TSTP Solution File: SEU975^5 by cocATP---0.2.0
View Problem
- Process Solution
%------------------------------------------------------------------------------
% File : cocATP---0.2.0
% Problem : SEU975^5 : TPTP v6.1.0. Released v4.0.0.
% Transfm : none
% Format : tptp:raw
% Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% Computer : n183.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 300.04s
% Output : None
% Verified :
% SZS Type : None (Parsing solution fails)
% Syntax : Number of formulae : 0
% Comments :
%------------------------------------------------------------------------------
%----NO SOLUTION OUTPUT BY SYSTEM
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% % Problem : SEU975^5 : TPTP v6.1.0. Released v4.0.0.
% % Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n183.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 11:47:16 CDT 2014
% % CPUTime : 300.04
% 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 0x23208c0>, <kernel.Type object at 0x2320320>) of role type named a_type
% Using role type
% Declaring a:Type
% FOF formula (<kernel.Constant object at 0x2320c20>, <kernel.DependentProduct object at 0x26fb8c0>) of role type named cR
% Using role type
% Declaring cR:(a->a)
% FOF formula (<kernel.Constant object at 0x21e8e18>, <kernel.DependentProduct object at 0x2320f38>) of role type named cP
% Using role type
% Declaring cP:(a->(a->a))
% FOF formula (<kernel.Constant object at 0x23208c0>, <kernel.DependentProduct object at 0x26fb758>) of role type named cL
% Using role type
% Declaring cL:(a->a)
% FOF formula (<kernel.Constant object at 0x2320c20>, <kernel.Constant object at 0x26fb878>) 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), (((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xx) Xy))) (forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu)))))))))->((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP (cL Xx)) (cL Xy)))) (forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu)))))))))))) of role conjecture named cPU_LEM2B_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), (((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xx) Xy))) (forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu)))))))))->((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP (cL Xx)) (cL Xy)))) (forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu)))))))))))):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), (((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xx) Xy))) (forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu)))))))))->((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP (cL Xx)) (cL Xy)))) (forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu))))))))))))']
% 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), (((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xx) Xy))) (forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu)))))))))->((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP (cL Xx)) (cL Xy)))) (forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu))))))))))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 (ex (a->Prop))):(((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xx) Xy))) (forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu)))))))))->((ex (a->Prop)) (fun (x:(a->Prop))=> ((and (x ((cP Xx) Xy))) (forall (Xt:a) (Xu:a), ((x ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (x ((cP (cL Xt)) (cL Xu))))) (x ((cP (cR Xt)) (cR Xu))))))))))
% Found (eta_expansion_dep000 (ex (a->Prop))) as proof of (P (fun (X:(a->Prop))=> ((and (X ((cP Xx) Xy))) (forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu)))))))))
% Found ((eta_expansion_dep00 (fun (X:(a->Prop))=> ((and (X ((cP Xx) Xy))) (forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu))))))))) (ex (a->Prop))) as proof of (P (fun (X:(a->Prop))=> ((and (X ((cP Xx) Xy))) (forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu)))))))))
% Found (((eta_expansion_dep0 (fun (x1:(a->Prop))=> Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xx) Xy))) (forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu))))))))) (ex (a->Prop))) as proof of (P (fun (X:(a->Prop))=> ((and (X ((cP Xx) Xy))) (forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu)))))))))
% Found ((((eta_expansion_dep (a->Prop)) (fun (x1:(a->Prop))=> Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xx) Xy))) (forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu))))))))) (ex (a->Prop))) as proof of (P (fun (X:(a->Prop))=> ((and (X ((cP Xx) Xy))) (forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu)))))))))
% Found ((((eta_expansion_dep (a->Prop)) (fun (x1:(a->Prop))=> Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xx) Xy))) (forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu))))))))) (ex (a->Prop))) as proof of (P (fun (X:(a->Prop))=> ((and (X ((cP Xx) Xy))) (forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu)))))))))
% Found eta_expansion0000:=(eta_expansion000 (ex (a->Prop))):(((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xx) Xy))) (forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu)))))))))->((ex (a->Prop)) (fun (x:(a->Prop))=> ((and (x ((cP Xx) Xy))) (forall (Xt:a) (Xu:a), ((x ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (x ((cP (cL Xt)) (cL Xu))))) (x ((cP (cR Xt)) (cR Xu))))))))))
% Found (eta_expansion000 (ex (a->Prop))) as proof of (P (fun (X:(a->Prop))=> ((and (X ((cP Xx) Xy))) (forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu)))))))))
% Found ((eta_expansion00 (fun (X:(a->Prop))=> ((and (X ((cP Xx) Xy))) (forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu))))))))) (ex (a->Prop))) as proof of (P (fun (X:(a->Prop))=> ((and (X ((cP Xx) Xy))) (forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu)))))))))
% Found (((eta_expansion0 Prop) (fun (X:(a->Prop))=> ((and (X ((cP Xx) Xy))) (forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu))))))))) (ex (a->Prop))) as proof of (P (fun (X:(a->Prop))=> ((and (X ((cP Xx) Xy))) (forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu)))))))))
% Found ((((eta_expansion (a->Prop)) Prop) (fun (X:(a->Prop))=> ((and (X ((cP Xx) Xy))) (forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu))))))))) (ex (a->Prop))) as proof of (P (fun (X:(a->Prop))=> ((and (X ((cP Xx) Xy))) (forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu)))))))))
% Found ((((eta_expansion (a->Prop)) Prop) (fun (X:(a->Prop))=> ((and (X ((cP Xx) Xy))) (forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu))))))))) (ex (a->Prop))) as proof of (P (fun (X:(a->Prop))=> ((and (X ((cP Xx) Xy))) (forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu)))))))))
% Found eq_ref000:=(eq_ref00 (ex (a->Prop))):(((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xx) Xy))) (forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu)))))))))->((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xx) Xy))) (forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu))))))))))
% Found (eq_ref00 (ex (a->Prop))) as proof of (P (fun (X:(a->Prop))=> ((and (X ((cP Xx) Xy))) (forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu)))))))))
% Found ((eq_ref0 (fun (X:(a->Prop))=> ((and (X ((cP Xx) Xy))) (forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu))))))))) (ex (a->Prop))) as proof of (P (fun (X:(a->Prop))=> ((and (X ((cP Xx) Xy))) (forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu)))))))))
% Found (((eq_ref ((a->Prop)->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xx) Xy))) (forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu))))))))) (ex (a->Prop))) as proof of (P (fun (X:(a->Prop))=> ((and (X ((cP Xx) Xy))) (forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu)))))))))
