%------------------------------------------------------------------------------
% File     : cocATP---0.2.0
% Problem  : SEU982^5 : TPTP v6.1.0. Released v4.0.0.
% Transfm  : none
% Format   : tptp:raw
% Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p

% Computer : n108.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 13:33:28 EDT 2014

% Result   : Timeout 300.01s
% Output   : None 
% Verified : 
% SZS Type : None (Parsing solution fails)
% Syntax   : Number of formulae    : 0

% Comments : 
%------------------------------------------------------------------------------
%----NO SOLUTION OUTPUT BY SYSTEM
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% % Problem  : SEU982^5 : TPTP v6.1.0. Released v4.0.0.
% % Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n108.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 11:47:36 CDT 2014
% % CPUTime  : 300.01 
% 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 0x1d74680>, <kernel.Type object at 0x1d74878>) of role type named a_type
% Using role type
% Declaring a:Type
% FOF formula (<kernel.Constant object at 0x21acf38>, <kernel.DependentProduct object at 0x1f4c200>) of role type named cR
% Using role type
% Declaring cR:(a->a)
% FOF formula (<kernel.Constant object at 0x1d74ea8>, <kernel.DependentProduct object at 0x1d74440>) of role type named cP
% Using role type
% Declaring cP:(a->(a->a))
% FOF formula (<kernel.Constant object at 0x1d74638>, <kernel.DependentProduct object at 0x1f4c1b8>) of role type named cL
% Using role type
% Declaring cL:(a->a)
% FOF formula (<kernel.Constant object at 0x1d74440>, <kernel.Constant object at 0x1d74ea8>) 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))))))->((and ((and (forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx:a) (Xy:a), (((and (X Xx)) (X Xy))->(X ((cP Xx) Xy)))))->(X cZ)))) (forall (Xx:a), ((forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy:a), (((and (X Xx0)) (X Xy))->(X ((cP Xx0) Xy)))))->(X Xx)))->(forall (Xy:a), (((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xy) Xx))) (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)))))))))->(forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xy))))))))) (forall (Xx:a) (Xy:a), (((and (forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xx)))) (forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xy))))->((ex a) (fun (Xz:a)=> ((forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xz)))->((and ((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xx) Xz))) (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 Xy) Xz))) (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_LEM3C_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))))))->((and ((and (forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx:a) (Xy:a), (((and (X Xx)) (X Xy))->(X ((cP Xx) Xy)))))->(X cZ)))) (forall (Xx:a), ((forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy:a), (((and (X Xx0)) (X Xy))->(X ((cP Xx0) Xy)))))->(X Xx)))->(forall (Xy:a), (((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xy) Xx))) (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)))))))))->(forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xy))))))))) (forall (Xx:a) (Xy:a), (((and (forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xx)))) (forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xy))))->((ex a) (fun (Xz:a)=> ((forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xz)))->((and ((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xx) Xz))) (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 Xy) Xz))) (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))))))->((and ((and (forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx:a) (Xy:a), (((and (X Xx)) (X Xy))->(X ((cP Xx) Xy)))))->(X cZ)))) (forall (Xx:a), ((forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy:a), (((and (X Xx0)) (X Xy))->(X ((cP Xx0) Xy)))))->(X Xx)))->(forall (Xy:a), (((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xy) Xx))) (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)))))))))->(forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xy))))))))) (forall (Xx:a) (Xy:a), (((and (forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xx)))) (forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xy))))->((ex a) (fun (Xz:a)=> ((forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xz)))->((and ((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xx) Xz))) (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 Xy) Xz))) (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))))))->((and ((and (forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx:a) (Xy:a), (((and (X Xx)) (X Xy))->(X ((cP Xx) Xy)))))->(X cZ)))) (forall (Xx:a), ((forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy:a), (((and (X Xx0)) (X Xy))->(X ((cP Xx0) Xy)))))->(X Xx)))->(forall (Xy:a), (((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xy) Xx))) (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)))))))))->(forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xy))))))))) (forall (Xx:a) (Xy:a), (((and (forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xx)))) (forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xy))))->((ex a) (fun (Xz:a)=> ((forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xz)))->((and ((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xx) Xz))) (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 Xy) Xz))) (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 (forall (Xx:a) (Xy:a), (((and (forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xx)))) (forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xy))))->((ex a) (fun (Xz:a)=> ((forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xz)))->((and ((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xx) Xz))) (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 Xy) Xz))) (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 Prop) (forall (Xx:a) (Xy:a), (((and (forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xx)))) (forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xy))))->((ex