, <kernel.DependentProduct object at 0x23d66c8>) of role type named cL
% Using role type
% Declaring cL:(a->a)
% FOF formula (<kernel.Constant object at 0x23d6f80>, <kernel.Constant object at 0x23d66c8>) of role type named cZ
% Using role type
% Declaring cZ:a
% FOF formula (((and ((and ((and ((and ((and (((eq a) (cL cZ)) cZ)) (((eq a) (cR cZ)) cZ))) (forall (Xx:a) (Xy:a), (((eq a) (cL ((cP Xx) Xy))) Xx)))) (forall (Xx:a) (Xy:a), (((eq a) (cR ((cP Xx) Xy))) Xy)))) (forall (Xt:a), ((iff (not (((eq a) Xt) cZ))) (((eq a) Xt) ((cP (cL Xt)) (cR Xt))))))) (forall (X:(a->Prop)), ((forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((iff (((eq a) Xt) cZ)) (((eq a) Xu) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu))))))->(forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->(((eq a) Xt) Xu))))))->(forall (Xx:a) (Xy:a) (Xu:a) (Xv:a), (((and ((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xx) Xy))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0)))))))))) ((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xu) Xv))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0))))))))))->((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP ((cP Xx) Xu)) ((cP Xy) Xv)))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0)))))))))))) of role conjecture named cPU_LEM2D_pme
% Conjecture to prove = (((and ((and ((and ((and ((and (((eq a) (cL cZ)) cZ)) (((eq a) (cR cZ)) cZ))) (forall (Xx:a) (Xy:a), (((eq a) (cL ((cP Xx) Xy))) Xx)))) (forall (Xx:a) (Xy:a), (((eq a) (cR ((cP Xx) Xy))) Xy)))) (forall (Xt:a), ((iff (not (((eq a) Xt) cZ))) (((eq a) Xt) ((cP (cL Xt)) (cR Xt))))))) (forall (X:(a->Prop)), ((forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((iff (((eq a) Xt) cZ)) (((eq a) Xu) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu))))))->(forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->(((eq a) Xt) Xu))))))->(forall (Xx:a) (Xy:a) (Xu:a) (Xv:a), (((and ((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xx) Xy))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0)))))))))) ((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xu) Xv))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0))))))))))->((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP ((cP Xx) Xu)) ((cP Xy) Xv)))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0)))))))))))):Prop
% We need to prove ['(((and ((and ((and ((and ((and (((eq a) (cL cZ)) cZ)) (((eq a) (cR cZ)) cZ))) (forall (Xx:a) (Xy:a), (((eq a) (cL ((cP Xx) Xy))) Xx)))) (forall (Xx:a) (Xy:a), (((eq a) (cR ((cP Xx) Xy))) Xy)))) (forall (Xt:a), ((iff (not (((eq a) Xt) cZ))) (((eq a) Xt) ((cP (cL Xt)) (cR Xt))))))) (forall (X:(a->Prop)), ((forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((iff (((eq a) Xt) cZ)) (((eq a) Xu) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu))))))->(forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->(((eq a) Xt) Xu))))))->(forall (Xx:a) (Xy:a) (Xu:a) (Xv:a), (((and ((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xx) Xy))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0)))))))))) ((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xu) Xv))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0))))))))))->((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP ((cP Xx) Xu)) ((cP Xy) Xv)))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0))))))))))))']
% Parameter a:Type.
% Parameter cR:(a->a).
% Parameter cP:(a->(a->a)).
% Parameter cL:(a->a).
% Parameter cZ:a.
% Trying to prove (((and ((and ((and ((and ((and (((eq a) (cL cZ)) cZ)) (((eq a) (cR cZ)) cZ))) (forall (Xx:a) (Xy:a), (((eq a) (cL ((cP Xx) Xy))) Xx)))) (forall (Xx:a) (Xy:a), (((eq a) (cR ((cP Xx) Xy))) Xy)))) (forall (Xt:a), ((iff (not (((eq a) Xt) cZ))) (((eq a) Xt) ((cP (cL Xt)) (cR Xt))))))) (forall (X:(a->Prop)), ((forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((iff (((eq a) Xt) cZ)) (((eq a) Xu) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu))))))->(forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->(((eq a) Xt) Xu))))))->(forall (Xx:a) (Xy:a) (Xu:a) (Xv:a), (((and ((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xx) Xy))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0)))))))))) ((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xu) Xv))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0))))))))))->((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP ((cP Xx) Xu)) ((cP Xy) Xv)))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0))))))))))))
% Found eq_ref000:=(eq_ref00 (ex (a->Prop))):(((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xu) Xv))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0)))))))))->((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xu) Xv))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0))))))))))
% Found (eq_ref00 (ex (a->Prop))) as proof of (P (fun (X:(a->Prop))=> ((and (X ((cP Xu) Xv))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0)))))))))
% Found ((eq_ref0 (fun (X:(a->Prop))=> ((and (X ((cP Xu) Xv))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0))))))))) (ex (a->Prop))) as proof of (P (fun (X:(a->Prop))=> ((and (X ((cP Xu) Xv))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0)))))))))