% Found (((eq_ref ((a->Prop)->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xx) Xy))) (forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu))))))))) (ex (a->Prop))) as proof of (P (fun (X:(a->Prop))=> ((and (X ((cP Xx) Xy))) (forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu)))))))))
% Found x4:(forall (Xt:a) (Xu:a), ((x1 ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (x1 ((cP (cL Xt)) (cL Xu))))) (x1 ((cP (cR Xt)) (cR Xu))))))
% Instantiate: x5:=x1:(a->Prop)
% Found x4 as proof of (forall (Xt:a) (Xu:a), ((x5 ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (x5 ((cP (cL Xt)) (cL Xu))))) (x5 ((cP (cR Xt)) (cR Xu))))))
% Found x5:(forall (Xt:a) (Xu:a), ((x2 ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (x2 ((cP (cL Xt)) (cL Xu))))) (x2 ((cP (cR Xt)) (cR Xu))))))
% Instantiate: x1:=x2:(a->Prop)
% Found x5 as proof of (forall (Xt:a) (Xu:a), ((x1 ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (x1 ((cP (cL Xt)) (cL Xu))))) (x1 ((cP (cR Xt)) (cR Xu))))))
% Found x5:(forall (Xt:a) (Xu:a), ((x1 ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (x1 ((cP (cL Xt)) (cL Xu))))) (x1 ((cP (cR Xt)) (cR Xu))))))
% Instantiate: x3:=x1:(a->Prop)
% Found x5 as proof of (forall (Xt:a) (Xu:a), ((x3 ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (x3 ((cP (cL Xt)) (cL Xu))))) (x3 ((cP (cR Xt)) (cR Xu))))))
% Found eq_substitution0000:=(fun (a0:a)=> ((eq_substitution000 a0) (cP (cL Xx)))):(forall (a0:a), ((((eq a) a0) (cL Xy))->(((eq a) ((cP (cL Xx)) a0)) ((cP (cL Xx)) (cL Xy)))))
% Instantiate: x1:=(fun (x2:a)=> (forall (a0:a), ((((eq a) a0) (cL Xy))->(((eq a) ((cP (cL Xx)) a0)) x2)))):(a->Prop)
% Found (fun (a0:a)=> ((eq_substitution000 a0) (cP (cL Xx)))) as proof of (x1 ((cP (cL Xx)) (cL Xy)))
% Found (fun (a0:a)=> (((fun (a0:a)=> ((eq_substitution00 a0) (cL Xy))) a0) (cP (cL Xx)))) as proof of (x1 ((cP (cL Xx)) (cL Xy)))
% Found (fun (a0:a)=> (((fun (a0:a)=> (((eq_substitution0 a) a0) (cL Xy))) a0) (cP (cL Xx)))) as proof of (x1 ((cP (cL Xx)) (cL Xy)))
% Found (fun (a0:a)=> (((fun (a0:a)=> ((((eq_substitution a) a) a0) (cL Xy))) a0) (cP (cL Xx)))) as proof of (x1 ((cP (cL Xx)) (cL Xy)))
% Found (fun (a0:a)=> (((fun (a0:a)=> ((((eq_substitution a) a) a0) (cL Xy))) a0) (cP (cL Xx)))) as proof of (x1 ((cP (cL Xx)) (cL Xy)))
% Found x0:((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xx) Xy))) (forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu)))))))))
% Instantiate: b:=(fun (X:(a->Prop))=> ((and (X ((cP Xx) Xy))) (forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu)))))))):((a->Prop)->Prop)
% Found x0 as proof of (P b)
% Found eq_ref00:=(eq_ref0 (fun (X:(a->Prop))=> ((and (X ((cP (cL Xx)) (cL Xy)))) (forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu))))))))):(((eq ((a->Prop)->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP (cL Xx)) (cL Xy)))) (forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu))))))))) (fun (X:(a->Prop))=> ((and (X ((cP (cL Xx)) (cL Xy)))) (forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu)))))))))
% Found (eq_ref0 (fun (X:(a->Prop))=> ((and (X ((cP (cL Xx)) (cL Xy)))) (forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu))))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP (cL Xx)) (cL Xy)))) (forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu))))))))) b)
% Found ((eq_ref ((a->Prop)->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP (cL Xx)) (cL Xy)))) (forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu))))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP (cL Xx)) (cL Xy)))) (forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu))))))))) b)
% Found ((eq_ref ((a->Prop)->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP (cL Xx)) (cL Xy)))) (forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu))))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP (cL Xx)) (cL Xy)))) (forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu))))))))) b)
% Found ((eq_ref ((a->Prop)->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP (cL Xx)) (cL Xy)))) (forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu))))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP (cL Xx)) (cL Xy)))) (forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu))))))))) b)
% Found x0:((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xx) Xy))) (forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu)))))))))
% Instantiate: f:=(fun (X:(a->Prop))=> ((and (X ((cP Xx) Xy))) (forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu)))))))):((a->Prop)->Prop)
% Found x0 as proof of (P f)
% Found x0:((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xx) Xy))) (forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu)))))))))
% Instantiate: f:=(fun (X:(a->Prop))=> ((and (X ((cP Xx) Xy))) (forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu)))))))):((a->Prop)->Prop)
% Found x0 as proof of (P f)
% 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 (cL Xx)) (cL Xy)))) (forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu)))))))))
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) (fun (X:(a->Prop))=> ((and (X ((cP (cL Xx)) (cL Xy)))) (forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu)))))))))
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) (fun (X:(a->Prop))=> ((and (X ((cP (cL Xx)) (cL Xy)))) (forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu)))))))))
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) (fun (X:(a->Prop))=> ((and (X ((cP (cL Xx)) (cL Xy)))) (forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu)))))))))
% Found eta_expansion000:=(eta_expansion00 (fun (X:(a->Prop))=> ((and (X ((cP Xx) Xy))) (forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu))))))))):(((eq ((a->Prop)->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xx) Xy))) (forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu))))))))) (fun (x:(a->Prop))=> ((and (x ((cP Xx) Xy))) (forall (Xt:a) (Xu:a), ((x ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (x ((cP (cL Xt)) (cL Xu))))) (x ((cP (cR Xt)) (cR Xu)))))))))
% Found (eta_expansion00 (fun (X:(a->Prop))=> ((and (X ((cP Xx) Xy))) (forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu))))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xx) Xy))) (forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu))))))))) b)
% Found ((eta_expansion0 Prop) (fun (X:(a->Prop))=> ((and (X ((cP Xx) Xy))) (forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu))))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xx) Xy))) (forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu))))))))) b)
% Found (((eta_expansion (a->Prop)) Prop) (fun (X:(a->Prop))=> ((and (X ((cP Xx) Xy))) (forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu))))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xx) Xy))) (forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu))))))))) b)
% Found (((eta_expansion (a->Prop)) Prop) (fun (X:(a->Prop))=> ((and (X ((cP Xx) Xy))) (forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu))))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xx) Xy))) (forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu))))))))) b)
% Found (((eta_expansion (a->Prop)) Prop) (fun (X:(a->Prop))=> ((and (X ((cP Xx) Xy))) (forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu))))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xx) Xy))) (forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu))))))))) b)
% Found eta_expansion0000:=(eta_expansion000 (ex (a->Prop))):(((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xx) Xy))) (forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu)))))))))->((ex (a->Prop)) (fun (x:(a->Prop))=> ((and (x ((cP Xx) Xy))) (forall (Xt:a) (Xu:a), ((x ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (x ((cP (cL Xt)) (cL Xu))))) (x ((cP (cR Xt)) (cR Xu))))))))))