a) (fun (Xz:a)=> ((forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xz)))->((and ((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xx) Xz))) (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 Xy) Xz))) (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)))))))))))))))) (forall (Xx:a) (Xy:a), (((and (forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xx)))) (forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xy))))->((ex a) (fun (Xz:a)=> ((forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xz)))->((and ((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xx) Xz))) (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 Xy) Xz))) (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 (forall (Xx:a) (Xy:a), (((and (forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xx)))) (forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xy))))->((ex a) (fun (Xz:a)=> ((forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xz)))->((and ((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xx) Xz))) (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 Xy) Xz))) (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 Prop) (forall (Xx:a) (Xy:a), (((and (forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xx)))) (forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xy))))->((ex a) (fun (Xz:a)=> ((forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xz)))->((and ((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xx) Xz))) (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 Xy) Xz))) (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 Prop) (forall (Xx:a) (Xy:a), (((and (forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xx)))) (forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xy))))->((ex a) (fun (Xz:a)=> ((forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xz)))->((and ((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xx) Xz))) (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 Xy) Xz))) (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 Prop) (forall (Xx:a) (Xy:a), (((and (forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xx)))) (forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xy))))->((ex a) (fun (Xz:a)=> ((forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xz)))->((and ((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xx) Xz))) (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 Xy) Xz))) (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 Prop) (forall (Xx:a) (Xy:a), (((and (forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xx)))) (forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xy))))->((ex a) (fun (Xz:a)=> ((forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xz)))->((and ((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xx) Xz))) (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 Xy) Xz))) (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 Prop) (forall (Xx:a) (Xy:a), (((and (forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xx)))) (forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xy))))->((ex a) (fun (Xz:a)=> ((forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xz)))->((and ((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xx) Xz))) (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 Xy) Xz))) (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 Prop) (forall (Xx:a) (Xy:a), (((and (forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xx)))) (forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xy))))->((ex a) (fun (Xz:a)=> ((forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xz)))->((and ((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xx) Xz))) (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 Xy) Xz))) (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 Prop) (forall (Xx:a) (Xy:a), (((and (forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xx)))) (forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xy))))->((ex a) (fun (Xz:a)=> ((forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xz)))->((and ((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xx) Xz))) (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 Xy) Xz))) (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 conj:(forall (A:Prop) (B:Prop), (A->(B->((and A) B))))
% Instantiate: b:=(forall (A:Prop) (B:Prop), (A->(B->((and A) B)))):Prop
% Found conj as proof of b
% Found eq_ref00:=(eq_ref0 (forall (Xx:a) (Xy:a), (((and (forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xx)))) (forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xy))))->((ex a) (fun (Xz:a)=> ((forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xz)))->((and ((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xx) Xz))) (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 Xy) Xz))) (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 Prop) (forall (Xx:a) (Xy:a), (((and (forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xx)))) (forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xy))))->((ex a) (fun (Xz:a)=> ((forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xz)))->((and ((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xx) Xz))) (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 Xy) Xz))) (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)))))))))))))))) (forall (Xx:a) (Xy:a), (((and (forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xx)))) (forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xy))))->((ex a) (fun (Xz:a)=> ((forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xz)))->((and ((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xx) Xz))) (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 Xy) Xz))) (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 (forall (Xx:a) (Xy:a), (((and (forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xx)))) (forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xy))))->((ex a) (fun (Xz:a)=> ((forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xz)))->((and ((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xx) Xz))) (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 Xy) Xz))) (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 Prop) (forall (Xx:a) (Xy:a), (((and (forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xx)))) (forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xy))))->((ex a) (fun (Xz:a)=> ((forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xz)))->((and ((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xx) Xz))) (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 Xy) Xz))) (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 Prop) (forall (Xx:a) (Xy:a), (((and (forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xx)))) (forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xy))))->((ex a) (fun (Xz:a)=> ((forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xz)))->((and ((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xx) Xz))) (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 Xy) Xz))) (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 Prop) (forall (Xx:a) (Xy:a), (((and (forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xx)))) (forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xy))))->((ex a) (fun (Xz:a)=> ((forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xz)))->((and ((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xx) Xz))) (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 Xy) Xz))) (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 Prop) (forall (Xx:a) (Xy:a), (((and (forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xx)))) (forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xy))))->((ex a) (fun (Xz:a)=> ((forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xz)))->((and ((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xx) Xz))) (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 Xy) Xz))) (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 Prop) (forall (Xx:a) (Xy:a), (((and (forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xx)))) (forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xy))))->((ex a) (fun (Xz:a)=> ((forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xz)))->((and ((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xx) Xz))) (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 Xy) Xz))) (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 Prop) (forall (Xx:a) (Xy:a), (((and (forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xx)))) (forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xy))))->((ex a) (fun (Xz:a)=> ((forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xz)))->((and ((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xx) Xz))) (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 Xy) Xz))) (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 Prop) (forall (Xx:a) (Xy:a), (((and (forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xx)))) (forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xy))))->((ex a) (fun (Xz:a)=> ((forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xz)))->((and ((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xx) Xz))) (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 Xy) Xz))) (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 x1:(X cZ)
% Found (fun (x2:(forall (Xx:a) (Xy:a), (((and (X Xx)) (X Xy))->(X ((cP Xx) Xy)))))=> x1) as proof of (X cZ)
% Found (fun (x1:(X cZ)) (x2:(forall (Xx:a) (Xy:a), (((and (X Xx)) (X Xy))->(X ((cP Xx) Xy)))))=> x1) as proof of ((forall (Xx:a) (Xy:a), (((and (X Xx)) (X Xy))->(X ((cP Xx) Xy))))->(X cZ))
% Found (fun (x1:(X cZ)) (x2:(forall (Xx:a) (Xy:a), (((and (X Xx)) (X Xy))->(X ((cP Xx) Xy)))))=> x1) as proof of ((X cZ)->((forall (Xx:a) (Xy:a), (((and (X Xx)) (X Xy))->(X ((cP Xx) Xy))))->(X cZ)))
% Found (and_rect00 (fun (x1:(X cZ)) (x2:(forall (Xx:a) (Xy:a), (((and (X Xx)) (X Xy))->(X ((cP Xx) Xy)))))=> x1)) as proof of (X cZ)
% Found ((and_rect0 (X cZ)) (fun (x1:(X cZ)) (x2:(forall (Xx:a) (Xy:a), (((and (X Xx)) (X Xy))->(X ((cP Xx) Xy)))))=> x1)) as proof of (X cZ)
% Found (((fun (P:Type) (x1:((X cZ)->((forall (Xx:a) (Xy:a), (((and (X Xx)) (X Xy))->(X ((cP Xx) Xy))))->P)))=> (((((and_rect (X cZ)) (forall (Xx:a) (Xy:a), (((and (X Xx)) (X Xy))->(X ((cP Xx) Xy))))) P) x1) x0)) (X cZ)) (fun (x1:(X cZ)) (x2:(forall (Xx:a) (Xy:a), (((and (X Xx)) (X Xy))->(X ((cP Xx) Xy)))))=> x1)) as proof of (X cZ)
% Found (fun (x0:((and (X cZ)) (forall (Xx:a) (Xy:a), (((and (X Xx)) (X Xy))->(X ((cP Xx) Xy))))))=> (((fun (P:Type) (x1:((X cZ)->((forall (Xx:a) (Xy:a), (((and (X Xx)) (X Xy))->(X ((cP Xx) Xy))))->P)))=> (((((and_rect (X cZ)) (forall (Xx:a) (Xy:a), (((and (X Xx)) (X Xy))->(X ((cP Xx) Xy))))) P) x1) x0)) (X cZ)) (fun (x1:(X cZ)) (x2:(forall (Xx:a) (Xy:a), (((and (X Xx)) (X Xy))->(X ((cP Xx) Xy)))))=> x1))) as proof of (X cZ)
% Found (fun (X:(a->Prop)) (x0:((and (X cZ)) (forall (Xx:a) (Xy:a), (((and (X Xx)) (X Xy))->(X ((cP Xx) Xy))))))=> (((fun (P:Type) (x1:((X cZ)->((forall (Xx:a) (Xy:a), (((and (X Xx)) (X Xy))->(X ((cP Xx) Xy))))->P)))=> (((((and_rect (X cZ)) (forall (Xx:a) (Xy:a), (((and (X Xx)) (X Xy))->(X ((cP Xx) Xy))))) P) x1) x0)) (X cZ)) (fun (x1:(X cZ)) (x2:(forall (Xx:a) (Xy:a), (((and (X Xx)) (X Xy))->(X ((cP Xx) Xy)))))=> x1))) as proof of (((and (X cZ)) (forall (Xx:a) (Xy:a), (((and (X Xx)) (X Xy))->(X ((cP Xx) Xy)))))->(X cZ))
% Found (fun (X:(a->Prop)) (x0:((and (X cZ)) (forall (Xx:a) (Xy:a), (((and (X Xx)) (X Xy))->(X ((cP Xx) Xy))))))=> (((fun (P:Type) (x1:((X cZ)->((forall (Xx:a) (Xy:a), (((and (X Xx)) (X Xy))->(X ((cP Xx) Xy))))->P)))=> (((((and_rect (X cZ)) (forall (Xx:a) (Xy:a), (((and (X Xx)) (X Xy))->(X ((cP Xx) Xy))))) P) x1) x0)) (X cZ)) (fun (x1:(X cZ)) (x2:(forall (Xx:a) (Xy:a), (((and (X Xx)) (X Xy))->(X ((cP Xx) Xy)))))=> x1))) as proof of (forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx:a) (Xy:a), (((and (X Xx)) (X Xy))->(X ((cP Xx) Xy)))))->(X cZ)))
% Found iff_refl:=(fun (A:Prop)=> ((((conj (A->A)) (A->A)) (fun (H:A)=> H)) (fun (H:A)=> H))):(forall (P:Prop), ((iff P) P))