% Found (((eq_ref ((a->Prop)->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xu) Xv))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0))))))))) (ex (a->Prop))) as proof of (P (fun (X:(a->Prop))=> ((and (X ((cP Xu) Xv))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0)))))))))
% Found (((eq_ref ((a->Prop)->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xu) Xv))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0))))))))) (ex (a->Prop))) as proof of (P (fun (X:(a->Prop))=> ((and (X ((cP Xu) Xv))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0)))))))))
% Found eq_ref000:=(eq_ref00 (ex (a->Prop))):(((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xu) Xv))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0)))))))))->((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xu) Xv))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0))))))))))
% Found (eq_ref00 (ex (a->Prop))) as proof of (P (fun (X:(a->Prop))=> ((and (X ((cP Xu) Xv))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0)))))))))
% Found ((eq_ref0 (fun (X:(a->Prop))=> ((and (X ((cP Xu) Xv))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0))))))))) (ex (a->Prop))) as proof of (P (fun (X:(a->Prop))=> ((and (X ((cP Xu) Xv))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0)))))))))
% Found (((eq_ref ((a->Prop)->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xu) Xv))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0))))))))) (ex (a->Prop))) as proof of (P (fun (X:(a->Prop))=> ((and (X ((cP Xu) Xv))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0)))))))))
% Found (((eq_ref ((a->Prop)->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xu) Xv))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0))))))))) (ex (a->Prop))) as proof of (P (fun (X:(a->Prop))=> ((and (X ((cP Xu) Xv))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0)))))))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 (ex (a->Prop))):(((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xu) Xv))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0)))))))))->((ex (a->Prop)) (fun (x:(a->Prop))=> ((and (x ((cP Xu) Xv))) (forall (Xt:a) (Xu0:a), ((x ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (x ((cP (cL Xt)) (cL Xu0))))) (x ((cP (cR Xt)) (cR Xu0))))))))))
% Found (eta_expansion_dep000 (ex (a->Prop))) as proof of (P (fun (X:(a->Prop))=> ((and (X ((cP Xu) Xv))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0)))))))))
% Found ((eta_expansion_dep00 (fun (X:(a->Prop))=> ((and (X ((cP Xu) Xv))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0))))))))) (ex (a->Prop))) as proof of (P (fun (X:(a->Prop))=> ((and (X ((cP Xu) Xv))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0)))))))))
% Found (((eta_expansion_dep0 (fun (x3:(a->Prop))=> Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xu) Xv))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0))))))))) (ex (a->Prop))) as proof of (P (fun (X:(a->Prop))=> ((and (X ((cP Xu) Xv))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0)))))))))
% Found ((((eta_expansion_dep (a->Prop)) (fun (x3:(a->Prop))=> Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xu) Xv))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0))))))))) (ex (a->Prop))) as proof of (P (fun (X:(a->Prop))=> ((and (X ((cP Xu) Xv))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0)))))))))
% Found ((((eta_expansion_dep (a->Prop)) (fun (x3:(a->Prop))=> Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xu) Xv))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0))))))))) (ex (a->Prop))) as proof of (P (fun (X:(a->Prop))=> ((and (X ((cP Xu) Xv))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0)))))))))
% Found eq_ref000:=(eq_ref00 (ex (a->Prop))):(((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xu) Xv))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0)))))))))->((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xu) Xv))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0))))))))))
% Found (eq_ref00 (ex (a->Prop))) as proof of (P (fun (X:(a->Prop))=> ((and (X ((cP Xu) Xv))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0)))))))))
% Found ((eq_ref0 (fun (X:(a->Prop))=> ((and (X ((cP Xu) Xv))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0))))))))) (ex (a->Prop))) as proof of (P (fun (X:(a->Prop))=> ((and (X ((cP Xu) Xv))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0)))))))))
% Found (((eq_ref ((a->Prop)->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xu) Xv))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0))))))))) (ex (a->Prop))) as proof of (P (fun (X:(a->Prop))=> ((and (X ((cP Xu) Xv))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0)))))))))
% Found (((eq_ref ((a->Prop)->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xu) Xv))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0))))))))) (ex (a->Prop))) as proof of (P (fun (X:(a->Prop))=> ((and (X ((cP Xu) Xv))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0)))))))))
