TSTP Solution File: ALG296^5 by cocATP---0.2.0
View Problem
- Process Solution
%------------------------------------------------------------------------------
% File : cocATP---0.2.0
% Problem : ALG296^5 : TPTP v6.1.0. Released v4.0.0.
% Transfm : none
% Format : tptp:raw
% Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% Computer : n105.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 13:18:23 EDT 2014
% Result : Timeout 300.10s
% Output : None
% Verified :
% SZS Type : None (Parsing solution fails)
% Syntax : Number of formulae : 0
% Comments :
%------------------------------------------------------------------------------
%----NO SOLUTION OUTPUT BY SYSTEM
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% % Problem : ALG296^5 : TPTP v6.1.0. Released v4.0.0.
% % Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n105.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 09:10:36 CDT 2014
% % CPUTime : 300.10
% 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 0x1cc9a28>, <kernel.Type object at 0x1a74bd8>) of role type named a_type
% Using role type
% Declaring a:Type
% FOF formula (<kernel.Constant object at 0x1a73c68>, <kernel.DependentProduct object at 0x1a74b90>) of role type named cP
% Using role type
% Declaring cP:(a->(a->a))
% FOF formula (<kernel.Constant object at 0x1cc9f38>, <kernel.DependentProduct object at 0x1a74638>) of role type named cR
% Using role type
% Declaring cR:(a->a)
% FOF formula (<kernel.Constant object at 0x1cc9a28>, <kernel.Constant object at 0x1a74638>) of role type named cZ
% Using role type
% Declaring cZ:a
% FOF formula (<kernel.Constant object at 0x1cc9a28>, <kernel.DependentProduct object at 0x1a74050>) of role type named cL
% Using role type
% Declaring cL:(a->a)
% FOF formula (((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 (Xt:a) (Xb:a), ((forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx:a), ((X Xx)->((and (X ((cP Xx) cZ))) (X ((cP Xx) ((cP cZ) cZ)))))))->(X Xb)))->((ex a) (fun (Xu:a)=> ((and ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xb) Xu)) ((cP Xb_0) Xu_0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X ((cP Xc) Xv))->((and (X ((cP ((cP Xc) cZ)) (cL Xv)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X ((cP Xb_0) Xu_0)))))))))) (forall (Xv:a), (((ex a) (fun (Xb_1:a)=> ((ex a) (fun (Xu0:a)=> ((and (((eq a) ((cP Xb) Xv)) ((cP Xb_1) Xu0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv0:a), ((X ((cP Xc) Xv0))->((and (X ((cP ((cP Xc) cZ)) (cL Xv0)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv0)))))))->(X ((cP Xb_1) Xu0)))))))))->(((eq a) Xu) Xv))))))))) of role conjecture named cPU_LEM6_pme
% Conjecture to prove = (((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 (Xt:a) (Xb:a), ((forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx:a), ((X Xx)->((and (X ((cP Xx) cZ))) (X ((cP Xx) ((cP cZ) cZ)))))))->(X Xb)))->((ex a) (fun (Xu:a)=> ((and ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xb) Xu)) ((cP Xb_0) Xu_0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X ((cP Xc) Xv))->((and (X ((cP ((cP Xc) cZ)) (cL Xv)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X ((cP Xb_0) Xu_0)))))))))) (forall (Xv:a), (((ex a) (fun (Xb_1:a)=> ((ex a) (fun (Xu0:a)=> ((and (((eq a) ((cP Xb) Xv)) ((cP Xb_1) Xu0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv0:a), ((X ((cP Xc) Xv0))->((and (X ((cP ((cP Xc) cZ)) (cL Xv0)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv0)))))))->(X ((cP Xb_1) Xu0)))))))))->(((eq a) Xu) Xv))))))))):Prop
% We need to prove ['(((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 (Xt:a) (Xb:a), ((forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx:a), ((X Xx)->((and (X ((cP Xx) cZ))) (X ((cP Xx) ((cP cZ) cZ)))))))->(X Xb)))->((ex a) (fun (Xu:a)=> ((and ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xb) Xu)) ((cP Xb_0) Xu_0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X ((cP Xc) Xv))->((and (X ((cP ((cP Xc) cZ)) (cL Xv)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X ((cP Xb_0) Xu_0)))))))))) (forall (Xv:a), (((ex a) (fun (Xb_1:a)=> ((ex a) (fun (Xu0:a)=> ((and (((eq a) ((cP Xb) Xv)) ((cP Xb_1) Xu0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv0:a), ((X ((cP Xc) Xv0))->((and (X ((cP ((cP Xc) cZ)) (cL Xv0)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv0)))))))->(X ((cP Xb_1) Xu0)))))))))->(((eq a) Xu) Xv)))))))))']
% Parameter a:Type.
% Parameter cP:(a->(a->a)).
% Parameter cR:(a->a).
% Parameter cZ:a.
% Parameter cL:(a->a).
% Trying to prove (((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 (Xt:a) (Xb:a), ((forall (X:(a->Prop)), (((and (X cZ)) (forall (Xx:a), ((X Xx)->((and (X ((cP Xx) cZ))) (X ((cP Xx) ((cP cZ) cZ)))))))->(X Xb)))->((ex a) (fun (Xu:a)=> ((and ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xb) Xu)) ((cP Xb_0) Xu_0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X ((cP Xc) Xv))->((and (X ((cP ((cP Xc) cZ)) (cL Xv)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X ((cP Xb_0) Xu_0)))))))))) (forall (Xv:a), (((ex a) (fun (Xb_1:a)=> ((ex a) (fun (Xu0:a)=> ((and (((eq a) ((cP Xb) Xv)) ((cP Xb_1) Xu0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv0:a), ((X ((cP Xc) Xv0))->((and (X ((cP ((cP Xc) cZ)) (cL Xv0)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv0)))))))->(X ((cP Xb_1) Xu0)))))))))->(((eq a) Xu) Xv)))))))))
% Found eq_ref00:=(eq_ref0 x1):(((eq a) x1) x1)
% Found (eq_ref0 x1) as proof of (((eq a) x1) Xv)
% Found ((eq_ref a) x1) as proof of (((eq a) x1) Xv)
% Found ((eq_ref a) x1) as proof of (((eq a) x1) Xv)
% Found (fun (x00:((ex a) (fun (Xb_1:a)=> ((ex a) (fun (Xu0:a)=> ((and (((eq a) ((cP Xb) Xv)) ((cP Xb_1) Xu0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv0:a), ((X ((cP Xc) Xv0))->((and (X ((cP ((cP Xc) cZ)) (cL Xv0)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv0)))))))->(X ((cP Xb_1) Xu0))))))))))=> ((eq_ref a) x1)) as proof of (((eq a) x1) Xv)
% Found eq_ref000:=(eq_ref00 P):((P x1)->(P x1))
% Found (eq_ref00 P) as proof of ((P x1)->(P Xv))
% Found ((eq_ref0 x1) P) as proof of ((P x1)->(P Xv))
% Found (((eq_ref a) x1) P) as proof of ((P x1)->(P Xv))
% Found (((eq_ref a) x1) P) as proof of ((P x1)->(P Xv))
% Found (fun (P:(a->Prop))=> (((eq_ref a) x1) P)) as proof of ((P x1)->(P Xv))
% Found (fun (x00:((ex a) (fun (Xb_1:a)=> ((ex a) (fun (Xu0:a)=> ((and (((eq a) ((cP Xb) Xv)) ((cP Xb_1) Xu0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv0:a), ((X ((cP Xc) Xv0))->((and (X ((cP ((cP Xc) cZ)) (cL Xv0)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv0)))))))->(X ((cP Xb_1) Xu0)))))))))) (P:(a->Prop))=> (((eq_ref a) x1) P)) as proof of (((eq a) x1) Xv)
% Found x2:(P x1)
% Instantiate: x1:=Xv:a
% Found (fun (x2:(P x1))=> x2) as proof of (P Xv)
% Found (fun (P:(a->Prop)) (x2:(P x1))=> x2) as proof of ((P x1)->(P Xv))
% Found (fun (x00:((ex a) (fun (Xb_1:a)=> ((ex a) (fun (Xu0:a)=> ((and (((eq a) ((cP Xb) Xv)) ((cP Xb_1) Xu0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv0:a), ((X ((cP Xc) Xv0))->((and (X ((cP ((cP Xc) cZ)) (cL Xv0)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv0)))))))->(X ((cP Xb_1) Xu0)))))))))) (P:(a->Prop)) (x2:(P x1))=> x2) as proof of (((eq a) x1) Xv)
% Found eq_ref00:=(eq_ref0 x1):(((eq a) x1) x1)
% Found (eq_ref0 x1) as proof of (((eq a) x1) x')
% Found ((eq_ref a) x1) as proof of (((eq a) x1) x')
% Found ((eq_ref a) x1) as proof of (((eq a) x1) x')
% Found (fun (x00:((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xb) x')) ((cP Xb_0) Xu_0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X ((cP Xc) Xv))->((and (X ((cP ((cP Xc) cZ)) (cL Xv)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X ((cP Xb_0) Xu_0))))))))))=> ((eq_ref a) x1)) as proof of (((eq a) x1) x')
% Found x2:(P x1)
% Instantiate: x1:=x':a
% Found (fun (x2:(P x1))=> x2) as proof of (P x')
% Found (fun (P:(a->Prop)) (x2:(P x1))=> x2) as proof of ((P x1)->(P x'))
% Found (fun (x00:((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xb) x')) ((cP Xb_0) Xu_0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X ((cP Xc) Xv))->((and (X ((cP ((cP Xc) cZ)) (cL Xv)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X ((cP Xb_0) Xu_0)))))))))) (P:(a->Prop)) (x2:(P x1))=> x2) as proof of (((eq a) x1) x')
% Found eq_ref000:=(eq_ref00 P):((P x1)->(P x1))
% Found (eq_ref00 P) as proof of ((P x1)->(P x'))
% Found ((eq_ref0 x1) P) as proof of ((P x1)->(P x'))
% Found (((eq_ref a) x1) P) as proof of ((P x1)->(P x'))
% Found (((eq_ref a) x1) P) as proof of ((P x1)->(P x'))
% Found (fun (P:(a->Prop))=> (((eq_ref a) x1) P)) as proof of ((P x1)->(P x'))
% Found (fun (x00:((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xb) x')) ((cP Xb_0) Xu_0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X ((cP Xc) Xv))->((and (X ((cP ((cP Xc) cZ)) (cL Xv)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X ((cP Xb_0) Xu_0)))))))))) (P:(a->Prop))=> (((eq_ref a) x1) P)) as proof of (((eq a) x1) x')
% Found eq_ref00:=(eq_ref0 x3):(((eq a) x3) x3)
% Found (eq_ref0 x3) as proof of (((eq a) x3) Xv)
% Found ((eq_ref a) x3) as proof of (((eq a) x3) Xv)
% Found ((eq_ref a) x3) as proof of (((eq a) x3) Xv)
% Found (fun (x00:((ex a) (fun (Xb_1:a)=> ((ex a) (fun (Xu0:a)=> ((and (((eq a) ((cP Xb) Xv)) ((cP Xb_1) Xu0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv0:a), ((X ((cP Xc) Xv0))->((and (X ((cP ((cP Xc) cZ)) (cL Xv0)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv0)))))))->(X ((cP Xb_1) Xu0))))))))))=> ((eq_ref a) x3)) as proof of (((eq a) x3) Xv)
% Found eq_ref00:=(eq_ref0 x1):(((eq a) x1) x1)
% Found (eq_ref0 x1) as proof of (((eq a) x1) Xv)
% Found ((eq_ref a) x1) as proof of (((eq a) x1) Xv)
% Found ((eq_ref a) x1) as proof of (((eq a) x1) Xv)
% Found (fun (x00:((ex a) (fun (Xb_1:a)=> ((ex a) (fun (Xu0:a)=> ((and (((eq a) ((cP Xb) Xv)) ((cP Xb_1) Xu0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv0:a), ((X ((cP Xc) Xv0))->((and (X ((cP ((cP Xc) cZ)) (cL Xv0)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv0)))))))->(X ((cP Xb_1) Xu0))))))))))=> ((eq_ref a) x1)) as proof of (((eq a) x1) Xv)
% Found x4:(P x3)
% Instantiate: x3:=Xv:a
% Found (fun (x4:(P x3))=> x4) as proof of (P Xv)
% Found (fun (P:(a->Prop)) (x4:(P x3))=> x4) as proof of ((P x3)->(P Xv))
% Found (fun (x00:((ex a) (fun (Xb_1:a)=> ((ex a) (fun (Xu0:a)=> ((and (((eq a) ((cP Xb) Xv)) ((cP Xb_1) Xu0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv0:a), ((X ((cP Xc) Xv0))->((and (X ((cP ((cP Xc) cZ)) (cL Xv0)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv0)))))))->(X ((cP Xb_1) Xu0)))))))))) (P:(a->Prop)) (x4:(P x3))=> x4) as proof of (((eq a) x3) Xv)
% Found eq_ref000:=(eq_ref00 P):((P x3)->(P x3))
% Found (eq_ref00 P) as proof of ((P x3)->(P Xv))
% Found ((eq_ref0 x3) P) as proof of ((P x3)->(P Xv))
% Found (((eq_ref a) x3) P) as proof of ((P x3)->(P Xv))
% Found (((eq_ref a) x3) P) as proof of ((P x3)->(P Xv))
% Found (fun (P:(a->Prop))=> (((eq_ref a) x3) P)) as proof of ((P x3)->(P Xv))
% Found (fun (x00:((ex a) (fun (Xb_1:a)=> ((ex a) (fun (Xu0:a)=> ((and (((eq a) ((cP Xb) Xv)) ((cP Xb_1) Xu0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv0:a), ((X ((cP Xc) Xv0))->((and (X ((cP ((cP Xc) cZ)) (cL Xv0)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv0)))))))->(X ((cP Xb_1) Xu0)))))))))) (P:(a->Prop))=> (((eq_ref a) x3) P)) as proof of (((eq a) x3) Xv)
% Found eq_ref00:=(eq_ref0 x3):(((eq a) x3) x3)
% Found (eq_ref0 x3) as proof of (((eq a) x3) x')
% Found ((eq_ref a) x3) as proof of (((eq a) x3) x')
% Found ((eq_ref a) x3) as proof of (((eq a) x3) x')
% Found (fun (x00:((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xb) x')) ((cP Xb_0) Xu_0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X ((cP Xc) Xv))->((and (X ((cP ((cP Xc) cZ)) (cL Xv)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X ((cP Xb_0) Xu_0))))))))))=> ((eq_ref a) x3)) as proof of (((eq a) x3) x')
% Found eq_ref000:=(eq_ref00 P):((P x1)->(P x1))
% Found (eq_ref00 P) as proof of ((P x1)->(P Xv))
% Found ((eq_ref0 x1) P) as proof of ((P x1)->(P Xv))
% Found (((eq_ref a) x1) P) as proof of ((P x1)->(P Xv))
% Found (((eq_ref a) x1) P) as proof of ((P x1)->(P Xv))
% Found (fun (P:(a->Prop))=> (((eq_ref a) x1) P)) as proof of ((P x1)->(P Xv))
% Found (fun (x00:((ex a) (fun (Xb_1:a)=> ((ex a) (fun (Xu0:a)=> ((and (((eq a) ((cP Xb) Xv)) ((cP Xb_1) Xu0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv0:a), ((X ((cP Xc) Xv0))->((and (X ((cP ((cP Xc) cZ)) (cL Xv0)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv0)))))))->(X ((cP Xb_1) Xu0)))))))))) (P:(a->Prop))=> (((eq_ref a) x1) P)) as proof of (((eq a) x1) Xv)
% Found x4:(P x1)
% Instantiate: x1:=Xv:a
% Found (fun (x4:(P x1))=> x4) as proof of (P Xv)
% Found (fun (P:(a->Prop)) (x4:(P x1))=> x4) as proof of ((P x1)->(P Xv))
% Found (fun (x00:((ex a) (fun (Xb_1:a)=> ((ex a) (fun (Xu0:a)=> ((and (((eq a) ((cP Xb) Xv)) ((cP Xb_1) Xu0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv0:a), ((X ((cP Xc) Xv0))->((and (X ((cP ((cP Xc) cZ)) (cL Xv0)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv0)))))))->(X ((cP Xb_1) Xu0)))))))))) (P:(a->Prop)) (x4:(P x1))=> x4) as proof of (((eq a) x1) Xv)
% Found eq_ref00:=(eq_ref0 x1):(((eq a) x1) x1)
% Found (eq_ref0 x1) as proof of (((eq a) x1) x')
% Found ((eq_ref a) x1) as proof of (((eq a) x1) x')
% Found ((eq_ref a) x1) as proof of (((eq a) x1) x')
% Found (fun (x00:((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xb) x')) ((cP Xb_0) Xu_0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X ((cP Xc) Xv))->((and (X ((cP ((cP Xc) cZ)) (cL Xv)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X ((cP Xb_0) Xu_0))))))))))=> ((eq_ref a) x1)) as proof of (((eq a) x1) x')
% Found x4:(P x3)
% Instantiate: x3:=x':a
% Found (fun (x4:(P x3))=> x4) as proof of (P x')
% Found (fun (P:(a->Prop)) (x4:(P x3))=> x4) as proof of ((P x3)->(P x'))
% Found (fun (x00:((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xb) x')) ((cP Xb_0) Xu_0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X ((cP Xc) Xv))->((and (X ((cP ((cP Xc) cZ)) (cL Xv)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X ((cP Xb_0) Xu_0)))))))))) (P:(a->Prop)) (x4:(P x3))=> x4) as proof of (((eq a) x3) x')
% Found eq_ref000:=(eq_ref00 P):((P x3)->(P x3))
% Found (eq_ref00 P) as proof of ((P x3)->(P x'))
% Found ((eq_ref0 x3) P) as proof of ((P x3)->(P x'))
% Found (((eq_ref a) x3) P) as proof of ((P x3)->(P x'))
% Found (((eq_ref a) x3) P) as proof of ((P x3)->(P x'))
% Found (fun (P:(a->Prop))=> (((eq_ref a) x3) P)) as proof of ((P x3)->(P x'))
% Found (fun (x00:((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xb) x')) ((cP Xb_0) Xu_0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X ((cP Xc) Xv))->((and (X ((cP ((cP Xc) cZ)) (cL Xv)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X ((cP Xb_0) Xu_0)))))))))) (P:(a->Prop))=> (((eq_ref a) x3) P)) as proof of (((eq a) x3) x')
% Found eq_ref00:=(eq_ref0 x1):(((eq a) x1) x1)
% Found (eq_ref0 x1) as proof of (((eq a) x1) Xv)
% Found ((eq_ref a) x1) as proof of (((eq a) x1) Xv)
% Found ((eq_ref a) x1) as proof of (((eq a) x1) Xv)
% Found (fun (x3:((ex a) (fun (Xu0:a)=> ((and (((eq a) ((cP Xb) Xv)) ((cP x2) Xu0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv0:a), ((X ((cP Xc) Xv0))->((and (X ((cP ((cP Xc) cZ)) (cL Xv0)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv0)))))))->(X ((cP x2) Xu0))))))))=> ((eq_ref a) x1)) as proof of (((eq a) x1) Xv)
% Found (fun (x2:a) (x3:((ex a) (fun (Xu0:a)=> ((and (((eq a) ((cP Xb) Xv)) ((cP x2) Xu0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv0:a), ((X ((cP Xc) Xv0))->((and (X ((cP ((cP Xc) cZ)) (cL Xv0)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv0)))))))->(X ((cP x2) Xu0))))))))=> ((eq_ref a) x1)) as proof of (((ex a) (fun (Xu0:a)=> ((and (((eq a) ((cP Xb) Xv)) ((cP x2) Xu0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv0:a), ((X ((cP Xc) Xv0))->((and (X ((cP ((cP Xc) cZ)) (cL Xv0)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv0)))))))->(X ((cP x2) Xu0)))))))->(((eq a) x1) Xv))
% Found (fun (x2:a) (x3:((ex a) (fun (Xu0:a)=> ((and (((eq a) ((cP Xb) Xv)) ((cP x2) Xu0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv0:a), ((X ((cP Xc) Xv0))->((and (X ((cP ((cP Xc) cZ)) (cL Xv0)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv0)))))))->(X ((cP x2) Xu0))))))))=> ((eq_ref a) x1)) as proof of (forall (x:a), (((ex a) (fun (Xu0:a)=> ((and (((eq a) ((cP Xb) Xv)) ((cP x) Xu0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv0:a), ((X ((cP Xc) Xv0))->((and (X ((cP ((cP Xc) cZ)) (cL Xv0)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv0)))))))->(X ((cP x) Xu0)))))))->(((eq a) x1) Xv)))
% Found (ex_ind00 (fun (x2:a) (x3:((ex a) (fun (Xu0:a)=> ((and (((eq a) ((cP Xb) Xv)) ((cP x2) Xu0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv0:a), ((X ((cP Xc) Xv0))->((and (X ((cP ((cP Xc) cZ)) (cL Xv0)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv0)))))))->(X ((cP x2) Xu0))))))))=> ((eq_ref a) x1))) as proof of (((eq a) x1) Xv)
% Found ((ex_ind0 (((eq a) x1) Xv)) (fun (x2:a) (x3:((ex a) (fun (Xu0:a)=> ((and (((eq a) ((cP Xb) Xv)) ((cP x2) Xu0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv0:a), ((X ((cP Xc) Xv0))->((and (X ((cP ((cP Xc) cZ)) (cL Xv0)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv0)))))))->(X ((cP x2) Xu0))))))))=> ((eq_ref a) x1))) as proof of (((eq a) x1) Xv)
% Found (((fun (P:Prop) (x2:(forall (x:a), (((ex a) (fun (Xu0:a)=> ((and (((eq a) ((cP Xb) Xv)) ((cP x) Xu0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv0:a), ((X ((cP Xc) Xv0))->((and (X ((cP ((cP Xc) cZ)) (cL Xv0)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv0)))))))->(X ((cP x) Xu0)))))))->P)))=> (((((ex_ind a) (fun (Xb_1:a)=> ((ex a) (fun (Xu0:a)=> ((and (((eq a) ((cP Xb) Xv)) ((cP Xb_1) Xu0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv0:a), ((X ((cP Xc) Xv0))->((and (X ((cP ((cP Xc) cZ)) (cL Xv0)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv0)))))))->(X ((cP Xb_1) Xu0))))))))) P) x2) x00)) (((eq a) x1) Xv)) (fun (x2:a) (x3:((ex a) (fun (Xu0:a)=> ((and (((eq a) ((cP Xb) Xv)) ((cP x2) Xu0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv0:a), ((X ((cP Xc) Xv0))->((and (X ((cP ((cP Xc) cZ)) (cL Xv0)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv0)))))))->(X ((cP x2) Xu0))))))))=> ((eq_ref a) x1))) as proof of (((eq a) x1) Xv)
% Found (fun (x00:((ex a) (fun (Xb_1:a)=> ((ex a) (fun (Xu0:a)=> ((and (((eq a) ((cP Xb) Xv)) ((cP Xb_1) Xu0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv0:a), ((X ((cP Xc) Xv0))->((and (X ((cP ((cP Xc) cZ)) (cL Xv0)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv0)))))))->(X ((cP Xb_1) Xu0))))))))))=> (((fun (P:Prop) (x2:(forall (x:a), (((ex a) (fun (Xu0:a)=> ((and (((eq a) ((cP Xb) Xv)) ((cP x) Xu0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv0:a), ((X ((cP Xc) Xv0))->((and (X ((cP ((cP Xc) cZ)) (cL Xv0)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv0)))))))->(X ((cP x) Xu0)))))))->P)))=> (((((ex_ind a) (fun (Xb_1:a)=> ((ex a) (fun (Xu0:a)=> ((and (((eq a) ((cP Xb) Xv)) ((cP Xb_1) Xu0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv0:a), ((X ((cP Xc) Xv0))->((and (X ((cP ((cP Xc) cZ)) (cL Xv0)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv0)))))))->(X ((cP Xb_1) Xu0))))))))) P) x2) x00)) (((eq a) x1) Xv)) (fun (x2:a) (x3:((ex a) (fun (Xu0:a)=> ((and (((eq a) ((cP Xb) Xv)) ((cP x2) Xu0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv0:a), ((X ((cP Xc) Xv0))->((and (X ((cP ((cP Xc) cZ)) (cL Xv0)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv0)))))))->(X ((cP x2) Xu0))))))))=> ((eq_ref a) x1)))) as proof of (((eq a) x1) Xv)
% Found eq_ref00:=(eq_ref0 Xv):(((eq a) Xv) Xv)
% Found (eq_ref0 Xv) as proof of (((eq a) Xv) x1)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) x1)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) x1)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) x1)
% Found (eq_sym000 ((eq_ref a) Xv)) as proof of (((eq a) x1) Xv)
% Found ((eq_sym00 x1) ((eq_ref a) Xv)) as proof of (((eq a) x1) Xv)
% Found (((eq_sym0 Xv) x1) ((eq_ref a) Xv)) as proof of (((eq a) x1) Xv)
% Found ((((eq_sym a) Xv) x1) ((eq_ref a) Xv)) as proof of (((eq a) x1) Xv)
% Found (fun (x00:((ex a) (fun (Xb_1:a)=> ((ex a) (fun (Xu0:a)=> ((and (((eq a) ((cP Xb) Xv)) ((cP Xb_1) Xu0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv0:a), ((X ((cP Xc) Xv0))->((and (X ((cP ((cP Xc) cZ)) (cL Xv0)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv0)))))))->(X ((cP Xb_1) Xu0))))))))))=> ((((eq_sym a) Xv) x1) ((eq_ref a) Xv))) as proof of (((eq a) x1) Xv)
% Found eq_ref000:=(eq_ref00 P):((P x1)->(P x1))
% Found (eq_ref00 P) as proof of ((P x1)->(P x'))
% Found ((eq_ref0 x1) P) as proof of ((P x1)->(P x'))
% Found (((eq_ref a) x1) P) as proof of ((P x1)->(P x'))
% Found (((eq_ref a) x1) P) as proof of ((P x1)->(P x'))