% Found (eta_expansion000 (ex (a->Prop))) as proof of (P (fun (X:(a->Prop))=> ((and (X ((cP Xx) Xy))) (forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu)))))))))
% Found ((eta_expansion00 (fun (X:(a->Prop))=> ((and (X ((cP Xx) Xy))) (forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu))))))))) (ex (a->Prop))) as proof of (P (fun (X:(a->Prop))=> ((and (X ((cP Xx) Xy))) (forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu)))))))))
% Found (((eta_expansion0 Prop) (fun (X:(a->Prop))=> ((and (X ((cP Xx) Xy))) (forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu))))))))) (ex (a->Prop))) as proof of (P (fun (X:(a->Prop))=> ((and (X ((cP Xx) Xy))) (forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu)))))))))
% Found ((((eta_expansion (a->Prop)) Prop) (fun (X:(a->Prop))=> ((and (X ((cP Xx) Xy))) (forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu))))))))) (ex (a->Prop))) as proof of (P (fun (X:(a->Prop))=> ((and (X ((cP Xx) Xy))) (forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu)))))))))
% Found ((((eta_expansion (a->Prop)) Prop) (fun (X:(a->Prop))=> ((and (X ((cP Xx) Xy))) (forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu))))))))) (ex (a->Prop))) as proof of (P (fun (X:(a->Prop))=> ((and (X ((cP Xx) Xy))) (forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu)))))))))
% 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 (cL Xx)) (cL Xy)))) (forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu)))))))))
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) (fun (X:(a->Prop))=> ((and (X ((cP (cL Xx)) (cL Xy)))) (forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu)))))))))
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) (fun (X:(a->Prop))=> ((and (X ((cP (cL Xx)) (cL Xy)))) (forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu)))))))))
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) (fun (X:(a->Prop))=> ((and (X ((cP (cL Xx)) (cL Xy)))) (forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu)))))))))
% Found eta_expansion000:=(eta_expansion00 (fun (X:(a->Prop))=> ((and (X ((cP Xx) Xy))) (forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu))))))))):(((eq ((a->Prop)->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xx) Xy))) (forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu))))))))) (fun (x:(a->Prop))=> ((and (x ((cP Xx) Xy))) (forall (Xt:a) (Xu:a), ((x ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (x ((cP (cL Xt)) (cL Xu))))) (x ((cP (cR Xt)) (cR Xu)))))))))
% Found (eta_expansion00 (fun (X:(a->Prop))=> ((and (X ((cP Xx) Xy))) (forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu))))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xx) Xy))) (forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu))))))))) b)
% Found ((eta_expansion0 Prop) (fun (X:(a->Prop))=> ((and (X ((cP Xx) Xy))) (forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu))))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xx) Xy))) (forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu))))))))) b)
% Found (((eta_expansion (a->Prop)) Prop) (fun (X:(a->Prop))=> ((and (X ((cP Xx) Xy))) (forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu))))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xx) Xy))) (forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu))))))))) b)
% Found (((eta_expansion (a->Prop)) Prop) (fun (X:(a->Prop))=> ((and (X ((cP Xx) Xy))) (forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu))))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xx) Xy))) (forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu))))))))) b)
% Found (((eta_expansion (a->Prop)) Prop) (fun (X:(a->Prop))=> ((and (X ((cP Xx) Xy))) (forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu))))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xx) Xy))) (forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu))))))))) b)
% Found x6:(forall (Xt:a) (Xu:a), ((x3 ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (x3 ((cP (cL Xt)) (cL Xu))))) (x3 ((cP (cR Xt)) (cR Xu))))))
% Instantiate: x7:=x3:(a->Prop)
% Found x6 as proof of (forall (Xt:a) (Xu:a), ((x7 ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (x7 ((cP (cL Xt)) (cL Xu))))) (x7 ((cP (cR Xt)) (cR Xu))))))
% Found x0:((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xx) Xy))) (forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu)))))))))
% Instantiate: b:=(fun (X:(a->Prop))=> ((and (X ((cP Xx) Xy))) (forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu)))))))):((a->Prop)->Prop)
% Found x0 as proof of (P b)
% Found eq_ref00:=(eq_ref0 (fun (X:(a->Prop))=> ((and (X ((cP (cL Xx)) (cL Xy)))) (forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu))))))))):(((eq ((a->Prop)->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP (cL Xx)) (cL Xy)))) (forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu))))))))) (fun (X:(a->Prop))=> ((and (X ((cP (cL Xx)) (cL Xy)))) (forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu)))))))))
% Found (eq_ref0 (fun (X:(a->Prop))=> ((and (X ((cP (cL Xx)) (cL Xy)))) (forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu))))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP (cL Xx)) (cL Xy)))) (forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu))))))))) b)
% Found ((eq_ref ((a->Prop)->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP (cL Xx)) (cL Xy)))) (forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu))))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP (cL Xx)) (cL Xy)))) (forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu))))))))) b)
% Found ((eq_ref ((a->Prop)->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP (cL Xx)) (cL Xy)))) (forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu))))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP (cL Xx)) (cL Xy)))) (forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu))))))))) b)
% Found ((eq_ref ((a->Prop)->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP (cL Xx)) (cL Xy)))) (forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu))))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP (cL Xx)) (cL Xy)))) (forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu))))))))) b)
% Found eq_substitution0000:=(fun (a0:a)=> ((eq_substitution000 a0) (cP (cL Xx)))):(forall (a0:a), ((((eq a) a0) (cL Xy))->(((eq a) ((cP (cL Xx)) a0)) ((cP (cL Xx)) (cL Xy)))))
% Instantiate: x3:=(fun (x4:a)=> (forall (a0:a), ((((eq a) a0) (cL Xy))->(((eq a) ((cP (cL Xx)) a0)) x4)))):(a->Prop)
% Found (fun (a0:a)=> ((eq_substitution000 a0) (cP (cL Xx)))) as proof of (x3 ((cP (cL Xx)) (cL Xy)))
% Found (fun (a0:a)=> (((fun (a0:a)=> ((eq_substitution00 a0) (cL Xy))) a0) (cP (cL Xx)))) as proof of (x3 ((cP (cL Xx)) (cL Xy)))
% Found (fun (a0:a)=> (((fun (a0:a)=> (((eq_substitution0 a) a0) (cL Xy))) a0) (cP (cL Xx)))) as proof of (x3 ((cP (cL Xx)) (cL Xy)))
% Found (fun (a0:a)=> (((fun (a0:a)=> ((((eq_substitution a) a) a0) (cL Xy))) a0) (cP (cL Xx)))) as proof of (x3 ((cP (cL Xx)) (cL Xy)))
% Found (fun (a0:a)=> (((fun (a0:a)=> ((((eq_substitution a) a) a0) (cL Xy))) a0) (cP (cL Xx)))) as proof of (x3 ((cP (cL Xx)) (cL Xy)))
% Found x5000:=(x500 x00):((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (x2 ((cP (cL Xt)) (cL Xu))))) (x2 ((cP (cR Xt)) (cR Xu))))
% Found (x500 x00) as proof of ((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (x1 ((cP (cL Xt)) (cL Xu))))) (x1 ((cP (cR Xt)) (cR Xu))))
% Found ((x50 Xu) x00) as proof of ((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (x1 ((cP (cL Xt)) (cL Xu))))) (x1 ((cP (cR Xt)) (cR Xu))))
% Found (((x5 Xt) Xu) x00) as proof of ((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (x1 ((cP (cL Xt)) (cL Xu))))) (x1 ((cP (cR Xt)) (cR Xu))))