% Found eta_expansion0000:=(eta_expansion000 (ex (a->Prop))):(((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xu) Xv))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0)))))))))->((ex (a->Prop)) (fun (x:(a->Prop))=> ((and (x ((cP Xu) Xv))) (forall (Xt:a) (Xu0:a), ((x ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (x ((cP (cL Xt)) (cL Xu0))))) (x ((cP (cR Xt)) (cR Xu0))))))))))
% Found (eta_expansion000 (ex (a->Prop))) as proof of (P (fun (X:(a->Prop))=> ((and (X ((cP Xu) Xv))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0)))))))))
% Found ((eta_expansion00 (fun (X:(a->Prop))=> ((and (X ((cP Xu) Xv))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0))))))))) (ex (a->Prop))) as proof of (P (fun (X:(a->Prop))=> ((and (X ((cP Xu) Xv))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0)))))))))
% Found (((eta_expansion0 Prop) (fun (X:(a->Prop))=> ((and (X ((cP Xu) Xv))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0))))))))) (ex (a->Prop))) as proof of (P (fun (X:(a->Prop))=> ((and (X ((cP Xu) Xv))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0)))))))))
% Found ((((eta_expansion (a->Prop)) Prop) (fun (X:(a->Prop))=> ((and (X ((cP Xu) Xv))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0))))))))) (ex (a->Prop))) as proof of (P (fun (X:(a->Prop))=> ((and (X ((cP Xu) Xv))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0)))))))))
% Found ((((eta_expansion (a->Prop)) Prop) (fun (X:(a->Prop))=> ((and (X ((cP Xu) Xv))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0))))))))) (ex (a->Prop))) as proof of (P (fun (X:(a->Prop))=> ((and (X ((cP Xu) Xv))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0)))))))))
% Found eq_ref000:=(eq_ref00 (ex (a->Prop))):(((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xu) Xv))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0)))))))))->((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xu) Xv))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0))))))))))
% Found (eq_ref00 (ex (a->Prop))) as proof of (P (fun (X:(a->Prop))=> ((and (X ((cP Xu) Xv))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0)))))))))
% Found ((eq_ref0 (fun (X:(a->Prop))=> ((and (X ((cP Xu) Xv))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0))))))))) (ex (a->Prop))) as proof of (P (fun (X:(a->Prop))=> ((and (X ((cP Xu) Xv))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0)))))))))
% Found (((eq_ref ((a->Prop)->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xu) Xv))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0))))))))) (ex (a->Prop))) as proof of (P (fun (X:(a->Prop))=> ((and (X ((cP Xu) Xv))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0)))))))))
% Found (((eq_ref ((a->Prop)->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xu) Xv))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0))))))))) (ex (a->Prop))) as proof of (P (fun (X:(a->Prop))=> ((and (X ((cP Xu) Xv))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0)))))))))
% Found eq_substitution0000:=(fun (a0:a)=> ((eq_substitution000 a0) (cP ((cP Xx) Xu)))):(forall (a0:a), ((((eq a) a0) ((cP Xy) Xv))->(((eq a) ((cP ((cP Xx) Xu)) a0)) ((cP ((cP Xx) Xu)) ((cP Xy) Xv)))))
% Instantiate: x1:=(fun (x2:a)=> (forall (a0:a), ((((eq a) a0) ((cP Xy) Xv))->(((eq a) ((cP ((cP Xx) Xu)) a0)) x2)))):(a->Prop)
% Found (fun (a0:a)=> ((eq_substitution000 a0) (cP ((cP Xx) Xu)))) as proof of (x1 ((cP ((cP Xx) Xu)) ((cP Xy) Xv)))
% Found (fun (a0:a)=> (((fun (a0:a)=> ((eq_substitution00 a0) ((cP Xy) Xv))) a0) (cP ((cP Xx) Xu)))) as proof of (x1 ((cP ((cP Xx) Xu)) ((cP Xy) Xv)))
% Found (fun (a0:a)=> (((fun (a0:a)=> (((eq_substitution0 a) a0) ((cP Xy) Xv))) a0) (cP ((cP Xx) Xu)))) as proof of (x1 ((cP ((cP Xx) Xu)) ((cP Xy) Xv)))
% Found (fun (a0:a)=> (((fun (a0:a)=> ((((eq_substitution a) a) a0) ((cP Xy) Xv))) a0) (cP ((cP Xx) Xu)))) as proof of (x1 ((cP ((cP Xx) Xu)) ((cP Xy) Xv)))
% Found (fun (a0:a)=> (((fun (a0:a)=> ((((eq_substitution a) a) a0) ((cP Xy) Xv))) a0) (cP ((cP Xx) Xu)))) as proof of (x1 ((cP ((cP Xx) Xu)) ((cP Xy) Xv)))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (X:(a->Prop))=> ((and (X ((cP ((cP Xx) Xu)) ((cP Xy) Xv)))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0))))))))):(((eq ((a->Prop)->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP ((cP Xx) Xu)) ((cP Xy) Xv)))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0))))))))) (fun (x:(a->Prop))=> ((and (x ((cP ((cP Xx) Xu)) ((cP Xy) Xv)))) (forall (Xt:a) (Xu0:a), ((x ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (x ((cP (cL Xt)) (cL Xu0))))) (x ((cP (cR Xt)) (cR Xu0)))))))))
% Found (eta_expansion_dep00 (fun (X:(a->Prop))=> ((and (X ((cP ((cP Xx) Xu)) ((cP Xy) Xv)))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0))))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP ((cP Xx) Xu)) ((cP Xy) Xv)))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0))))))))) b)