% Found (fun (P:(a->Prop))=> (((eq_ref a) x1) P)) as proof of ((P x1)->(P x'))
% Found (fun (x00:((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xb) x')) ((cP Xb_0) Xu_0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X ((cP Xc) Xv))->((and (X ((cP ((cP Xc) cZ)) (cL Xv)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X ((cP Xb_0) Xu_0)))))))))) (P:(a->Prop))=> (((eq_ref a) x1) P)) as proof of (((eq a) x1) x')
% Found x4:(P x1)
% Instantiate: x1:=x':a
% Found (fun (x4:(P x1))=> x4) as proof of (P x')
% Found (fun (P:(a->Prop)) (x4:(P x1))=> x4) as proof of ((P x1)->(P x'))
% Found (fun (x00:((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xb) x')) ((cP Xb_0) Xu_0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X ((cP Xc) Xv))->((and (X ((cP ((cP Xc) cZ)) (cL Xv)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X ((cP Xb_0) Xu_0)))))))))) (P:(a->Prop)) (x4:(P x1))=> x4) as proof of (((eq a) x1) x')
% Found eq_ref00:=(eq_ref0 x5):(((eq a) x5) x5)
% Found (eq_ref0 x5) as proof of (((eq a) x5) Xv)
% Found ((eq_ref a) x5) as proof of (((eq a) x5) Xv)
% Found ((eq_ref a) x5) as proof of (((eq a) x5) Xv)
% Found (fun (x00:((ex a) (fun (Xb_1:a)=> ((ex a) (fun (Xu0:a)=> ((and (((eq a) ((cP Xb) Xv)) ((cP Xb_1) Xu0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv0:a), ((X ((cP Xc) Xv0))->((and (X ((cP ((cP Xc) cZ)) (cL Xv0)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv0)))))))->(X ((cP Xb_1) Xu0))))))))))=> ((eq_ref a) x5)) as proof of (((eq a) x5) Xv)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xu:a)=> ((and ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xb) Xu)) ((cP Xb_0) Xu_0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X ((cP Xc) Xv))->((and (X ((cP ((cP Xc) cZ)) (cL Xv)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X ((cP Xb_0) Xu_0)))))))))) (forall (Xv:a), (((ex a) (fun (Xb_1:a)=> ((ex a) (fun (Xu0:a)=> ((and (((eq a) ((cP Xb) Xv)) ((cP Xb_1) Xu0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv0:a), ((X ((cP Xc) Xv0))->((and (X ((cP ((cP Xc) cZ)) (cL Xv0)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv0)))))))->(X ((cP Xb_1) Xu0)))))))))->(((eq a) Xu) Xv)))))):(((eq (a->Prop)) (fun (Xu:a)=> ((and ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xb) Xu)) ((cP Xb_0) Xu_0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X ((cP Xc) Xv))->((and (X ((cP ((cP Xc) cZ)) (cL Xv)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X ((cP Xb_0) Xu_0)))))))))) (forall (Xv:a), (((ex a) (fun (Xb_1:a)=> ((ex a) (fun (Xu0:a)=> ((and (((eq a) ((cP Xb) Xv)) ((cP Xb_1) Xu0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv0:a), ((X ((cP Xc) Xv0))->((and (X ((cP ((cP Xc) cZ)) (cL Xv0)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv0)))))))->(X ((cP Xb_1) Xu0)))))))))->(((eq a) Xu) Xv)))))) (fun (x:a)=> ((and ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xb) x)) ((cP Xb_0) Xu_0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X ((cP Xc) Xv))->((and (X ((cP ((cP Xc) cZ)) (cL Xv)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X ((cP Xb_0) Xu_0)))))))))) (forall (Xv:a), (((ex a) (fun (Xb_1:a)=> ((ex a) (fun (Xu0:a)=> ((and (((eq a) ((cP Xb) Xv)) ((cP Xb_1) Xu0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv0:a), ((X ((cP Xc) Xv0))->((and (X ((cP ((cP Xc) cZ)) (cL Xv0)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv0)))))))->(X ((cP Xb_1) Xu0)))))))))->(((eq a) x) Xv))))))
% Found (eta_expansion_dep00 (fun (Xu:a)=> ((and ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xb) Xu)) ((cP Xb_0) Xu_0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X ((cP Xc) Xv))->((and (X ((cP ((cP Xc) cZ)) (cL Xv)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X ((cP Xb_0) Xu_0)))))))))) (forall (Xv:a), (((ex a) (fun (Xb_1:a)=> ((ex a) (fun (Xu0:a)=> ((and (((eq a) ((cP Xb) Xv)) ((cP Xb_1) Xu0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv0:a), ((X ((cP Xc) Xv0))->((and (X ((cP ((cP Xc) cZ)) (cL Xv0)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv0)))))))->(X ((cP Xb_1) Xu0)))))))))->(((eq a) Xu) Xv)))))) as proof of (((eq (a->Prop)) (fun (Xu:a)=> ((and ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xb) Xu)) ((cP Xb_0) Xu_0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X ((cP Xc) Xv))->((and (X ((cP ((cP Xc) cZ)) (cL Xv)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X ((cP Xb_0) Xu_0)))))))))) (forall (Xv:a), (((ex a) (fun (Xb_1:a)=> ((ex a) (fun (Xu0:a)=> ((and (((eq a) ((cP Xb) Xv)) ((cP Xb_1) Xu0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv0:a), ((X ((cP Xc) Xv0))->((and (X ((cP ((cP Xc) cZ)) (cL Xv0)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv0)))))))->(X ((cP Xb_1) Xu0)))))))))->(((eq a) Xu) Xv)))))) b)
% Found ((eta_expansion_dep0 (fun (x2:a)=> Prop)) (fun (Xu:a)=> ((and ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xb) Xu)) ((cP Xb_0) Xu_0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X ((cP Xc) Xv))->((and (X ((cP ((cP Xc) cZ)) (cL Xv)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X ((cP Xb_0) Xu_0)))))))))) (forall (Xv:a), (((ex a) (fun (Xb_1:a)=> ((ex a) (fun (Xu0:a)=> ((and (((eq a) ((cP Xb) Xv)) ((cP Xb_1) Xu0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv0:a), ((X ((cP Xc) Xv0))->((and (X ((cP ((cP Xc) cZ)) (cL Xv0)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv0)))))))->(X ((cP Xb_1) Xu0)))))))))->(((eq a) Xu) Xv)))))) as proof of (((eq (a->Prop)) (fun (Xu:a)=> ((and ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xb) Xu)) ((cP Xb_0) Xu_0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X ((cP Xc) Xv))->((and (X ((cP ((cP Xc) cZ)) (cL Xv)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X ((cP Xb_0) Xu_0)))))))))) (forall (Xv:a), (((ex a) (fun (Xb_1:a)=> ((ex a) (fun (Xu0:a)=> ((and (((eq a) ((cP Xb) Xv)) ((cP Xb_1) Xu0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv0:a), ((X ((cP Xc) Xv0))->((and (X ((cP ((cP Xc) cZ)) (cL Xv0)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv0)))))))->(X ((cP Xb_1) Xu0)))))))))->(((eq a) Xu) Xv)))))) b)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xu:a)=> ((and ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xb) Xu)) ((cP Xb_0) Xu_0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X ((cP Xc) Xv))->((and (X ((cP ((cP Xc) cZ)) (cL Xv)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X ((cP Xb_0) Xu_0)))))))))) (forall (Xv:a), (((ex a) (fun (Xb_1:a)=> ((ex a) (fun (Xu0:a)=> ((and (((eq a) ((cP Xb) Xv)) ((cP Xb_1) Xu0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv0:a), ((X ((cP Xc) Xv0))->((and (X ((cP ((cP Xc) cZ)) (cL Xv0)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv0)))))))->(X ((cP Xb_1) Xu0)))))))))->(((eq a) Xu) Xv)))))) as proof of (((eq (a->Prop)) (fun (Xu:a)=> ((and ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xb) Xu)) ((cP Xb_0) Xu_0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X ((cP Xc) Xv))->((and (X ((cP ((cP Xc) cZ)) (cL Xv)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X ((cP Xb_0) Xu_0)))))))))) (forall (Xv:a), (((ex a) (fun (Xb_1:a)=> ((ex a) (fun (Xu0:a)=> ((and (((eq a) ((cP Xb) Xv)) ((cP Xb_1) Xu0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv0:a), ((X ((cP Xc) Xv0))->((and (X ((cP ((cP Xc) cZ)) (cL Xv0)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv0)))))))->(X ((cP Xb_1) Xu0)))))))))->(((eq a) Xu) Xv)))))) b)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xu:a)=> ((and ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xb) Xu)) ((cP Xb_0) Xu_0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X ((cP Xc) Xv))->((and (X ((cP ((cP Xc) cZ)) (cL Xv)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X ((cP Xb_0) Xu_0)))))))))) (forall (Xv:a), (((ex a) (fun (Xb_1:a)=> ((ex a) (fun (Xu0:a)=> ((and (((eq a) ((cP Xb) Xv)) ((cP Xb_1) Xu0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv0:a), ((X ((cP Xc) Xv0))->((and (X ((cP ((cP Xc) cZ)) (cL Xv0)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv0)))))))->(X ((cP Xb_1) Xu0)))))))))->(((eq a) Xu) Xv)))))) as proof of (((eq (a->Prop)) (fun (Xu:a)=> ((and ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xb) Xu)) ((cP Xb_0) Xu_0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X ((cP Xc) Xv))->((and (X ((cP ((cP Xc) cZ)) (cL Xv)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X ((cP Xb_0) Xu_0)))))))))) (forall (Xv:a), (((ex a) (fun (Xb_1:a)=> ((ex a) (fun (Xu0:a)=> ((and (((eq a) ((cP Xb) Xv)) ((cP Xb_1) Xu0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv0:a), ((X ((cP Xc) Xv0))->((and (X ((cP ((cP Xc) cZ)) (cL Xv0)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv0)))))))->(X ((cP Xb_1) Xu0)))))))))->(((eq a) Xu) Xv)))))) b)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xu:a)=> ((and ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xb) Xu)) ((cP Xb_0) Xu_0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X ((cP Xc) Xv))->((and (X ((cP ((cP Xc) cZ)) (cL Xv)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X ((cP Xb_0) Xu_0)))))))))) (forall (Xv:a), (((ex a) (fun (Xb_1:a)=> ((ex a) (fun (Xu0:a)=> ((and (((eq a) ((cP Xb) Xv)) ((cP Xb_1) Xu0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv0:a), ((X ((cP Xc) Xv0))->((and (X ((cP ((cP Xc) cZ)) (cL Xv0)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv0)))))))->(X ((cP Xb_1) Xu0)))))))))->(((eq a) Xu) Xv)))))) as proof of (((eq (a->Prop)) (fun (Xu:a)=> ((and ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xb) Xu)) ((cP Xb_0) Xu_0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X ((cP Xc) Xv))->((and (X ((cP ((cP Xc) cZ)) (cL Xv)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X ((cP Xb_0) Xu_0)))))))))) (forall (Xv:a), (((ex a) (fun (Xb_1:a)=> ((ex a) (fun (Xu0:a)=> ((and (((eq a) ((cP Xb) Xv)) ((cP Xb_1) Xu0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv0:a), ((X ((cP Xc) Xv0))->((and (X ((cP ((cP Xc) cZ)) (cL Xv0)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv0)))))))->(X ((cP Xb_1) Xu0)))))))))->(((eq a) Xu) Xv)))))) b)
% Found eq_ref00:=(eq_ref0 x1):(((eq a) x1) x1)
% Found (eq_ref0 x1) as proof of (((eq a) x1) x')
% Found ((eq_ref a) x1) as proof of (((eq a) x1) x')
% Found ((eq_ref a) x1) as proof of (((eq a) x1) x')
% Found (fun (x3:((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xb) x')) ((cP x2) Xu_0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X ((cP Xc) Xv))->((and (X ((cP ((cP Xc) cZ)) (cL Xv)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X ((cP x2) Xu_0))))))))=> ((eq_ref a) x1)) as proof of (((eq a) x1) x')
% Found (fun (x2:a) (x3:((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xb) x')) ((cP x2) Xu_0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X ((cP Xc) Xv))->((and (X ((cP ((cP Xc) cZ)) (cL Xv)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X ((cP x2) Xu_0))))))))=> ((eq_ref a) x1)) as proof of (((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xb) x')) ((cP x2) Xu_0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X ((cP Xc) Xv))->((and (X ((cP ((cP Xc) cZ)) (cL Xv)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X ((cP x2) Xu_0)))))))->(((eq a) x1) x'))
% Found (fun (x2:a) (x3:((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xb) x')) ((cP x2) Xu_0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X ((cP Xc) Xv))->((and (X ((cP ((cP Xc) cZ)) (cL Xv)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X ((cP x2) Xu_0))))))))=> ((eq_ref a) x1)) as proof of (forall (x:a), (((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xb) x')) ((cP x) Xu_0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X ((cP Xc) Xv))->((and (X ((cP ((cP Xc) cZ)) (cL Xv)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X ((cP x) Xu_0)))))))->(((eq a) x1) x')))
% Found (ex_ind00 (fun (x2:a) (x3:((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xb) x')) ((cP x2) Xu_0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X ((cP Xc) Xv))->((and (X ((cP ((cP Xc) cZ)) (cL Xv)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X ((cP x2) Xu_0))))))))=> ((eq_ref a) x1))) as proof of (((eq a) x1) x')
% Found ((ex_ind0 (((eq a) x1) x')) (fun (x2:a) (x3:((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xb) x')) ((cP x2) Xu_0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X ((cP Xc) Xv))->((and (X ((cP ((cP Xc) cZ)) (cL Xv)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X ((cP x2) Xu_0))))))))=> ((eq_ref a) x1))) as proof of (((eq a) x1) x')
% Found (((fun (P:Prop) (x2:(forall (x:a), (((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xb) x')) ((cP x) Xu_0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X ((cP Xc) Xv))->((and (X ((cP ((cP Xc) cZ)) (cL Xv)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X ((cP x) Xu_0)))))))->P)))=> (((((ex_ind a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xb) x')) ((cP Xb_0) Xu_0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X ((cP Xc) Xv))->((and (X ((cP ((cP Xc) cZ)) (cL Xv)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X ((cP Xb_0) Xu_0))))))))) P) x2) x00)) (((eq a) x1) x')) (fun (x2:a) (x3:((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xb) x')) ((cP x2) Xu_0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X ((cP Xc) Xv))->((and (X ((cP ((cP Xc) cZ)) (cL Xv)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X ((cP x2) Xu_0))))))))=> ((eq_ref a) x1))) as proof of (((eq a) x1) x')
% Found (fun (x00:((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xb) x')) ((cP Xb_0) Xu_0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X ((cP Xc) Xv))->((and (X ((cP ((cP Xc) cZ)) (cL Xv)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X ((cP Xb_0) Xu_0))))))))))=> (((fun (P:Prop) (x2:(forall (x:a), (((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xb) x')) ((cP x) Xu_0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X ((cP Xc) Xv))->((and (X ((cP ((cP Xc) cZ)) (cL Xv)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X ((cP x) Xu_0)))))))->P)))=> (((((ex_ind a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xb) x')) ((cP Xb_0) Xu_0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X ((cP Xc) Xv))->((and (X ((cP ((cP Xc) cZ)) (cL Xv)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X ((cP Xb_0) Xu_0))))))))) P) x2) x00)) (((eq a) x1) x')) (fun (x2:a) (x3:((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xb) x')) ((cP x2) Xu_0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X ((cP Xc) Xv))->((and (X ((cP ((cP Xc) cZ)) (cL Xv)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X ((cP x2) Xu_0))))))))=> ((eq_ref a) x1)))) as proof of (((eq a) x1) x')
% Found eq_ref00:=(eq_ref0 x'):(((eq a) x') x')
% Found (eq_ref0 x') as proof of (((eq a) x') x1)
% Found ((eq_ref a) x') as proof of (((eq a) x') x1)
% Found ((eq_ref a) x') as proof of (((eq a) x') x1)
% Found ((eq_ref a) x') as proof of (((eq a) x') x1)
% Found (eq_sym000 ((eq_ref a) x')) as proof of (((eq a) x1) x')
% Found ((eq_sym00 x1) ((eq_ref a) x')) as proof of (((eq a) x1) x')
% Found (((eq_sym0 x') x1) ((eq_ref a) x')) as proof of (((eq a) x1) x')
% Found ((((eq_sym a) x') x1) ((eq_ref a) x')) as proof of (((eq a) x1) x')
% Found (fun (x00:((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xb) x')) ((cP Xb_0) Xu_0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X ((cP Xc) Xv))->((and (X ((cP ((cP Xc) cZ)) (cL Xv)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X ((cP Xb_0) Xu_0))))))))))=> ((((eq_sym a) x') x1) ((eq_ref a) x'))) as proof of (((eq a) x1) x')
% Found eq_ref00:=(eq_ref0 x1):(((eq a) x1) x1)
% Found (eq_ref0 x1) as proof of (((eq a) x1) Xv)
% Found ((eq_ref a) x1) as proof of (((eq a) x1) Xv)
% Found ((eq_ref a) x1) as proof of (((eq a) x1) Xv)
% Found (fun (x00:((ex a) (fun (Xb_1:a)=> ((ex a) (fun (Xu0:a)=> ((and (((eq a) ((cP Xb) Xv)) ((cP Xb_1) Xu0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv0:a), ((X ((cP Xc) Xv0))->((and (X ((cP ((cP Xc) cZ)) (cL Xv0)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv0)))))))->(X ((cP Xb_1) Xu0))))))))))=> ((eq_ref a) x1)) as proof of (((eq a) x1) Xv)
% Found eq_ref00:=(eq_ref0 x3):(((eq a) x3) x3)
% Found (eq_ref0 x3) as proof of (((eq a) x3) Xv)
% Found ((eq_ref a) x3) as proof of (((eq a) x3) Xv)
% Found ((eq_ref a) x3) as proof of (((eq a) x3) Xv)
% Found (fun (x00:((ex a) (fun (Xb_1:a)=> ((ex a) (fun (Xu0:a)=> ((and (((eq a) ((cP Xb) Xv)) ((cP Xb_1) Xu0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv0:a), ((X ((cP Xc) Xv0))->((and (X ((cP ((cP Xc) cZ)) (cL Xv0)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv0)))))))->(X ((cP Xb_1) Xu0))))))))))=> ((eq_ref a) x3)) as proof of (((eq a) x3) Xv)
% Found eta_expansion_dep000:=(eta_expansion_dep00 ((unique a) (fun (x2:a)=> ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xb) x2)) ((cP Xb_0) Xu_0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X ((cP Xc) Xv))->((and (X ((cP ((cP Xc) cZ)) (cL Xv)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X ((cP Xb_0) Xu_0)))))))))))):(((eq (a->Prop)) ((unique a) (fun (x2:a)=> ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xb) x2)) ((cP Xb_0) Xu_0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X ((cP Xc) Xv))->((and (X ((cP ((cP Xc) cZ)) (cL Xv)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X ((cP Xb_0) Xu_0)))))))))))) (fun (x:a)=> (((unique a) (fun (x2:a)=> ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xb) x2)) ((cP Xb_0) Xu_0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X ((cP Xc) Xv))->((and (X ((cP ((cP Xc) cZ)) (cL Xv)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X ((cP Xb_0) Xu_0))))))))))) x)))
% Found (eta_expansion_dep00 ((unique a) (fun (x2:a)=> ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xb) x2)) ((cP Xb_0) Xu_0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X ((cP Xc) Xv))->((and (X ((cP ((cP Xc) cZ)) (cL Xv)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X ((cP Xb_0) Xu_0)))))))))))) as proof of (((eq (a->Prop)) ((unique a) (fun (x2:a)=> ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xb) x2)) ((cP Xb_0) Xu_0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X ((cP Xc) Xv))->((and (X ((cP ((cP Xc) cZ)) (cL Xv)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X ((cP Xb_0) Xu_0)))))))))))) b)
% Found ((eta_expansion_dep0 (fun (x2:a)=> Prop)) ((unique a) (fun (x2:a)=> ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xb) x2)) ((cP Xb_0) Xu_0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X ((cP Xc) Xv))->((and (X ((cP ((cP Xc) cZ)) (cL Xv)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X ((cP Xb_0) Xu_0)))))))))))) as proof of (((eq (a->Prop)) ((unique a) (fun (x2:a)=> ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xb) x2)) ((cP Xb_0) Xu_0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X ((cP Xc) Xv))->((and (X ((cP ((cP Xc) cZ)) (cL Xv)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X ((cP Xb_0) Xu_0)))))))))))) b)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) ((unique a) (fun (x2:a)=> ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xb) x2)) ((cP Xb_0) Xu_0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X ((cP Xc) Xv))->((and (X ((cP ((cP Xc) cZ)) (cL Xv)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X ((cP Xb_0) Xu_0)))))))))))) as proof of (((eq (a->Prop)) ((unique a) (fun (x2:a)=> ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xb) x2)) ((cP Xb_0) Xu_0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X ((cP Xc) Xv))->((and (X ((cP ((cP Xc) cZ)) (cL Xv)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X ((cP Xb_0) Xu_0)))))))))))) b)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) ((unique a) (fun (x2:a)=> ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xb) x2)) ((cP Xb_0) Xu_0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X ((cP Xc) Xv))->((and (X ((cP ((cP Xc) cZ)) (cL Xv)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X ((cP Xb_0) Xu_0)))))))))))) as proof of (((eq (a->Prop)) ((unique a) (fun (x2:a)=> ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xb) x2)) ((cP Xb_0) Xu_0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X ((cP Xc) Xv))->((and (X ((cP ((cP Xc) cZ)) (cL Xv)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X ((cP Xb_0) Xu_0)))))))))))) b)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) ((unique a) (fun (x2:a)=> ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xb) x2)) ((cP Xb_0) Xu_0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X ((cP Xc) Xv))->((and (X ((cP ((cP Xc) cZ)) (cL Xv)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X ((cP Xb_0) Xu_0)))))))))))) as proof of (((eq (a->Prop)) ((unique a) (fun (x2:a)=> ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xb) x2)) ((cP Xb_0) Xu_0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X ((cP Xc) Xv))->((and (X ((cP ((cP Xc) cZ)) (cL Xv)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X ((cP Xb_0) Xu_0)))))))))))) b)
% Found eq_ref000:=(eq_ref00 P):((P x5)->(P x5))