% Found (((x5 Xt) Xu) x00) as proof of ((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (x1 ((cP (cL Xt)) (cL Xu))))) (x1 ((cP (cR Xt)) (cR Xu))))
% Found (fun (x5:(forall (Xt0:a) (Xu0:a), ((x2 ((cP Xt0) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt0) cZ))) (x2 ((cP (cL Xt0)) (cL Xu0))))) (x2 ((cP (cR Xt0)) (cR Xu0)))))))=> (((x5 Xt) Xu) x00)) as proof of ((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (x1 ((cP (cL Xt)) (cL Xu))))) (x1 ((cP (cR Xt)) (cR Xu))))
% Found (fun (x4:(x2 ((cP Xx) Xy))) (x5:(forall (Xt0:a) (Xu0:a), ((x2 ((cP Xt0) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt0) cZ))) (x2 ((cP (cL Xt0)) (cL Xu0))))) (x2 ((cP (cR Xt0)) (cR Xu0)))))))=> (((x5 Xt) Xu) x00)) as proof of ((forall (Xt0:a) (Xu0:a), ((x2 ((cP Xt0) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt0) cZ))) (x2 ((cP (cL Xt0)) (cL Xu0))))) (x2 ((cP (cR Xt0)) (cR Xu0))))))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (x1 ((cP (cL Xt)) (cL Xu))))) (x1 ((cP (cR Xt)) (cR Xu)))))
% Found (fun (x4:(x2 ((cP Xx) Xy))) (x5:(forall (Xt0:a) (Xu0:a), ((x2 ((cP Xt0) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt0) cZ))) (x2 ((cP (cL Xt0)) (cL Xu0))))) (x2 ((cP (cR Xt0)) (cR Xu0)))))))=> (((x5 Xt) Xu) x00)) as proof of ((x2 ((cP Xx) Xy))->((forall (Xt0:a) (Xu0:a), ((x2 ((cP Xt0) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt0) cZ))) (x2 ((cP (cL Xt0)) (cL Xu0))))) (x2 ((cP (cR Xt0)) (cR Xu0))))))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (x1 ((cP (cL Xt)) (cL Xu))))) (x1 ((cP (cR Xt)) (cR Xu))))))
% Found (and_rect00 (fun (x4:(x2 ((cP Xx) Xy))) (x5:(forall (Xt0:a) (Xu0:a), ((x2 ((cP Xt0) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt0) cZ))) (x2 ((cP (cL Xt0)) (cL Xu0))))) (x2 ((cP (cR Xt0)) (cR Xu0)))))))=> (((x5 Xt) Xu) x00))) as proof of ((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (x1 ((cP (cL Xt)) (cL Xu))))) (x1 ((cP (cR Xt)) (cR Xu))))
% Found ((and_rect0 ((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (x1 ((cP (cL Xt)) (cL Xu))))) (x1 ((cP (cR Xt)) (cR Xu))))) (fun (x4:(x2 ((cP Xx) Xy))) (x5:(forall (Xt0:a) (Xu0:a), ((x2 ((cP Xt0) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt0) cZ))) (x2 ((cP (cL Xt0)) (cL Xu0))))) (x2 ((cP (cR Xt0)) (cR Xu0)))))))=> (((x5 Xt) Xu) x00))) as proof of ((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (x1 ((cP (cL Xt)) (cL Xu))))) (x1 ((cP (cR Xt)) (cR Xu))))
% Found (((fun (P:Type) (x4:((x2 ((cP Xx) Xy))->((forall (Xt0:a) (Xu0:a), ((x2 ((cP Xt0) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt0) cZ))) (x2 ((cP (cL Xt0)) (cL Xu0))))) (x2 ((cP (cR Xt0)) (cR Xu0))))))->P)))=> (((((and_rect (x2 ((cP Xx) Xy))) (forall (Xt0:a) (Xu0:a), ((x2 ((cP Xt0) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt0) cZ))) (x2 ((cP (cL Xt0)) (cL Xu0))))) (x2 ((cP (cR Xt0)) (cR Xu0))))))) P) x4) x3)) ((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (x1 ((cP (cL Xt)) (cL Xu))))) (x1 ((cP (cR Xt)) (cR Xu))))) (fun (x4:(x2 ((cP Xx) Xy))) (x5:(forall (Xt0:a) (Xu0:a), ((x2 ((cP Xt0) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt0) cZ))) (x2 ((cP (cL Xt0)) (cL Xu0))))) (x2 ((cP (cR Xt0)) (cR Xu0)))))))=> (((x5 Xt) Xu) x00))) as proof of ((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (x1 ((cP (cL Xt)) (cL Xu))))) (x1 ((cP (cR Xt)) (cR Xu))))
% Found (fun (x3:((and (x2 ((cP Xx) Xy))) (forall (Xt0:a) (Xu0:a), ((x2 ((cP Xt0) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt0) cZ))) (x2 ((cP (cL Xt0)) (cL Xu0))))) (x2 ((cP (cR Xt0)) (cR Xu0))))))))=> (((fun (P:Type) (x4:((x2 ((cP Xx) Xy))->((forall (Xt0:a) (Xu0:a), ((x2 ((cP Xt0) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt0) cZ))) (x2 ((cP (cL Xt0)) (cL Xu0))))) (x2 ((cP (cR Xt0)) (cR Xu0))))))->P)))=> (((((and_rect (x2 ((cP Xx) Xy))) (forall (Xt0:a) (Xu0:a), ((x2 ((cP Xt0) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt0) cZ))) (x2 ((cP (cL Xt0)) (cL Xu0))))) (x2 ((cP (cR Xt0)) (cR Xu0))))))) P) x4) x3)) ((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (x1 ((cP (cL Xt)) (cL Xu))))) (x1 ((cP (cR Xt)) (cR Xu))))) (fun (x4:(x2 ((cP Xx) Xy))) (x5:(forall (Xt0:a) (Xu0:a), ((x2 ((cP Xt0) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt0) cZ))) (x2 ((cP (cL Xt0)) (cL Xu0))))) (x2 ((cP (cR Xt0)) (cR Xu0)))))))=> (((x5 Xt) Xu) x00)))) as proof of ((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (x1 ((cP (cL Xt)) (cL Xu))))) (x1 ((cP (cR Xt)) (cR Xu))))
% Found x5000:=(x500 x00):((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (x1 ((cP (cL Xt)) (cL Xu))))) (x1 ((cP (cR Xt)) (cR Xu))))
% Found (x500 x00) as proof of ((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (x3 ((cP (cL Xt)) (cL Xu))))) (x3 ((cP (cR Xt)) (cR Xu))))
% Found ((x50 Xu) x00) as proof of ((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (x3 ((cP (cL Xt)) (cL Xu))))) (x3 ((cP (cR Xt)) (cR Xu))))
% Found (((x5 Xt) Xu) x00) as proof of ((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (x3 ((cP (cL Xt)) (cL Xu))))) (x3 ((cP (cR Xt)) (cR Xu))))
% Found (((x5 Xt) Xu) x00) as proof of ((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (x3 ((cP (cL Xt)) (cL Xu))))) (x3 ((cP (cR Xt)) (cR Xu))))
% Found (fun (x5:(forall (Xt0:a) (Xu0:a), ((x1 ((cP Xt0) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt0) cZ))) (x1 ((cP (cL Xt0)) (cL Xu0))))) (x1 ((cP (cR Xt0)) (cR Xu0)))))))=> (((x5 Xt) Xu) x00)) as proof of ((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (x3 ((cP (cL Xt)) (cL Xu))))) (x3 ((cP (cR Xt)) (cR Xu))))
% Found (fun (x4:(x1 ((cP Xx) Xy))) (x5:(forall (Xt0:a) (Xu0:a), ((x1 ((cP Xt0) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt0) cZ))) (x1 ((cP (cL Xt0)) (cL Xu0))))) (x1 ((cP (cR Xt0)) (cR Xu0)))))))=> (((x5 Xt) Xu) x00)) as proof of ((forall (Xt0:a) (Xu0:a), ((x1 ((cP Xt0) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt0) cZ))) (x1 ((cP (cL Xt0)) (cL Xu0))))) (x1 ((cP (cR Xt0)) (cR Xu0))))))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (x3 ((cP (cL Xt)) (cL Xu))))) (x3 ((cP (cR Xt)) (cR Xu)))))
% Found (fun (x4:(x1 ((cP Xx) Xy))) (x5:(forall (Xt0:a) (Xu0:a), ((x1 ((cP Xt0) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt0) cZ))) (x1 ((cP (cL Xt0)) (cL Xu0))))) (x1 ((cP (cR Xt0)) (cR Xu0)))))))=> (((x5 Xt) Xu) x00)) as proof of ((x1 ((cP Xx) Xy))->((forall (Xt0:a) (Xu0:a), ((x1 ((cP Xt0) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt0) cZ))) (x1 ((cP (cL Xt0)) (cL Xu0))))) (x1 ((cP (cR Xt0)) (cR Xu0))))))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (x3 ((cP (cL Xt)) (cL Xu))))) (x3 ((cP (cR Xt)) (cR Xu))))))
% Found (and_rect00 (fun (x4:(x1 ((cP Xx) Xy))) (x5:(forall (Xt0:a) (Xu0:a), ((x1 ((cP Xt0) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt0) cZ))) (x1 ((cP (cL Xt0)) (cL Xu0))))) (x1 ((cP (cR Xt0)) (cR Xu0)))))))=> (((x5 Xt) Xu) x00))) as proof of ((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (x3 ((cP (cL Xt)) (cL Xu))))) (x3 ((cP (cR Xt)) (cR Xu))))
% Found ((and_rect0 ((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (x3 ((cP (cL Xt)) (cL Xu))))) (x3 ((cP (cR Xt)) (cR Xu))))) (fun (x4:(x1 ((cP Xx) Xy))) (x5:(forall (Xt0:a) (Xu0:a), ((x1 ((cP Xt0) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt0) cZ))) (x1 ((cP (cL Xt0)) (cL Xu0))))) (x1 ((cP (cR Xt0)) (cR Xu0)))))))=> (((x5 Xt) Xu) x00))) as proof of ((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (x3 ((cP (cL Xt)) (cL Xu))))) (x3 ((cP (cR Xt)) (cR Xu))))