% Instantiate: b:=(forall (P:Prop), ((iff P) P)):Prop
% Found iff_refl as proof of b
% Found eq_ref00:=(eq_ref0 (forall (Xx:a) (Xy:a), (((and (forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xx)))) (forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xy))))->((ex a) (fun (Xz:a)=> ((forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xz)))->((and ((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xx) Xz))) (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 Xy) Xz))) (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 Prop) (forall (Xx:a) (Xy:a), (((and (forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xx)))) (forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xy))))->((ex a) (fun (Xz:a)=> ((forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xz)))->((and ((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xx) Xz))) (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 Xy) Xz))) (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)))))))))))))))) (forall (Xx:a) (Xy:a), (((and (forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xx)))) (forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xy))))->((ex a) (fun (Xz:a)=> ((forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xz)))->((and ((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xx) Xz))) (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 Xy) Xz))) (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 (forall (Xx:a) (Xy:a), (((and (forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xx)))) (forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xy))))->((ex a) (fun (Xz:a)=> ((forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xz)))->((and ((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xx) Xz))) (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 Xy) Xz))) (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 Prop) (forall (Xx:a) (Xy:a), (((and (forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xx)))) (forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xy))))->((ex a) (fun (Xz:a)=> ((forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xz)))->((and ((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xx) Xz))) (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 Xy) Xz))) (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 Prop) (forall (Xx:a) (Xy:a), (((and (forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xx)))) (forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xy))))->((ex a) (fun (Xz:a)=> ((forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xz)))->((and ((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xx) Xz))) (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 Xy) Xz))) (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 Prop) (forall (Xx:a) (Xy:a), (((and (forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xx)))) (forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xy))))->((ex a) (fun (Xz:a)=> ((forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xz)))->((and ((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xx) Xz))) (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 Xy) Xz))) (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 Prop) (forall (Xx:a) (Xy:a), (((and (forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xx)))) (forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xy))))->((ex a) (fun (Xz:a)=> ((forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xz)))->((and ((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xx) Xz))) (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 Xy) Xz))) (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 Prop) (forall (Xx:a) (Xy:a), (((and (forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xx)))) (forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xy))))->((ex a) (fun (Xz:a)=> ((forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xz)))->((and ((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xx) Xz))) (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 Xy) Xz))) (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 Prop) (forall (Xx:a) (Xy:a), (((and (forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xx)))) (forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xy))))->((ex a) (fun (Xz:a)=> ((forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xz)))->((and ((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xx) Xz))) (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 Xy) Xz))) (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 Prop) (forall (Xx:a) (Xy:a), (((and (forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xx)))) (forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xy))))->((ex a) (fun (Xz:a)=> ((forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xz)))->((and ((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xx) Xz))) (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 Xy) Xz))) (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 iff_refl:=(fun (A:Prop)=> ((((conj (A->A)) (A->A)) (fun (H:A)=> H)) (fun (H:A)=> H))):(forall (P:Prop), ((iff P) P))
% Instantiate: b:=(forall (P:Prop), ((iff P) P)):Prop
% Found iff_refl as proof of b
% Found x3:(X cZ)
% Found (fun (x4:(forall (Xx:a) (Xy:a), (((and (X Xx)) (X Xy))->(X ((cP Xx) Xy)))))=> x3) as proof of (X cZ)
% Found (fun (x3:(X cZ)) (x4:(forall (Xx:a) (Xy:a), (((and (X Xx)) (X Xy))->(X ((cP Xx) Xy)))))=> x3) as proof of ((forall (Xx:a) (Xy:a), (((and (X Xx)) (X Xy))->(X ((cP Xx) Xy))))->(X cZ))
% Found (fun (x3:(X cZ)) (x4:(forall (Xx:a) (Xy:a), (((and (X Xx)) (X Xy))->(X ((cP Xx) Xy)))))=> x3) as proof of ((X cZ)->((forall (Xx:a) (Xy:a), (((and (X Xx)) (X Xy))->(X ((cP Xx) Xy))))->(X cZ)))
% Found (and_rect10 (fun (x3:(X cZ)) (x4:(forall (Xx:a) (Xy:a), (((and (X Xx)) (X Xy))->(X ((cP Xx) Xy)))))=> x3)) as proof of (X cZ)
% Found ((and_rect1 (X cZ)) (fun (x3:(X cZ)) (x4:(forall (Xx:a) (Xy:a), (((and (X Xx)) (X Xy))->(X ((cP Xx) Xy)))))=> x3)) as proof of (X cZ)