% Found (eq_ref00 P) as proof of ((P x5)->(P Xv))
% Found ((eq_ref0 x5) P) as proof of ((P x5)->(P Xv))
% Found (((eq_ref a) x5) P) as proof of ((P x5)->(P Xv))
% Found (((eq_ref a) x5) P) as proof of ((P x5)->(P Xv))
% Found (fun (P:(a->Prop))=> (((eq_ref a) x5) P)) as proof of ((P x5)->(P Xv))
% Found (fun (x00:((ex a) (fun (Xb_1:a)=> ((ex a) (fun (Xu0:a)=> ((and (((eq a) ((cP Xb) Xv)) ((cP Xb_1) Xu0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv0:a), ((X ((cP Xc) Xv0))->((and (X ((cP ((cP Xc) cZ)) (cL Xv0)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv0)))))))->(X ((cP Xb_1) Xu0)))))))))) (P:(a->Prop))=> (((eq_ref a) x5) P)) as proof of (((eq a) x5) Xv)
% Found x6:(P x5)
% Instantiate: x5:=Xv:a
% Found (fun (x6:(P x5))=> x6) as proof of (P Xv)
% Found (fun (P:(a->Prop)) (x6:(P x5))=> x6) as proof of ((P x5)->(P Xv))
% Found (fun (x00:((ex a) (fun (Xb_1:a)=> ((ex a) (fun (Xu0:a)=> ((and (((eq a) ((cP Xb) Xv)) ((cP Xb_1) Xu0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv0:a), ((X ((cP Xc) Xv0))->((and (X ((cP ((cP Xc) cZ)) (cL Xv0)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv0)))))))->(X ((cP Xb_1) Xu0)))))))))) (P:(a->Prop)) (x6:(P x5))=> x6) as proof of (((eq a) x5) Xv)
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) ((and ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xb) x1)) ((cP Xb_0) Xu_0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X ((cP Xc) Xv))->((and (X ((cP ((cP Xc) cZ)) (cL Xv)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X ((cP Xb_0) Xu_0)))))))))) (forall (Xv:a), (((ex a) (fun (Xb_1:a)=> ((ex a) (fun (Xu0:a)=> ((and (((eq a) ((cP Xb) Xv)) ((cP Xb_1) Xu0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv0:a), ((X ((cP Xc) Xv0))->((and (X ((cP ((cP Xc) cZ)) (cL Xv0)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv0)))))))->(X ((cP Xb_1) Xu0)))))))))->(((eq a) x1) Xv)))))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xb) x1)) ((cP Xb_0) Xu_0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X ((cP Xc) Xv))->((and (X ((cP ((cP Xc) cZ)) (cL Xv)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X ((cP Xb_0) Xu_0)))))))))) (forall (Xv:a), (((ex a) (fun (Xb_1:a)=> ((ex a) (fun (Xu0:a)=> ((and (((eq a) ((cP Xb) Xv)) ((cP Xb_1) Xu0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv0:a), ((X ((cP Xc) Xv0))->((and (X ((cP ((cP Xc) cZ)) (cL Xv0)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv0)))))))->(X ((cP Xb_1) Xu0)))))))))->(((eq a) x1) Xv)))))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xb) x1)) ((cP Xb_0) Xu_0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X ((cP Xc) Xv))->((and (X ((cP ((cP Xc) cZ)) (cL Xv)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X ((cP Xb_0) Xu_0)))))))))) (forall (Xv:a), (((ex a) (fun (Xb_1:a)=> ((ex a) (fun (Xu0:a)=> ((and (((eq a) ((cP Xb) Xv)) ((cP Xb_1) Xu0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv0:a), ((X ((cP Xc) Xv0))->((and (X ((cP ((cP Xc) cZ)) (cL Xv0)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv0)))))))->(X ((cP Xb_1) Xu0)))))))))->(((eq a) x1) Xv)))))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) ((and ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xb) x1)) ((cP Xb_0) Xu_0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X ((cP Xc) Xv))->((and (X ((cP ((cP Xc) cZ)) (cL Xv)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X ((cP Xb_0) Xu_0)))))))))) (forall (Xv:a), (((ex a) (fun (Xb_1:a)=> ((ex a) (fun (Xu0:a)=> ((and (((eq a) ((cP Xb) Xv)) ((cP Xb_1) Xu0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv0:a), ((X ((cP Xc) Xv0))->((and (X ((cP ((cP Xc) cZ)) (cL Xv0)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv0)))))))->(X ((cP Xb_1) Xu0)))))))))->(((eq a) x1) Xv)))))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:a), (((eq Prop) (f x)) ((and ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xb) x)) ((cP Xb_0) Xu_0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X ((cP Xc) Xv))->((and (X ((cP ((cP Xc) cZ)) (cL Xv)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X ((cP Xb_0) Xu_0)))))))))) (forall (Xv:a), (((ex a) (fun (Xb_1:a)=> ((ex a) (fun (Xu0:a)=> ((and (((eq a) ((cP Xb) Xv)) ((cP Xb_1) Xu0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv0:a), ((X ((cP Xc) Xv0))->((and (X ((cP ((cP Xc) cZ)) (cL Xv0)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv0)))))))->(X ((cP Xb_1) Xu0)))))))))->(((eq a) x) Xv))))))
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) ((and ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xb) x1)) ((cP Xb_0) Xu_0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X ((cP Xc) Xv))->((and (X ((cP ((cP Xc) cZ)) (cL Xv)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X ((cP Xb_0) Xu_0)))))))))) (forall (Xv:a), (((ex a) (fun (Xb_1:a)=> ((ex a) (fun (Xu0:a)=> ((and (((eq a) ((cP Xb) Xv)) ((cP Xb_1) Xu0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv0:a), ((X ((cP Xc) Xv0))->((and (X ((cP ((cP Xc) cZ)) (cL Xv0)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv0)))))))->(X ((cP Xb_1) Xu0)))))))))->(((eq a) x1) Xv)))))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xb) x1)) ((cP Xb_0) Xu_0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X ((cP Xc) Xv))->((and (X ((cP ((cP Xc) cZ)) (cL Xv)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X ((cP Xb_0) Xu_0)))))))))) (forall (Xv:a), (((ex a) (fun (Xb_1:a)=> ((ex a) (fun (Xu0:a)=> ((and (((eq a) ((cP Xb) Xv)) ((cP Xb_1) Xu0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv0:a), ((X ((cP Xc) Xv0))->((and (X ((cP ((cP Xc) cZ)) (cL Xv0)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv0)))))))->(X ((cP Xb_1) Xu0)))))))))->(((eq a) x1) Xv)))))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xb) x1)) ((cP Xb_0) Xu_0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X ((cP Xc) Xv))->((and (X ((cP ((cP Xc) cZ)) (cL Xv)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X ((cP Xb_0) Xu_0)))))))))) (forall (Xv:a), (((ex a) (fun (Xb_1:a)=> ((ex a) (fun (Xu0:a)=> ((and (((eq a) ((cP Xb) Xv)) ((cP Xb_1) Xu0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv0:a), ((X ((cP Xc) Xv0))->((and (X ((cP ((cP Xc) cZ)) (cL Xv0)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv0)))))))->(X ((cP Xb_1) Xu0)))))))))->(((eq a) x1) Xv)))))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) ((and ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xb) x1)) ((cP Xb_0) Xu_0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X ((cP Xc) Xv))->((and (X ((cP ((cP Xc) cZ)) (cL Xv)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X ((cP Xb_0) Xu_0)))))))))) (forall (Xv:a), (((ex a) (fun (Xb_1:a)=> ((ex a) (fun (Xu0:a)=> ((and (((eq a) ((cP Xb) Xv)) ((cP Xb_1) Xu0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv0:a), ((X ((cP Xc) Xv0))->((and (X ((cP ((cP Xc) cZ)) (cL Xv0)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv0)))))))->(X ((cP Xb_1) Xu0)))))))))->(((eq a) x1) Xv)))))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:a), (((eq Prop) (f x)) ((and ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xb) x)) ((cP Xb_0) Xu_0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X ((cP Xc) Xv))->((and (X ((cP ((cP Xc) cZ)) (cL Xv)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X ((cP Xb_0) Xu_0)))))))))) (forall (Xv:a), (((ex a) (fun (Xb_1:a)=> ((ex a) (fun (Xu0:a)=> ((and (((eq a) ((cP Xb) Xv)) ((cP Xb_1) Xu0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv0:a), ((X ((cP Xc) Xv0))->((and (X ((cP ((cP Xc) cZ)) (cL Xv0)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv0)))))))->(X ((cP Xb_1) Xu0)))))))))->(((eq a) x) Xv))))))
% Found eq_ref00:=(eq_ref0 x5):(((eq a) x5) x5)
% Found (eq_ref0 x5) as proof of (((eq a) x5) x')
% Found ((eq_ref a) x5) as proof of (((eq a) x5) x')
% Found ((eq_ref a) x5) as proof of (((eq a) x5) x')
% Found (fun (x00:((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xb) x')) ((cP Xb_0) Xu_0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X ((cP Xc) Xv))->((and (X ((cP ((cP Xc) cZ)) (cL Xv)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X ((cP Xb_0) Xu_0))))))))))=> ((eq_ref a) x5)) as proof of (((eq a) x5) x')
% Found eq_ref00:=(eq_ref0 x1):(((eq a) x1) x1)
% Found (eq_ref0 x1) as proof of (((eq a) x1) Xv)
% Found ((eq_ref a) x1) as proof of (((eq a) x1) Xv)
% Found ((eq_ref a) x1) as proof of (((eq a) x1) Xv)
% Found (fun (x3:(forall (Xt:a), ((iff (not (((eq a) Xt) cZ))) (((eq a) Xt) ((cP (cL Xt)) (cR Xt))))))=> ((eq_ref a) x1)) as proof of (((eq a) x1) Xv)
% Found (fun (x2:((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)))) (x3:(forall (Xt:a), ((iff (not (((eq a) Xt) cZ))) (((eq a) Xt) ((cP (cL Xt)) (cR Xt))))))=> ((eq_ref a) x1)) as proof of ((forall (Xt:a), ((iff (not (((eq a) Xt) cZ))) (((eq a) Xt) ((cP (cL Xt)) (cR Xt)))))->(((eq a) x1) Xv))
% Found (fun (x2:((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)))) (x3:(forall (Xt:a), ((iff (not (((eq a) Xt) cZ))) (((eq a) Xt) ((cP (cL Xt)) (cR Xt))))))=> ((eq_ref a) x1)) as proof of (((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)))))->(((eq a) x1) Xv)))
% Found (and_rect00 (fun (x2:((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)))) (x3:(forall (Xt:a), ((iff (not (((eq a) Xt) cZ))) (((eq a) Xt) ((cP (cL Xt)) (cR Xt))))))=> ((eq_ref a) x1))) as proof of (((eq a) x1) Xv)
% Found ((and_rect0 (((eq a) x1) Xv)) (fun (x2:((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)))) (x3:(forall (Xt:a), ((iff (not (((eq a) Xt) cZ))) (((eq a) Xt) ((cP (cL Xt)) (cR Xt))))))=> ((eq_ref a) x1))) as proof of (((eq a) x1) Xv)
% Found (((fun (P:Type) (x2:(((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)))))->P)))=> (((((and_rect ((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)))))) P) x2) x)) (((eq a) x1) Xv)) (fun (x2:((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)))) (x3:(forall (Xt:a), ((iff (not (((eq a) Xt) cZ))) (((eq a) Xt) ((cP (cL Xt)) (cR Xt))))))=> ((eq_ref a) x1))) as proof of (((eq a) x1) Xv)
% Found (fun (x00:((ex a) (fun (Xb_1:a)=> ((ex a) (fun (Xu0:a)=> ((and (((eq a) ((cP Xb) Xv)) ((cP Xb_1) Xu0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv0:a), ((X ((cP Xc) Xv0))->((and (X ((cP ((cP Xc) cZ)) (cL Xv0)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv0)))))))->(X ((cP Xb_1) Xu0))))))))))=> (((fun (P:Type) (x2:(((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)))))->P)))=> (((((and_rect ((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)))))) P) x2) x)) (((eq a) x1) Xv)) (fun (x2:((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)))) (x3:(forall (Xt:a), ((iff (not (((eq a) Xt) cZ))) (((eq a) Xt) ((cP (cL Xt)) (cR Xt))))))=> ((eq_ref a) x1)))) as proof of (((eq a) x1) Xv)
% Found eq_ref000:=(eq_ref00 P):((P x1)->(P x1))
% Found (eq_ref00 P) as proof of ((P x1)->(P Xv))
% Found ((eq_ref0 x1) P) as proof of ((P x1)->(P Xv))
% Found (((eq_ref a) x1) P) as proof of ((P x1)->(P Xv))
% Found (((eq_ref a) x1) P) as proof of ((P x1)->(P Xv))
% Found (fun (P:(a->Prop))=> (((eq_ref a) x1) P)) as proof of ((P x1)->(P Xv))
% Found (fun (x00:((ex a) (fun (Xb_1:a)=> ((ex a) (fun (Xu0:a)=> ((and (((eq a) ((cP Xb) Xv)) ((cP Xb_1) Xu0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv0:a), ((X ((cP Xc) Xv0))->((and (X ((cP ((cP Xc) cZ)) (cL Xv0)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv0)))))))->(X ((cP Xb_1) Xu0)))))))))) (P:(a->Prop))=> (((eq_ref a) x1) P)) as proof of (((eq a) x1) Xv)
% Found x6:(P x1)
% Instantiate: x1:=Xv:a
% Found (fun (x6:(P x1))=> x6) as proof of (P Xv)
% Found (fun (P:(a->Prop)) (x6:(P x1))=> x6) as proof of ((P x1)->(P Xv))
% Found (fun (x00:((ex a) (fun (Xb_1:a)=> ((ex a) (fun (Xu0:a)=> ((and (((eq a) ((cP Xb) Xv)) ((cP Xb_1) Xu0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv0:a), ((X ((cP Xc) Xv0))->((and (X ((cP ((cP Xc) cZ)) (cL Xv0)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv0)))))))->(X ((cP Xb_1) Xu0)))))))))) (P:(a->Prop)) (x6:(P x1))=> x6) as proof of (((eq a) x1) Xv)
% Found eq_ref000:=(eq_ref00 P):((P x3)->(P x3))
% Found (eq_ref00 P) as proof of ((P x3)->(P Xv))
% Found ((eq_ref0 x3) P) as proof of ((P x3)->(P Xv))
% Found (((eq_ref a) x3) P) as proof of ((P x3)->(P Xv))
% Found (((eq_ref a) x3) P) as proof of ((P x3)->(P Xv))
% Found (fun (P:(a->Prop))=> (((eq_ref a) x3) P)) as proof of ((P x3)->(P Xv))
% Found (fun (x00:((ex a) (fun (Xb_1:a)=> ((ex a) (fun (Xu0:a)=> ((and (((eq a) ((cP Xb) Xv)) ((cP Xb_1) Xu0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv0:a), ((X ((cP Xc) Xv0))->((and (X ((cP ((cP Xc) cZ)) (cL Xv0)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv0)))))))->(X ((cP Xb_1) Xu0)))))))))) (P:(a->Prop))=> (((eq_ref a) x3) P)) as proof of (((eq a) x3) Xv)
% Found x6:(P x3)
% Instantiate: x3:=Xv:a
% Found (fun (x6:(P x3))=> x6) as proof of (P Xv)
% Found (fun (P:(a->Prop)) (x6:(P x3))=> x6) as proof of ((P x3)->(P Xv))
% Found (fun (x00:((ex a) (fun (Xb_1:a)=> ((ex a) (fun (Xu0:a)=> ((and (((eq a) ((cP Xb) Xv)) ((cP Xb_1) Xu0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv0:a), ((X ((cP Xc) Xv0))->((and (X ((cP ((cP Xc) cZ)) (cL Xv0)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv0)))))))->(X ((cP Xb_1) Xu0)))))))))) (P:(a->Prop)) (x6:(P x3))=> x6) as proof of (((eq a) x3) Xv)
% Found eq_ref00:=(eq_ref0 x1):(((eq a) x1) x1)
% Found (eq_ref0 x1) as proof of (((eq a) x1) x')
% Found ((eq_ref a) x1) as proof of (((eq a) x1) x')
% Found ((eq_ref a) x1) as proof of (((eq a) x1) x')
% Found (fun (x00:((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xb) x')) ((cP Xb_0) Xu_0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X ((cP Xc) Xv))->((and (X ((cP ((cP Xc) cZ)) (cL Xv)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X ((cP Xb_0) Xu_0))))))))))=> ((eq_ref a) x1)) as proof of (((eq a) x1) x')
% Found eq_ref00:=(eq_ref0 (forall (Xx:a), (((ex a) ((unique a) (fun (x2:a)=> ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xx) x2)) ((cP Xb_0) Xu_0))) (forall (X0:(a->Prop)), (((and (X0 ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X0 ((cP Xc) Xv))->((and (X0 ((cP ((cP Xc) cZ)) (cL Xv)))) (X0 ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X0 ((cP Xb_0) Xu_0))))))))))))->((and ((ex a) ((unique a) (fun (x2:a)=> ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP ((cP Xx) cZ)) x2)) ((cP Xb_0) Xu_0))) (forall (X0:(a->Prop)), (((and (X0 ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X0 ((cP Xc) Xv))->((and (X0 ((cP ((cP Xc) cZ)) (cL Xv)))) (X0 ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X0 ((cP Xb_0) Xu_0))))))))))))) ((ex a) ((unique a) (fun (x2:a)=> ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP ((cP Xx) ((cP cZ) cZ))) x2)) ((cP Xb_0) Xu_0))) (forall (X0:(a->Prop)), (((and (X0 ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X0 ((cP Xc) Xv))->((and (X0 ((cP ((cP Xc) cZ)) (cL Xv)))) (X0 ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X0 ((cP Xb_0) Xu_0)))))))))))))))):(((eq Prop) (forall (Xx:a), (((ex a) ((unique a) (fun (x2:a)=> ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xx) x2)) ((cP Xb_0) Xu_0))) (forall (X0:(a->Prop)), (((and (X0 ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X0 ((cP Xc) Xv))->((and (X0 ((cP ((cP Xc) cZ)) (cL Xv)))) (X0 ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X0 ((cP Xb_0) Xu_0))))))))))))->((and ((ex a) ((unique a) (fun (x2:a)=> ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP ((cP Xx) cZ)) x2)) ((cP Xb_0) Xu_0))) (forall (X0:(a->Prop)), (((and (X0 ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X0 ((cP Xc) Xv))->((and (X0 ((cP ((cP Xc) cZ)) (cL Xv)))) (X0 ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X0 ((cP Xb_0) Xu_0))))))))))))) ((ex a) ((unique a) (fun (x2:a)=> ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP ((cP Xx) ((cP cZ) cZ))) x2)) ((cP Xb_0) Xu_0))) (forall (X0:(a->Prop)), (((and (X0 ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X0 ((cP Xc) Xv))->((and (X0 ((cP ((cP Xc) cZ)) (cL Xv)))) (X0 ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X0 ((cP Xb_0) Xu_0)))))))))))))))) (forall (Xx:a), (((ex a) ((unique a) (fun (x2:a)=> ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xx) x2)) ((cP Xb_0) Xu_0))) (forall (X0:(a->Prop)), (((and (X0 ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X0 ((cP Xc) Xv))->((and (X0 ((cP ((cP Xc) cZ)) (cL Xv)))) (X0 ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X0 ((cP Xb_0) Xu_0))))))))))))->((and ((ex a) ((unique a) (fun (x2:a)=> ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP ((cP Xx) cZ)) x2)) ((cP Xb_0) Xu_0))) (forall (X0:(a->Prop)), (((and (X0 ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X0 ((cP Xc) Xv))->((and (X0 ((cP ((cP Xc) cZ)) (cL Xv)))) (X0 ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X0 ((cP Xb_0) Xu_0))))))))))))) ((ex a) ((unique a) (fun (x2:a)=> ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP ((cP Xx) ((cP cZ) cZ))) x2)) ((cP Xb_0) Xu_0))) (forall (X0:(a->Prop)), (((and (X0 ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X0 ((cP Xc) Xv))->((and (X0 ((cP ((cP Xc) cZ)) (cL Xv)))) (X0 ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X0 ((cP Xb_0) Xu_0))))))))))))))))
% Found (eq_ref0 (forall (Xx:a), (((ex a) ((unique a) (fun (x2:a)=> ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xx) x2)) ((cP Xb_0) Xu_0))) (forall (X0:(a->Prop)), (((and (X0 ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X0 ((cP Xc) Xv))->((and (X0 ((cP ((cP Xc) cZ)) (cL Xv)))) (X0 ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X0 ((cP Xb_0) Xu_0))))))))))))->((and ((ex a) ((unique a) (fun (x2:a)=> ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP ((cP Xx) cZ)) x2)) ((cP Xb_0) Xu_0))) (forall (X0:(a->Prop)), (((and (X0 ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X0 ((cP Xc) Xv))->((and (X0 ((cP ((cP Xc) cZ)) (cL Xv)))) (X0 ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X0 ((cP Xb_0) Xu_0))))))))))))) ((ex a) ((unique a) (fun (x2:a)=> ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP ((cP Xx) ((cP cZ) cZ))) x2)) ((cP Xb_0) Xu_0))) (forall (X0:(a->Prop)), (((and (X0 ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X0 ((cP Xc) Xv))->((and (X0 ((cP ((cP Xc) cZ)) (cL Xv)))) (X0 ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X0 ((cP Xb_0) Xu_0)))))))))))))))) as proof of (((eq Prop) (forall (Xx:a), (((ex a) ((unique a) (fun (x2:a)=> ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xx) x2)) ((cP Xb_0) Xu_0))) (forall (X0:(a->Prop)), (((and (X0 ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X0 ((cP Xc) Xv))->((and (X0 ((cP ((cP Xc) cZ)) (cL Xv)))) (X0 ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X0 ((cP Xb_0) Xu_0))))))))))))->((and ((ex a) ((unique a) (fun (x2:a)=> ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP ((cP Xx) cZ)) x2)) ((cP Xb_0) Xu_0))) (forall (X0:(a->Prop)), (((and (X0 ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X0 ((cP Xc) Xv))->((and (X0 ((cP ((cP Xc) cZ)) (cL Xv)))) (X0 ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X0 ((cP Xb_0) Xu_0))))))))))))) ((ex a) ((unique a) (fun (x2:a)=> ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP ((cP Xx) ((cP cZ) cZ))) x2)) ((cP Xb_0) Xu_0))) (forall (X0:(a->Prop)), (((and (X0 ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X0 ((cP Xc) Xv))->((and (X0 ((cP ((cP Xc) cZ)) (cL Xv)))) (X0 ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X0 ((cP Xb_0) Xu_0)))))))))))))))) b)
% Found ((eq_ref Prop) (forall (Xx:a), (((ex a) ((unique a) (fun (x2:a)=> ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xx) x2)) ((cP Xb_0) Xu_0))) (forall (X0:(a->Prop)), (((and (X0 ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X0 ((cP Xc) Xv))->((and (X0 ((cP ((cP Xc) cZ)) (cL Xv)))) (X0 ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X0 ((cP Xb_0) Xu_0))))))))))))->((and ((ex a) ((unique a) (fun (x2:a)=> ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP ((cP Xx) cZ)) x2)) ((cP Xb_0) Xu_0))) (forall (X0:(a->Prop)), (((and (X0 ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X0 ((cP Xc) Xv))->((and (X0 ((cP ((cP Xc) cZ)) (cL Xv)))) (X0 ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X0 ((cP Xb_0) Xu_0))))))))))))) ((ex a) ((unique a) (fun (x2:a)=> ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP ((cP Xx) ((cP cZ) cZ))) x2)) ((cP Xb_0) Xu_0))) (forall (X0:(a->Prop)), (((and (X0 ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X0 ((cP Xc) Xv))->((and (X0 ((cP ((cP Xc) cZ)) (cL Xv)))) (X0 ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X0 ((cP Xb_0) Xu_0)))))))))))))))) as proof of (((eq Prop) (forall (Xx:a), (((ex a) ((unique a) (fun (x2:a)=> ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xx) x2)) ((cP Xb_0) Xu_0))) (forall (X0:(a->Prop)), (((and (X0 ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X0 ((cP Xc) Xv))->((and (X0 ((cP ((cP Xc) cZ)) (cL Xv)))) (X0 ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X0 ((cP Xb_0) Xu_0))))))))))))->((and ((ex a) ((unique a) (fun (x2:a)=> ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP ((cP Xx) cZ)) x2)) ((cP Xb_0) Xu_0))) (forall (X0:(a->Prop)), (((and (X0 ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X0 ((cP Xc) Xv))->((and (X0 ((cP ((cP Xc) cZ)) (cL Xv)))) (X0 ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X0 ((cP Xb_0) Xu_0))))))))))))) ((ex a) ((unique a) (fun (x2:a)=> ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP ((cP Xx) ((cP cZ) cZ))) x2)) ((cP Xb_0) Xu_0))) (forall (X0:(a->Prop)), (((and (X0 ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X0 ((cP Xc) Xv))->((and (X0 ((cP ((cP Xc) cZ)) (cL Xv)))) (X0 ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X0 ((cP Xb_0) Xu_0)))))))))))))))) b)