% Found ((eta_expansion_dep0 (fun (x2:(a->Prop))=> Prop)) (fun (X:(a->Prop))=> ((and (X ((cP ((cP Xx) Xu)) ((cP Xy) Xv)))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0))))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP ((cP Xx) Xu)) ((cP Xy) Xv)))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0))))))))) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x2:(a->Prop))=> Prop)) (fun (X:(a->Prop))=> ((and (X ((cP ((cP Xx) Xu)) ((cP Xy) Xv)))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0))))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP ((cP Xx) Xu)) ((cP Xy) Xv)))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0))))))))) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x2:(a->Prop))=> Prop)) (fun (X:(a->Prop))=> ((and (X ((cP ((cP Xx) Xu)) ((cP Xy) Xv)))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0))))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP ((cP Xx) Xu)) ((cP Xy) Xv)))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0))))))))) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x2:(a->Prop))=> Prop)) (fun (X:(a->Prop))=> ((and (X ((cP ((cP Xx) Xu)) ((cP Xy) Xv)))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0))))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP ((cP Xx) Xu)) ((cP Xy) Xv)))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0))))))))) b)
% Found eta_expansion0000:=(eta_expansion000 (ex (a->Prop))):(((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xu) Xv))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0)))))))))->((ex (a->Prop)) (fun (x:(a->Prop))=> ((and (x ((cP Xu) Xv))) (forall (Xt:a) (Xu0:a), ((x ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (x ((cP (cL Xt)) (cL Xu0))))) (x ((cP (cR Xt)) (cR Xu0))))))))))
% Found (eta_expansion000 (ex (a->Prop))) as proof of (P (fun (X:(a->Prop))=> ((and (X ((cP Xu) Xv))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0)))))))))
% Found ((eta_expansion00 (fun (X:(a->Prop))=> ((and (X ((cP Xu) Xv))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0))))))))) (ex (a->Prop))) as proof of (P (fun (X:(a->Prop))=> ((and (X ((cP Xu) Xv))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0)))))))))
% Found (((eta_expansion0 Prop) (fun (X:(a->Prop))=> ((and (X ((cP Xu) Xv))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0))))))))) (ex (a->Prop))) as proof of (P (fun (X:(a->Prop))=> ((and (X ((cP Xu) Xv))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0)))))))))
% Found ((((eta_expansion (a->Prop)) Prop) (fun (X:(a->Prop))=> ((and (X ((cP Xu) Xv))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0))))))))) (ex (a->Prop))) as proof of (P (fun (X:(a->Prop))=> ((and (X ((cP Xu) Xv))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0)))))))))
% Found ((((eta_expansion (a->Prop)) Prop) (fun (X:(a->Prop))=> ((and (X ((cP Xu) Xv))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0))))))))) (ex (a->Prop))) as proof of (P (fun (X:(a->Prop))=> ((and (X ((cP Xu) Xv))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0)))))))))
% Found x6:(forall (Xt:a) (Xu0:a), ((x3 ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (x3 ((cP (cL Xt)) (cL Xu0))))) (x3 ((cP (cR Xt)) (cR Xu0))))))
% Instantiate: x7:=x3:(a->Prop)
% Found x6 as proof of (forall (Xt:a) (Xu0:a), ((x7 ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (x7 ((cP (cL Xt)) (cL Xu0))))) (x7 ((cP (cR Xt)) (cR Xu0))))))
% Found x6:(forall (Xt:a) (Xu0:a), ((x3 ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (x3 ((cP (cL Xt)) (cL Xu0))))) (x3 ((cP (cR Xt)) (cR Xu0))))))
% Instantiate: x7:=x3:(a->Prop)
% Found x6 as proof of (forall (Xt:a) (Xu0:a), ((x7 ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (x7 ((cP (cL Xt)) (cL Xu0))))) (x7 ((cP (cR Xt)) (cR Xu0))))))
% Found eta_expansion0000:=(eta_expansion000 (ex (a->Prop))):(((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xu) Xv))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0)))))))))->((ex (a->Prop)) (fun (x:(a->Prop))=> ((and (x ((cP Xu) Xv))) (forall (Xt:a) (Xu0:a), ((x ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (x ((cP (cL Xt)) (cL Xu0))))) (x ((cP (cR Xt)) (cR Xu0))))))))))
% Found (eta_expansion000 (ex (a->Prop))) as proof of (P (fun (X:(a->Prop))=> ((and (X ((cP Xu) Xv))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0)))))))))
% Found ((eta_expansion00 (fun (X:(a->Prop))=> ((and (X ((cP Xu) Xv))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0))))))))) (ex (a->Prop))) as proof of (P (fun (X:(a->Prop))=> ((and (X ((cP Xu) Xv))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0)))))))))
% Found (((eta_expansion0 Prop) (fun (X:(a->Prop))=> ((and (X ((cP Xu) Xv))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0))))))))) (ex (a->Prop))) as proof of (P (fun (X:(a->Prop))=> ((and (X ((cP Xu) Xv))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0)))))))))
% Found ((((eta_expansion (a->Prop)) Prop) (fun (X:(a->Prop))=> ((and (X ((cP Xu) Xv))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0))))))))) (ex (a->Prop))) as proof of (P (fun (X:(a->Prop))=> ((and (X ((cP Xu) Xv))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0)))))))))
% Found ((((eta_expansion (a->Prop)) Prop) (fun (X:(a->Prop))=> ((and (X ((cP Xu) Xv))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0))))))))) (ex (a->Prop))) as proof of (P (fun (X:(a->Prop))=> ((and (X ((cP Xu) Xv))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0)))))))))