% Found (((fun (P:Type) (x3:((X cZ)->((forall (Xx:a) (Xy:a), (((and (X Xx)) (X Xy))->(X ((cP Xx) Xy))))->P)))=> (((((and_rect (X cZ)) (forall (Xx:a) (Xy:a), (((and (X Xx)) (X Xy))->(X ((cP Xx) Xy))))) P) x3) x2)) (X cZ)) (fun (x3:(X cZ)) (x4:(forall (Xx:a) (Xy:a), (((and (X Xx)) (X Xy))->(X ((cP Xx) Xy)))))=> x3)) as proof of (X cZ)
% Found (fun (x2:((and (X cZ)) (forall (Xx:a) (Xy:a), (((and (X Xx)) (X Xy))->(X ((cP Xx) Xy))))))=> (((fun (P:Type) (x3:((X cZ)->((forall (Xx:a) (Xy:a), (((and (X Xx)) (X Xy))->(X ((cP Xx) Xy))))->P)))=> (((((and_rect (X cZ)) (forall (Xx:a) (Xy:a), (((and (X Xx)) (X Xy))->(X ((cP Xx) Xy))))) P) x3) x2)) (X cZ)) (fun (x3:(X cZ)) (x4:(forall (Xx:a) (Xy:a), (((and (X Xx)) (X Xy))->(X ((cP Xx) Xy)))))=> x3))) as proof of (X cZ)
% Found (fun (X:(a->Prop)) (x2:((and (X cZ)) (forall (Xx:a) (Xy:a), (((and (X Xx)) (X Xy))->(X ((cP Xx) Xy))))))=> (((fun (P:Type) (x3:((X cZ)->((forall (Xx:a) (Xy:a), (((and (X Xx)) (X Xy))->(X ((cP Xx) Xy))))->P)))=> (((((and_rect (X cZ)) (forall (Xx:a) (Xy:a), (((and (X Xx)) (X Xy))->(X ((cP Xx) Xy))))) P) x3) x2)) (X cZ)) (fun (x3:(X cZ)) (x4:(forall (Xx:a) (Xy:a), (((and (X Xx)) (X Xy))->(X ((cP Xx) Xy)))))=> x3))) as proof of (((and (X cZ)) (forall (Xx:a) (Xy:a), (((and (X Xx)) (X Xy))->(X ((cP Xx) Xy)))))->(X cZ))
% Found (fun (X:(a->Prop)) (x2:((and (X cZ)) (forall (Xx:a) (Xy:a), (((and (X Xx)) (X Xy))->(X ((cP Xx) Xy))))))=> (((fun (P:Type) (x3:((X cZ)->((forall (Xx:a) (Xy:a), (((and (X Xx)) (X Xy))->(X ((cP Xx) Xy))))->P)))=> (((((and_rect (X cZ)) (forall (Xx:a) (Xy:a), (((and (X Xx)) (X Xy))->(X ((cP Xx) Xy))))) P) x3) x2)) (X cZ)) (fun (x3:(X cZ)) (x4:(forall (Xx:a) (Xy:a), (((and (X Xx)) (X Xy))->(X ((cP Xx) Xy)))))=> x3))) as proof of (forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx:a) (Xy:a), (((and (X Xx)) (X Xy))->(X ((cP Xx) Xy)))))->(X cZ)))
% Found or_ind:(forall (A:Prop) (B:Prop) (P:Prop), ((A->P)->((B->P)->(((or A) B)->P))))
% Instantiate: b:=(forall (A:Prop) (B:Prop) (P:Prop), ((A->P)->((B->P)->(((or A) B)->P)))):Prop
% Found or_ind as proof of b
% Found x6:(((eq a) Xt) cZ)
% Found x6 as proof of (((eq a) Xt) cZ)
% Found eq_ref00:=(eq_ref0 (forall (Xx:a), ((forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy:a), (((and (X Xx0)) (X Xy))->(X ((cP Xx0) Xy)))))->(X Xx)))->(forall (Xy:a), (((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xy) Xx))) (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)))))))))->(forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xy)))))))):(((eq Prop) (forall (Xx:a), ((forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy:a), (((and (X Xx0)) (X Xy))->(X ((cP Xx0) Xy)))))->(X Xx)))->(forall (Xy:a), (((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xy) Xx))) (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)))))))))->(forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xy)))))))) (forall (Xx:a), ((forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy:a), (((and (X Xx0)) (X Xy))->(X ((cP Xx0) Xy)))))->(X Xx)))->(forall (Xy:a), (((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xy) Xx))) (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)))))))))->(forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xy))))))))
% Found (eq_ref0 (forall (Xx:a), ((forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy:a), (((and (X Xx0)) (X Xy))->(X ((cP Xx0) Xy)))))->(X Xx)))->(forall (Xy:a), (((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xy) Xx))) (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)))))))))->(forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xy)))))))) as proof of (((eq Prop) (forall (Xx:a), ((forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy:a), (((and (X Xx0)) (X Xy))->(X ((cP Xx0) Xy)))))->(X Xx)))->(forall (Xy:a), (((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xy) Xx))) (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)))))))))->(forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xy)))))))) b)
% Found ((eq_ref Prop) (forall (Xx:a), ((forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy:a), (((and (X Xx0)) (X Xy))->(X ((cP Xx0) Xy)))))->(X Xx)))->(forall (Xy:a), (((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xy) Xx))) (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)))))))))->(forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xy)))))))) as proof of (((eq Prop) (forall (Xx:a), ((forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy:a), (((and (X Xx0)) (X Xy))->(X ((cP Xx0) Xy)))))->(X Xx)))->(forall (Xy:a), (((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xy) Xx))) (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)))))))))->(forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xy)))))))) b)
% Found ((eq_ref Prop) (forall (Xx:a), ((forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy:a), (((and (X Xx0)) (X Xy))->(X ((cP Xx0) Xy)))))->(X Xx)))->(forall (Xy:a), (((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xy) Xx))) (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)))))))))->(forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xy)))))))) as proof of (((eq Prop) (forall (Xx:a), ((forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy:a), (((and (X Xx0)) (X Xy))->(X ((cP Xx0) Xy)))))->(X Xx)))->(forall (Xy:a), (((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xy) Xx))) (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)))))))))->(forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xy)))))))) b)
% Found ((eq_ref Prop) (forall (Xx:a), ((forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy:a), (((and (X Xx0)) (X Xy))->(X ((cP Xx0) Xy)))))->(X Xx)))->(forall (Xy:a), (((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xy) Xx))) (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)))))))))->(forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xy)))))))) as proof of (((eq Prop) (forall (Xx:a), ((forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy:a), (((and (X Xx0)) (X Xy))->(X ((cP Xx0) Xy)))))->(X Xx)))->(forall (Xy:a), (((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xy) Xx))) (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)))))))))->(forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xy)))))))) b)