% Found (((fun (P:Type) (x4:((x1 ((cP Xx) Xy))->((forall (Xt:a) (Xu:a), ((x1 ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (x1 ((cP (cL Xt)) (cL Xu))))) (x1 ((cP (cR Xt)) (cR Xu))))))->P)))=> (((((and_rect (x1 ((cP Xx) Xy))) (forall (Xt:a) (Xu:a), ((x1 ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (x1 ((cP (cL Xt)) (cL Xu))))) (x1 ((cP (cR Xt)) (cR Xu))))))) P) x4) x2)) ((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (x3 ((cP (cL Xt)) (cL Xu))))) (x3 ((cP (cR Xt)) (cR Xu))))) (fun (x4:(x1 ((cP Xx) Xy))) (x5:(forall (Xt0:a) (Xu0:a), ((x1 ((cP Xt0) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt0) cZ))) (x1 ((cP (cL Xt0)) (cL Xu0))))) (x1 ((cP (cR Xt0)) (cR Xu0)))))))=> (((x5 Xt) Xu) x00))) as proof of ((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (x3 ((cP (cL Xt)) (cL Xu))))) (x3 ((cP (cR Xt)) (cR Xu))))
% Found (fun (x00:(x3 ((cP Xt) Xu)))=> (((fun (P:Type) (x4:((x1 ((cP Xx) Xy))->((forall (Xt:a) (Xu:a), ((x1 ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (x1 ((cP (cL Xt)) (cL Xu))))) (x1 ((cP (cR Xt)) (cR Xu))))))->P)))=> (((((and_rect (x1 ((cP Xx) Xy))) (forall (Xt:a) (Xu:a), ((x1 ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (x1 ((cP (cL Xt)) (cL Xu))))) (x1 ((cP (cR Xt)) (cR Xu))))))) P) x4) x2)) ((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (x3 ((cP (cL Xt)) (cL Xu))))) (x3 ((cP (cR Xt)) (cR Xu))))) (fun (x4:(x1 ((cP Xx) Xy))) (x5:(forall (Xt0:a) (Xu0:a), ((x1 ((cP Xt0) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt0) cZ))) (x1 ((cP (cL Xt0)) (cL Xu0))))) (x1 ((cP (cR Xt0)) (cR Xu0)))))))=> (((x5 Xt) Xu) x00)))) as proof of ((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (x3 ((cP (cL Xt)) (cL Xu))))) (x3 ((cP (cR Xt)) (cR Xu))))
% Found (fun (Xu:a) (x00:(x3 ((cP Xt) Xu)))=> (((fun (P:Type) (x4:((x1 ((cP Xx) Xy))->((forall (Xt:a) (Xu:a), ((x1 ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (x1 ((cP (cL Xt)) (cL Xu))))) (x1 ((cP (cR Xt)) (cR Xu))))))->P)))=> (((((and_rect (x1 ((cP Xx) Xy))) (forall (Xt:a) (Xu:a), ((x1 ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (x1 ((cP (cL Xt)) (cL Xu))))) (x1 ((cP (cR Xt)) (cR Xu))))))) P) x4) x2)) ((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (x3 ((cP (cL Xt)) (cL Xu))))) (x3 ((cP (cR Xt)) (cR Xu))))) (fun (x4:(x1 ((cP Xx) Xy))) (x5:(forall (Xt0:a) (Xu0:a), ((x1 ((cP Xt0) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt0) cZ))) (x1 ((cP (cL Xt0)) (cL Xu0))))) (x1 ((cP (cR Xt0)) (cR Xu0)))))))=> (((x5 Xt) Xu) x00)))) as proof of ((x3 ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (x3 ((cP (cL Xt)) (cL Xu))))) (x3 ((cP (cR Xt)) (cR Xu)))))
% Found (fun (Xt:a) (Xu:a) (x00:(x3 ((cP Xt) Xu)))=> (((fun (P:Type) (x4:((x1 ((cP Xx) Xy))->((forall (Xt:a) (Xu:a), ((x1 ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (x1 ((cP (cL Xt)) (cL Xu))))) (x1 ((cP (cR Xt)) (cR Xu))))))->P)))=> (((((and_rect (x1 ((cP Xx) Xy))) (forall (Xt:a) (Xu:a), ((x1 ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (x1 ((cP (cL Xt)) (cL Xu))))) (x1 ((cP (cR Xt)) (cR Xu))))))) P) x4) x2)) ((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (x3 ((cP (cL Xt)) (cL Xu))))) (x3 ((cP (cR Xt)) (cR Xu))))) (fun (x4:(x1 ((cP Xx) Xy))) (x5:(forall (Xt0:a) (Xu0:a), ((x1 ((cP Xt0) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt0) cZ))) (x1 ((cP (cL Xt0)) (cL Xu0))))) (x1 ((cP (cR Xt0)) (cR Xu0)))))))=> (((x5 Xt) Xu) x00)))) as proof of (forall (Xu:a), ((x3 ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (x3 ((cP (cL Xt)) (cL Xu))))) (x3 ((cP (cR Xt)) (cR Xu))))))
% Found (fun (Xt:a) (Xu:a) (x00:(x3 ((cP Xt) Xu)))=> (((fun (P:Type) (x4:((x1 ((cP Xx) Xy))->((forall (Xt:a) (Xu:a), ((x1 ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (x1 ((cP (cL Xt)) (cL Xu))))) (x1 ((cP (cR Xt)) (cR Xu))))))->P)))=> (((((and_rect (x1 ((cP Xx) Xy))) (forall (Xt:a) (Xu:a), ((x1 ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (x1 ((cP (cL Xt)) (cL Xu))))) (x1 ((cP (cR Xt)) (cR Xu))))))) P) x4) x2)) ((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (x3 ((cP (cL Xt)) (cL Xu))))) (x3 ((cP (cR Xt)) (cR Xu))))) (fun (x4:(x1 ((cP Xx) Xy))) (x5:(forall (Xt0:a) (Xu0:a), ((x1 ((cP Xt0) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt0) cZ))) (x1 ((cP (cL Xt0)) (cL Xu0))))) (x1 ((cP (cR Xt0)) (cR Xu0)))))))=> (((x5 Xt) Xu) x00)))) as proof of (forall (Xt:a) (Xu:a), ((x3 ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (x3 ((cP (cL Xt)) (cL Xu))))) (x3 ((cP (cR Xt)) (cR Xu))))))
% Found eq_substitution0000:=(fun (a0:a)=> ((eq_substitution000 a0) (cP (cL Xx)))):(forall (a0:a), ((((eq a) a0) (cL Xy))->(((eq a) ((cP (cL Xx)) a0)) ((cP (cL Xx)) (cL Xy)))))
% Instantiate: x5:=(fun (x6:a)=> (forall (a0:a), ((((eq a) a0) (cL Xy))->(((eq a) ((cP (cL Xx)) a0)) x6)))):(a->Prop)
% Found (fun (a0:a)=> ((eq_substitution000 a0) (cP (cL Xx)))) as proof of (x5 ((cP (cL Xx)) (cL Xy)))
% Found (fun (a0:a)=> (((fun (a0:a)=> ((eq_substitution00 a0) (cL Xy))) a0) (cP (cL Xx)))) as proof of (x5 ((cP (cL Xx)) (cL Xy)))
% Found (fun (a0:a)=> (((fun (a0:a)=> (((eq_substitution0 a) a0) (cL Xy))) a0) (cP (cL Xx)))) as proof of (x5 ((cP (cL Xx)) (cL Xy)))
% Found (fun (a0:a)=> (((fun (a0:a)=> ((((eq_substitution a) a) a0) (cL Xy))) a0) (cP (cL Xx)))) as proof of (x5 ((cP (cL Xx)) (cL Xy)))
% Found (fun (a0:a)=> (((fun (a0:a)=> ((((eq_substitution a) a) a0) (cL Xy))) a0) (cP (cL Xx)))) as proof of (x5 ((cP (cL Xx)) (cL Xy)))
% Found x0:((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xx) Xy))) (forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu)))))))))
% Instantiate: b:=(fun (X:(a->Prop))=> ((and (X ((cP Xx) Xy))) (forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu)))))))):((a->Prop)->Prop)
% Found x0 as proof of (P b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (X:(a->Prop))=> ((and (X ((cP (cL Xx)) (cL Xy)))) (forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu))))))))):(((eq ((a->Prop)->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP (cL Xx)) (cL Xy)))) (forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu))))))))) (fun (x:(a->Prop))=> ((and (x ((cP (cL Xx)) (cL Xy)))) (forall (Xt:a) (Xu:a), ((x ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (x ((cP (cL Xt)) (cL Xu))))) (x ((cP (cR Xt)) (cR Xu)))))))))
% Found (eta_expansion_dep00 (fun (X:(a->Prop))=> ((and (X ((cP (cL Xx)) (cL Xy)))) (forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu))))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP (cL Xx)) (cL Xy)))) (forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu))))))))) b)
% Found ((eta_expansion_dep0 (fun (x4:(a->Prop))=> Prop)) (fun (X:(a->Prop))=> ((and (X ((cP (cL Xx)) (cL Xy)))) (forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu))))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP (cL Xx)) (cL Xy)))) (forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu))))))))) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x4:(a->Prop))=> Prop)) (fun (X:(a->Prop))=> ((and (X ((cP (cL Xx)) (cL Xy)))) (forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu))))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP (cL Xx)) (cL Xy)))) (forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu))))))))) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x4:(a->Prop))=> Prop)) (fun (X:(a->Prop))=> ((and (X ((cP (cL Xx)) (cL Xy)))) (forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu))))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP (cL Xx)) (cL Xy)))) (forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu))))))))) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x4:(a->Prop))=> Prop)) (fun (X:(a->Prop))=> ((and (X ((cP (cL Xx)) (cL Xy)))) (forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu))))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP (cL Xx)) (cL Xy)))) (forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu))))))))) b)
% Found x0:((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xx) Xy))) (forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu)))))))))
% Instantiate: f:=(fun (X:(a->Prop))=> ((and (X ((cP Xx) Xy))) (forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu)))))))):((a->Prop)->Prop)