% Found eta_expansion0000:=(eta_expansion000 (ex (a->Prop))):(((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xu) Xv))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0)))))))))->((ex (a->Prop)) (fun (x:(a->Prop))=> ((and (x ((cP Xu) Xv))) (forall (Xt:a) (Xu0:a), ((x ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (x ((cP (cL Xt)) (cL Xu0))))) (x ((cP (cR Xt)) (cR Xu0))))))))))
% Found (eta_expansion000 (ex (a->Prop))) as proof of (P (fun (X:(a->Prop))=> ((and (X ((cP Xu) Xv))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0)))))))))
% Found ((eta_expansion00 (fun (X:(a->Prop))=> ((and (X ((cP Xu) Xv))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0))))))))) (ex (a->Prop))) as proof of (P (fun (X:(a->Prop))=> ((and (X ((cP Xu) Xv))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0)))))))))
% Found (((eta_expansion0 Prop) (fun (X:(a->Prop))=> ((and (X ((cP Xu) Xv))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0))))))))) (ex (a->Prop))) as proof of (P (fun (X:(a->Prop))=> ((and (X ((cP Xu) Xv))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0)))))))))
% Found ((((eta_expansion (a->Prop)) Prop) (fun (X:(a->Prop))=> ((and (X ((cP Xu) Xv))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0))))))))) (ex (a->Prop))) as proof of (P (fun (X:(a->Prop))=> ((and (X ((cP Xu) Xv))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0)))))))))
% Found ((((eta_expansion (a->Prop)) Prop) (fun (X:(a->Prop))=> ((and (X ((cP Xu) Xv))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0))))))))) (ex (a->Prop))) as proof of (P (fun (X:(a->Prop))=> ((and (X ((cP Xu) Xv))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0)))))))))
% Found x7:(forall (Xt:a) (Xu0:a), ((x4 ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (x4 ((cP (cL Xt)) (cL Xu0))))) (x4 ((cP (cR Xt)) (cR Xu0))))))
% Instantiate: x1:=x4:(a->Prop)
% Found x7 as proof of (forall (Xt:a) (Xu0:a), ((x1 ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (x1 ((cP (cL Xt)) (cL Xu0))))) (x1 ((cP (cR Xt)) (cR Xu0))))))
% Found x7:(forall (Xt:a) (Xu0:a), ((x4 ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (x4 ((cP (cL Xt)) (cL Xu0))))) (x4 ((cP (cR Xt)) (cR Xu0))))))
% Instantiate: x3:=x4:(a->Prop)
% Found x7 as proof of (forall (Xt:a) (Xu0:a), ((x3 ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (x3 ((cP (cL Xt)) (cL Xu0))))) (x3 ((cP (cR Xt)) (cR Xu0))))))
% Found x7:(forall (Xt:a) (Xu0:a), ((x3 ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (x3 ((cP (cL Xt)) (cL Xu0))))) (x3 ((cP (cR Xt)) (cR Xu0))))))
% Instantiate: x5:=x3:(a->Prop)
% Found x7 as proof of (forall (Xt:a) (Xu0:a), ((x5 ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (x5 ((cP (cL Xt)) (cL Xu0))))) (x5 ((cP (cR Xt)) (cR Xu0))))))
% Found x7:(forall (Xt:a) (Xu0:a), ((x4 ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (x4 ((cP (cL Xt)) (cL Xu0))))) (x4 ((cP (cR Xt)) (cR Xu0))))))
% Instantiate: x1:=x4:(a->Prop)
% Found x7 as proof of (forall (Xt:a) (Xu0:a), ((x1 ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (x1 ((cP (cL Xt)) (cL Xu0))))) (x1 ((cP (cR Xt)) (cR Xu0))))))
% Found x7:(forall (Xt:a) (Xu0:a), ((x4 ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (x4 ((cP (cL Xt)) (cL Xu0))))) (x4 ((cP (cR Xt)) (cR Xu0))))))
% Instantiate: x3:=x4:(a->Prop)
% Found x7 as proof of (forall (Xt:a) (Xu0:a), ((x3 ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (x3 ((cP (cL Xt)) (cL Xu0))))) (x3 ((cP (cR Xt)) (cR Xu0))))))
% Found x7:(forall (Xt:a) (Xu0:a), ((x3 ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (x3 ((cP (cL Xt)) (cL Xu0))))) (x3 ((cP (cR Xt)) (cR Xu0))))))
% Instantiate: x5:=x3:(a->Prop)
% Found x7 as proof of (forall (Xt:a) (Xu0:a), ((x5 ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (x5 ((cP (cL Xt)) (cL Xu0))))) (x5 ((cP (cR Xt)) (cR Xu0))))))
% Found x2:((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xu) Xv))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0)))))))))
% Instantiate: b:=(fun (X:(a->Prop))=> ((and (X ((cP Xu) Xv))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0)))))))):((a->Prop)->Prop)
% Found x2 as proof of (P b)
% Found eta_expansion000:=(eta_expansion00 (fun (X:(a->Prop))=> ((and (X ((cP ((cP Xx) Xu)) ((cP Xy) Xv)))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0))))))))):(((eq ((a->Prop)->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP ((cP Xx) Xu)) ((cP Xy) Xv)))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0))))))))) (fun (x:(a->Prop))=> ((and (x ((cP ((cP Xx) Xu)) ((cP Xy) Xv)))) (forall (Xt:a) (Xu0:a), ((x ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (x ((cP (cL Xt)) (cL Xu0))))) (x ((cP (cR Xt)) (cR Xu0)))))))))
% Found (eta_expansion00 (fun (X:(a->Prop))=> ((and (X ((cP ((cP Xx) Xu)) ((cP Xy) Xv)))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0))))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP ((cP Xx) Xu)) ((cP Xy) Xv)))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0))))))))) b)
% Found ((eta_expansion0 Prop) (fun (X:(a->Prop))=> ((and (X ((cP ((cP Xx) Xu)) ((cP Xy) Xv)))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0))))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP ((cP Xx) Xu)) ((cP Xy) Xv)))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0))))))))) b)