% Found eq_ref00:=(eq_ref0 (forall (Xx:a) (Xy:a), (((and (forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xx)))) (forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xy))))->((ex a) (fun (Xz:a)=> ((forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xz)))->((and ((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xx) Xz))) (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 Xy) Xz))) (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 Prop) (forall (Xx:a) (Xy:a), (((and (forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xx)))) (forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xy))))->((ex a) (fun (Xz:a)=> ((forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xz)))->((and ((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xx) Xz))) (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 Xy) Xz))) (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)))))))))))))))) (forall (Xx:a) (Xy:a), (((and (forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xx)))) (forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xy))))->((ex a) (fun (Xz:a)=> ((forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xz)))->((and ((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xx) Xz))) (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 Xy) Xz))) (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 (forall (Xx:a) (Xy:a), (((and (forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xx)))) (forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xy))))->((ex a) (fun (Xz:a)=> ((forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xz)))->((and ((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xx) Xz))) (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 Xy) Xz))) (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 Prop) (forall (Xx:a) (Xy:a), (((and (forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xx)))) (forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xy))))->((ex a) (fun (Xz:a)=> ((forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xz)))->((and ((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xx) Xz))) (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 Xy) Xz))) (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 Prop) (forall (Xx:a) (Xy:a), (((and (forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xx)))) (forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xy))))->((ex a) (fun (Xz:a)=> ((forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xz)))->((and ((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xx) Xz))) (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 Xy) Xz))) (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 Prop) (forall (Xx:a) (Xy:a), (((and (forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xx)))) (forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xy))))->((ex a) (fun (Xz:a)=> ((forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xz)))->((and ((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xx) Xz))) (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 Xy) Xz))) (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 Prop) (forall (Xx:a) (Xy:a), (((and (forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xx)))) (forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xy))))->((ex a) (fun (Xz:a)=> ((forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xz)))->((and ((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xx) Xz))) (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 Xy) Xz))) (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 Prop) (forall (Xx:a) (Xy:a), (((and (forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xx)))) (forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xy))))->((ex a) (fun (Xz:a)=> ((forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xz)))->((and ((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xx) Xz))) (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 Xy) Xz))) (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 Prop) (forall (Xx:a) (Xy:a), (((and (forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xx)))) (forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xy))))->((ex a) (fun (Xz:a)=> ((forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xz)))->((and ((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xx) Xz))) (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 Xy) Xz))) (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 Prop) (forall (Xx:a) (Xy:a), (((and (forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xx)))) (forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xy))))->((ex a) (fun (Xz:a)=> ((forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xz)))->((and ((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xx) Xz))) (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 Xy) Xz))) (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 or_ind:(forall (A:Prop) (B:Prop) (P:Prop), ((A->P)->((B->P)->(((or A) B)->P))))
% Instantiate: b:=(forall (A:Prop) (B:Prop) (P:Prop), ((A->P)->((B->P)->(((or A) B)->P)))):Prop
% Found or_ind as proof of b
% Found or_ind:(forall (A:Prop) (B:Prop) (P:Prop), ((A->P)->((B->P)->(((or A) B)->P))))
% Instantiate: b:=(forall (A:Prop) (B:Prop) (P:Prop), ((A->P)->((B->P)->(((or A) B)->P)))):Prop
% Found or_ind as proof of b
% Found x5:(X cZ)
% Found (fun (x6:(forall (Xx:a) (Xy:a), (((and (X Xx)) (X Xy))->(X ((cP Xx) Xy)))))=> x5) as proof of (X cZ)
% Found (fun (x5:(X cZ)) (x6:(forall (Xx:a) (Xy:a), (((and (X Xx)) (X Xy))->(X ((cP Xx) Xy)))))=> x5) as proof of ((forall (Xx:a) (Xy:a), (((and (X Xx)) (X Xy))->(X ((cP Xx) Xy))))->(X cZ))
% Found (fun (x5:(X cZ)) (x6:(forall (Xx:a) (Xy:a), (((and (X Xx)) (X Xy))->(X ((cP Xx) Xy)))))=> x5) as proof of ((X cZ)->((forall (Xx:a) (Xy:a), (((and (X Xx)) (X Xy))->(X ((cP Xx) Xy))))->(X cZ)))