% Found ((eq_ref Prop) (forall (Xx:a), (((ex a) ((unique a) (fun (x2:a)=> ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xx) x2)) ((cP Xb_0) Xu_0))) (forall (X0:(a->Prop)), (((and (X0 ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X0 ((cP Xc) Xv))->((and (X0 ((cP ((cP Xc) cZ)) (cL Xv)))) (X0 ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X0 ((cP Xb_0) Xu_0))))))))))))->((and ((ex a) ((unique a) (fun (x2:a)=> ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP ((cP Xx) cZ)) x2)) ((cP Xb_0) Xu_0))) (forall (X0:(a->Prop)), (((and (X0 ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X0 ((cP Xc) Xv))->((and (X0 ((cP ((cP Xc) cZ)) (cL Xv)))) (X0 ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X0 ((cP Xb_0) Xu_0))))))))))))) ((ex a) ((unique a) (fun (x2:a)=> ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP ((cP Xx) ((cP cZ) cZ))) x2)) ((cP Xb_0) Xu_0))) (forall (X0:(a->Prop)), (((and (X0 ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X0 ((cP Xc) Xv))->((and (X0 ((cP ((cP Xc) cZ)) (cL Xv)))) (X0 ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X0 ((cP Xb_0) Xu_0)))))))))))))))) as proof of (((eq Prop) (forall (Xx:a), (((ex a) ((unique a) (fun (x2:a)=> ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xx) x2)) ((cP Xb_0) Xu_0))) (forall (X0:(a->Prop)), (((and (X0 ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X0 ((cP Xc) Xv))->((and (X0 ((cP ((cP Xc) cZ)) (cL Xv)))) (X0 ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X0 ((cP Xb_0) Xu_0))))))))))))->((and ((ex a) ((unique a) (fun (x2:a)=> ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP ((cP Xx) cZ)) x2)) ((cP Xb_0) Xu_0))) (forall (X0:(a->Prop)), (((and (X0 ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X0 ((cP Xc) Xv))->((and (X0 ((cP ((cP Xc) cZ)) (cL Xv)))) (X0 ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X0 ((cP Xb_0) Xu_0))))))))))))) ((ex a) ((unique a) (fun (x2:a)=> ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP ((cP Xx) ((cP cZ) cZ))) x2)) ((cP Xb_0) Xu_0))) (forall (X0:(a->Prop)), (((and (X0 ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X0 ((cP Xc) Xv))->((and (X0 ((cP ((cP Xc) cZ)) (cL Xv)))) (X0 ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X0 ((cP Xb_0) Xu_0)))))))))))))))) b)
% Found ((eq_ref Prop) (forall (Xx:a), (((ex a) ((unique a) (fun (x2:a)=> ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xx) x2)) ((cP Xb_0) Xu_0))) (forall (X0:(a->Prop)), (((and (X0 ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X0 ((cP Xc) Xv))->((and (X0 ((cP ((cP Xc) cZ)) (cL Xv)))) (X0 ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X0 ((cP Xb_0) Xu_0))))))))))))->((and ((ex a) ((unique a) (fun (x2:a)=> ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP ((cP Xx) cZ)) x2)) ((cP Xb_0) Xu_0))) (forall (X0:(a->Prop)), (((and (X0 ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X0 ((cP Xc) Xv))->((and (X0 ((cP ((cP Xc) cZ)) (cL Xv)))) (X0 ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X0 ((cP Xb_0) Xu_0))))))))))))) ((ex a) ((unique a) (fun (x2:a)=> ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP ((cP Xx) ((cP cZ) cZ))) x2)) ((cP Xb_0) Xu_0))) (forall (X0:(a->Prop)), (((and (X0 ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X0 ((cP Xc) Xv))->((and (X0 ((cP ((cP Xc) cZ)) (cL Xv)))) (X0 ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X0 ((cP Xb_0) Xu_0)))))))))))))))) as proof of (((eq Prop) (forall (Xx:a), (((ex a) ((unique a) (fun (x2:a)=> ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xx) x2)) ((cP Xb_0) Xu_0))) (forall (X0:(a->Prop)), (((and (X0 ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X0 ((cP Xc) Xv))->((and (X0 ((cP ((cP Xc) cZ)) (cL Xv)))) (X0 ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X0 ((cP Xb_0) Xu_0))))))))))))->((and ((ex a) ((unique a) (fun (x2:a)=> ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP ((cP Xx) cZ)) x2)) ((cP Xb_0) Xu_0))) (forall (X0:(a->Prop)), (((and (X0 ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X0 ((cP Xc) Xv))->((and (X0 ((cP ((cP Xc) cZ)) (cL Xv)))) (X0 ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X0 ((cP Xb_0) Xu_0))))))))))))) ((ex a) ((unique a) (fun (x2:a)=> ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP ((cP Xx) ((cP cZ) cZ))) x2)) ((cP Xb_0) Xu_0))) (forall (X0:(a->Prop)), (((and (X0 ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X0 ((cP Xc) Xv))->((and (X0 ((cP ((cP Xc) cZ)) (cL Xv)))) (X0 ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X0 ((cP Xb_0) Xu_0)))))))))))))))) b)
% Found eq_ref00:=(eq_ref0 x3):(((eq a) x3) x3)
% Found (eq_ref0 x3) as proof of (((eq a) x3) x')
% Found ((eq_ref a) x3) as proof of (((eq a) x3) x')
% Found ((eq_ref a) x3) as proof of (((eq a) x3) x')
% Found (fun (x00:((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xb) x')) ((cP Xb_0) Xu_0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X ((cP Xc) Xv))->((and (X ((cP ((cP Xc) cZ)) (cL Xv)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X ((cP Xb_0) Xu_0))))))))))=> ((eq_ref a) x3)) as proof of (((eq a) x3) x')
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) (((unique a) (fun (x2:a)=> ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xb) x2)) ((cP Xb_0) Xu_0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X ((cP Xc) Xv))->((and (X ((cP ((cP Xc) cZ)) (cL Xv)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X ((cP Xb_0) Xu_0))))))))))) x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (((unique a) (fun (x2:a)=> ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xb) x2)) ((cP Xb_0) Xu_0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X ((cP Xc) Xv))->((and (X ((cP ((cP Xc) cZ)) (cL Xv)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X ((cP Xb_0) Xu_0))))))))))) x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (((unique a) (fun (x2:a)=> ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xb) x2)) ((cP Xb_0) Xu_0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X ((cP Xc) Xv))->((and (X ((cP ((cP Xc) cZ)) (cL Xv)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X ((cP Xb_0) Xu_0))))))))))) x1))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) (((unique a) (fun (x2:a)=> ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xb) x2)) ((cP Xb_0) Xu_0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X ((cP Xc) Xv))->((and (X ((cP ((cP Xc) cZ)) (cL Xv)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X ((cP Xb_0) Xu_0))))))))))) x1))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:a), (((eq Prop) (f x)) (((unique a) (fun (x2:a)=> ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xb) x2)) ((cP Xb_0) Xu_0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X ((cP Xc) Xv))->((and (X ((cP ((cP Xc) cZ)) (cL Xv)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X ((cP Xb_0) Xu_0))))))))))) x)))
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) (((unique a) (fun (x2:a)=> ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xb) x2)) ((cP Xb_0) Xu_0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X ((cP Xc) Xv))->((and (X ((cP ((cP Xc) cZ)) (cL Xv)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X ((cP Xb_0) Xu_0))))))))))) x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (((unique a) (fun (x2:a)=> ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xb) x2)) ((cP Xb_0) Xu_0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X ((cP Xc) Xv))->((and (X ((cP ((cP Xc) cZ)) (cL Xv)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X ((cP Xb_0) Xu_0))))))))))) x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (((unique a) (fun (x2:a)=> ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xb) x2)) ((cP Xb_0) Xu_0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X ((cP Xc) Xv))->((and (X ((cP ((cP Xc) cZ)) (cL Xv)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X ((cP Xb_0) Xu_0))))))))))) x1))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) (((unique a) (fun (x2:a)=> ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xb) x2)) ((cP Xb_0) Xu_0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X ((cP Xc) Xv))->((and (X ((cP ((cP Xc) cZ)) (cL Xv)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X ((cP Xb_0) Xu_0))))))))))) x1))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:a), (((eq Prop) (f x)) (((unique a) (fun (x2:a)=> ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xb) x2)) ((cP Xb_0) Xu_0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X ((cP Xc) Xv))->((and (X ((cP ((cP Xc) cZ)) (cL Xv)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X ((cP Xb_0) Xu_0))))))))))) x)))
% Found eq_ref000:=(eq_ref00 P):((P x5)->(P x5))
% Found (eq_ref00 P) as proof of ((P x5)->(P x'))
% Found ((eq_ref0 x5) P) as proof of ((P x5)->(P x'))
% Found (((eq_ref a) x5) P) as proof of ((P x5)->(P x'))
% Found (((eq_ref a) x5) P) as proof of ((P x5)->(P x'))
% Found (fun (P:(a->Prop))=> (((eq_ref a) x5) P)) as proof of ((P x5)->(P x'))
% Found (fun (x00:((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xb) x')) ((cP Xb_0) Xu_0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X ((cP Xc) Xv))->((and (X ((cP ((cP Xc) cZ)) (cL Xv)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X ((cP Xb_0) Xu_0)))))))))) (P:(a->Prop))=> (((eq_ref a) x5) P)) as proof of (((eq a) x5) x')
% Found x6:(P x5)
% Instantiate: x5:=x':a
% Found (fun (x6:(P x5))=> x6) as proof of (P x')
% Found (fun (P:(a->Prop)) (x6:(P x5))=> x6) as proof of ((P x5)->(P x'))
% Found (fun (x00:((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xb) x')) ((cP Xb_0) Xu_0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X ((cP Xc) Xv))->((and (X ((cP ((cP Xc) cZ)) (cL Xv)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X ((cP Xb_0) Xu_0)))))))))) (P:(a->Prop)) (x6:(P x5))=> x6) as proof of (((eq a) x5) x')
% Found eq_ref00:=(eq_ref0 x1):(((eq a) x1) x1)
% Found (eq_ref0 x1) as proof of (((eq a) x1) x')
% Found ((eq_ref a) x1) as proof of (((eq a) x1) x')
% Found ((eq_ref a) x1) as proof of (((eq a) x1) x')
% Found (fun (x3:(forall (Xt:a), ((iff (not (((eq a) Xt) cZ))) (((eq a) Xt) ((cP (cL Xt)) (cR Xt))))))=> ((eq_ref a) x1)) as proof of (((eq a) x1) x')
% Found (fun (x2:((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)))) (x3:(forall (Xt:a), ((iff (not (((eq a) Xt) cZ))) (((eq a) Xt) ((cP (cL Xt)) (cR Xt))))))=> ((eq_ref a) x1)) as proof of ((forall (Xt:a), ((iff (not (((eq a) Xt) cZ))) (((eq a) Xt) ((cP (cL Xt)) (cR Xt)))))->(((eq a) x1) x'))
% Found (fun (x2:((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)))) (x3:(forall (Xt:a), ((iff (not (((eq a) Xt) cZ))) (((eq a) Xt) ((cP (cL Xt)) (cR Xt))))))=> ((eq_ref a) x1)) as proof of (((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)))))->(((eq a) x1) x')))
% Found (and_rect00 (fun (x2:((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)))) (x3:(forall (Xt:a), ((iff (not (((eq a) Xt) cZ))) (((eq a) Xt) ((cP (cL Xt)) (cR Xt))))))=> ((eq_ref a) x1))) as proof of (((eq a) x1) x')
% Found ((and_rect0 (((eq a) x1) x')) (fun (x2:((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)))) (x3:(forall (Xt:a), ((iff (not (((eq a) Xt) cZ))) (((eq a) Xt) ((cP (cL Xt)) (cR Xt))))))=> ((eq_ref a) x1))) as proof of (((eq a) x1) x')
% Found (((fun (P:Type) (x2:(((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)))))->P)))=> (((((and_rect ((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)))))) P) x2) x)) (((eq a) x1) x')) (fun (x2:((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)))) (x3:(forall (Xt:a), ((iff (not (((eq a) Xt) cZ))) (((eq a) Xt) ((cP (cL Xt)) (cR Xt))))))=> ((eq_ref a) x1))) as proof of (((eq a) x1) x')
% Found (fun (x00:((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xb) x')) ((cP Xb_0) Xu_0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X ((cP Xc) Xv))->((and (X ((cP ((cP Xc) cZ)) (cL Xv)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X ((cP Xb_0) Xu_0))))))))))=> (((fun (P:Type) (x2:(((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)))))->P)))=> (((((and_rect ((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)))))) P) x2) x)) (((eq a) x1) x')) (fun (x2:((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)))) (x3:(forall (Xt:a), ((iff (not (((eq a) Xt) cZ))) (((eq a) Xt) ((cP (cL Xt)) (cR Xt))))))=> ((eq_ref a) x1)))) as proof of (((eq a) x1) x')
% Found eta_expansion000:=(eta_expansion00 (fun (Xu:a)=> ((and ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xb) Xu)) ((cP Xb_0) Xu_0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X ((cP Xc) Xv))->((and (X ((cP ((cP Xc) cZ)) (cL Xv)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X ((cP Xb_0) Xu_0)))))))))) (forall (Xv:a), (((ex a) (fun (Xb_1:a)=> ((ex a) (fun (Xu0:a)=> ((and (((eq a) ((cP Xb) Xv)) ((cP Xb_1) Xu0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv0:a), ((X ((cP Xc) Xv0))->((and (X ((cP ((cP Xc) cZ)) (cL Xv0)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv0)))))))->(X ((cP Xb_1) Xu0)))))))))->(((eq a) Xu) Xv)))))):(((eq (a->Prop)) (fun (Xu:a)=> ((and ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xb) Xu)) ((cP Xb_0) Xu_0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X ((cP Xc) Xv))->((and (X ((cP ((cP Xc) cZ)) (cL Xv)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X ((cP Xb_0) Xu_0)))))))))) (forall (Xv:a), (((ex a) (fun (Xb_1:a)=> ((ex a) (fun (Xu0:a)=> ((and (((eq a) ((cP Xb) Xv)) ((cP Xb_1) Xu0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv0:a), ((X ((cP Xc) Xv0))->((and (X ((cP ((cP Xc) cZ)) (cL Xv0)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv0)))))))->(X ((cP Xb_1) Xu0)))))))))->(((eq a) Xu) Xv)))))) (fun (x:a)=> ((and ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xb) x)) ((cP Xb_0) Xu_0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X ((cP Xc) Xv))->((and (X ((cP ((cP Xc) cZ)) (cL Xv)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X ((cP Xb_0) Xu_0)))))))))) (forall (Xv:a), (((ex a) (fun (Xb_1:a)=> ((ex a) (fun (Xu0:a)=> ((and (((eq a) ((cP Xb) Xv)) ((cP Xb_1) Xu0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv0:a), ((X ((cP Xc) Xv0))->((and (X ((cP ((cP Xc) cZ)) (cL Xv0)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv0)))))))->(X ((cP Xb_1) Xu0)))))))))->(((eq a) x) Xv))))))
% Found (eta_expansion00 (fun (Xu:a)=> ((and ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xb) Xu)) ((cP Xb_0) Xu_0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X ((cP Xc) Xv))->((and (X ((cP ((cP Xc) cZ)) (cL Xv)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X ((cP Xb_0) Xu_0)))))))))) (forall (Xv:a), (((ex a) (fun (Xb_1:a)=> ((ex a) (fun (Xu0:a)=> ((and (((eq a) ((cP Xb) Xv)) ((cP Xb_1) Xu0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv0:a), ((X ((cP Xc) Xv0))->((and (X ((cP ((cP Xc) cZ)) (cL Xv0)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv0)))))))->(X ((cP Xb_1) Xu0)))))))))->(((eq a) Xu) Xv)))))) as proof of (((eq (a->Prop)) (fun (Xu:a)=> ((and ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xb) Xu)) ((cP Xb_0) Xu_0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X ((cP Xc) Xv))->((and (X ((cP ((cP Xc) cZ)) (cL Xv)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X ((cP Xb_0) Xu_0)))))))))) (forall (Xv:a), (((ex a) (fun (Xb_1:a)=> ((ex a) (fun (Xu0:a)=> ((and (((eq a) ((cP Xb) Xv)) ((cP Xb_1) Xu0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv0:a), ((X ((cP Xc) Xv0))->((and (X ((cP ((cP Xc) cZ)) (cL Xv0)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv0)))))))->(X ((cP Xb_1) Xu0)))))))))->(((eq a) Xu) Xv)))))) b)
% Found ((eta_expansion0 Prop) (fun (Xu:a)=> ((and ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xb) Xu)) ((cP Xb_0) Xu_0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X ((cP Xc) Xv))->((and (X ((cP ((cP Xc) cZ)) (cL Xv)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X ((cP Xb_0) Xu_0)))))))))) (forall (Xv:a), (((ex a) (fun (Xb_1:a)=> ((ex a) (fun (Xu0:a)=> ((and (((eq a) ((cP Xb) Xv)) ((cP Xb_1) Xu0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv0:a), ((X ((cP Xc) Xv0))->((and (X ((cP ((cP Xc) cZ)) (cL Xv0)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv0)))))))->(X ((cP Xb_1) Xu0)))))))))->(((eq a) Xu) Xv)))))) as proof of (((eq (a->Prop)) (fun (Xu:a)=> ((and ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xb) Xu)) ((cP Xb_0) Xu_0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X ((cP Xc) Xv))->((and (X ((cP ((cP Xc) cZ)) (cL Xv)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X ((cP Xb_0) Xu_0)))))))))) (forall (Xv:a), (((ex a) (fun (Xb_1:a)=> ((ex a) (fun (Xu0:a)=> ((and (((eq a) ((cP Xb) Xv)) ((cP Xb_1) Xu0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv0:a), ((X ((cP Xc) Xv0))->((and (X ((cP ((cP Xc) cZ)) (cL Xv0)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv0)))))))->(X ((cP Xb_1) Xu0)))))))))->(((eq a) Xu) Xv)))))) b)
% Found (((eta_expansion a) Prop) (fun (Xu:a)=> ((and ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xb) Xu)) ((cP Xb_0) Xu_0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X ((cP Xc) Xv))->((and (X ((cP ((cP Xc) cZ)) (cL Xv)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X ((cP Xb_0) Xu_0)))))))))) (forall (Xv:a), (((ex a) (fun (Xb_1:a)=> ((ex a) (fun (Xu0:a)=> ((and (((eq a) ((cP Xb) Xv)) ((cP Xb_1) Xu0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv0:a), ((X ((cP Xc) Xv0))->((and (X ((cP ((cP Xc) cZ)) (cL Xv0)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv0)))))))->(X ((cP Xb_1) Xu0)))))))))->(((eq a) Xu) Xv)))))) as proof of (((eq (a->Prop)) (fun (Xu:a)=> ((and ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xb) Xu)) ((cP Xb_0) Xu_0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X ((cP Xc) Xv))->((and (X ((cP ((cP Xc) cZ)) (cL Xv)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X ((cP Xb_0) Xu_0)))))))))) (forall (Xv:a), (((ex a) (fun (Xb_1:a)=> ((ex a) (fun (Xu0:a)=> ((and (((eq a) ((cP Xb) Xv)) ((cP Xb_1) Xu0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv0:a), ((X ((cP Xc) Xv0))->((and (X ((cP ((cP Xc) cZ)) (cL Xv0)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv0)))))))->(X ((cP Xb_1) Xu0)))))))))->(((eq a) Xu) Xv)))))) b)
% Found (((eta_expansion a) Prop) (fun (Xu:a)=> ((and ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xb) Xu)) ((cP Xb_0) Xu_0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X ((cP Xc) Xv))->((and (X ((cP ((cP Xc) cZ)) (cL Xv)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X ((cP Xb_0) Xu_0)))))))))) (forall (Xv:a), (((ex a) (fun (Xb_1:a)=> ((ex a) (fun (Xu0:a)=> ((and (((eq a) ((cP Xb) Xv)) ((cP Xb_1) Xu0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv0:a), ((X ((cP Xc) Xv0))->((and (X ((cP ((cP Xc) cZ)) (cL Xv0)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv0)))))))->(X ((cP Xb_1) Xu0)))))))))->(((eq a) Xu) Xv)))))) as proof of (((eq (a->Prop)) (fun (Xu:a)=> ((and ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xb) Xu)) ((cP Xb_0) Xu_0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X ((cP Xc) Xv))->((and (X ((cP ((cP Xc) cZ)) (cL Xv)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X ((cP Xb_0) Xu_0)))))))))) (forall (Xv:a), (((ex a) (fun (Xb_1:a)=> ((ex a) (fun (Xu0:a)=> ((and (((eq a) ((cP Xb) Xv)) ((cP Xb_1) Xu0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv0:a), ((X ((cP Xc) Xv0))->((and (X ((cP ((cP Xc) cZ)) (cL Xv0)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv0)))))))->(X ((cP Xb_1) Xu0)))))))))->(((eq a) Xu) Xv)))))) b)
% Found (((eta_expansion a) Prop) (fun (Xu:a)=> ((and ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xb) Xu)) ((cP Xb_0) Xu_0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X ((cP Xc) Xv))->((and (X ((cP ((cP Xc) cZ)) (cL Xv)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X ((cP Xb_0) Xu_0)))))))))) (forall (Xv:a), (((ex a) (fun (Xb_1:a)=> ((ex a) (fun (Xu0:a)=> ((and (((eq a) ((cP Xb) Xv)) ((cP Xb_1) Xu0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv0:a), ((X ((cP Xc) Xv0))->((and (X ((cP ((cP Xc) cZ)) (cL Xv0)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv0)))))))->(X ((cP Xb_1) Xu0)))))))))->(((eq a) Xu) Xv)))))) as proof of (((eq (a->Prop)) (fun (Xu:a)=> ((and ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xb) Xu)) ((cP Xb_0) Xu_0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X ((cP Xc) Xv))->((and (X ((cP ((cP Xc) cZ)) (cL Xv)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X ((cP Xb_0) Xu_0)))))))))) (forall (Xv:a), (((ex a) (fun (Xb_1:a)=> ((ex a) (fun (Xu0:a)=> ((and (((eq a) ((cP Xb) Xv)) ((cP Xb_1) Xu0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv0:a), ((X ((cP Xc) Xv0))->((and (X ((cP ((cP Xc) cZ)) (cL Xv0)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv0)))))))->(X ((cP Xb_1) Xu0)))))))))->(((eq a) Xu) Xv)))))) b)