% Found (((eta_expansion (a->Prop)) Prop) (fun (X:(a->Prop))=> ((and (X ((cP ((cP Xx) Xu)) ((cP Xy) Xv)))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0))))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP ((cP Xx) Xu)) ((cP Xy) Xv)))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0))))))))) b)
% Found (((eta_expansion (a->Prop)) Prop) (fun (X:(a->Prop))=> ((and (X ((cP ((cP Xx) Xu)) ((cP Xy) Xv)))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0))))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP ((cP Xx) Xu)) ((cP Xy) Xv)))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0))))))))) b)
% Found (((eta_expansion (a->Prop)) Prop) (fun (X:(a->Prop))=> ((and (X ((cP ((cP Xx) Xu)) ((cP Xy) Xv)))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0))))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP ((cP Xx) Xu)) ((cP Xy) Xv)))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0))))))))) b)
% Found eq_ref000:=(eq_ref00 (ex (a->Prop))):(((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xu) Xv))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0)))))))))->((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xu) Xv))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0))))))))))
% Found (eq_ref00 (ex (a->Prop))) as proof of (((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xu) Xv))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0)))))))))->(P b))
% Found ((eq_ref0 (fun (X:(a->Prop))=> ((and (X ((cP Xu) Xv))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0))))))))) (ex (a->Prop))) as proof of (((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xu) Xv))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0)))))))))->(P b))
% Found (((eq_ref ((a->Prop)->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xu) Xv))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0))))))))) (ex (a->Prop))) as proof of (((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xu) Xv))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0)))))))))->(P b))
% Found (((eq_ref ((a->Prop)->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xu) Xv))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0))))))))) (ex (a->Prop))) as proof of (((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xu) Xv))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0)))))))))->(P b))
% Found (fun (x1:((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xx) Xy))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0))))))))))=> (((eq_ref ((a->Prop)->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xu) Xv))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0))))))))) (ex (a->Prop)))) as proof of (((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xu) Xv))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0)))))))))->(P b))
% Found (fun (x1:((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xx) Xy))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0))))))))))=> (((eq_ref ((a->Prop)->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xu) Xv))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0))))))))) (ex (a->Prop)))) as proof of (((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xx) Xy))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0)))))))))->(((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xu) Xv))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0)))))))))->(P b)))
% Found (and_rect00 (fun (x1:((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xx) Xy))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0))))))))))=> (((eq_ref ((a->Prop)->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xu) Xv))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0))))))))) (ex (a->Prop))))) as proof of (P b)
% Found ((and_rect0 (P b)) (fun (x1:((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xx) Xy))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0))))))))))=> (((eq_ref ((a->Prop)->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xu) Xv))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0))))))))) (ex (a->Prop))))) as proof of (P b)
% Found (((fun (P0:Type) (x1:(((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xx) Xy))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0)))))))))->(((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xu) Xv))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0)))))))))->P0)))=> (((((and_rect ((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xx) Xy))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0)))))))))) ((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xu) Xv))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0)))))))))) P0) x1) x0)) (P b)) (fun (x1:((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xx) Xy))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0))))))))))=> (((eq_ref ((a->Prop)->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xu) Xv))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0))))))))) (ex (a->Prop))))) as proof of (P b)