% Found x0 as proof of (P f)
% Found x0:((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xx) Xy))) (forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu)))))))))
% Instantiate: f:=(fun (X:(a->Prop))=> ((and (X ((cP Xx) Xy))) (forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu)))))))):((a->Prop)->Prop)
% Found x0 as proof of (P f)
% Found conj0100:=(conj010 x00):((and (x1 ((cP (cL Xx)) (cL Xy)))) (x1 ((cP (cL Xt)) (cL Xu))))
% Found (conj010 x00) as proof of ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (x1 ((cP (cL Xt)) (cL Xu))))
% Found ((conj01 (x1 ((cP (cL Xt)) (cL Xu)))) x00) as proof of ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (x1 ((cP (cL Xt)) (cL Xu))))
% Found (((fun (B:Prop)=> ((conj0 B) x00)) (x1 ((cP (cL Xt)) (cL Xu)))) x00) as proof of ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (x1 ((cP (cL Xt)) (cL Xu))))
% Found (((fun (B:Prop)=> ((conj0 B) x00)) (x1 ((cP (cL Xt)) (cL Xu)))) x00) as proof of ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (x1 ((cP (cL Xt)) (cL Xu))))
% Found (((fun (B:Prop)=> ((conj0 B) x00)) (x1 ((cP (cL Xt)) (cL Xu)))) x00) as proof of ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (x1 ((cP (cL Xt)) (cL Xu))))
% Found x00:(x1 ((cP Xt) Xu))
% Found x00 as proof of (x1 ((cP (cR Xt)) (cR Xu)))
% Found ((conj10 (((fun (B:Prop)=> ((conj0 B) x00)) (x1 ((cP (cL Xt)) (cL Xu)))) x00)) x00) as proof of ((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (x1 ((cP (cL Xt)) (cL Xu))))) (x1 ((cP (cR Xt)) (cR Xu))))
% Found (((conj1 (x1 ((cP (cR Xt)) (cR Xu)))) (((fun (B:Prop)=> ((conj0 B) x00)) (x1 ((cP (cL Xt)) (cL Xu)))) x00)) x00) as proof of ((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (x1 ((cP (cL Xt)) (cL Xu))))) (x1 ((cP (cR Xt)) (cR Xu))))
% Found ((((conj ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (x1 ((cP (cL Xt)) (cL Xu))))) (x1 ((cP (cR Xt)) (cR Xu)))) (((fun (B:Prop)=> ((conj0 B) x00)) (x1 ((cP (cL Xt)) (cL Xu)))) x00)) x00) as proof of ((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (x1 ((cP (cL Xt)) (cL Xu))))) (x1 ((cP (cR Xt)) (cR Xu))))
% Found (fun (x00:(x1 ((cP Xt) Xu)))=> ((((conj ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (x1 ((cP (cL Xt)) (cL Xu))))) (x1 ((cP (cR Xt)) (cR Xu)))) (((fun (B:Prop)=> ((conj0 B) x00)) (x1 ((cP (cL Xt)) (cL Xu)))) x00)) x00)) as proof of ((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (x1 ((cP (cL Xt)) (cL Xu))))) (x1 ((cP (cR Xt)) (cR Xu))))
% Found eq_substitution0000:=(fun (a0:a)=> ((eq_substitution000 a0) (cP (cL Xx)))):(forall (a0:a), ((((eq a) a0) (cL Xy))->(((eq a) ((cP (cL Xx)) a0)) ((cP (cL Xx)) (cL Xy)))))
% Instantiate: x3:=(fun (x4:a)=> (forall (a0:a), ((((eq a) a0) (cL Xy))->(((eq a) ((cP (cL Xx)) a0)) x4)))):(a->Prop)
% Found (fun (a0:a)=> ((eq_substitution000 a0) (cP (cL Xx)))) as proof of (x3 ((cP (cL Xx)) (cL Xy)))
% Found (fun (a0:a)=> (((fun (a0:a)=> ((eq_substitution00 a0) (cL Xy))) a0) (cP (cL Xx)))) as proof of (x3 ((cP (cL Xx)) (cL Xy)))
% Found (fun (a0:a)=> (((fun (a0:a)=> (((eq_substitution0 a) a0) (cL Xy))) a0) (cP (cL Xx)))) as proof of (x3 ((cP (cL Xx)) (cL Xy)))
% Found (fun (a0:a)=> (((fun (a0:a)=> ((((eq_substitution a) a) a0) (cL Xy))) a0) (cP (cL Xx)))) as proof of (x3 ((cP (cL Xx)) (cL Xy)))
% Found (fun (a0:a)=> (((fun (a0:a)=> ((((eq_substitution a) a) a0) (cL Xy))) a0) (cP (cL Xx)))) as proof of (x3 ((cP (cL Xx)) (cL Xy)))
% Found x7:(forall (Xt:a) (Xu:a), ((x4 ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (x4 ((cP (cL Xt)) (cL Xu))))) (x4 ((cP (cR Xt)) (cR Xu))))))
% Instantiate: x3:=x4:(a->Prop)
% Found x7 as proof of (forall (Xt:a) (Xu:a), ((x3 ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (x3 ((cP (cL Xt)) (cL Xu))))) (x3 ((cP (cR Xt)) (cR Xu))))))
% Found x7:(forall (Xt:a) (Xu:a), ((x3 ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (x3 ((cP (cL Xt)) (cL Xu))))) (x3 ((cP (cR Xt)) (cR Xu))))))
% Instantiate: x5:=x3:(a->Prop)
% Found x7 as proof of (forall (Xt:a) (Xu:a), ((x5 ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (x5 ((cP (cL Xt)) (cL Xu))))) (x5 ((cP (cR Xt)) (cR Xu))))))
% Found x4:(forall (Xt:a) (Xu:a), ((x1 ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (x1 ((cP (cL Xt)) (cL Xu))))) (x1 ((cP (cR Xt)) (cR Xu))))))
% Instantiate: x7:=x1:(a->Prop)
% Found x4 as proof of (forall (Xt:a) (Xu:a), ((x7 ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (x7 ((cP (cL Xt)) (cL Xu))))) (x7 ((cP (cR Xt)) (cR Xu))))))
% Found x6:(forall (Xt:a) (Xu:a), ((x1 ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (x1 ((cP (cL Xt)) (cL Xu))))) (x1 ((cP (cR Xt)) (cR Xu))))))
% Instantiate: x7:=x1:(a->Prop)
% Found x6 as proof of (forall (Xt:a) (Xu:a), ((x7 ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (x7 ((cP (cL Xt)) (cL Xu))))) (x7 ((cP (cR Xt)) (cR Xu))))))
% Found eq_substitution0000:=(fun (a0:a)=> ((eq_substitution000 a0) (cP (cL Xx)))):(forall (a0:a), ((((eq a) a0) (cL Xy))->(((eq a) ((cP (cL Xx)) a0)) ((cP (cL Xx)) (cL Xy)))))
% Instantiate: x1:=(fun (x4:a)=> (forall (a0:a), ((((eq a) a0) (cL Xy))->(((eq a) ((cP (cL Xx)) a0)) x4)))):(a->Prop)
% Found (fun (a0:a)=> ((eq_substitution000 a0) (cP (cL Xx)))) as proof of (x1 ((cP (cL Xx)) (cL Xy)))
% Found (fun (a0:a)=> (((fun (a0:a)=> ((eq_substitution00 a0) (cL Xy))) a0) (cP (cL Xx)))) as proof of (x1 ((cP (cL Xx)) (cL Xy)))
% Found (fun (a0:a)=> (((fun (a0:a)=> (((eq_substitution0 a) a0) (cL Xy))) a0) (cP (cL Xx)))) as proof of (x1 ((cP (cL Xx)) (cL Xy)))
% Found (fun (a0:a)=> (((fun (a0:a)=> ((((eq_substitution a) a) a0) (cL Xy))) a0) (cP (cL Xx)))) as proof of (x1 ((cP (cL Xx)) (cL Xy)))
% Found (fun (a0:a)=> (((fun (a0:a)=> ((((eq_substitution a) a) a0) (cL Xy))) a0) (cP (cL Xx)))) as proof of (x1 ((cP (cL Xx)) (cL Xy)))
% Found x5000:=(x500 x00):((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (x2 ((cP (cL Xt)) (cL Xu))))) (x2 ((cP (cR Xt)) (cR Xu))))
% Found (x500 x00) as proof of ((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (x1 ((cP (cL Xt)) (cL Xu))))) (x1 ((cP (cR Xt)) (cR Xu))))
% Found ((x50 Xu) x00) as proof of ((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (x1 ((cP (cL Xt)) (cL Xu))))) (x1 ((cP (cR Xt)) (cR Xu))))
% Found (((x5 Xt) Xu) x00) as proof of ((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (x1 ((cP (cL Xt)) (cL Xu))))) (x1 ((cP (cR Xt)) (cR Xu))))
% Found (((x5 Xt) Xu) x00) as proof of ((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (x1 ((cP (cL Xt)) (cL Xu))))) (x1 ((cP (cR Xt)) (cR Xu))))
% Found (fun (x5:(forall (Xt0:a) (Xu0:a), ((x2 ((cP Xt0) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt0) cZ))) (x2 ((cP (cL Xt0)) (cL Xu0))))) (x2 ((cP (cR Xt0)) (cR Xu0)))))))=> (((x5 Xt) Xu) x00)) as proof of ((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (x1 ((cP (cL Xt)) (cL Xu))))) (x1 ((cP (cR Xt)) (cR Xu))))
% Found (fun (x4:(x2 ((cP Xx) Xy))) (x5:(forall (Xt0:a) (Xu0:a), ((x2 ((cP Xt0) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt0) cZ))) (x2 ((cP (cL Xt0)) (cL Xu0))))) (x2 ((cP (cR Xt0)) (cR Xu0)))))))=> (((x5 Xt) Xu) x00)) as proof of ((forall (Xt0:a) (Xu0:a), ((x2 ((cP Xt0) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt0) cZ))) (x2 ((cP (cL Xt0)) (cL Xu0))))) (x2 ((cP (cR Xt0)) (cR Xu0))))))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (x1 ((cP (cL Xt)) (cL Xu))))) (x1 ((cP (cR Xt)) (cR Xu)))))
% Found (fun (x4:(x2 ((cP Xx) Xy))) (x5:(forall (Xt0:a) (Xu0:a), ((x2 ((cP Xt0) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt0) cZ))) (x2 ((cP (cL Xt0)) (cL Xu0))))) (x2 ((cP (cR Xt0)) (cR Xu0)))))))=> (((x5 Xt) Xu) x00)) as proof of ((x2 ((cP Xx) Xy))->((forall (Xt0:a) (Xu0:a), ((x2 ((cP Xt0) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt0) cZ))) (x2 ((cP (cL Xt0)) (cL Xu0))))) (x2 ((cP (cR Xt0)) (cR Xu0))))))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (x1 ((cP (cL Xt)) (cL Xu))))) (x1 ((cP (cR Xt)) (cR Xu))))))