% Found eq_ref00:=(eq_ref0 x3):(((eq a) x3) x3)
% Found (eq_ref0 x3) as proof of (((eq a) x3) Xv)
% Found ((eq_ref a) x3) as proof of (((eq a) x3) Xv)
% Found ((eq_ref a) x3) as proof of (((eq a) x3) Xv)
% Found (fun (x5:((ex a) (fun (Xu0:a)=> ((and (((eq a) ((cP Xb) Xv)) ((cP x4) Xu0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv0:a), ((X ((cP Xc) Xv0))->((and (X ((cP ((cP Xc) cZ)) (cL Xv0)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv0)))))))->(X ((cP x4) Xu0))))))))=> ((eq_ref a) x3)) as proof of (((eq a) x3) Xv)
% Found (fun (x4:a) (x5:((ex a) (fun (Xu0:a)=> ((and (((eq a) ((cP Xb) Xv)) ((cP x4) Xu0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv0:a), ((X ((cP Xc) Xv0))->((and (X ((cP ((cP Xc) cZ)) (cL Xv0)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv0)))))))->(X ((cP x4) Xu0))))))))=> ((eq_ref a) x3)) as proof of (((ex a) (fun (Xu0:a)=> ((and (((eq a) ((cP Xb) Xv)) ((cP x4) Xu0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv0:a), ((X ((cP Xc) Xv0))->((and (X ((cP ((cP Xc) cZ)) (cL Xv0)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv0)))))))->(X ((cP x4) Xu0)))))))->(((eq a) x3) Xv))
% Found (fun (x4:a) (x5:((ex a) (fun (Xu0:a)=> ((and (((eq a) ((cP Xb) Xv)) ((cP x4) Xu0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv0:a), ((X ((cP Xc) Xv0))->((and (X ((cP ((cP Xc) cZ)) (cL Xv0)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv0)))))))->(X ((cP x4) Xu0))))))))=> ((eq_ref a) x3)) as proof of (forall (x:a), (((ex a) (fun (Xu0:a)=> ((and (((eq a) ((cP Xb) Xv)) ((cP x) Xu0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv0:a), ((X ((cP Xc) Xv0))->((and (X ((cP ((cP Xc) cZ)) (cL Xv0)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv0)))))))->(X ((cP x) Xu0)))))))->(((eq a) x3) Xv)))
% Found (ex_ind00 (fun (x4:a) (x5:((ex a) (fun (Xu0:a)=> ((and (((eq a) ((cP Xb) Xv)) ((cP x4) Xu0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv0:a), ((X ((cP Xc) Xv0))->((and (X ((cP ((cP Xc) cZ)) (cL Xv0)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv0)))))))->(X ((cP x4) Xu0))))))))=> ((eq_ref a) x3))) as proof of (((eq a) x3) Xv)
% Found ((ex_ind0 (((eq a) x3) Xv)) (fun (x4:a) (x5:((ex a) (fun (Xu0:a)=> ((and (((eq a) ((cP Xb) Xv)) ((cP x4) Xu0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv0:a), ((X ((cP Xc) Xv0))->((and (X ((cP ((cP Xc) cZ)) (cL Xv0)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv0)))))))->(X ((cP x4) Xu0))))))))=> ((eq_ref a) x3))) as proof of (((eq a) x3) Xv)
% Found (((fun (P:Prop) (x4:(forall (x:a), (((ex a) (fun (Xu0:a)=> ((and (((eq a) ((cP Xb) Xv)) ((cP x) Xu0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv0:a), ((X ((cP Xc) Xv0))->((and (X ((cP ((cP Xc) cZ)) (cL Xv0)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv0)))))))->(X ((cP x) Xu0)))))))->P)))=> (((((ex_ind a) (fun (Xb_1:a)=> ((ex a) (fun (Xu0:a)=> ((and (((eq a) ((cP Xb) Xv)) ((cP Xb_1) Xu0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv0:a), ((X ((cP Xc) Xv0))->((and (X ((cP ((cP Xc) cZ)) (cL Xv0)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv0)))))))->(X ((cP Xb_1) Xu0))))))))) P) x4) x00)) (((eq a) x3) Xv)) (fun (x4:a) (x5:((ex a) (fun (Xu0:a)=> ((and (((eq a) ((cP Xb) Xv)) ((cP x4) Xu0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv0:a), ((X ((cP Xc) Xv0))->((and (X ((cP ((cP Xc) cZ)) (cL Xv0)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv0)))))))->(X ((cP x4) Xu0))))))))=> ((eq_ref a) x3))) as proof of (((eq a) x3) Xv)
% Found (fun (x00:((ex a) (fun (Xb_1:a)=> ((ex a) (fun (Xu0:a)=> ((and (((eq a) ((cP Xb) Xv)) ((cP Xb_1) Xu0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv0:a), ((X ((cP Xc) Xv0))->((and (X ((cP ((cP Xc) cZ)) (cL Xv0)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv0)))))))->(X ((cP Xb_1) Xu0))))))))))=> (((fun (P:Prop) (x4:(forall (x:a), (((ex a) (fun (Xu0:a)=> ((and (((eq a) ((cP Xb) Xv)) ((cP x) Xu0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv0:a), ((X ((cP Xc) Xv0))->((and (X ((cP ((cP Xc) cZ)) (cL Xv0)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv0)))))))->(X ((cP x) Xu0)))))))->P)))=> (((((ex_ind a) (fun (Xb_1:a)=> ((ex a) (fun (Xu0:a)=> ((and (((eq a) ((cP Xb) Xv)) ((cP Xb_1) Xu0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv0:a), ((X ((cP Xc) Xv0))->((and (X ((cP ((cP Xc) cZ)) (cL Xv0)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv0)))))))->(X ((cP Xb_1) Xu0))))))))) P) x4) x00)) (((eq a) x3) Xv)) (fun (x4:a) (x5:((ex a) (fun (Xu0:a)=> ((and (((eq a) ((cP Xb) Xv)) ((cP x4) Xu0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv0:a), ((X ((cP Xc) Xv0))->((and (X ((cP ((cP Xc) cZ)) (cL Xv0)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv0)))))))->(X ((cP x4) Xu0))))))))=> ((eq_ref a) x3)))) as proof of (((eq a) x3) Xv)
% Found eq_ref00:=(eq_ref0 Xv):(((eq a) Xv) Xv)
% Found (eq_ref0 Xv) as proof of (((eq a) Xv) x3)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) x3)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) x3)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) x3)
% Found (eq_sym000 ((eq_ref a) Xv)) as proof of (((eq a) x3) Xv)
% Found ((eq_sym00 x3) ((eq_ref a) Xv)) as proof of (((eq a) x3) Xv)
% Found (((eq_sym0 Xv) x3) ((eq_ref a) Xv)) as proof of (((eq a) x3) Xv)
% Found ((((eq_sym a) Xv) x3) ((eq_ref a) Xv)) as proof of (((eq a) x3) Xv)
% Found (fun (x00:((ex a) (fun (Xb_1:a)=> ((ex a) (fun (Xu0:a)=> ((and (((eq a) ((cP Xb) Xv)) ((cP Xb_1) Xu0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv0:a), ((X ((cP Xc) Xv0))->((and (X ((cP ((cP Xc) cZ)) (cL Xv0)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv0)))))))->(X ((cP Xb_1) Xu0))))))))))=> ((((eq_sym a) Xv) x3) ((eq_ref a) Xv))) as proof of (((eq a) x3) Xv)
% Found x6:(P x1)
% Instantiate: x1:=x':a
% Found (fun (x6:(P x1))=> x6) as proof of (P x')
% Found (fun (P:(a->Prop)) (x6:(P x1))=> x6) as proof of ((P x1)->(P x'))
% Found (fun (x00:((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xb) x')) ((cP Xb_0) Xu_0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X ((cP Xc) Xv))->((and (X ((cP ((cP Xc) cZ)) (cL Xv)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X ((cP Xb_0) Xu_0)))))))))) (P:(a->Prop)) (x6:(P x1))=> x6) as proof of (((eq a) x1) x')
% Found eq_ref000:=(eq_ref00 P):((P x1)->(P x1))
% Found (eq_ref00 P) as proof of ((P x1)->(P x'))
% Found ((eq_ref0 x1) P) as proof of ((P x1)->(P x'))
% Found (((eq_ref a) x1) P) as proof of ((P x1)->(P x'))
% Found (((eq_ref a) x1) P) as proof of ((P x1)->(P x'))
% Found (fun (P:(a->Prop))=> (((eq_ref a) x1) P)) as proof of ((P x1)->(P x'))
% Found (fun (x00:((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xb) x')) ((cP Xb_0) Xu_0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X ((cP Xc) Xv))->((and (X ((cP ((cP Xc) cZ)) (cL Xv)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X ((cP Xb_0) Xu_0)))))))))) (P:(a->Prop))=> (((eq_ref a) x1) P)) as proof of (((eq a) x1) x')
% Found eq_ref000:=(eq_ref00 P):((P x3)->(P x3))
% Found (eq_ref00 P) as proof of ((P x3)->(P x'))
% Found ((eq_ref0 x3) P) as proof of ((P x3)->(P x'))
% Found (((eq_ref a) x3) P) as proof of ((P x3)->(P x'))
% Found (((eq_ref a) x3) P) as proof of ((P x3)->(P x'))
% Found (fun (P:(a->Prop))=> (((eq_ref a) x3) P)) as proof of ((P x3)->(P x'))
% Found (fun (x00:((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xb) x')) ((cP Xb_0) Xu_0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X ((cP Xc) Xv))->((and (X ((cP ((cP Xc) cZ)) (cL Xv)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X ((cP Xb_0) Xu_0)))))))))) (P:(a->Prop))=> (((eq_ref a) x3) P)) as proof of (((eq a) x3) x')
% Found x6:(P x3)
% Instantiate: x3:=x':a
% Found (fun (x6:(P x3))=> x6) as proof of (P x')
% Found (fun (P:(a->Prop)) (x6:(P x3))=> x6) as proof of ((P x3)->(P x'))
% Found (fun (x00:((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xb) x')) ((cP Xb_0) Xu_0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X ((cP Xc) Xv))->((and (X ((cP ((cP Xc) cZ)) (cL Xv)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X ((cP Xb_0) Xu_0)))))))))) (P:(a->Prop)) (x6:(P x3))=> x6) as proof of (((eq a) x3) x')
% Found eq_ref00:=(eq_ref0 x1):(((eq a) x1) x1)
% Found (eq_ref0 x1) as proof of (((eq a) x1) x')
% Found ((eq_ref a) x1) as proof of (((eq a) x1) x')
% Found ((eq_ref a) x1) as proof of (((eq a) x1) x')
% Found (fun (x01:((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP cZ) x')) ((cP Xb_0) Xu_0))) (forall (X0:(a->Prop)), (((and (X0 ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X0 ((cP Xc) Xv))->((and (X0 ((cP ((cP Xc) cZ)) (cL Xv)))) (X0 ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X0 ((cP Xb_0) Xu_0))))))))))=> ((eq_ref a) x1)) as proof of (((eq a) x1) x')
% Found eq_ref00:=(eq_ref0 x1):(((eq a) x1) x1)
% Found (eq_ref0 x1) as proof of (((eq a) x1) Xv)
% Found ((eq_ref a) x1) as proof of (((eq a) x1) Xv)
% Found ((eq_ref a) x1) as proof of (((eq a) x1) Xv)
% Found (fun (x5:((ex a) (fun (Xu0:a)=> ((and (((eq a) ((cP Xb) Xv)) ((cP x4) Xu0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv0:a), ((X ((cP Xc) Xv0))->((and (X ((cP ((cP Xc) cZ)) (cL Xv0)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv0)))))))->(X ((cP x4) Xu0))))))))=> ((eq_ref a) x1)) as proof of (((eq a) x1) Xv)
% Found (fun (x4:a) (x5:((ex a) (fun (Xu0:a)=> ((and (((eq a) ((cP Xb) Xv)) ((cP x4) Xu0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv0:a), ((X ((cP Xc) Xv0))->((and (X ((cP ((cP Xc) cZ)) (cL Xv0)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv0)))))))->(X ((cP x4) Xu0))))))))=> ((eq_ref a) x1)) as proof of (((ex a) (fun (Xu0:a)=> ((and (((eq a) ((cP Xb) Xv)) ((cP x4) Xu0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv0:a), ((X ((cP Xc) Xv0))->((and (X ((cP ((cP Xc) cZ)) (cL Xv0)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv0)))))))->(X ((cP x4) Xu0)))))))->(((eq a) x1) Xv))
% Found (fun (x4:a) (x5:((ex a) (fun (Xu0:a)=> ((and (((eq a) ((cP Xb) Xv)) ((cP x4) Xu0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv0:a), ((X ((cP Xc) Xv0))->((and (X ((cP ((cP Xc) cZ)) (cL Xv0)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv0)))))))->(X ((cP x4) Xu0))))))))=> ((eq_ref a) x1)) as proof of (forall (x:a), (((ex a) (fun (Xu0:a)=> ((and (((eq a) ((cP Xb) Xv)) ((cP x) Xu0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv0:a), ((X ((cP Xc) Xv0))->((and (X ((cP ((cP Xc) cZ)) (cL Xv0)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv0)))))))->(X ((cP x) Xu0)))))))->(((eq a) x1) Xv)))
% Found (ex_ind00 (fun (x4:a) (x5:((ex a) (fun (Xu0:a)=> ((and (((eq a) ((cP Xb) Xv)) ((cP x4) Xu0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv0:a), ((X ((cP Xc) Xv0))->((and (X ((cP ((cP Xc) cZ)) (cL Xv0)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv0)))))))->(X ((cP x4) Xu0))))))))=> ((eq_ref a) x1))) as proof of (((eq a) x1) Xv)
% Found ((ex_ind0 (((eq a) x1) Xv)) (fun (x4:a) (x5:((ex a) (fun (Xu0:a)=> ((and (((eq a) ((cP Xb) Xv)) ((cP x4) Xu0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv0:a), ((X ((cP Xc) Xv0))->((and (X ((cP ((cP Xc) cZ)) (cL Xv0)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv0)))))))->(X ((cP x4) Xu0))))))))=> ((eq_ref a) x1))) as proof of (((eq a) x1) Xv)
% Found (((fun (P:Prop) (x4:(forall (x:a), (((ex a) (fun (Xu0:a)=> ((and (((eq a) ((cP Xb) Xv)) ((cP x) Xu0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv0:a), ((X ((cP Xc) Xv0))->((and (X ((cP ((cP Xc) cZ)) (cL Xv0)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv0)))))))->(X ((cP x) Xu0)))))))->P)))=> (((((ex_ind a) (fun (Xb_1:a)=> ((ex a) (fun (Xu0:a)=> ((and (((eq a) ((cP Xb) Xv)) ((cP Xb_1) Xu0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv0:a), ((X ((cP Xc) Xv0))->((and (X ((cP ((cP Xc) cZ)) (cL Xv0)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv0)))))))->(X ((cP Xb_1) Xu0))))))))) P) x4) x00)) (((eq a) x1) Xv)) (fun (x4:a) (x5:((ex a) (fun (Xu0:a)=> ((and (((eq a) ((cP Xb) Xv)) ((cP x4) Xu0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv0:a), ((X ((cP Xc) Xv0))->((and (X ((cP ((cP Xc) cZ)) (cL Xv0)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv0)))))))->(X ((cP x4) Xu0))))))))=> ((eq_ref a) x1))) as proof of (((eq a) x1) Xv)
% Found (fun (x00:((ex a) (fun (Xb_1:a)=> ((ex a) (fun (Xu0:a)=> ((and (((eq a) ((cP Xb) Xv)) ((cP Xb_1) Xu0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv0:a), ((X ((cP Xc) Xv0))->((and (X ((cP ((cP Xc) cZ)) (cL Xv0)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv0)))))))->(X ((cP Xb_1) Xu0))))))))))=> (((fun (P:Prop) (x4:(forall (x:a), (((ex a) (fun (Xu0:a)=> ((and (((eq a) ((cP Xb) Xv)) ((cP x) Xu0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv0:a), ((X ((cP Xc) Xv0))->((and (X ((cP ((cP Xc) cZ)) (cL Xv0)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv0)))))))->(X ((cP x) Xu0)))))))->P)))=> (((((ex_ind a) (fun (Xb_1:a)=> ((ex a) (fun (Xu0:a)=> ((and (((eq a) ((cP Xb) Xv)) ((cP Xb_1) Xu0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv0:a), ((X ((cP Xc) Xv0))->((and (X ((cP ((cP Xc) cZ)) (cL Xv0)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv0)))))))->(X ((cP Xb_1) Xu0))))))))) P) x4) x00)) (((eq a) x1) Xv)) (fun (x4:a) (x5:((ex a) (fun (Xu0:a)=> ((and (((eq a) ((cP Xb) Xv)) ((cP x4) Xu0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv0:a), ((X ((cP Xc) Xv0))->((and (X ((cP ((cP Xc) cZ)) (cL Xv0)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv0)))))))->(X ((cP x4) Xu0))))))))=> ((eq_ref a) x1)))) as proof of (((eq a) x1) Xv)
% Found eta_expansion000:=(eta_expansion00 ((unique a) (fun (x20:a)=> ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xb) x20)) ((cP Xb_0) Xu_0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X ((cP Xc) Xv))->((and (X ((cP ((cP Xc) cZ)) (cL Xv)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X ((cP Xb_0) Xu_0)))))))))))):(((eq (a->Prop)) ((unique a) (fun (x20:a)=> ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xb) x20)) ((cP Xb_0) Xu_0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X ((cP Xc) Xv))->((and (X ((cP ((cP Xc) cZ)) (cL Xv)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X ((cP Xb_0) Xu_0)))))))))))) (fun (x:a)=> (((unique a) (fun (x20:a)=> ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xb) x20)) ((cP Xb_0) Xu_0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X ((cP Xc) Xv))->((and (X ((cP ((cP Xc) cZ)) (cL Xv)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X ((cP Xb_0) Xu_0))))))))))) x)))
% Found (eta_expansion00 ((unique a) (fun (x20:a)=> ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xb) x20)) ((cP Xb_0) Xu_0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X ((cP Xc) Xv))->((and (X ((cP ((cP Xc) cZ)) (cL Xv)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X ((cP Xb_0) Xu_0)))))))))))) as proof of (((eq (a->Prop)) ((unique a) (fun (x20:a)=> ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xb) x20)) ((cP Xb_0) Xu_0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X ((cP Xc) Xv))->((and (X ((cP ((cP Xc) cZ)) (cL Xv)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X ((cP Xb_0) Xu_0)))))))))))) b)
% Found ((eta_expansion0 Prop) ((unique a) (fun (x20:a)=> ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xb) x20)) ((cP Xb_0) Xu_0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X ((cP Xc) Xv))->((and (X ((cP ((cP Xc) cZ)) (cL Xv)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X ((cP Xb_0) Xu_0)))))))))))) as proof of (((eq (a->Prop)) ((unique a) (fun (x20:a)=> ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xb) x20)) ((cP Xb_0) Xu_0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X ((cP Xc) Xv))->((and (X ((cP ((cP Xc) cZ)) (cL Xv)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X ((cP Xb_0) Xu_0)))))))))))) b)
% Found (((eta_expansion a) Prop) ((unique a) (fun (x20:a)=> ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xb) x20)) ((cP Xb_0) Xu_0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X ((cP Xc) Xv))->((and (X ((cP ((cP Xc) cZ)) (cL Xv)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X ((cP Xb_0) Xu_0)))))))))))) as proof of (((eq (a->Prop)) ((unique a) (fun (x20:a)=> ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xb) x20)) ((cP Xb_0) Xu_0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X ((cP Xc) Xv))->((and (X ((cP ((cP Xc) cZ)) (cL Xv)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X ((cP Xb_0) Xu_0)))))))))))) b)
% Found (((eta_expansion a) Prop) ((unique a) (fun (x20:a)=> ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xb) x20)) ((cP Xb_0) Xu_0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X ((cP Xc) Xv))->((and (X ((cP ((cP Xc) cZ)) (cL Xv)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X ((cP Xb_0) Xu_0)))))))))))) as proof of (((eq (a->Prop)) ((unique a) (fun (x20:a)=> ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xb) x20)) ((cP Xb_0) Xu_0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X ((cP Xc) Xv))->((and (X ((cP ((cP Xc) cZ)) (cL Xv)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X ((cP Xb_0) Xu_0)))))))))))) b)
% Found (((eta_expansion a) Prop) ((unique a) (fun (x20:a)=> ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xb) x20)) ((cP Xb_0) Xu_0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X ((cP Xc) Xv))->((and (X ((cP ((cP Xc) cZ)) (cL Xv)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X ((cP Xb_0) Xu_0)))))))))))) as proof of (((eq (a->Prop)) ((unique a) (fun (x20:a)=> ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xb) x20)) ((cP Xb_0) Xu_0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X ((cP Xc) Xv))->((and (X ((cP ((cP Xc) cZ)) (cL Xv)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X ((cP Xb_0) Xu_0)))))))))))) b)
% Found eq_ref00:=(eq_ref0 Xv):(((eq a) Xv) Xv)
% Found (eq_ref0 Xv) as proof of (((eq a) Xv) x1)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) x1)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) x1)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) x1)
% Found (eq_sym000 ((eq_ref a) Xv)) as proof of (((eq a) x1) Xv)
% Found ((eq_sym00 x1) ((eq_ref a) Xv)) as proof of (((eq a) x1) Xv)
% Found (((eq_sym0 Xv) x1) ((eq_ref a) Xv)) as proof of (((eq a) x1) Xv)
% Found ((((eq_sym a) Xv) x1) ((eq_ref a) Xv)) as proof of (((eq a) x1) Xv)
% Found (fun (x00:((ex a) (fun (Xb_1:a)=> ((ex a) (fun (Xu0:a)=> ((and (((eq a) ((cP Xb) Xv)) ((cP Xb_1) Xu0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv0:a), ((X ((cP Xc) Xv0))->((and (X ((cP ((cP Xc) cZ)) (cL Xv0)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv0)))))))->(X ((cP Xb_1) Xu0))))))))))=> ((((eq_sym a) Xv) x1) ((eq_ref a) Xv))) as proof of (((eq a) x1) Xv)
% Found eq_ref00:=(eq_ref0 x7):(((eq a) x7) x7)
% Found (eq_ref0 x7) as proof of (((eq a) x7) Xv)
% Found ((eq_ref a) x7) as proof of (((eq a) x7) Xv)
% Found ((eq_ref a) x7) as proof of (((eq a) x7) Xv)
% Found (fun (x00:((ex a) (fun (Xb_1:a)=> ((ex a) (fun (Xu0:a)=> ((and (((eq a) ((cP Xb) Xv)) ((cP Xb_1) Xu0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv0:a), ((X ((cP Xc) Xv0))->((and (X ((cP ((cP Xc) cZ)) (cL Xv0)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv0)))))))->(X ((cP Xb_1) Xu0))))))))))=> ((eq_ref a) x7)) as proof of (((eq a) x7) Xv)
% 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 (f x3)):(((eq Prop) (f x3)) (f x3))