% Found (and_rect20 (fun (x5:(X cZ)) (x6:(forall (Xx:a) (Xy:a), (((and (X Xx)) (X Xy))->(X ((cP Xx) Xy)))))=> x5)) as proof of (X cZ)
% Found ((and_rect2 (X cZ)) (fun (x5:(X cZ)) (x6:(forall (Xx:a) (Xy:a), (((and (X Xx)) (X Xy))->(X ((cP Xx) Xy)))))=> x5)) as proof of (X cZ)
% Found (((fun (P:Type) (x5:((X cZ)->((forall (Xx:a) (Xy:a), (((and (X Xx)) (X Xy))->(X ((cP Xx) Xy))))->P)))=> (((((and_rect (X cZ)) (forall (Xx:a) (Xy:a), (((and (X Xx)) (X Xy))->(X ((cP Xx) Xy))))) P) x5) x4)) (X cZ)) (fun (x5:(X cZ)) (x6:(forall (Xx:a) (Xy:a), (((and (X Xx)) (X Xy))->(X ((cP Xx) Xy)))))=> x5)) as proof of (X cZ)
% Found (fun (x4:((and (X cZ)) (forall (Xx:a) (Xy:a), (((and (X Xx)) (X Xy))->(X ((cP Xx) Xy))))))=> (((fun (P:Type) (x5:((X cZ)->((forall (Xx:a) (Xy:a), (((and (X Xx)) (X Xy))->(X ((cP Xx) Xy))))->P)))=> (((((and_rect (X cZ)) (forall (Xx:a) (Xy:a), (((and (X Xx)) (X Xy))->(X ((cP Xx) Xy))))) P) x5) x4)) (X cZ)) (fun (x5:(X cZ)) (x6:(forall (Xx:a) (Xy:a), (((and (X Xx)) (X Xy))->(X ((cP Xx) Xy)))))=> x5))) as proof of (X cZ)
% Found (fun (X:(a->Prop)) (x4:((and (X cZ)) (forall (Xx:a) (Xy:a), (((and (X Xx)) (X Xy))->(X ((cP Xx) Xy))))))=> (((fun (P:Type) (x5:((X cZ)->((forall (Xx:a) (Xy:a), (((and (X Xx)) (X Xy))->(X ((cP Xx) Xy))))->P)))=> (((((and_rect (X cZ)) (forall (Xx:a) (Xy:a), (((and (X Xx)) (X Xy))->(X ((cP Xx) Xy))))) P) x5) x4)) (X cZ)) (fun (x5:(X cZ)) (x6:(forall (Xx:a) (Xy:a), (((and (X Xx)) (X Xy))->(X ((cP Xx) Xy)))))=> x5))) as proof of (((and (X cZ)) (forall (Xx:a) (Xy:a), (((and (X Xx)) (X Xy))->(X ((cP Xx) Xy)))))->(X cZ))
% Found (fun (X:(a->Prop)) (x4:((and (X cZ)) (forall (Xx:a) (Xy:a), (((and (X Xx)) (X Xy))->(X ((cP Xx) Xy))))))=> (((fun (P:Type) (x5:((X cZ)->((forall (Xx:a) (Xy:a), (((and (X Xx)) (X Xy))->(X ((cP Xx) Xy))))->P)))=> (((((and_rect (X cZ)) (forall (Xx:a) (Xy:a), (((and (X Xx)) (X Xy))->(X ((cP Xx) Xy))))) P) x5) x4)) (X cZ)) (fun (x5:(X cZ)) (x6:(forall (Xx:a) (Xy:a), (((and (X Xx)) (X Xy))->(X ((cP Xx) Xy)))))=> x5))) as proof of (forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx:a) (Xy:a), (((and (X Xx)) (X Xy))->(X ((cP Xx) Xy)))))->(X cZ)))
% Found x3:(X cZ)
% Found (fun (x4:(forall (Xx:a) (Xy:a), (((and (X Xx)) (X Xy))->(X ((cP Xx) Xy)))))=> x3) as proof of (X cZ)
% Found (fun (x3:(X cZ)) (x4:(forall (Xx:a) (Xy:a), (((and (X Xx)) (X Xy))->(X ((cP Xx) Xy)))))=> x3) as proof of ((forall (Xx:a) (Xy:a), (((and (X Xx)) (X Xy))->(X ((cP Xx) Xy))))->(X cZ))
% Found (fun (x3:(X cZ)) (x4:(forall (Xx:a) (Xy:a), (((and (X Xx)) (X Xy))->(X ((cP Xx) Xy)))))=> x3) as proof of ((X cZ)->((forall (Xx:a) (Xy:a), (((and (X Xx)) (X Xy))->(X ((cP Xx) Xy))))->(X cZ)))
% Found (and_rect10 (fun (x3:(X cZ)) (x4:(forall (Xx:a) (Xy:a), (((and (X Xx)) (X Xy))->(X ((cP Xx) Xy)))))=> x3)) as proof of (X cZ)
% Found ((and_rect1 (X cZ)) (fun (x3:(X cZ)) (x4:(forall (Xx:a) (Xy:a), (((and (X Xx)) (X Xy))->(X ((cP Xx) Xy)))))=> x3)) as proof of (X cZ)
% Found (((fun (P:Type) (x3:((X cZ)->((forall (Xx:a) (Xy:a), (((and (X Xx)) (X Xy))->(X ((cP Xx) Xy))))->P)))=> (((((and_rect (X cZ)) (forall (Xx:a) (Xy:a), (((and (X Xx)) (X Xy))->(X ((cP Xx) Xy))))) P) x3) x2)) (X cZ)) (fun (x3:(X cZ)) (x4:(forall (Xx:a) (Xy:a), (((and (X Xx)) (X Xy))->(X ((cP Xx) Xy)))))=> x3)) as proof of (X cZ)
% Found (fun (x2:((and (X cZ)) (forall (Xx:a) (Xy:a), (((and (X Xx)) (X Xy))->(X ((cP Xx) Xy))))))=> (((fun (P:Type) (x3:((X cZ)->((forall (Xx:a) (Xy:a), (((and (X Xx)) (X Xy))->(X ((cP Xx) Xy))))->P)))=> (((((and_rect (X cZ)) (forall (Xx:a) (Xy:a), (((and (X Xx)) (X Xy))->(X ((cP Xx) Xy))))) P) x3) x2)) (X cZ)) (fun (x3:(X cZ)) (x4:(forall (Xx:a) (Xy:a), (((and (X Xx)) (X Xy))->(X ((cP Xx) Xy)))))=> x3))) as proof of (X cZ)
% Found (fun (X:(a->Prop)) (x2:((and (X cZ)) (forall (Xx:a) (Xy:a), (((and (X Xx)) (X Xy))->(X ((cP Xx) Xy))))))=> (((fun (P:Type) (x3:((X cZ)->((forall (Xx:a) (Xy:a), (((and (X Xx)) (X Xy))->(X ((cP Xx) Xy))))->P)))=> (((((and_rect (X cZ)) (forall (Xx:a) (Xy:a), (((and (X Xx)) (X Xy))->(X ((cP Xx) Xy))))) P) x3) x2)) (X cZ)) (fun (x3:(X cZ)) (x4:(forall (Xx:a) (Xy:a), (((and (X Xx)) (X Xy))->(X ((cP Xx) Xy)))))=> x3))) as proof of (((and (X cZ)) (forall (Xx:a) (Xy:a), (((and (X Xx)) (X Xy))->(X ((cP Xx) Xy)))))->(X cZ))
% Found (fun (X:(a->Prop)) (x2:((and (X cZ)) (forall (Xx:a) (Xy:a), (((and (X Xx)) (X Xy))->(X ((cP Xx) Xy))))))=> (((fun (P:Type) (x3:((X cZ)->((forall (Xx:a) (Xy:a), (((and (X Xx)) (X Xy))->(X ((cP Xx) Xy))))->P)))=> (((((and_rect (X cZ)) (forall (Xx:a) (Xy:a), (((and (X Xx)) (X Xy))->(X ((cP Xx) Xy))))) P) x3) x2)) (X cZ)) (fun (x3:(X cZ)) (x4:(forall (Xx:a) (Xy:a), (((and (X Xx)) (X Xy))->(X ((cP Xx) Xy)))))=> x3))) as proof of (forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx:a) (Xy:a), (((and (X Xx)) (X Xy))->(X ((cP Xx) Xy)))))->(X cZ)))
% Found eq_ref00:=(eq_ref0 (forall (Xx:a), ((forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy:a), (((and (X Xx0)) (X Xy))->(X ((cP Xx0) Xy)))))->(X Xx)))->(forall (Xy:a), (((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xy) Xx))) (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)))))))))->(forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xy)))))))):(((eq Prop) (forall (Xx:a), ((forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy:a), (((and (X Xx0)) (X Xy))->(X ((cP Xx0) Xy)))))->(X Xx)))->(forall (Xy:a), (((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xy) Xx))) (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)))))))))->(forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xy)))))))) (forall (Xx:a), ((forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy:a), (((and (X Xx0)) (X Xy))->(X ((cP Xx0) Xy)))))->(X Xx)))->(forall (Xy:a), (((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xy) Xx))) (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)))))))))->(forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xy))))))))