% Found (((fun (P0:Type) (x1:(((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xx) Xy))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0)))))))))->(((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xu) Xv))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0)))))))))->P0)))=> (((((and_rect ((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xx) Xy))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0)))))))))) ((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xu) Xv))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0)))))))))) P0) x1) x0)) (P b)) (fun (x1:((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xx) Xy))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0))))))))))=> (((eq_ref ((a->Prop)->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xu) Xv))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0))))))))) (ex (a->Prop))))) as proof of (P b)
% Found eq_substitution0000:=(fun (a0:a)=> ((eq_substitution000 a0) (cP ((cP Xx) Xu)))):(forall (a0:a), ((((eq a) a0) ((cP Xy) Xv))->(((eq a) ((cP ((cP Xx) Xu)) a0)) ((cP ((cP Xx) Xu)) ((cP Xy) Xv)))))
% Instantiate: x3:=(fun (x4:a)=> (forall (a0:a), ((((eq a) a0) ((cP Xy) Xv))->(((eq a) ((cP ((cP Xx) Xu)) a0)) x4)))):(a->Prop)
% Found (fun (a0:a)=> ((eq_substitution000 a0) (cP ((cP Xx) Xu)))) as proof of (x3 ((cP ((cP Xx) Xu)) ((cP Xy) Xv)))
% Found (fun (a0:a)=> (((fun (a0:a)=> ((eq_substitution00 a0) ((cP Xy) Xv))) a0) (cP ((cP Xx) Xu)))) as proof of (x3 ((cP ((cP Xx) Xu)) ((cP Xy) Xv)))
% Found (fun (a0:a)=> (((fun (a0:a)=> (((eq_substitution0 a) a0) ((cP Xy) Xv))) a0) (cP ((cP Xx) Xu)))) as proof of (x3 ((cP ((cP Xx) Xu)) ((cP Xy) Xv)))
% Found (fun (a0:a)=> (((fun (a0:a)=> ((((eq_substitution a) a) a0) ((cP Xy) Xv))) a0) (cP ((cP Xx) Xu)))) as proof of (x3 ((cP ((cP Xx) Xu)) ((cP Xy) Xv)))
% Found (fun (a0:a)=> (((fun (a0:a)=> ((((eq_substitution a) a) a0) ((cP Xy) Xv))) a0) (cP ((cP Xx) Xu)))) as proof of (x3 ((cP ((cP Xx) Xu)) ((cP Xy) Xv)))
% Found x2:((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xu) Xv))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0)))))))))
% Instantiate: f:=(fun (X:(a->Prop))=> ((and (X ((cP Xu) Xv))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0)))))))):((a->Prop)->Prop)
% Found x2 as proof of (P f)
% Found x2:((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xu) Xv))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0)))))))))
% Instantiate: f:=(fun (X:(a->Prop))=> ((and (X ((cP Xu) Xv))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0)))))))):((a->Prop)->Prop)
% Found x2 as proof of (P f)
% Found eta_expansion0000:=(eta_expansion000 (ex (a->Prop))):(((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xu) Xv))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0)))))))))->((ex (a->Prop)) (fun (x:(a->Prop))=> ((and (x ((cP Xu) Xv))) (forall (Xt:a) (Xu0:a), ((x ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (x ((cP (cL Xt)) (cL Xu0))))) (x ((cP (cR Xt)) (cR Xu0))))))))))
% Found (eta_expansion000 (ex (a->Prop))) as proof of (P (fun (X:(a->Prop))=> ((and (X ((cP Xu) Xv))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0)))))))))
% Found ((eta_expansion00 (fun (X:(a->Prop))=> ((and (X ((cP Xu) Xv))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0))))))))) (ex (a->Prop))) as proof of (P (fun (X:(a->Prop))=> ((and (X ((cP Xu) Xv))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0)))))))))
% Found (((eta_expansion0 Prop) (fun (X:(a->Prop))=> ((and (X ((cP Xu) Xv))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0))))))))) (ex (a->Prop))) as proof of (P (fun (X:(a->Prop))=> ((and (X ((cP Xu) Xv))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0)))))))))
% Found ((((eta_expansion (a->Prop)) Prop) (fun (X:(a->Prop))=> ((and (X ((cP Xu) Xv))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0))))))))) (ex (a->Prop))) as proof of (P (fun (X:(a->Prop))=> ((and (X ((cP Xu) Xv))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0)))))))))
% Found ((((eta_expansion (a->Prop)) Prop) (fun (X:(a->Prop))=> ((and (X ((cP Xu) Xv))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0))))))))) (ex (a->Prop))) as proof of (P (fun (X:(a->Prop))=> ((and (X ((cP Xu) Xv))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0)))))))))
% Found eq_ref00:=(eq_ref0 b):(((eq ((a->Prop)->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq ((a->Prop)->Prop)) b) (fun (X:(a->Prop))=> ((and (X ((cP ((cP Xx) Xu)) ((cP Xy) Xv)))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0)))))))))
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) (fun (X:(a->Prop))=> ((and (X ((cP ((cP Xx) Xu)) ((cP Xy) Xv)))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0)))))))))
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) (fun (X:(a->Prop))=> ((and (X ((cP ((cP Xx) Xu)) ((cP Xy) Xv)))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0)))))))))
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) (fun (X:(a->Prop))=> ((and (X ((cP ((cP Xx) Xu)) ((cP Xy) Xv)))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0)))))))))