% Found (eq_ref0 (f x3)) as proof of (((eq Prop) (f x3)) ((and ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xb) x3)) ((cP Xb_0) Xu_0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X ((cP Xc) Xv))->((and (X ((cP ((cP Xc) cZ)) (cL Xv)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X ((cP Xb_0) Xu_0)))))))))) (forall (Xv:a), (((ex a) (fun (Xb_1:a)=> ((ex a) (fun (Xu0:a)=> ((and (((eq a) ((cP Xb) Xv)) ((cP Xb_1) Xu0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv0:a), ((X ((cP Xc) Xv0))->((and (X ((cP ((cP Xc) cZ)) (cL Xv0)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv0)))))))->(X ((cP Xb_1) Xu0)))))))))->(((eq a) x3) Xv)))))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((and ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xb) x3)) ((cP Xb_0) Xu_0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X ((cP Xc) Xv))->((and (X ((cP ((cP Xc) cZ)) (cL Xv)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X ((cP Xb_0) Xu_0)))))))))) (forall (Xv:a), (((ex a) (fun (Xb_1:a)=> ((ex a) (fun (Xu0:a)=> ((and (((eq a) ((cP Xb) Xv)) ((cP Xb_1) Xu0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv0:a), ((X ((cP Xc) Xv0))->((and (X ((cP ((cP Xc) cZ)) (cL Xv0)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv0)))))))->(X ((cP Xb_1) Xu0)))))))))->(((eq a) x3) Xv)))))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((and ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xb) x3)) ((cP Xb_0) Xu_0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X ((cP Xc) Xv))->((and (X ((cP ((cP Xc) cZ)) (cL Xv)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X ((cP Xb_0) Xu_0)))))))))) (forall (Xv:a), (((ex a) (fun (Xb_1:a)=> ((ex a) (fun (Xu0:a)=> ((and (((eq a) ((cP Xb) Xv)) ((cP Xb_1) Xu0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv0:a), ((X ((cP Xc) Xv0))->((and (X ((cP ((cP Xc) cZ)) (cL Xv0)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv0)))))))->(X ((cP Xb_1) Xu0)))))))))->(((eq a) x3) Xv)))))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (((eq Prop) (f x3)) ((and ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xb) x3)) ((cP Xb_0) Xu_0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X ((cP Xc) Xv))->((and (X ((cP ((cP Xc) cZ)) (cL Xv)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X ((cP Xb_0) Xu_0)))))))))) (forall (Xv:a), (((ex a) (fun (Xb_1:a)=> ((ex a) (fun (Xu0:a)=> ((and (((eq a) ((cP Xb) Xv)) ((cP Xb_1) Xu0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv0:a), ((X ((cP Xc) Xv0))->((and (X ((cP ((cP Xc) cZ)) (cL Xv0)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv0)))))))->(X ((cP Xb_1) Xu0)))))))))->(((eq a) x3) Xv)))))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (forall (x:a), (((eq Prop) (f x)) ((and ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xb) x)) ((cP Xb_0) Xu_0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X ((cP Xc) Xv))->((and (X ((cP ((cP Xc) cZ)) (cL Xv)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X ((cP Xb_0) Xu_0)))))))))) (forall (Xv:a), (((ex a) (fun (Xb_1:a)=> ((ex a) (fun (Xu0:a)=> ((and (((eq a) ((cP Xb) Xv)) ((cP Xb_1) Xu0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv0:a), ((X ((cP Xc) Xv0))->((and (X ((cP ((cP Xc) cZ)) (cL Xv0)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv0)))))))->(X ((cP Xb_1) Xu0)))))))))->(((eq a) x) Xv))))))
% Found eq_ref00:=(eq_ref0 (f x3)):(((eq Prop) (f x3)) (f x3))
% Found (eq_ref0 (f x3)) as proof of (((eq Prop) (f x3)) ((and ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xb) x3)) ((cP Xb_0) Xu_0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X ((cP Xc) Xv))->((and (X ((cP ((cP Xc) cZ)) (cL Xv)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X ((cP Xb_0) Xu_0)))))))))) (forall (Xv:a), (((ex a) (fun (Xb_1:a)=> ((ex a) (fun (Xu0:a)=> ((and (((eq a) ((cP Xb) Xv)) ((cP Xb_1) Xu0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv0:a), ((X ((cP Xc) Xv0))->((and (X ((cP ((cP Xc) cZ)) (cL Xv0)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv0)))))))->(X ((cP Xb_1) Xu0)))))))))->(((eq a) x3) Xv)))))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((and ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xb) x3)) ((cP Xb_0) Xu_0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X ((cP Xc) Xv))->((and (X ((cP ((cP Xc) cZ)) (cL Xv)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X ((cP Xb_0) Xu_0)))))))))) (forall (Xv:a), (((ex a) (fun (Xb_1:a)=> ((ex a) (fun (Xu0:a)=> ((and (((eq a) ((cP Xb) Xv)) ((cP Xb_1) Xu0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv0:a), ((X ((cP Xc) Xv0))->((and (X ((cP ((cP Xc) cZ)) (cL Xv0)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv0)))))))->(X ((cP Xb_1) Xu0)))))))))->(((eq a) x3) Xv)))))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((and ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xb) x3)) ((cP Xb_0) Xu_0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X ((cP Xc) Xv))->((and (X ((cP ((cP Xc) cZ)) (cL Xv)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X ((cP Xb_0) Xu_0)))))))))) (forall (Xv:a), (((ex a) (fun (Xb_1:a)=> ((ex a) (fun (Xu0:a)=> ((and (((eq a) ((cP Xb) Xv)) ((cP Xb_1) Xu0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv0:a), ((X ((cP Xc) Xv0))->((and (X ((cP ((cP Xc) cZ)) (cL Xv0)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv0)))))))->(X ((cP Xb_1) Xu0)))))))))->(((eq a) x3) Xv)))))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (((eq Prop) (f x3)) ((and ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xb) x3)) ((cP Xb_0) Xu_0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X ((cP Xc) Xv))->((and (X ((cP ((cP Xc) cZ)) (cL Xv)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X ((cP Xb_0) Xu_0)))))))))) (forall (Xv:a), (((ex a) (fun (Xb_1:a)=> ((ex a) (fun (Xu0:a)=> ((and (((eq a) ((cP Xb) Xv)) ((cP Xb_1) Xu0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv0:a), ((X ((cP Xc) Xv0))->((and (X ((cP ((cP Xc) cZ)) (cL Xv0)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv0)))))))->(X ((cP Xb_1) Xu0)))))))))->(((eq a) x3) Xv)))))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (forall (x:a), (((eq Prop) (f x)) ((and ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xb) x)) ((cP Xb_0) Xu_0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X ((cP Xc) Xv))->((and (X ((cP ((cP Xc) cZ)) (cL Xv)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X ((cP Xb_0) Xu_0)))))))))) (forall (Xv:a), (((ex a) (fun (Xb_1:a)=> ((ex a) (fun (Xu0:a)=> ((and (((eq a) ((cP Xb) Xv)) ((cP Xb_1) Xu0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv0:a), ((X ((cP Xc) Xv0))->((and (X ((cP ((cP Xc) cZ)) (cL Xv0)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv0)))))))->(X ((cP Xb_1) Xu0)))))))))->(((eq a) x) Xv))))))
% Found eq_ref00:=(eq_ref0 x3):(((eq a) x3) x3)
% Found (eq_ref0 x3) as proof of (((eq a) x3) x')
% Found ((eq_ref a) x3) as proof of (((eq a) x3) x')
% Found ((eq_ref a) x3) as proof of (((eq a) x3) x')
% Found (fun (x5:((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xb) x')) ((cP x4) Xu_0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X ((cP Xc) Xv))->((and (X ((cP ((cP Xc) cZ)) (cL Xv)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X ((cP x4) Xu_0))))))))=> ((eq_ref a) x3)) as proof of (((eq a) x3) x')
% Found (fun (x4:a) (x5:((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xb) x')) ((cP x4) Xu_0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X ((cP Xc) Xv))->((and (X ((cP ((cP Xc) cZ)) (cL Xv)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X ((cP x4) Xu_0))))))))=> ((eq_ref a) x3)) as proof of (((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xb) x')) ((cP x4) Xu_0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X ((cP Xc) Xv))->((and (X ((cP ((cP Xc) cZ)) (cL Xv)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X ((cP x4) Xu_0)))))))->(((eq a) x3) x'))
% Found (fun (x4:a) (x5:((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xb) x')) ((cP x4) Xu_0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X ((cP Xc) Xv))->((and (X ((cP ((cP Xc) cZ)) (cL Xv)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X ((cP x4) Xu_0))))))))=> ((eq_ref a) x3)) as proof of (forall (x:a), (((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xb) x')) ((cP x) Xu_0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X ((cP Xc) Xv))->((and (X ((cP ((cP Xc) cZ)) (cL Xv)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X ((cP x) Xu_0)))))))->(((eq a) x3) x')))
% Found (ex_ind00 (fun (x4:a) (x5:((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xb) x')) ((cP x4) Xu_0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X ((cP Xc) Xv))->((and (X ((cP ((cP Xc) cZ)) (cL Xv)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X ((cP x4) Xu_0))))))))=> ((eq_ref a) x3))) as proof of (((eq a) x3) x')
% Found ((ex_ind0 (((eq a) x3) x')) (fun (x4:a) (x5:((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xb) x')) ((cP x4) Xu_0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X ((cP Xc) Xv))->((and (X ((cP ((cP Xc) cZ)) (cL Xv)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X ((cP x4) Xu_0))))))))=> ((eq_ref a) x3))) as proof of (((eq a) x3) x')
% Found (((fun (P:Prop) (x4:(forall (x:a), (((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xb) x')) ((cP x) Xu_0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X ((cP Xc) Xv))->((and (X ((cP ((cP Xc) cZ)) (cL Xv)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X ((cP x) Xu_0)))))))->P)))=> (((((ex_ind a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xb) x')) ((cP Xb_0) Xu_0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X ((cP Xc) Xv))->((and (X ((cP ((cP Xc) cZ)) (cL Xv)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X ((cP Xb_0) Xu_0))))))))) P) x4) x00)) (((eq a) x3) x')) (fun (x4:a) (x5:((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xb) x')) ((cP x4) Xu_0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X ((cP Xc) Xv))->((and (X ((cP ((cP Xc) cZ)) (cL Xv)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X ((cP x4) Xu_0))))))))=> ((eq_ref a) x3))) as proof of (((eq a) x3) x')
% Found (fun (x00:((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xb) x')) ((cP Xb_0) Xu_0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X ((cP Xc) Xv))->((and (X ((cP ((cP Xc) cZ)) (cL Xv)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X ((cP Xb_0) Xu_0))))))))))=> (((fun (P:Prop) (x4:(forall (x:a), (((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xb) x')) ((cP x) Xu_0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X ((cP Xc) Xv))->((and (X ((cP ((cP Xc) cZ)) (cL Xv)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X ((cP x) Xu_0)))))))->P)))=> (((((ex_ind a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xb) x')) ((cP Xb_0) Xu_0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X ((cP Xc) Xv))->((and (X ((cP ((cP Xc) cZ)) (cL Xv)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X ((cP Xb_0) Xu_0))))))))) P) x4) x00)) (((eq a) x3) x')) (fun (x4:a) (x5:((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xb) x')) ((cP x4) Xu_0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X ((cP Xc) Xv))->((and (X ((cP ((cP Xc) cZ)) (cL Xv)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X ((cP x4) Xu_0))))))))=> ((eq_ref a) x3)))) as proof of (((eq a) x3) x')
% Found eq_ref00:=(eq_ref0 (forall (Xx:a), ((((and ((and ((and (((eq a) (cL cZ)) cZ)) (((eq a) (cR cZ)) cZ))) (forall (Xx0:a) (Xy:a), (((eq a) (cL ((cP Xx0) Xy))) Xx0)))) (forall (Xx0:a) (Xy:a), (((eq a) (cR ((cP Xx0) Xy))) Xy)))->((forall (Xt0:a), ((iff (not (((eq a) Xt0) cZ))) (((eq a) Xt0) ((cP (cL Xt0)) (cR Xt0)))))->((ex a) ((unique a) (fun (x20:a)=> ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xx) x20)) ((cP Xb_0) Xu_0))) (forall (X0:(a->Prop)), (((and (X0 ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X0 ((cP Xc) Xv))->((and (X0 ((cP ((cP Xc) cZ)) (cL Xv)))) (X0 ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X0 ((cP Xb_0) Xu_0))))))))))))))->((and (((and ((and ((and (((eq a) (cL cZ)) cZ)) (((eq a) (cR cZ)) cZ))) (forall (Xx0:a) (Xy:a), (((eq a) (cL ((cP Xx0) Xy))) Xx0)))) (forall (Xx0:a) (Xy:a), (((eq a) (cR ((cP Xx0) Xy))) Xy)))->((forall (Xt0:a), ((iff (not (((eq a) Xt0) cZ))) (((eq a) Xt0) ((cP (cL Xt0)) (cR Xt0)))))->((ex a) ((unique a) (fun (x20:a)=> ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP ((cP Xx) cZ)) x20)) ((cP Xb_0) Xu_0))) (forall (X0:(a->Prop)), (((and (X0 ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X0 ((cP Xc) Xv))->((and (X0 ((cP ((cP Xc) cZ)) (cL Xv)))) (X0 ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X0 ((cP Xb_0) Xu_0))))))))))))))) (((and ((and ((and (((eq a) (cL cZ)) cZ)) (((eq a) (cR cZ)) cZ))) (forall (Xx0:a) (Xy:a), (((eq a) (cL ((cP Xx0) Xy))) Xx0)))) (forall (Xx0:a) (Xy:a), (((eq a) (cR ((cP Xx0) Xy))) Xy)))->((forall (Xt0:a), ((iff (not (((eq a) Xt0) cZ))) (((eq a) Xt0) ((cP (cL Xt0)) (cR Xt0)))))->((ex a) ((unique a) (fun (x20:a)=> ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP ((cP Xx) ((cP cZ) cZ))) x20)) ((cP Xb_0) Xu_0))) (forall (X0:(a->Prop)), (((and (X0 ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X0 ((cP Xc) Xv))->((and (X0 ((cP ((cP Xc) cZ)) (cL Xv)))) (X0 ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X0 ((cP Xb_0) Xu_0)))))))))))))))))):(((eq Prop) (forall (Xx:a), ((((and ((and ((and (((eq a) (cL cZ)) cZ)) (((eq a) (cR cZ)) cZ))) (forall (Xx0:a) (Xy:a), (((eq a) (cL ((cP Xx0) Xy))) Xx0)))) (forall (Xx0:a) (Xy:a), (((eq a) (cR ((cP Xx0) Xy))) Xy)))->((forall (Xt0:a), ((iff (not (((eq a) Xt0) cZ))) (((eq a) Xt0) ((cP (cL Xt0)) (cR Xt0)))))->((ex a) ((unique a) (fun (x20:a)=> ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xx) x20)) ((cP Xb_0) Xu_0))) (forall (X0:(a->Prop)), (((and (X0 ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X0 ((cP Xc) Xv))->((and (X0 ((cP ((cP Xc) cZ)) (cL Xv)))) (X0 ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X0 ((cP Xb_0) Xu_0))))))))))))))->((and (((and ((and ((and (((eq a) (cL cZ)) cZ)) (((eq a) (cR cZ)) cZ))) (forall (Xx0:a) (Xy:a), (((eq a) (cL ((cP Xx0) Xy))) Xx0)))) (forall (Xx0:a) (Xy:a), (((eq a) (cR ((cP Xx0) Xy))) Xy)))->((forall (Xt0:a), ((iff (not (((eq a) Xt0) cZ))) (((eq a) Xt0) ((cP (cL Xt0)) (cR Xt0)))))->((ex a) ((unique a) (fun (x20:a)=> ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP ((cP Xx) cZ)) x20)) ((cP Xb_0) Xu_0))) (forall (X0:(a->Prop)), (((and (X0 ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X0 ((cP Xc) Xv))->((and (X0 ((cP ((cP Xc) cZ)) (cL Xv)))) (X0 ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X0 ((cP Xb_0) Xu_0))))))))))))))) (((and ((and ((and (((eq a) (cL cZ)) cZ)) (((eq a) (cR cZ)) cZ))) (forall (Xx0:a) (Xy:a), (((eq a) (cL ((cP Xx0) Xy))) Xx0)))) (forall (Xx0:a) (Xy:a), (((eq a) (cR ((cP Xx0) Xy))) Xy)))->((forall (Xt0:a), ((iff (not (((eq a) Xt0) cZ))) (((eq a) Xt0) ((cP (cL Xt0)) (cR Xt0)))))->((ex a) ((unique a) (fun (x20:a)=> ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP ((cP Xx) ((cP cZ) cZ))) x20)) ((cP Xb_0) Xu_0))) (forall (X0:(a->Prop)), (((and (X0 ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X0 ((cP Xc) Xv))->((and (X0 ((cP ((cP Xc) cZ)) (cL Xv)))) (X0 ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X0 ((cP Xb_0) Xu_0)))))))))))))))))) (forall (Xx:a), ((((and ((and ((and (((eq a) (cL cZ)) cZ)) (((eq a) (cR cZ)) cZ))) (forall (Xx0:a) (Xy:a), (((eq a) (cL ((cP Xx0) Xy))) Xx0)))) (forall (Xx0:a) (Xy:a), (((eq a) (cR ((cP Xx0) Xy))) Xy)))->((forall (Xt0:a), ((iff (not (((eq a) Xt0) cZ))) (((eq a) Xt0) ((cP (cL Xt0)) (cR Xt0)))))->((ex a) ((unique a) (fun (x20:a)=> ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xx) x20)) ((cP Xb_0) Xu_0))) (forall (X0:(a->Prop)), (((and (X0 ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X0 ((cP Xc) Xv))->((and (X0 ((cP ((cP Xc) cZ)) (cL Xv)))) (X0 ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X0 ((cP Xb_0) Xu_0))))))))))))))->((and (((and ((and ((and (((eq a) (cL cZ)) cZ)) (((eq a) (cR cZ)) cZ))) (forall (Xx0:a) (Xy:a), (((eq a) (cL ((cP Xx0) Xy))) Xx0)))) (forall (Xx0:a) (Xy:a), (((eq a) (cR ((cP Xx0) Xy))) Xy)))->((forall (Xt0:a), ((iff (not (((eq a) Xt0) cZ))) (((eq a) Xt0) ((cP (cL Xt0)) (cR Xt0)))))->((ex a) ((unique a) (fun (x20:a)=> ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP ((cP Xx) cZ)) x20)) ((cP Xb_0) Xu_0))) (forall (X0:(a->Prop)), (((and (X0 ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X0 ((cP Xc) Xv))->((and (X0 ((cP ((cP Xc) cZ)) (cL Xv)))) (X0 ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X0 ((cP Xb_0) Xu_0))))))))))))))) (((and ((and ((and (((eq a) (cL cZ)) cZ)) (((eq a) (cR cZ)) cZ))) (forall (Xx0:a) (Xy:a), (((eq a) (cL ((cP Xx0) Xy))) Xx0)))) (forall (Xx0:a) (Xy:a), (((eq a) (cR ((cP Xx0) Xy))) Xy)))->((forall (Xt0:a), ((iff (not (((eq a) Xt0) cZ))) (((eq a) Xt0) ((cP (cL Xt0)) (cR Xt0)))))->((ex a) ((unique a) (fun (x20:a)=> ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP ((cP Xx) ((cP cZ) cZ))) x20)) ((cP Xb_0) Xu_0))) (forall (X0:(a->Prop)), (((and (X0 ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X0 ((cP Xc) Xv))->((and (X0 ((cP ((cP Xc) cZ)) (cL Xv)))) (X0 ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X0 ((cP Xb_0) Xu_0))))))))))))))))))
% Found (eq_ref0 (forall (Xx:a), ((((and ((and ((and (((eq a) (cL cZ)) cZ)) (((eq a) (cR cZ)) cZ))) (forall (Xx0:a) (Xy:a), (((eq a) (cL ((cP Xx0) Xy))) Xx0)))) (forall (Xx0:a) (Xy:a), (((eq a) (cR ((cP Xx0) Xy))) Xy)))->((forall (Xt0:a), ((iff (not (((eq a) Xt0) cZ))) (((eq a) Xt0) ((cP (cL Xt0)) (cR Xt0)))))->((ex a) ((unique a) (fun (x20:a)=> ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xx) x20)) ((cP Xb_0) Xu_0))) (forall (X0:(a->Prop)), (((and (X0 ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X0 ((cP Xc) Xv))->((and (X0 ((cP ((cP Xc) cZ)) (cL Xv)))) (X0 ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X0 ((cP Xb_0) Xu_0))))))))))))))->((and (((and ((and ((and (((eq a) (cL cZ)) cZ)) (((eq a) (cR cZ)) cZ))) (forall (Xx0:a) (Xy:a), (((eq a) (cL ((cP Xx0) Xy))) Xx0)))) (forall (Xx0:a) (Xy:a), (((eq a) (cR ((cP Xx0) Xy))) Xy)))->((forall (Xt0:a), ((iff (not (((eq a) Xt0) cZ))) (((eq a) Xt0) ((cP (cL Xt0)) (cR Xt0)))))->((ex a) ((unique a) (fun (x20:a)=> ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP ((cP Xx) cZ)) x20)) ((cP Xb_0) Xu_0))) (forall (X0:(a->Prop)), (((and (X0 ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X0 ((cP Xc) Xv))->((and (X0 ((cP ((cP Xc) cZ)) (cL Xv)))) (X0 ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X0 ((cP Xb_0) Xu_0))))))))))))))) (((and ((and ((and (((eq a) (cL cZ)) cZ)) (((eq a) (cR cZ)) cZ))) (forall (Xx0:a) (Xy:a), (((eq a) (cL ((cP Xx0) Xy))) Xx0)))) (forall (Xx0:a) (Xy:a), (((eq a) (cR ((cP Xx0) Xy))) Xy)))->((forall (Xt0:a), ((iff (not (((eq a) Xt0) cZ))) (((eq a) Xt0) ((cP (cL Xt0)) (cR Xt0)))))->((ex a) ((unique a) (fun (x20:a)=> ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP ((cP Xx) ((cP cZ) cZ))) x20)) ((cP Xb_0) Xu_0))) (forall (X0:(a->Prop)), (((and (X0 ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X0 ((cP Xc) Xv))->((and (X0 ((cP ((cP Xc) cZ)) (cL Xv)))) (X0 ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X0 ((cP Xb_0) Xu_0)))))))))))))))))) as proof of (((eq Prop) (forall (Xx:a), ((((and ((and ((and (((eq a) (cL cZ)) cZ)) (((eq a) (cR cZ)) cZ))) (forall (Xx0:a) (Xy:a), (((eq a) (cL ((cP Xx0) Xy))) Xx0)))) (forall (Xx0:a) (Xy:a), (((eq a) (cR ((cP Xx0) Xy))) Xy)))->((forall (Xt0:a), ((iff (not (((eq a) Xt0) cZ))) (((eq a) Xt0) ((cP (cL Xt0)) (cR Xt0)))))->((ex a) ((unique a) (fun (x20:a)=> ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xx) x20)) ((cP Xb_0) Xu_0))) (forall (X0:(a->Prop)), (((and (X0 ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X0 ((cP Xc) Xv))->((and (X0 ((cP ((cP Xc) cZ)) (cL Xv)))) (X0 ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X0 ((cP Xb_0) Xu_0))))))))))))))->((and (((and ((and ((and (((eq a) (cL cZ)) cZ)) (((eq a) (cR cZ)) cZ))) (forall (Xx0:a) (Xy:a), (((eq a) (cL ((cP Xx0) Xy))) Xx0)))) (forall (Xx0:a) (Xy:a), (((eq a) (cR ((cP Xx0) Xy))) Xy)))->((forall (Xt0:a), ((iff (not (((eq a) Xt0) cZ))) (((eq a) Xt0) ((cP (cL Xt0)) (cR Xt0)))))->((ex a) ((unique a) (fun (x20:a)=> ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP ((cP Xx) cZ)) x20)) ((cP Xb_0) Xu_0))) (forall (X0:(a->Prop)), (((and (X0 ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X0 ((cP Xc) Xv))->((and (X0 ((cP ((cP Xc) cZ)) (cL Xv)))) (X0 ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X0 ((cP Xb_0) Xu_0))))))))))))))) (((and ((and ((and (((eq a) (cL cZ)) cZ)) (((eq a) (cR cZ)) cZ))) (forall (Xx0:a) (Xy:a), (((eq a) (cL ((cP Xx0) Xy))) Xx0)))) (forall (Xx0:a) (Xy:a), (((eq a) (cR ((cP Xx0) Xy))) Xy)))->((forall (Xt0:a), ((iff (not (((eq a) Xt0) cZ))) (((eq a) Xt0) ((cP (cL Xt0)) (cR Xt0)))))->((ex a) ((unique a) (fun (x20:a)=> ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP ((cP Xx) ((cP cZ) cZ))) x20)) ((cP Xb_0) Xu_0))) (forall (X0:(a->Prop)), (((and (X0 ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X0 ((cP Xc) Xv))->((and (X0 ((cP ((cP Xc) cZ)) (cL Xv)))) (X0 ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X0 ((cP Xb_0) Xu_0)))))))))))))))))) b)