% Found (eq_ref0 (forall (Xx:a), ((forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy:a), (((and (X Xx0)) (X Xy))->(X ((cP Xx0) Xy)))))->(X Xx)))->(forall (Xy:a), (((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xy) Xx))) (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)))))))))->(forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xy)))))))) as proof of (((eq Prop) (forall (Xx:a), ((forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy:a), (((and (X Xx0)) (X Xy))->(X ((cP Xx0) Xy)))))->(X Xx)))->(forall (Xy:a), (((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xy) Xx))) (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)))))))))->(forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xy)))))))) b)
% Found ((eq_ref Prop) (forall (Xx:a), ((forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy:a), (((and (X Xx0)) (X Xy))->(X ((cP Xx0) Xy)))))->(X Xx)))->(forall (Xy:a), (((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xy) Xx))) (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)))))))))->(forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xy)))))))) as proof of (((eq Prop) (forall (Xx:a), ((forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy:a), (((and (X Xx0)) (X Xy))->(X ((cP Xx0) Xy)))))->(X Xx)))->(forall (Xy:a), (((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xy) Xx))) (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)))))))))->(forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xy)))))))) b)
% Found ((eq_ref Prop) (forall (Xx:a), ((forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy:a), (((and (X Xx0)) (X Xy))->(X ((cP Xx0) Xy)))))->(X Xx)))->(forall (Xy:a), (((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xy) Xx))) (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)))))))))->(forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xy)))))))) as proof of (((eq Prop) (forall (Xx:a), ((forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy:a), (((and (X Xx0)) (X Xy))->(X ((cP Xx0) Xy)))))->(X Xx)))->(forall (Xy:a), (((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xy) Xx))) (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)))))))))->(forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xy)))))))) b)
% Found ((eq_ref Prop) (forall (Xx:a), ((forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy:a), (((and (X Xx0)) (X Xy))->(X ((cP Xx0) Xy)))))->(X Xx)))->(forall (Xy:a), (((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xy) Xx))) (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)))))))))->(forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xy)))))))) as proof of (((eq Prop) (forall (Xx:a), ((forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy:a), (((and (X Xx0)) (X Xy))->(X ((cP Xx0) Xy)))))->(X Xx)))->(forall (Xy:a), (((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xy) Xx))) (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)))))))))->(forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xy)))))))) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xz:a)=> ((forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xz)))->((and ((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xx) Xz))) (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 Xy) Xz))) (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)) (fun (Xz:a)=> ((forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xz)))->((and ((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xx) Xz))) (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 Xy) Xz))) (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)=> ((forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X x)))->((and ((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xx) x))) (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 Xy) x))) (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 (Xz:a)=> ((forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xz)))->((and ((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xx) Xz))) (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 Xy) Xz))) (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)) (fun (Xz:a)=> ((forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xz)))->((and ((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xx) Xz))) (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 Xy) Xz))) (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 (x2:a)=> Prop)) (fun (Xz:a)=> ((forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xz)))->((and ((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xx) Xz))) (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 Xy) Xz))) (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)) (fun (Xz:a)=> ((forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xz)))->((and ((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xx) Xz))) (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 Xy) Xz))) (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) (fun (x2:a)=> Prop)) (fun (Xz:a)=> ((forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xz)))->((and ((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xx) Xz))) (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 Xy) Xz))) (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)) (fun (Xz:a)=> ((forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xz)))->((and ((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xx) Xz))) (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 Xy) Xz))) (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) (fun (x2:a)=> Prop)) (fun (Xz:a)=> ((forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xz)))->((and ((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xx) Xz))) (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 Xy) Xz))) (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)) (fun (Xz:a)=> ((forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx0:a) (Xy0:a), (((and (X Xx0)) (X Xy0))->(X ((cP Xx0) Xy0)))))->(X Xz)))->((and ((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (X ((cP Xx) Xz))) (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 Xy) Xz))) (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 (c
% EOF
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