% Found (and_rect00 (fun (x4:(x2 ((cP Xx) Xy))) (x5:(forall (Xt0:a) (Xu0:a), ((x2 ((cP Xt0) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt0) cZ))) (x2 ((cP (cL Xt0)) (cL Xu0))))) (x2 ((cP (cR Xt0)) (cR Xu0)))))))=> (((x5 Xt) Xu) x00))) as proof of ((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (x1 ((cP (cL Xt)) (cL Xu))))) (x1 ((cP (cR Xt)) (cR Xu))))
% Found ((and_rect0 ((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (x1 ((cP (cL Xt)) (cL Xu))))) (x1 ((cP (cR Xt)) (cR Xu))))) (fun (x4:(x2 ((cP Xx) Xy))) (x5:(forall (Xt0:a) (Xu0:a), ((x2 ((cP Xt0) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt0) cZ))) (x2 ((cP (cL Xt0)) (cL Xu0))))) (x2 ((cP (cR Xt0)) (cR Xu0)))))))=> (((x5 Xt) Xu) x00))) as proof of ((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (x1 ((cP (cL Xt)) (cL Xu))))) (x1 ((cP (cR Xt)) (cR Xu))))
% Found (((fun (P:Type) (x4:((x2 ((cP Xx) Xy))->((forall (Xt:a) (Xu:a), ((x2 ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (x2 ((cP (cL Xt)) (cL Xu))))) (x2 ((cP (cR Xt)) (cR Xu))))))->P)))=> (((((and_rect (x2 ((cP Xx) Xy))) (forall (Xt:a) (Xu:a), ((x2 ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (x2 ((cP (cL Xt)) (cL Xu))))) (x2 ((cP (cR Xt)) (cR Xu))))))) P) x4) x3)) ((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (x1 ((cP (cL Xt)) (cL Xu))))) (x1 ((cP (cR Xt)) (cR Xu))))) (fun (x4:(x2 ((cP Xx) Xy))) (x5:(forall (Xt0:a) (Xu0:a), ((x2 ((cP Xt0) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt0) cZ))) (x2 ((cP (cL Xt0)) (cL Xu0))))) (x2 ((cP (cR Xt0)) (cR Xu0)))))))=> (((x5 Xt) Xu) x00))) as proof of ((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (x1 ((cP (cL Xt)) (cL Xu))))) (x1 ((cP (cR Xt)) (cR Xu))))
% Found (fun (x00:(x1 ((cP Xt) Xu)))=> (((fun (P:Type) (x4:((x2 ((cP Xx) Xy))->((forall (Xt:a) (Xu:a), ((x2 ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (x2 ((cP (cL Xt)) (cL Xu))))) (x2 ((cP (cR Xt)) (cR Xu))))))->P)))=> (((((and_rect (x2 ((cP Xx) Xy))) (forall (Xt:a) (Xu:a), ((x2 ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (x2 ((cP (cL Xt)) (cL Xu))))) (x2 ((cP (cR Xt)) (cR Xu))))))) P) x4) x3)) ((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (x1 ((cP (cL Xt)) (cL Xu))))) (x1 ((cP (cR Xt)) (cR Xu))))) (fun (x4:(x2 ((cP Xx) Xy))) (x5:(forall (Xt0:a) (Xu0:a), ((x2 ((cP Xt0) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt0) cZ))) (x2 ((cP (cL Xt0)) (cL Xu0))))) (x2 ((cP (cR Xt0)) (cR Xu0)))))))=> (((x5 Xt) Xu) x00)))) as proof of ((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (x1 ((cP (cL Xt)) (cL Xu))))) (x1 ((cP (cR Xt)) (cR Xu))))
% Found (fun (Xu:a) (x00:(x1 ((cP Xt) Xu)))=> (((fun (P:Type) (x4:((x2 ((cP Xx) Xy))->((forall (Xt:a) (Xu:a), ((x2 ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (x2 ((cP (cL Xt)) (cL Xu))))) (x2 ((cP (cR Xt)) (cR Xu))))))->P)))=> (((((and_rect (x2 ((cP Xx) Xy))) (forall (Xt:a) (Xu:a), ((x2 ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (x2 ((cP (cL Xt)) (cL Xu))))) (x2 ((cP (cR Xt)) (cR Xu))))))) P) x4) x3)) ((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (x1 ((cP (cL Xt)) (cL Xu))))) (x1 ((cP (cR Xt)) (cR Xu))))) (fun (x4:(x2 ((cP Xx) Xy))) (x5:(forall (Xt0:a) (Xu0:a), ((x2 ((cP Xt0) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt0) cZ))) (x2 ((cP (cL Xt0)) (cL Xu0))))) (x2 ((cP (cR Xt0)) (cR Xu0)))))))=> (((x5 Xt) Xu) x00)))) as proof of ((x1 ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (x1 ((cP (cL Xt)) (cL Xu))))) (x1 ((cP (cR Xt)) (cR Xu)))))
% Found (fun (Xt:a) (Xu:a) (x00:(x1 ((cP Xt) Xu)))=> (((fun (P:Type) (x4:((x2 ((cP Xx) Xy))->((forall (Xt:a) (Xu:a), ((x2 ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (x2 ((cP (cL Xt)) (cL Xu))))) (x2 ((cP (cR Xt)) (cR Xu))))))->P)))=> (((((and_rect (x2 ((cP Xx) Xy))) (forall (Xt:a) (Xu:a), ((x2 ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (x2 ((cP (cL Xt)) (cL Xu))))) (x2 ((cP (cR Xt)) (cR Xu))))))) P) x4) x3)) ((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (x1 ((cP (cL Xt)) (cL Xu))))) (x1 ((cP (cR Xt)) (cR Xu))))) (fun (x4:(x2 ((cP Xx) Xy))) (x5:(forall (Xt0:a) (Xu0:a), ((x2 ((cP Xt0) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt0) cZ))) (x2 ((cP (cL Xt0)) (cL Xu0))))) (x2 ((cP (cR Xt0)) (cR Xu0)))))))=> (((x5 Xt) Xu) x00)))) as proof of (forall (Xu:a), ((x1 ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (x1 ((cP (cL Xt)) (cL Xu))))) (x1 ((cP (cR Xt)) (cR Xu))))))
% Found (fun (Xt:a) (Xu:a) (x00:(x1 ((cP Xt) Xu)))=> (((fun (P:Type) (x4:((x2 ((cP Xx) Xy))->((forall (Xt:a) (Xu:a), ((x2 ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (x2 ((cP (cL Xt)) (cL Xu))))) (x2 ((cP (cR Xt)) (cR Xu))))))->P)))=> (((((and_rect (x2 ((cP Xx) Xy))) (forall (Xt:a) (Xu:a), ((x2 ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (x2 ((cP (cL Xt)) (cL Xu))))) (x2 ((cP (cR Xt)) (cR Xu))))))) P) x4) x3)) ((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (x1 ((cP (cL Xt)) (cL Xu))))) (x1 ((cP (cR Xt)) (cR Xu))))) (fun (x4:(x2 ((cP Xx) Xy))) (x5:(forall (Xt0:a) (Xu0:a), ((x2 ((cP Xt0) Xu0))->((and ((and ((((eq a) Xu0) cZ)->(((eq a) Xt0) cZ))) (x2 ((cP (cL Xt0)) (cL Xu0))))) (x2 ((cP (cR Xt0)) (cR Xu0)))))))=> (((x5 Xt) Xu) x00)))) as proof of (forall (Xt:a) (Xu:a), ((x1 ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (x1 ((cP (cL Xt)) (cL Xu))))) (x1 ((cP (cR Xt)) (cR Xu))))))
% Found x0:((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xx) Xy))) (forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu)))))))))
% Instantiate: f:=(fun (X:(a->Prop))=> ((and (X ((cP Xx) Xy))) (forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu)))))))):((a->Prop)->Prop)
% Found x0 as proof of (P f)
% Found x0:((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xx) Xy))) (forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu)))))))))
% Instantiate: f:=(fun (X:(a->Prop))=> ((and (X ((cP Xx) Xy))) (forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu)))))))):((a->Prop)->Prop)
% Found x0 as proof of (P f)
% Found eq_substitution0000:=(fun (a0:a)=> ((eq_substitution000 a0) (cP (cL Xx)))):(forall (a0:a), ((((eq a) a0) (cL Xy))->(((eq a) ((cP (cL Xx)) a0)) ((cP (cL Xx)) (cL Xy)))))
% Instantiate: x1:=(fun (x6:a)=> (forall (a0:a), ((((eq a) a0) (cL Xy))->(((eq a) ((cP (cL Xx)) a0)) x6)))):(a->Prop)
% Found (fun (a0:a)=> ((eq_substitution000 a0) (cP (cL Xx)))) as proof of (x1 ((cP (cL Xx)) (cL Xy)))
% Found (fun (a0:a)=> (((fun (a0:a)=> ((eq_substitution00 a0) (cL Xy))) a0) (cP (cL Xx)))) as proof of (x1 ((cP (cL Xx)) (cL Xy)))
% Found (fun (a0:a)=> (((fun (a0:a)=> (((eq_substitution0 a) a0) (cL Xy))) a0) (cP (cL Xx)))) as proof of (x1 ((cP (cL Xx)) (cL Xy)))
% Found (fun (a0:a)=> (((fun (a0:a)=> ((((eq_substitution a) a) a0) (cL Xy))) a0) (cP (cL Xx)))) as proof of (x1 ((cP (cL Xx)) (cL Xy)))
% Found (fun (a0:a)=> (((fun (a0:a)=> ((((eq_substitution a) a) a0) (cL Xy))) a0) (cP (cL Xx)))) as proof of (x1 ((cP (cL Xx)) (cL Xy)))
% Found eq_substitution0000:=(fun (a0:a)=> ((eq_substitution000 a0) (cP (cL Xx)))):(forall (a0:a), ((((eq a) a0) (cL Xy))->(((eq a) ((cP (cL Xx)) a0)) ((cP (cL Xx)) (cL Xy)))))
% Instantiate: x3:=(fun (x6:a)=> (forall (a0:a), ((((eq a) a0) (cL Xy))->(((eq a) ((cP (cL Xx)) a0)) x6)))):(a->Prop)
% Found (fun (a0:a)=> ((eq_substitution000 a0) (cP (cL Xx)))) as proof of (x3 ((cP (cL Xx)) (cL Xy)))
% Found (fun (a0:a)=> (((fun (a0:a)=> ((eq_substitution00 a0) (cL Xy))) a0) (cP (cL Xx)))) as proof of (x3 ((cP (cL Xx)) (cL Xy)))
% Found (fun (a0:a)=> (((fun (a0:a)=> (((eq_substitution0 a) a0) (cL Xy))) a0) (cP (cL Xx)))) as proof of (x3 ((cP (cL Xx)) (cL Xy)))