% Found ((eq_ref Prop) (forall (Xx:a), ((((and ((and ((and (((eq a) (cL cZ)) cZ)) (((eq a) (cR cZ)) cZ))) (forall (Xx0:a) (Xy:a), (((eq a) (cL ((cP Xx0) Xy))) Xx0)))) (forall (Xx0:a) (Xy:a), (((eq a) (cR ((cP Xx0) Xy))) Xy)))->((forall (Xt0:a), ((iff (not (((eq a) Xt0) cZ))) (((eq a) Xt0) ((cP (cL Xt0)) (cR Xt0)))))->((ex a) ((unique a) (fun (x20:a)=> ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xx) x20)) ((cP Xb_0) Xu_0))) (forall (X0:(a->Prop)), (((and (X0 ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X0 ((cP Xc) Xv))->((and (X0 ((cP ((cP Xc) cZ)) (cL Xv)))) (X0 ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X0 ((cP Xb_0) Xu_0))))))))))))))->((and (((and ((and ((and (((eq a) (cL cZ)) cZ)) (((eq a) (cR cZ)) cZ))) (forall (Xx0:a) (Xy:a), (((eq a) (cL ((cP Xx0) Xy))) Xx0)))) (forall (Xx0:a) (Xy:a), (((eq a) (cR ((cP Xx0) Xy))) Xy)))->((forall (Xt0:a), ((iff (not (((eq a) Xt0) cZ))) (((eq a) Xt0) ((cP (cL Xt0)) (cR Xt0)))))->((ex a) ((unique a) (fun (x20:a)=> ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP ((cP Xx) cZ)) x20)) ((cP Xb_0) Xu_0))) (forall (X0:(a->Prop)), (((and (X0 ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X0 ((cP Xc) Xv))->((and (X0 ((cP ((cP Xc) cZ)) (cL Xv)))) (X0 ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X0 ((cP Xb_0) Xu_0))))))))))))))) (((and ((and ((and (((eq a) (cL cZ)) cZ)) (((eq a) (cR cZ)) cZ))) (forall (Xx0:a) (Xy:a), (((eq a) (cL ((cP Xx0) Xy))) Xx0)))) (forall (Xx0:a) (Xy:a), (((eq a) (cR ((cP Xx0) Xy))) Xy)))->((forall (Xt0:a), ((iff (not (((eq a) Xt0) cZ))) (((eq a) Xt0) ((cP (cL Xt0)) (cR Xt0)))))->((ex a) ((unique a) (fun (x20:a)=> ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP ((cP Xx) ((cP cZ) cZ))) x20)) ((cP Xb_0) Xu_0))) (forall (X0:(a->Prop)), (((and (X0 ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X0 ((cP Xc) Xv))->((and (X0 ((cP ((cP Xc) cZ)) (cL Xv)))) (X0 ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X0 ((cP Xb_0) Xu_0)))))))))))))))))) as proof of (((eq Prop) (forall (Xx:a), ((((and ((and ((and (((eq a) (cL cZ)) cZ)) (((eq a) (cR cZ)) cZ))) (forall (Xx0:a) (Xy:a), (((eq a) (cL ((cP Xx0) Xy))) Xx0)))) (forall (Xx0:a) (Xy:a), (((eq a) (cR ((cP Xx0) Xy))) Xy)))->((forall (Xt0:a), ((iff (not (((eq a) Xt0) cZ))) (((eq a) Xt0) ((cP (cL Xt0)) (cR Xt0)))))->((ex a) ((unique a) (fun (x20:a)=> ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xx) x20)) ((cP Xb_0) Xu_0))) (forall (X0:(a->Prop)), (((and (X0 ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X0 ((cP Xc) Xv))->((and (X0 ((cP ((cP Xc) cZ)) (cL Xv)))) (X0 ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X0 ((cP Xb_0) Xu_0))))))))))))))->((and (((and ((and ((and (((eq a) (cL cZ)) cZ)) (((eq a) (cR cZ)) cZ))) (forall (Xx0:a) (Xy:a), (((eq a) (cL ((cP Xx0) Xy))) Xx0)))) (forall (Xx0:a) (Xy:a), (((eq a) (cR ((cP Xx0) Xy))) Xy)))->((forall (Xt0:a), ((iff (not (((eq a) Xt0) cZ))) (((eq a) Xt0) ((cP (cL Xt0)) (cR Xt0)))))->((ex a) ((unique a) (fun (x20:a)=> ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP ((cP Xx) cZ)) x20)) ((cP Xb_0) Xu_0))) (forall (X0:(a->Prop)), (((and (X0 ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X0 ((cP Xc) Xv))->((and (X0 ((cP ((cP Xc) cZ)) (cL Xv)))) (X0 ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X0 ((cP Xb_0) Xu_0))))))))))))))) (((and ((and ((and (((eq a) (cL cZ)) cZ)) (((eq a) (cR cZ)) cZ))) (forall (Xx0:a) (Xy:a), (((eq a) (cL ((cP Xx0) Xy))) Xx0)))) (forall (Xx0:a) (Xy:a), (((eq a) (cR ((cP Xx0) Xy))) Xy)))->((forall (Xt0:a), ((iff (not (((eq a) Xt0) cZ))) (((eq a) Xt0) ((cP (cL Xt0)) (cR Xt0)))))->((ex a) ((unique a) (fun (x20:a)=> ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP ((cP Xx) ((cP cZ) cZ))) x20)) ((cP Xb_0) Xu_0))) (forall (X0:(a->Prop)), (((and (X0 ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X0 ((cP Xc) Xv))->((and (X0 ((cP ((cP Xc) cZ)) (cL Xv)))) (X0 ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X0 ((cP Xb_0) Xu_0)))))))))))))))))) b)
% Found ((eq_ref Prop) (forall (Xx:a), ((((and ((and ((and (((eq a) (cL cZ)) cZ)) (((eq a) (cR cZ)) cZ))) (forall (Xx0:a) (Xy:a), (((eq a) (cL ((cP Xx0) Xy))) Xx0)))) (forall (Xx0:a) (Xy:a), (((eq a) (cR ((cP Xx0) Xy))) Xy)))->((forall (Xt0:a), ((iff (not (((eq a) Xt0) cZ))) (((eq a) Xt0) ((cP (cL Xt0)) (cR Xt0)))))->((ex a) ((unique a) (fun (x20:a)=> ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xx) x20)) ((cP Xb_0) Xu_0))) (forall (X0:(a->Prop)), (((and (X0 ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X0 ((cP Xc) Xv))->((and (X0 ((cP ((cP Xc) cZ)) (cL Xv)))) (X0 ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X0 ((cP Xb_0) Xu_0))))))))))))))->((and (((and ((and ((and (((eq a) (cL cZ)) cZ)) (((eq a) (cR cZ)) cZ))) (forall (Xx0:a) (Xy:a), (((eq a) (cL ((cP Xx0) Xy))) Xx0)))) (forall (Xx0:a) (Xy:a), (((eq a) (cR ((cP Xx0) Xy))) Xy)))->((forall (Xt0:a), ((iff (not (((eq a) Xt0) cZ))) (((eq a) Xt0) ((cP (cL Xt0)) (cR Xt0)))))->((ex a) ((unique a) (fun (x20:a)=> ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP ((cP Xx) cZ)) x20)) ((cP Xb_0) Xu_0))) (forall (X0:(a->Prop)), (((and (X0 ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X0 ((cP Xc) Xv))->((and (X0 ((cP ((cP Xc) cZ)) (cL Xv)))) (X0 ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X0 ((cP Xb_0) Xu_0))))))))))))))) (((and ((and ((and (((eq a) (cL cZ)) cZ)) (((eq a) (cR cZ)) cZ))) (forall (Xx0:a) (Xy:a), (((eq a) (cL ((cP Xx0) Xy))) Xx0)))) (forall (Xx0:a) (Xy:a), (((eq a) (cR ((cP Xx0) Xy))) Xy)))->((forall (Xt0:a), ((iff (not (((eq a) Xt0) cZ))) (((eq a) Xt0) ((cP (cL Xt0)) (cR Xt0)))))->((ex a) ((unique a) (fun (x20:a)=> ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP ((cP Xx) ((cP cZ) cZ))) x20)) ((cP Xb_0) Xu_0))) (forall (X0:(a->Prop)), (((and (X0 ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X0 ((cP Xc) Xv))->((and (X0 ((cP ((cP Xc) cZ)) (cL Xv)))) (X0 ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X0 ((cP Xb_0) Xu_0)))))))))))))))))) as proof of (((eq Prop) (forall (Xx:a), ((((and ((and ((and (((eq a) (cL cZ)) cZ)) (((eq a) (cR cZ)) cZ))) (forall (Xx0:a) (Xy:a), (((eq a) (cL ((cP Xx0) Xy))) Xx0)))) (forall (Xx0:a) (Xy:a), (((eq a) (cR ((cP Xx0) Xy))) Xy)))->((forall (Xt0:a), ((iff (not (((eq a) Xt0) cZ))) (((eq a) Xt0) ((cP (cL Xt0)) (cR Xt0)))))->((ex a) ((unique a) (fun (x20:a)=> ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xx) x20)) ((cP Xb_0) Xu_0))) (forall (X0:(a->Prop)), (((and (X0 ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X0 ((cP Xc) Xv))->((and (X0 ((cP ((cP Xc) cZ)) (cL Xv)))) (X0 ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X0 ((cP Xb_0) Xu_0))))))))))))))->((and (((and ((and ((and (((eq a) (cL cZ)) cZ)) (((eq a) (cR cZ)) cZ))) (forall (Xx0:a) (Xy:a), (((eq a) (cL ((cP Xx0) Xy))) Xx0)))) (forall (Xx0:a) (Xy:a), (((eq a) (cR ((cP Xx0) Xy))) Xy)))->((forall (Xt0:a), ((iff (not (((eq a) Xt0) cZ))) (((eq a) Xt0) ((cP (cL Xt0)) (cR Xt0)))))->((ex a) ((unique a) (fun (x20:a)=> ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP ((cP Xx) cZ)) x20)) ((cP Xb_0) Xu_0))) (forall (X0:(a->Prop)), (((and (X0 ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X0 ((cP Xc) Xv))->((and (X0 ((cP ((cP Xc) cZ)) (cL Xv)))) (X0 ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X0 ((cP Xb_0) Xu_0))))))))))))))) (((and ((and ((and (((eq a) (cL cZ)) cZ)) (((eq a) (cR cZ)) cZ))) (forall (Xx0:a) (Xy:a), (((eq a) (cL ((cP Xx0) Xy))) Xx0)))) (forall (Xx0:a) (Xy:a), (((eq a) (cR ((cP Xx0) Xy))) Xy)))->((forall (Xt0:a), ((iff (not (((eq a) Xt0) cZ))) (((eq a) Xt0) ((cP (cL Xt0)) (cR Xt0)))))->((ex a) ((unique a) (fun (x20:a)=> ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP ((cP Xx) ((cP cZ) cZ))) x20)) ((cP Xb_0) Xu_0))) (forall (X0:(a->Prop)), (((and (X0 ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X0 ((cP Xc) Xv))->((and (X0 ((cP ((cP Xc) cZ)) (cL Xv)))) (X0 ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X0 ((cP Xb_0) Xu_0)))))))))))))))))) b)
% Found ((eq_ref Prop) (forall (Xx:a), ((((and ((and ((and (((eq a) (cL cZ)) cZ)) (((eq a) (cR cZ)) cZ))) (forall (Xx0:a) (Xy:a), (((eq a) (cL ((cP Xx0) Xy))) Xx0)))) (forall (Xx0:a) (Xy:a), (((eq a) (cR ((cP Xx0) Xy))) Xy)))->((forall (Xt0:a), ((iff (not (((eq a) Xt0) cZ))) (((eq a) Xt0) ((cP (cL Xt0)) (cR Xt0)))))->((ex a) ((unique a) (fun (x20:a)=> ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xx) x20)) ((cP Xb_0) Xu_0))) (forall (X0:(a->Prop)), (((and (X0 ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X0 ((cP Xc) Xv))->((and (X0 ((cP ((cP Xc) cZ)) (cL Xv)))) (X0 ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X0 ((cP Xb_0) Xu_0))))))))))))))->((and (((and ((and ((and (((eq a) (cL cZ)) cZ)) (((eq a) (cR cZ)) cZ))) (forall (Xx0:a) (Xy:a), (((eq a) (cL ((cP Xx0) Xy))) Xx0)))) (forall (Xx0:a) (Xy:a), (((eq a) (cR ((cP Xx0) Xy))) Xy)))->((forall (Xt0:a), ((iff (not (((eq a) Xt0) cZ))) (((eq a) Xt0) ((cP (cL Xt0)) (cR Xt0)))))->((ex a) ((unique a) (fun (x20:a)=> ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP ((cP Xx) cZ)) x20)) ((cP Xb_0) Xu_0))) (forall (X0:(a->Prop)), (((and (X0 ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X0 ((cP Xc) Xv))->((and (X0 ((cP ((cP Xc) cZ)) (cL Xv)))) (X0 ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X0 ((cP Xb_0) Xu_0))))))))))))))) (((and ((and ((and (((eq a) (cL cZ)) cZ)) (((eq a) (cR cZ)) cZ))) (forall (Xx0:a) (Xy:a), (((eq a) (cL ((cP Xx0) Xy))) Xx0)))) (forall (Xx0:a) (Xy:a), (((eq a) (cR ((cP Xx0) Xy))) Xy)))->((forall (Xt0:a), ((iff (not (((eq a) Xt0) cZ))) (((eq a) Xt0) ((cP (cL Xt0)) (cR Xt0)))))->((ex a) ((unique a) (fun (x20:a)=> ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP ((cP Xx) ((cP cZ) cZ))) x20)) ((cP Xb_0) Xu_0))) (forall (X0:(a->Prop)), (((and (X0 ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X0 ((cP Xc) Xv))->((and (X0 ((cP ((cP Xc) cZ)) (cL Xv)))) (X0 ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X0 ((cP Xb_0) Xu_0)))))))))))))))))) as proof of (((eq Prop) (forall (Xx:a), ((((and ((and ((and (((eq a) (cL cZ)) cZ)) (((eq a) (cR cZ)) cZ))) (forall (Xx0:a) (Xy:a), (((eq a) (cL ((cP Xx0) Xy))) Xx0)))) (forall (Xx0:a) (Xy:a), (((eq a) (cR ((cP Xx0) Xy))) Xy)))->((forall (Xt0:a), ((iff (not (((eq a) Xt0) cZ))) (((eq a) Xt0) ((cP (cL Xt0)) (cR Xt0)))))->((ex a) ((unique a) (fun (x20:a)=> ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xx) x20)) ((cP Xb_0) Xu_0))) (forall (X0:(a->Prop)), (((and (X0 ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X0 ((cP Xc) Xv))->((and (X0 ((cP ((cP Xc) cZ)) (cL Xv)))) (X0 ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X0 ((cP Xb_0) Xu_0))))))))))))))->((and (((and ((and ((and (((eq a) (cL cZ)) cZ)) (((eq a) (cR cZ)) cZ))) (forall (Xx0:a) (Xy:a), (((eq a) (cL ((cP Xx0) Xy))) Xx0)))) (forall (Xx0:a) (Xy:a), (((eq a) (cR ((cP Xx0) Xy))) Xy)))->((forall (Xt0:a), ((iff (not (((eq a) Xt0) cZ))) (((eq a) Xt0) ((cP (cL Xt0)) (cR Xt0)))))->((ex a) ((unique a) (fun (x20:a)=> ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP ((cP Xx) cZ)) x20)) ((cP Xb_0) Xu_0))) (forall (X0:(a->Prop)), (((and (X0 ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X0 ((cP Xc) Xv))->((and (X0 ((cP ((cP Xc) cZ)) (cL Xv)))) (X0 ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X0 ((cP Xb_0) Xu_0))))))))))))))) (((and ((and ((and (((eq a) (cL cZ)) cZ)) (((eq a) (cR cZ)) cZ))) (forall (Xx0:a) (Xy:a), (((eq a) (cL ((cP Xx0) Xy))) Xx0)))) (forall (Xx0:a) (Xy:a), (((eq a) (cR ((cP Xx0) Xy))) Xy)))->((forall (Xt0:a), ((iff (not (((eq a) Xt0) cZ))) (((eq a) Xt0) ((cP (cL Xt0)) (cR Xt0)))))->((ex a) ((unique a) (fun (x20:a)=> ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP ((cP Xx) ((cP cZ) cZ))) x20)) ((cP Xb_0) Xu_0))) (forall (X0:(a->Prop)), (((and (X0 ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X0 ((cP Xc) Xv))->((and (X0 ((cP ((cP Xc) cZ)) (cL Xv)))) (X0 ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X0 ((cP Xb_0) Xu_0)))))))))))))))))) b)
% Found eq_ref00:=(eq_ref0 x'):(((eq a) x') x')
% Found (eq_ref0 x') as proof of (((eq a) x') x3)
% Found ((eq_ref a) x') as proof of (((eq a) x') x3)
% Found ((eq_ref a) x') as proof of (((eq a) x') x3)
% Found ((eq_ref a) x') as proof of (((eq a) x') x3)
% Found (eq_sym000 ((eq_ref a) x')) as proof of (((eq a) x3) x')
% Found ((eq_sym00 x3) ((eq_ref a) x')) as proof of (((eq a) x3) x')
% Found (((eq_sym0 x') x3) ((eq_ref a) x')) as proof of (((eq a) x3) x')
% Found ((((eq_sym a) x') x3) ((eq_ref a) x')) as proof of (((eq a) x3) x')
% Found (fun (x00:((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xb) x')) ((cP Xb_0) Xu_0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X ((cP Xc) Xv))->((and (X ((cP ((cP Xc) cZ)) (cL Xv)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X ((cP Xb_0) Xu_0))))))))))=> ((((eq_sym a) x') x3) ((eq_ref a) x'))) as proof of (((eq a) x3) x')
% Found x5:(((eq a) Xt0) cZ)
% Found x5 as proof of (((eq a) Xt0) cZ)
% Found eq_ref00:=(eq_ref0 x1):(((eq a) x1) x1)
% Found (eq_ref0 x1) as proof of (((eq a) x1) Xv)
% Found ((eq_ref a) x1) as proof of (((eq a) x1) Xv)
% Found ((eq_ref a) x1) as proof of (((eq a) x1) Xv)
% Found (fun (x00:((ex a) (fun (Xb_1:a)=> ((ex a) (fun (Xu0:a)=> ((and (((eq a) ((cP Xb) Xv)) ((cP Xb_1) Xu0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv0:a), ((X ((cP Xc) Xv0))->((and (X ((cP ((cP Xc) cZ)) (cL Xv0)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv0)))))))->(X ((cP Xb_1) Xu0))))))))))=> ((eq_ref a) x1)) as proof of (((eq a) x1) Xv)
% Found eq_ref00:=(eq_ref0 x3):(((eq a) x3) x3)
% Found (eq_ref0 x3) as proof of (((eq a) x3) Xv)
% Found ((eq_ref a) x3) as proof of (((eq a) x3) Xv)
% Found ((eq_ref a) x3) as proof of (((eq a) x3) Xv)
% Found (fun (x00:((ex a) (fun (Xb_1:a)=> ((ex a) (fun (Xu0:a)=> ((and (((eq a) ((cP Xb) Xv)) ((cP Xb_1) Xu0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv0:a), ((X ((cP Xc) Xv0))->((and (X ((cP ((cP Xc) cZ)) (cL Xv0)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv0)))))))->(X ((cP Xb_1) Xu0))))))))))=> ((eq_ref a) x3)) as proof of (((eq a) x3) Xv)
% Found eq_ref00:=(eq_ref0 x5):(((eq a) x5) x5)
% Found (eq_ref0 x5) as proof of (((eq a) x5) Xv)
% Found ((eq_ref a) x5) as proof of (((eq a) x5) Xv)
% Found ((eq_ref a) x5) as proof of (((eq a) x5) Xv)
% Found (fun (x00:((ex a) (fun (Xb_1:a)=> ((ex a) (fun (Xu0:a)=> ((and (((eq a) ((cP Xb) Xv)) ((cP Xb_1) Xu0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv0:a), ((X ((cP Xc) Xv0))->((and (X ((cP ((cP Xc) cZ)) (cL Xv0)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv0)))))))->(X ((cP Xb_1) Xu0))))))))))=> ((eq_ref a) x5)) as proof of (((eq a) x5) Xv)
% Found eq_ref00:=(eq_ref0 (forall (Xx:a), (((forall (Xt0:a), ((iff (not (((eq a) Xt0) cZ))) (((eq a) Xt0) ((cP (cL Xt0)) (cR Xt0)))))->((ex a) ((unique a) (fun (x200:a)=> ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xx) x200)) ((cP Xb_0) Xu_0))) (forall (X0:(a->Prop)), (((and (X0 ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X0 ((cP Xc) Xv))->((and (X0 ((cP ((cP Xc) cZ)) (cL Xv)))) (X0 ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X0 ((cP Xb_0) Xu_0)))))))))))))->((and ((forall (Xt0:a), ((iff (not (((eq a) Xt0) cZ))) (((eq a) Xt0) ((cP (cL Xt0)) (cR Xt0)))))->((ex a) ((unique a) (fun (x200:a)=> ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP ((cP Xx) cZ)) x200)) ((cP Xb_0) Xu_0))) (forall (X0:(a->Prop)), (((and (X0 ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X0 ((cP Xc) Xv))->((and (X0 ((cP ((cP Xc) cZ)) (cL Xv)))) (X0 ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X0 ((cP Xb_0) Xu_0)))))))))))))) ((forall (Xt0:a), ((iff (not (((eq a) Xt0) cZ))) (((eq a) Xt0) ((cP (cL Xt0)) (cR Xt0)))))->((ex a) ((unique a) (fun (x200:a)=> ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP ((cP Xx) ((cP cZ) cZ))) x200)) ((cP Xb_0) Xu_0))) (forall (X0:(a->Prop)), (((and (X0 ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X0 ((cP Xc) Xv))->((and (X0 ((cP ((cP Xc) cZ)) (cL Xv)))) (X0 ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X0 ((cP Xb_0) Xu_0))))))))))))))))):(((eq Prop) (forall (Xx:a), (((forall (Xt0:a), ((iff (not (((eq a) Xt0) cZ))) (((eq a) Xt0) ((cP (cL Xt0)) (cR Xt0)))))->((ex a) ((unique a) (fun (x200:a)=> ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xx) x200)) ((cP Xb_0) Xu_0))) (forall (X0:(a->Prop)), (((and (X0 ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X0 ((cP Xc) Xv))->((and (X0 ((cP ((cP Xc) cZ)) (cL Xv)))) (X0 ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X0 ((cP Xb_0) Xu_0)))))))))))))->((and ((forall (Xt0:a), ((iff (not (((eq a) Xt0) cZ))) (((eq a) Xt0) ((cP (cL Xt0)) (cR Xt0)))))->((ex a) ((unique a) (fun (x200:a)=> ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP ((cP Xx) cZ)) x200)) ((cP Xb_0) Xu_0))) (forall (X0:(a->Prop)), (((and (X0 ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X0 ((cP Xc) Xv))->((and (X0 ((cP ((cP Xc) cZ)) (cL Xv)))) (X0 ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X0 ((cP Xb_0) Xu_0)))))))))))))) ((forall (Xt0:a), ((iff (not (((eq a) Xt0) cZ))) (((eq a) Xt0) ((cP (cL Xt0)) (cR Xt0)))))->((ex a) ((unique a) (fun (x200:a)=> ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP ((cP Xx) ((cP cZ) cZ))) x200)) ((cP Xb_0) Xu_0))) (forall (X0:(a->Prop)), (((and (X0 ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X0 ((cP Xc) Xv))->((and (X0 ((cP ((cP Xc) cZ)) (cL Xv)))) (X0 ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X0 ((cP Xb_0) Xu_0))))))))))))))))) (forall (Xx:a), (((forall (Xt0:a), ((iff (not (((eq a) Xt0) cZ))) (((eq a) Xt0) ((cP (cL Xt0)) (cR Xt0)))))->((ex a) ((unique a) (fun (x200:a)=> ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xx) x200)) ((cP Xb_0) Xu_0))) (forall (X0:(a->Prop)), (((and (X0 ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X0 ((cP Xc) Xv))->((and (X0 ((cP ((cP Xc) cZ)) (cL Xv)))) (X0 ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X0 ((cP Xb_0) Xu_0)))))))))))))->((and ((forall (Xt0:a), ((iff (not (((eq a) Xt0) cZ))) (((eq a) Xt0) ((cP (cL Xt0)) (cR Xt0)))))->((ex a) ((unique a) (fun (x200:a)=> ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP ((cP Xx) cZ)) x200)) ((cP Xb_0) Xu_0))) (forall (X0:(a->Prop)), (((and (X0 ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X0 ((cP Xc) Xv))->((and (X0 ((cP ((cP Xc) cZ)) (cL Xv)))) (X0 ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X0 ((cP Xb_0) Xu_0)))))))))))))) ((forall (Xt0:a), ((iff (not (((eq a) Xt0) cZ))) (((eq a) Xt0) ((cP (cL Xt0)) (cR Xt0)))))->((ex a) ((unique a) (fun (x200:a)=> ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP ((cP Xx) ((cP cZ) cZ))) x200)) ((cP Xb_0) Xu_0))) (forall (X0:(a->Prop)), (((and (X0 ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X0 ((cP Xc) Xv))->((and (X0 ((cP ((cP Xc) cZ)) (cL Xv)))) (X0 ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X0 ((cP Xb_0) Xu_0)))))))))))))))))
% Found (eq_ref0 (forall (Xx:a), (((forall (Xt0:a), ((iff (not (((eq a) Xt0) cZ))) (((eq a) Xt0) ((cP (cL Xt0)) (cR Xt0)))))->((ex a) ((unique a) (fun (x200:a)=> ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xx) x200)) ((cP Xb_0) Xu_0))) (forall (X0:(a->Prop)), (((and (X0 ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X0 ((cP Xc) Xv))->((and (X0 ((cP ((cP Xc) cZ)) (cL Xv)))) (X0 ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X0 ((cP Xb_0) Xu_0)))))))))))))->((and ((forall (Xt0:a), ((iff (not (((eq a) Xt0) cZ))) (((eq a) Xt0) ((cP (cL Xt0)) (cR Xt0)))))->((ex a) ((unique a) (fun (x200:a)=> ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP ((cP Xx) cZ)) x200)) ((cP Xb_0) Xu_0))) (forall (X0:(a->Prop)), (((and (X0 ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X0 ((cP Xc) Xv))->((and (X0 ((cP ((cP Xc) cZ)) (cL Xv)))) (X0 ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X0 ((cP Xb_0) Xu_0)))))))))))))) ((forall (Xt0:a), ((iff (not (((eq a) Xt0) cZ))) (((eq a) Xt0) ((cP (cL Xt0)) (cR Xt0)))))->((ex a) ((unique a) (fun (x200:a)=> ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP ((cP Xx) ((cP cZ) cZ))) x200)) ((cP Xb_0) Xu_0))) (forall (X0:(a->Prop)), (((and (X0 ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X0 ((cP Xc) Xv))->((and (X0 ((cP ((cP Xc) cZ)) (cL Xv)))) (X0 ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X0 ((cP Xb_0) Xu_0))))))))))))))))) as proof of (((eq Prop) (forall (Xx:a), (((forall (Xt0:a), ((iff (not (((eq a) Xt0) cZ))) (((eq a) Xt0) ((cP (cL Xt0)) (cR Xt0)))))->((ex a) ((unique a) (fun (x200:a)=> ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xx) x200)) ((cP Xb_0) Xu_0))) (forall (X0:(a->Prop)), (((and (X0 ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X0 ((cP Xc) Xv))->((and (X0 ((cP ((cP Xc) cZ)) (cL Xv)))) (X0 ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X0 ((cP Xb_0) Xu_0)))))))))))))->((and ((forall (Xt0:a), ((iff (not (((eq a) Xt0) cZ))) (((eq a) Xt0) ((cP (cL Xt0)) (cR Xt0)))))->((ex a) ((unique a) (fun (x200:a)=> ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP ((cP Xx) cZ)) x200)) ((cP Xb_0) Xu_0))) (forall (X0:(a->Prop)), (((and (X0 ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X0 ((cP Xc) Xv))->((and (X0 ((cP ((cP Xc) cZ)) (cL Xv)))) (X0 ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X0 ((cP Xb_0) Xu_0)))))))))))))) ((forall (Xt0:a), ((iff (not (((eq a) Xt0) cZ))) (((eq a) Xt0) ((cP (cL Xt0)) (cR Xt0)))))->((ex a) ((unique a) (fun (x200:a)=> ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP ((cP Xx) ((cP cZ) cZ))) x200)) ((cP Xb_0) Xu_0))) (forall (X0:(a->Prop)), (((and (X0 ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X0 ((cP Xc) Xv))->((and (X0 ((cP ((cP Xc) cZ)) (cL Xv)))) (X0 ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X0 ((cP Xb_0) Xu_0))))))))))))))))) b)