% Found eta_expansion000:=(eta_expansion00 (fun (X:(a->Prop))=> ((and (X ((cP Xu) Xv))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0))))))))):(((eq ((a->Prop)->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xu) Xv))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0))))))))) (fun (x:(a->Prop))=> ((and (x ((cP Xu) Xv))) (forall (Xt:a) (Xu0:a), ((x ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (x ((cP (cL Xt)) (cL Xu0))))) (x ((cP (cR Xt)) (cR Xu0)))))))))
% Found (eta_expansion00 (fun (X:(a->Prop))=> ((and (X ((cP Xu) Xv))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0))))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xu) Xv))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0))))))))) b)
% Found ((eta_expansion0 Prop) (fun (X:(a->Prop))=> ((and (X ((cP Xu) Xv))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0))))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xu) Xv))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0))))))))) b)
% Found (((eta_expansion (a->Prop)) Prop) (fun (X:(a->Prop))=> ((and (X ((cP Xu) Xv))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0))))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xu) Xv))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0))))))))) b)
% Found (((eta_expansion (a->Prop)) Prop) (fun (X:(a->Prop))=> ((and (X ((cP Xu) Xv))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0))))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xu) Xv))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0))))))))) b)
% Found (((eta_expansion (a->Prop)) Prop) (fun (X:(a->Prop))=> ((and (X ((cP Xu) Xv))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0))))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xu) Xv))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0))))))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq ((a->Prop)->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq ((a->Prop)->Prop)) b) (fun (X:(a->Prop))=> ((and (X ((cP ((cP Xx) Xu)) ((cP Xy) Xv)))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0)))))))))
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) (fun (X:(a->Prop))=> ((and (X ((cP ((cP Xx) Xu)) ((cP Xy) Xv)))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0)))))))))
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) (fun (X:(a->Prop))=> ((and (X ((cP ((cP Xx) Xu)) ((cP Xy) Xv)))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0)))))))))
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) (fun (X:(a->Prop))=> ((and (X ((cP ((cP Xx) Xu)) ((cP Xy) Xv)))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0)))))))))
% Found eta_expansion000:=(eta_expansion00 (fun (X:(a->Prop))=> ((and (X ((cP Xu) Xv))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0))))))))):(((eq ((a->Prop)->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xu) Xv))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0))))))))) (fun (x:(a->Prop))=> ((and (x ((cP Xu) Xv))) (forall (Xt:a) (Xu0:a), ((x ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (x ((cP (cL Xt)) (cL Xu0))))) (x ((cP (cR Xt)) (cR Xu0)))))))))
% Found (eta_expansion00 (fun (X:(a->Prop))=> ((and (X ((cP Xu) Xv))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0))))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xu) Xv))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0))))))))) b)
% Found ((eta_expansion0 Prop) (fun (X:(a->Prop))=> ((and (X ((cP Xu) Xv))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0))))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xu) Xv))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0))))))))) b)
% Found (((eta_expansion (a->Prop)) Prop) (fun (X:(a->Prop))=> ((and (X ((cP Xu) Xv))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0))))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xu) Xv))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0))))))))) b)
% Found (((eta_expansion (a->Prop)) Prop) (fun (X:(a->Prop))=> ((and (X ((cP Xu) Xv))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0))))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xu) Xv))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0))))))))) b)
% Found (((eta_expansion (a->Prop)) Prop) (fun (X:(a->Prop))=> ((and (X ((cP Xu) Xv))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0))))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xu) Xv))) (forall (Xt:a) (Xu0:a), ((X ((cP Xt) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu0))))) (X ((cP (cR Xt)) (cR Xu0))))))))) b)
% Found eq_substitution0000:=(fun (a0:a)=> ((eq_substitution000 a0) (cP ((cP Xx) Xu)))):(forall (a0:a), ((((eq a) a0) ((cP Xy) Xv))->(((eq a) ((cP ((cP Xx) Xu)) a0)) ((cP ((cP Xx) Xu)) ((cP Xy) Xv)))))
% Instantiate: x1:=(fun (x4:a)=> (forall (a0:a), ((((eq a) a0) ((cP Xy) Xv))->(((eq a) ((cP ((cP Xx) Xu)) a0)) x4)))):(a->Prop)
% Found (fun (a0:a)=> ((eq_substitution000 a0) (cP ((cP Xx) Xu)))) as proof of (x1 ((cP ((cP Xx) Xu)) ((cP Xy) Xv)))
% Found (fun (a0:a)=> (((fun (a0:a)=> ((eq_substitution00 a0) ((cP Xy) Xv))) a0) (cP ((cP Xx) Xu)))) as proof of (x1 ((cP ((cP Xx) Xu)) ((cP Xy) Xv)))
% Found (fun (a0:a)=> (((fun (a0:a)=> (((eq_substitution0 a) a0) ((cP Xy) Xv))) a0) (cP ((cP Xx) Xu)))) as proof of (x1 ((cP ((cP Xx) Xu)) ((cP Xy) Xv)))
% Found (fun (a0:a)=> (((
% EOF
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