% Found (fun (a0:a)=> (((fun (a0:a)=> ((((eq_substitution a) a) a0) (cL Xy))) a0) (cP (cL Xx)))) as proof of (x3 ((cP (cL Xx)) (cL Xy)))
% Found (fun (a0:a)=> (((fun (a0:a)=> ((((eq_substitution a) a) a0) (cL Xy))) a0) (cP (cL Xx)))) as proof of (x3 ((cP (cL Xx)) (cL Xy)))
% Found eq_ref000:=(eq_ref00 P0):((P0 (fun (X:(a->Prop))=> ((and (X ((cP (cL Xx)) (cL Xy)))) (forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu)))))))))->(P0 (fun (X:(a->Prop))=> ((and (X ((cP (cL Xx)) (cL Xy)))) (forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu))))))))))
% Found (eq_ref00 P0) as proof of (P1 (fun (X:(a->Prop))=> ((and (X ((cP (cL Xx)) (cL Xy)))) (forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu)))))))))
% Found ((eq_ref0 (fun (X:(a->Prop))=> ((and (X ((cP (cL Xx)) (cL Xy)))) (forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu))))))))) P0) as proof of (P1 (fun (X:(a->Prop))=> ((and (X ((cP (cL Xx)) (cL Xy)))) (forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu)))))))))
% Found (((eq_ref ((a->Prop)->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP (cL Xx)) (cL Xy)))) (forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu))))))))) P0) as proof of (P1 (fun (X:(a->Prop))=> ((and (X ((cP (cL Xx)) (cL Xy)))) (forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu)))))))))
% Found (((eq_ref ((a->Prop)->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP (cL Xx)) (cL Xy)))) (forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu))))))))) P0) as proof of (P1 (fun (X:(a->Prop))=> ((and (X ((cP (cL Xx)) (cL Xy)))) (forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu)))))))))
% Found eq_ref000:=(eq_ref00 P0):((P0 (fun (X:(a->Prop))=> ((and (X ((cP (cL Xx)) (cL Xy)))) (forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu)))))))))->(P0 (fun (X:(a->Prop))=> ((and (X ((cP (cL Xx)) (cL Xy)))) (forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu))))))))))
% Found (eq_ref00 P0) as proof of (P1 (fun (X:(a->Prop))=> ((and (X ((cP (cL Xx)) (cL Xy)))) (forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu)))))))))
% Found ((eq_ref0 (fun (X:(a->Prop))=> ((and (X ((cP (cL Xx)) (cL Xy)))) (forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu))))))))) P0) as proof of (P1 (fun (X:(a->Prop))=> ((and (X ((cP (cL Xx)) (cL Xy)))) (forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu)))))))))
% Found (((eq_ref ((a->Prop)->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP (cL Xx)) (cL Xy)))) (forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu))))))))) P0) as proof of (P1 (fun (X:(a->Prop))=> ((and (X ((cP (cL Xx)) (cL Xy)))) (forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu)))))))))
% Found (((eq_ref ((a->Prop)->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP (cL Xx)) (cL Xy)))) (forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu))))))))) P0) as proof of (P1 (fun (X:(a->Prop))=> ((and (X ((cP (cL Xx)) (cL Xy)))) (forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu)))))))))
% Found eq_substitution0000:=(fun (a0:a)=> ((eq_substitution000 a0) (cP (cL Xx)))):(forall (a0:a), ((((eq a) a0) (cL Xy))->(((eq a) ((cP (cL Xx)) a0)) ((cP (cL Xx)) (cL Xy)))))
% Instantiate: x1:=(fun (x4:a)=> (forall (a0:a), ((((eq a) a0) (cL Xy))->(((eq a) ((cP (cL Xx)) a0)) x4)))):(a->Prop)
% Found (fun (a0:a)=> ((eq_substitution000 a0) (cP (cL Xx)))) as proof of (x1 ((cP (cL Xx)) (cL Xy)))
% Found (fun (a0:a)=> (((fun (a0:a)=> ((eq_substitution00 a0) (cL Xy))) a0) (cP (cL Xx)))) as proof of (x1 ((cP (cL Xx)) (cL Xy)))
% Found (fun (a0:a)=> (((fun (a0:a)=> (((eq_substitution0 a) a0) (cL Xy))) a0) (cP (cL Xx)))) as proof of (x1 ((cP (cL Xx)) (cL Xy)))
% Found (fun (a0:a)=> (((fun (a0:a)=> ((((eq_substitution a) a) a0) (cL Xy))) a0) (cP (cL Xx)))) as proof of (x1 ((cP (cL Xx)) (cL Xy)))
% Found (fun (a0:a)=> (((fun (a0:a)=> ((((eq_substitution a) a) a0) (cL Xy))) a0) (cP (cL Xx)))) as proof of (x1 ((cP (cL Xx)) (cL Xy)))
% Found x7:(forall (Xt:a) (Xu:a), ((x4 ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (x4 ((cP (cL Xt)) (cL Xu))))) (x4 ((cP (cR Xt)) (cR Xu))))))
% Instantiate: x1:=x4:(a->Prop)
% Found x7 as proof of (forall (Xt:a) (Xu:a), ((x1 ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (x1 ((cP (cL Xt)) (cL Xu))))) (x1 ((cP (cR Xt)) (cR Xu))))))
% Found x4:(forall (Xt:a) (Xu:a), ((x1 ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (x1 ((cP (cL Xt)) (cL Xu))))) (x1 ((cP (cR Xt)) (cR Xu))))))
% Instantiate: x5:=x1:(a->Prop)
% Found x4 as proof of (forall (Xt:a) (Xu:a), ((x5 ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (x5 ((cP (cL Xt)) (cL Xu))))) (x5 ((cP (cR Xt)) (cR Xu))))))
% Found x7:(forall (Xt:a) (Xu:a), ((x1 ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (x1 ((cP (cL Xt)) (cL Xu))))) (x1 ((cP (cR Xt)) (cR Xu))))))
% Instantiate: x5:=x1:(a->Prop)
% Found x7 as proof of (forall (Xt:a) (Xu:a), ((x5 ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (x5 ((cP (cL Xt)) (cL Xu))))) (x5 ((cP (cR Xt)) (cR Xu))))))
% Found x60:=(fun (Xx0:a)=> ((x6 Xx0) ((cP (cL Xx)) (cL Xy)))):(forall (Xx0:a), (((eq a) (cR ((cP Xx0) ((cP (cL Xx)) (cL Xy))))) ((cP (cL Xx)) (cL Xy))))
% Instantiate: x7:=(fun (x8:a)=> (forall (Xx0:a), (((eq a) (cR ((cP Xx0) x8))) x8))):(a->Prop)
% Found (fun (Xx0:a)=> ((x6 Xx0) ((cP (cL Xx)) (cL Xy)))) as proof of (x7 ((cP (cL Xx)) (cL Xy)))
% Found (fun (Xx0:a)=> ((x6 Xx0) ((cP (cL Xx)) (cL Xy)))) as proof of (x7 ((cP (cL Xx)) (cL Xy)))
% 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 Xx) Xy))) (forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu)))))))))
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) (fun (X:(a->Prop))=> ((and (X ((cP Xx) Xy))) (forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu)))))))))
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) (fun (X:(a->Prop))=> ((and (X ((cP Xx) Xy))) (forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu)))))))))
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) (fun (X:(a->Prop))=> ((and (X ((cP Xx) Xy))) (forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu)))))))))
% Found eq_ref00:=(eq_ref0 (fun (X:(a->Prop))=> ((and (X ((cP (cL Xx)) (cL Xy)))) (forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu))))))))):(((eq ((a->Prop)->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP (cL Xx)) (cL Xy)))) (forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu))))))))) (fun (X:(a->Prop))=> ((and (X ((cP (cL Xx)) (cL Xy)))) (forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu)))))))))
% Found (eq_ref0 (fun (X:(a->Prop))=> ((and (X ((cP (cL Xx)) (cL Xy)))) (forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu))))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP (cL Xx)) (cL Xy)))) (forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu))))))))) b)
% Found ((eq_ref ((a->Prop)->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP (cL Xx)) (cL Xy)))) (forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu))))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP (cL Xx)) (cL Xy)))) (forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu))))))))) b)
% Found ((eq_ref ((a->Prop)->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP (cL Xx)) (cL Xy)))) (forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu))))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP (cL Xx)) (cL Xy)))) (forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu))))))))) b)
% Found ((eq_ref ((a->Prop)->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP (cL Xx)) (cL Xy)))) (forall (Xt:a) (Xu:a), ((X ((cP Xt) Xu))->((and ((and ((((eq a) Xu) cZ)->(((eq a) Xt) cZ))) (X ((cP (cL Xt)) (cL Xu))))) (X ((cP (cR Xt)) (cR Xu))))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (X:(a->Prop))
% EOF
%------------------------------------------------------------------------------