% Found ((eq_ref Prop) (forall (Xx:a), (((forall (Xt0:a), ((iff (not (((eq a) Xt0) cZ))) (((eq a) Xt0) ((cP (cL Xt0)) (cR Xt0)))))->((ex a) ((unique a) (fun (x200:a)=> ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xx) x200)) ((cP Xb_0) Xu_0))) (forall (X0:(a->Prop)), (((and (X0 ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X0 ((cP Xc) Xv))->((and (X0 ((cP ((cP Xc) cZ)) (cL Xv)))) (X0 ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X0 ((cP Xb_0) Xu_0)))))))))))))->((and ((forall (Xt0:a), ((iff (not (((eq a) Xt0) cZ))) (((eq a) Xt0) ((cP (cL Xt0)) (cR Xt0)))))->((ex a) ((unique a) (fun (x200:a)=> ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP ((cP Xx) cZ)) x200)) ((cP Xb_0) Xu_0))) (forall (X0:(a->Prop)), (((and (X0 ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X0 ((cP Xc) Xv))->((and (X0 ((cP ((cP Xc) cZ)) (cL Xv)))) (X0 ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X0 ((cP Xb_0) Xu_0)))))))))))))) ((forall (Xt0:a), ((iff (not (((eq a) Xt0) cZ))) (((eq a) Xt0) ((cP (cL Xt0)) (cR Xt0)))))->((ex a) ((unique a) (fun (x200:a)=> ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP ((cP Xx) ((cP cZ) cZ))) x200)) ((cP Xb_0) Xu_0))) (forall (X0:(a->Prop)), (((and (X0 ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X0 ((cP Xc) Xv))->((and (X0 ((cP ((cP Xc) cZ)) (cL Xv)))) (X0 ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X0 ((cP Xb_0) Xu_0))))))))))))))))) as proof of (((eq Prop) (forall (Xx:a), (((forall (Xt0:a), ((iff (not (((eq a) Xt0) cZ))) (((eq a) Xt0) ((cP (cL Xt0)) (cR Xt0)))))->((ex a) ((unique a) (fun (x200:a)=> ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xx) x200)) ((cP Xb_0) Xu_0))) (forall (X0:(a->Prop)), (((and (X0 ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X0 ((cP Xc) Xv))->((and (X0 ((cP ((cP Xc) cZ)) (cL Xv)))) (X0 ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X0 ((cP Xb_0) Xu_0)))))))))))))->((and ((forall (Xt0:a), ((iff (not (((eq a) Xt0) cZ))) (((eq a) Xt0) ((cP (cL Xt0)) (cR Xt0)))))->((ex a) ((unique a) (fun (x200:a)=> ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP ((cP Xx) cZ)) x200)) ((cP Xb_0) Xu_0))) (forall (X0:(a->Prop)), (((and (X0 ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X0 ((cP Xc) Xv))->((and (X0 ((cP ((cP Xc) cZ)) (cL Xv)))) (X0 ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X0 ((cP Xb_0) Xu_0)))))))))))))) ((forall (Xt0:a), ((iff (not (((eq a) Xt0) cZ))) (((eq a) Xt0) ((cP (cL Xt0)) (cR Xt0)))))->((ex a) ((unique a) (fun (x200:a)=> ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP ((cP Xx) ((cP cZ) cZ))) x200)) ((cP Xb_0) Xu_0))) (forall (X0:(a->Prop)), (((and (X0 ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X0 ((cP Xc) Xv))->((and (X0 ((cP ((cP Xc) cZ)) (cL Xv)))) (X0 ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X0 ((cP Xb_0) Xu_0))))))))))))))))) b)
% Found ((eq_ref Prop) (forall (Xx:a), (((forall (Xt0:a), ((iff (not (((eq a) Xt0) cZ))) (((eq a) Xt0) ((cP (cL Xt0)) (cR Xt0)))))->((ex a) ((unique a) (fun (x200:a)=> ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xx) x200)) ((cP Xb_0) Xu_0))) (forall (X0:(a->Prop)), (((and (X0 ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X0 ((cP Xc) Xv))->((and (X0 ((cP ((cP Xc) cZ)) (cL Xv)))) (X0 ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X0 ((cP Xb_0) Xu_0)))))))))))))->((and ((forall (Xt0:a), ((iff (not (((eq a) Xt0) cZ))) (((eq a) Xt0) ((cP (cL Xt0)) (cR Xt0)))))->((ex a) ((unique a) (fun (x200:a)=> ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP ((cP Xx) cZ)) x200)) ((cP Xb_0) Xu_0))) (forall (X0:(a->Prop)), (((and (X0 ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X0 ((cP Xc) Xv))->((and (X0 ((cP ((cP Xc) cZ)) (cL Xv)))) (X0 ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X0 ((cP Xb_0) Xu_0)))))))))))))) ((forall (Xt0:a), ((iff (not (((eq a) Xt0) cZ))) (((eq a) Xt0) ((cP (cL Xt0)) (cR Xt0)))))->((ex a) ((unique a) (fun (x200:a)=> ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP ((cP Xx) ((cP cZ) cZ))) x200)) ((cP Xb_0) Xu_0))) (forall (X0:(a->Prop)), (((and (X0 ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X0 ((cP Xc) Xv))->((and (X0 ((cP ((cP Xc) cZ)) (cL Xv)))) (X0 ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X0 ((cP Xb_0) Xu_0))))))))))))))))) as proof of (((eq Prop) (forall (Xx:a), (((forall (Xt0:a), ((iff (not (((eq a) Xt0) cZ))) (((eq a) Xt0) ((cP (cL Xt0)) (cR Xt0)))))->((ex a) ((unique a) (fun (x200:a)=> ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xx) x200)) ((cP Xb_0) Xu_0))) (forall (X0:(a->Prop)), (((and (X0 ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X0 ((cP Xc) Xv))->((and (X0 ((cP ((cP Xc) cZ)) (cL Xv)))) (X0 ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X0 ((cP Xb_0) Xu_0)))))))))))))->((and ((forall (Xt0:a), ((iff (not (((eq a) Xt0) cZ))) (((eq a) Xt0) ((cP (cL Xt0)) (cR Xt0)))))->((ex a) ((unique a) (fun (x200:a)=> ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP ((cP Xx) cZ)) x200)) ((cP Xb_0) Xu_0))) (forall (X0:(a->Prop)), (((and (X0 ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X0 ((cP Xc) Xv))->((and (X0 ((cP ((cP Xc) cZ)) (cL Xv)))) (X0 ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X0 ((cP Xb_0) Xu_0)))))))))))))) ((forall (Xt0:a), ((iff (not (((eq a) Xt0) cZ))) (((eq a) Xt0) ((cP (cL Xt0)) (cR Xt0)))))->((ex a) ((unique a) (fun (x200:a)=> ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP ((cP Xx) ((cP cZ) cZ))) x200)) ((cP Xb_0) Xu_0))) (forall (X0:(a->Prop)), (((and (X0 ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X0 ((cP Xc) Xv))->((and (X0 ((cP ((cP Xc) cZ)) (cL Xv)))) (X0 ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X0 ((cP Xb_0) Xu_0))))))))))))))))) b)
% Found ((eq_ref Prop) (forall (Xx:a), (((forall (Xt0:a), ((iff (not (((eq a) Xt0) cZ))) (((eq a) Xt0) ((cP (cL Xt0)) (cR Xt0)))))->((ex a) ((unique a) (fun (x200:a)=> ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xx) x200)) ((cP Xb_0) Xu_0))) (forall (X0:(a->Prop)), (((and (X0 ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X0 ((cP Xc) Xv))->((and (X0 ((cP ((cP Xc) cZ)) (cL Xv)))) (X0 ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X0 ((cP Xb_0) Xu_0)))))))))))))->((and ((forall (Xt0:a), ((iff (not (((eq a) Xt0) cZ))) (((eq a) Xt0) ((cP (cL Xt0)) (cR Xt0)))))->((ex a) ((unique a) (fun (x200:a)=> ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP ((cP Xx) cZ)) x200)) ((cP Xb_0) Xu_0))) (forall (X0:(a->Prop)), (((and (X0 ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X0 ((cP Xc) Xv))->((and (X0 ((cP ((cP Xc) cZ)) (cL Xv)))) (X0 ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X0 ((cP Xb_0) Xu_0)))))))))))))) ((forall (Xt0:a), ((iff (not (((eq a) Xt0) cZ))) (((eq a) Xt0) ((cP (cL Xt0)) (cR Xt0)))))->((ex a) ((unique a) (fun (x200:a)=> ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP ((cP Xx) ((cP cZ) cZ))) x200)) ((cP Xb_0) Xu_0))) (forall (X0:(a->Prop)), (((and (X0 ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X0 ((cP Xc) Xv))->((and (X0 ((cP ((cP Xc) cZ)) (cL Xv)))) (X0 ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X0 ((cP Xb_0) Xu_0))))))))))))))))) as proof of (((eq Prop) (forall (Xx:a), (((forall (Xt0:a), ((iff (not (((eq a) Xt0) cZ))) (((eq a) Xt0) ((cP (cL Xt0)) (cR Xt0)))))->((ex a) ((unique a) (fun (x200:a)=> ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xx) x200)) ((cP Xb_0) Xu_0))) (forall (X0:(a->Prop)), (((and (X0 ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X0 ((cP Xc) Xv))->((and (X0 ((cP ((cP Xc) cZ)) (cL Xv)))) (X0 ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X0 ((cP Xb_0) Xu_0)))))))))))))->((and ((forall (Xt0:a), ((iff (not (((eq a) Xt0) cZ))) (((eq a) Xt0) ((cP (cL Xt0)) (cR Xt0)))))->((ex a) ((unique a) (fun (x200:a)=> ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP ((cP Xx) cZ)) x200)) ((cP Xb_0) Xu_0))) (forall (X0:(a->Prop)), (((and (X0 ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X0 ((cP Xc) Xv))->((and (X0 ((cP ((cP Xc) cZ)) (cL Xv)))) (X0 ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X0 ((cP Xb_0) Xu_0)))))))))))))) ((forall (Xt0:a), ((iff (not (((eq a) Xt0) cZ))) (((eq a) Xt0) ((cP (cL Xt0)) (cR Xt0)))))->((ex a) ((unique a) (fun (x200:a)=> ((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP ((cP Xx) ((cP cZ) cZ))) x200)) ((cP Xb_0) Xu_0))) (forall (X0:(a->Prop)), (((and (X0 ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X0 ((cP Xc) Xv))->((and (X0 ((cP ((cP Xc) cZ)) (cL Xv)))) (X0 ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X0 ((cP Xb_0) Xu_0))))))))))))))))) b)
% Found eq_ref00:=(eq_ref0 x1):(((eq a) x1) x1)
% Found (eq_ref0 x1) as proof of (((eq a) x1) Xv)
% Found ((eq_ref a) x1) as proof of (((eq a) x1) Xv)
% Found ((eq_ref a) x1) as proof of (((eq a) x1) Xv)
% Found (fun (x5:(forall (Xx:a) (Xy:a), (((eq a) (cR ((cP Xx) Xy))) Xy)))=> ((eq_ref a) x1)) as proof of (((eq a) x1) Xv)
% Found (fun (x4:((and ((and (((eq a) (cL cZ)) cZ)) (((eq a) (cR cZ)) cZ))) (forall (Xx:a) (Xy:a), (((eq a) (cL ((cP Xx) Xy))) Xx)))) (x5:(forall (Xx:a) (Xy:a), (((eq a) (cR ((cP Xx) Xy))) Xy)))=> ((eq_ref a) x1)) as proof of ((forall (Xx:a) (Xy:a), (((eq a) (cR ((cP Xx) Xy))) Xy))->(((eq a) x1) Xv))
% Found (fun (x4:((and ((and (((eq a) (cL cZ)) cZ)) (((eq a) (cR cZ)) cZ))) (forall (Xx:a) (Xy:a), (((eq a) (cL ((cP Xx) Xy))) Xx)))) (x5:(forall (Xx:a) (Xy:a), (((eq a) (cR ((cP Xx) Xy))) Xy)))=> ((eq_ref a) x1)) as proof of (((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))->(((eq a) x1) Xv)))
% Found (and_rect10 (fun (x4:((and ((and (((eq a) (cL cZ)) cZ)) (((eq a) (cR cZ)) cZ))) (forall (Xx:a) (Xy:a), (((eq a) (cL ((cP Xx) Xy))) Xx)))) (x5:(forall (Xx:a) (Xy:a), (((eq a) (cR ((cP Xx) Xy))) Xy)))=> ((eq_ref a) x1))) as proof of (((eq a) x1) Xv)
% Found ((and_rect1 (((eq a) x1) Xv)) (fun (x4:((and ((and (((eq a) (cL cZ)) cZ)) (((eq a) (cR cZ)) cZ))) (forall (Xx:a) (Xy:a), (((eq a) (cL ((cP Xx) Xy))) Xx)))) (x5:(forall (Xx:a) (Xy:a), (((eq a) (cR ((cP Xx) Xy))) Xy)))=> ((eq_ref a) x1))) as proof of (((eq a) x1) Xv)
% Found (((fun (P:Type) (x4:(((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))->P)))=> (((((and_rect ((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))) P) x4) x2)) (((eq a) x1) Xv)) (fun (x4:((and ((and (((eq a) (cL cZ)) cZ)) (((eq a) (cR cZ)) cZ))) (forall (Xx:a) (Xy:a), (((eq a) (cL ((cP Xx) Xy))) Xx)))) (x5:(forall (Xx:a) (Xy:a), (((eq a) (cR ((cP Xx) Xy))) Xy)))=> ((eq_ref a) x1))) as proof of (((eq a) x1) Xv)
% Found (fun (x00:((ex a) (fun (Xb_1:a)=> ((ex a) (fun (Xu0:a)=> ((and (((eq a) ((cP Xb) Xv)) ((cP Xb_1) Xu0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv0:a), ((X ((cP Xc) Xv0))->((and (X ((cP ((cP Xc) cZ)) (cL Xv0)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv0)))))))->(X ((cP Xb_1) Xu0))))))))))=> (((fun (P:Type) (x4:(((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))->P)))=> (((((and_rect ((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))) P) x4) x2)) (((eq a) x1) Xv)) (fun (x4:((and ((and (((eq a) (cL cZ)) cZ)) (((eq a) (cR cZ)) cZ))) (forall (Xx:a) (Xy:a), (((eq a) (cL ((cP Xx) Xy))) Xx)))) (x5:(forall (Xx:a) (Xy:a), (((eq a) (cR ((cP Xx) Xy))) Xy)))=> ((eq_ref a) x1)))) as proof of (((eq a) x1) Xv)
% Found eq_ref00:=(eq_ref0 x3):(((eq a) x3) x3)
% Found (eq_ref0 x3) as proof of (((eq a) x3) Xv)
% Found ((eq_ref a) x3) as proof of (((eq a) x3) Xv)
% Found ((eq_ref a) x3) as proof of (((eq a) x3) Xv)
% Found (fun (x5:(forall (Xx:a) (Xy:a), (((eq a) (cR ((cP Xx) Xy))) Xy)))=> ((eq_ref a) x3)) as proof of (((eq a) x3) Xv)
% Found (fun (x4:((and ((and (((eq a) (cL cZ)) cZ)) (((eq a) (cR cZ)) cZ))) (forall (Xx:a) (Xy:a), (((eq a) (cL ((cP Xx) Xy))) Xx)))) (x5:(forall (Xx:a) (Xy:a), (((eq a) (cR ((cP Xx) Xy))) Xy)))=> ((eq_ref a) x3)) as proof of ((forall (Xx:a) (Xy:a), (((eq a) (cR ((cP Xx) Xy))) Xy))->(((eq a) x3) Xv))
% Found (fun (x4:((and ((and (((eq a) (cL cZ)) cZ)) (((eq a) (cR cZ)) cZ))) (forall (Xx:a) (Xy:a), (((eq a) (cL ((cP Xx) Xy))) Xx)))) (x5:(forall (Xx:a) (Xy:a), (((eq a) (cR ((cP Xx) Xy))) Xy)))=> ((eq_ref a) x3)) as proof of (((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))->(((eq a) x3) Xv)))
% Found (and_rect10 (fun (x4:((and ((and (((eq a) (cL cZ)) cZ)) (((eq a) (cR cZ)) cZ))) (forall (Xx:a) (Xy:a), (((eq a) (cL ((cP Xx) Xy))) Xx)))) (x5:(forall (Xx:a) (Xy:a), (((eq a) (cR ((cP Xx) Xy))) Xy)))=> ((eq_ref a) x3))) as proof of (((eq a) x3) Xv)
% Found ((and_rect1 (((eq a) x3) Xv)) (fun (x4:((and ((and (((eq a) (cL cZ)) cZ)) (((eq a) (cR cZ)) cZ))) (forall (Xx:a) (Xy:a), (((eq a) (cL ((cP Xx) Xy))) Xx)))) (x5:(forall (Xx:a) (Xy:a), (((eq a) (cR ((cP Xx) Xy))) Xy)))=> ((eq_ref a) x3))) as proof of (((eq a) x3) Xv)
% Found (((fun (P:Type) (x4:(((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))->P)))=> (((((and_rect ((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))) P) x4) x1)) (((eq a) x3) Xv)) (fun (x4:((and ((and (((eq a) (cL cZ)) cZ)) (((eq a) (cR cZ)) cZ))) (forall (Xx:a) (Xy:a), (((eq a) (cL ((cP Xx) Xy))) Xx)))) (x5:(forall (Xx:a) (Xy:a), (((eq a) (cR ((cP Xx) Xy))) Xy)))=> ((eq_ref a) x3))) as proof of (((eq a) x3) Xv)
% Found (fun (x00:((ex a) (fun (Xb_1:a)=> ((ex a) (fun (Xu0:a)=> ((and (((eq a) ((cP Xb) Xv)) ((cP Xb_1) Xu0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv0:a), ((X ((cP Xc) Xv0))->((and (X ((cP ((cP Xc) cZ)) (cL Xv0)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv0)))))))->(X ((cP Xb_1) Xu0))))))))))=> (((fun (P:Type) (x4:(((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))->P)))=> (((((and_rect ((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))) P) x4) x1)) (((eq a) x3) Xv)) (fun (x4:((and ((and (((eq a) (cL cZ)) cZ)) (((eq a) (cR cZ)) cZ))) (forall (Xx:a) (Xy:a), (((eq a) (cL ((cP Xx) Xy))) Xx)))) (x5:(forall (Xx:a) (Xy:a), (((eq a) (cR ((cP Xx) Xy))) Xy)))=> ((eq_ref a) x3)))) as proof of (((eq a) x3) Xv)
% Found eq_ref000:=(eq_ref00 P):((P x1)->(P x1))
% Found (eq_ref00 P) as proof of ((P x1)->(P x'))
% Found ((eq_ref0 x1) P) as proof of ((P x1)->(P x'))
% Found (((eq_ref a) x1) P) as proof of ((P x1)->(P x'))
% Found (((eq_ref a) x1) P) as proof of ((P x1)->(P x'))
% Found (fun (P:(a->Prop))=> (((eq_ref a) x1) P)) as proof of ((P x1)->(P x'))
% Found (fun (x01:((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP cZ) x')) ((cP Xb_0) Xu_0))) (forall (X0:(a->Prop)), (((and (X0 ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X0 ((cP Xc) Xv))->((and (X0 ((cP ((cP Xc) cZ)) (cL Xv)))) (X0 ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X0 ((cP Xb_0) Xu_0)))))))))) (P:(a->Prop))=> (((eq_ref a) x1) P)) as proof of (((eq a) x1) x')
% Found x2:(P x1)
% Instantiate: x1:=x':a
% Found (fun (x2:(P x1))=> x2) as proof of (P x')
% Found (fun (P:(a->Prop)) (x2:(P x1))=> x2) as proof of ((P x1)->(P x'))
% Found (fun (x01:((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP cZ) x')) ((cP Xb_0) Xu_0))) (forall (X0:(a->Prop)), (((and (X0 ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X0 ((cP Xc) Xv))->((and (X0 ((cP ((cP Xc) cZ)) (cL Xv)))) (X0 ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X0 ((cP Xb_0) Xu_0)))))))))) (P:(a->Prop)) (x2:(P x1))=> x2) as proof of (((eq a) x1) x')
% Found eq_ref00:=(eq_ref0 x1):(((eq a) x1) x1)
% Found (eq_ref0 x1) as proof of (((eq a) x1) x')
% Found ((eq_ref a) x1) as proof of (((eq a) x1) x')
% Found ((eq_ref a) x1) as proof of (((eq a) x1) x')
% Found (fun (x5:((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xb) x')) ((cP x4) Xu_0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X ((cP Xc) Xv))->((and (X ((cP ((cP Xc) cZ)) (cL Xv)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X ((cP x4) Xu_0))))))))=> ((eq_ref a) x1)) as proof of (((eq a) x1) x')
% Found (fun (x4:a) (x5:((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xb) x')) ((cP x4) Xu_0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X ((cP Xc) Xv))->((and (X ((cP ((cP Xc) cZ)) (cL Xv)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X ((cP x4) Xu_0))))))))=> ((eq_ref a) x1)) as proof of (((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xb) x')) ((cP x4) Xu_0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X ((cP Xc) Xv))->((and (X ((cP ((cP Xc) cZ)) (cL Xv)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X ((cP x4) Xu_0)))))))->(((eq a) x1) x'))
% Found (fun (x4:a) (x5:((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xb) x')) ((cP x4) Xu_0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X ((cP Xc) Xv))->((and (X ((cP ((cP Xc) cZ)) (cL Xv)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X ((cP x4) Xu_0))))))))=> ((eq_ref a) x1)) as proof of (forall (x:a), (((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xb) x')) ((cP x) Xu_0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X ((cP Xc) Xv))->((and (X ((cP ((cP Xc) cZ)) (cL Xv)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X ((cP x) Xu_0)))))))->(((eq a) x1) x')))
% Found (ex_ind00 (fun (x4:a) (x5:((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xb) x')) ((cP x4) Xu_0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X ((cP Xc) Xv))->((and (X ((cP ((cP Xc) cZ)) (cL Xv)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X ((cP x4) Xu_0))))))))=> ((eq_ref a) x1))) as proof of (((eq a) x1) x')
% Found ((ex_ind0 (((eq a) x1) x')) (fun (x4:a) (x5:((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xb) x')) ((cP x4) Xu_0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X ((cP Xc) Xv))->((and (X ((cP ((cP Xc) cZ)) (cL Xv)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X ((cP x4) Xu_0))))))))=> ((eq_ref a) x1))) as proof of (((eq a) x1) x')
% Found (((fun (P:Prop) (x4:(forall (x:a), (((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xb) x')) ((cP x) Xu_0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X ((cP Xc) Xv))->((and (X ((cP ((cP Xc) cZ)) (cL Xv)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X ((cP x) Xu_0)))))))->P)))=> (((((ex_ind a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xb) x')) ((cP Xb_0) Xu_0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X ((cP Xc) Xv))->((and (X ((cP ((cP Xc) cZ)) (cL Xv)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X ((cP Xb_0) Xu_0))))))))) P) x4) x00)) (((eq a) x1) x')) (fun (x4:a) (x5:((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xb) x')) ((cP x4) Xu_0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X ((cP Xc) Xv))->((and (X ((cP ((cP Xc) cZ)) (cL Xv)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X ((cP x4) Xu_0))))))))=> ((eq_ref a) x1))) as proof of (((eq a) x1) x')
% Found (fun (x00:((ex a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xb) x')) ((cP Xb_0) Xu_0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X ((cP Xc) Xv))->((and (X ((cP ((cP Xc) cZ)) (cL Xv)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X ((cP Xb_0) Xu_0))))))))))=> (((fun (P:Prop) (x4:(forall (x:a), (((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xb) x')) ((cP x) Xu_0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X ((cP Xc) Xv))->((and (X ((cP ((cP Xc) cZ)) (cL Xv)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X ((cP x) Xu_0)))))))->P)))=> (((((ex_ind a) (fun (Xb_0:a)=> ((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xb) x')) ((cP Xb_0) Xu_0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X ((cP Xc) Xv))->((and (X ((cP ((cP Xc) cZ)) (cL Xv)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X ((cP Xb_0) Xu_0))))))))) P) x4) x00)) (((eq a) x1) x')) (fun (x4:a) (x5:((ex a) (fun (Xu_0:a)=> ((and (((eq a) ((cP Xb) x')) ((cP x4) Xu_0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv:a), ((X ((cP Xc) Xv))->((and (X ((cP ((cP Xc) cZ)) (cL Xv)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv)))))))->(X ((cP x4) Xu_0))))))))=> ((eq_ref a) x1)))) as proof of (((eq a) x1) x')
% Found eq_ref000:=(eq_ref00 P):((P x7)->(P x7))
% Found (eq_ref00 P) as proof of ((P x7)->(P Xv))
% Found ((eq_ref0 x7) P) as proof of ((P x7)->(P Xv))
% Found (((eq_ref a) x7) P) as proof of ((P x7)->(P Xv))
% Found (((eq_ref a) x7) P) as proof of ((P x7)->(P Xv))
% Found (fun (P:(a->Prop))=> (((eq_ref a) x7) P)) as proof of ((P x7)->(P Xv))
% Found (fun (x00:((ex a) (fun (Xb_1:a)=> ((ex a) (fun (Xu0:a)=> ((and (((eq a) ((cP Xb) Xv)) ((cP Xb_1) Xu0))) (forall (X:(a->Prop)), (((and (X ((cP cZ) Xt))) (forall (Xc:a) (Xv0:a), ((X ((cP Xc) Xv0))->((and (X ((cP ((cP Xc) cZ)) (cL Xv0)))) (X ((cP ((cP Xc) ((cP cZ) cZ))) (cR Xv0)))))))->(X ((cP Xb_1) Xu0)))))))))) (P:(a->Prop))=> (((eq_ref a) x7) P)) as proof of (((eq a) x7) Xv)
% Found x8:(P x7)
% Instantiate: x7:=Xv:a
% Found (fun (x8:(P x7))=> x8) as proof of (P Xv)
% Found (fun (P:(a->Prop)) (x8:(P x7))=> x8) as proof of ((P x7)->(P Xv))
% Found (fun (x00
% EOF
%------------------------------------------------------------------------------