TSTP Solution File: ALG290^5 by cocATP---0.2.0
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- Process Solution
%------------------------------------------------------------------------------
% File : cocATP---0.2.0
% Problem : ALG290^5 : TPTP v6.1.0. Released v4.0.0.
% Transfm : none
% Format : tptp:raw
% Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% Computer : n102.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 300.06s
% Output : None
% Verified :
% SZS Type : None (Parsing solution fails)
% Syntax : Number of formulae : 0
% Comments :
%------------------------------------------------------------------------------
%----NO SOLUTION OUTPUT BY SYSTEM
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% % Problem : ALG290^5 : TPTP v6.1.0. Released v4.0.0.
% % Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n102.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 09:09:41 CDT 2014
% % CPUTime : 300.06
% 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 0x23bdc20>, <kernel.Type object at 0x23bc680>) of role type named a_type
% Using role type
% Declaring a:Type
% FOF formula (<kernel.Constant object at 0x23bb518>, <kernel.DependentProduct object at 0x23bc1b8>) of role type named cP
% Using role type
% Declaring cP:(a->(a->a))
% FOF formula (<kernel.Constant object at 0x23bd4d0>, <kernel.DependentProduct object at 0x23bc638>) of role type named cG
% Using role type
% Declaring cG:(a->Prop)
% FOF formula (<kernel.Constant object at 0x23bdc20>, <kernel.DependentProduct object at 0x23bc050>) of role type named cX
% Using role type
% Declaring cX:(a->Prop)
% FOF formula (<kernel.Constant object at 0x23bdc20>, <kernel.DependentProduct object at 0x23bc290>) of role type named cR
% Using role type
% Declaring cR:(a->a)
% FOF formula (<kernel.Constant object at 0x23bc638>, <kernel.DependentProduct object at 0x23bc248>) of role type named cL
% Using role type
% Declaring cL:(a->a)
% FOF formula (<kernel.Constant object at 0x23bc050>, <kernel.DependentProduct object at 0x23bccb0>) of role type named cF
% Using role type
% Declaring cF:(a->Prop)
% FOF formula (<kernel.Constant object at 0x23bc290>, <kernel.Constant object at 0x23bccb0>) of role type named cZ
% Using role type
% Declaring cZ:a
% FOF formula (((and ((and ((and ((and ((and (((eq a) (cL cZ)) cZ)) (((eq a) (cR cZ)) cZ))) (forall (Xx:a) (Xy:a), (((eq a) (cL ((cP Xx) Xy))) Xx)))) (forall (Xx:a) (Xy:a), (((eq a) (cR ((cP Xx) Xy))) Xy)))) (forall (Xt:a), ((iff (not (((eq a) Xt) cZ))) (((eq a) Xt) ((cP (cL Xt)) (cR Xt))))))) (forall (X0:(a->Prop)), (((ex a) (fun (Xt:a)=> ((and (X0 Xt)) (forall (Xu:a), ((X0 Xu)->(X0 (cL Xu)))))))->(X0 cZ))))->(((eq (a->Prop)) (fun (Xy:a)=> ((ex a) (fun (Xx:a)=> ((and (forall (Xx_17:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz:a), ((X0 Xz)->(X0 (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_17))))))->(cX Xx_17)))) ((or (cF ((cP Xx) Xy))) (cG ((cP Xx) Xy)))))))) (fun (Xz:a)=> ((or ((ex a) (fun (Xx:a)=> ((and (forall (Xx_18:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_18))))))->(cX Xx_18)))) (cF ((cP Xx) Xz)))))) ((ex a) (fun (Xx:a)=> ((and (forall (Xx_19:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_19))))))->(cX Xx_19)))) (cG ((cP Xx) Xz))))))))) of role conjecture named cPU_X2310A_pme
% Conjecture to prove = (((and ((and ((and ((and ((and (((eq a) (cL cZ)) cZ)) (((eq a) (cR cZ)) cZ))) (forall (Xx:a) (Xy:a), (((eq a) (cL ((cP Xx) Xy))) Xx)))) (forall (Xx:a) (Xy:a), (((eq a) (cR ((cP Xx) Xy))) Xy)))) (forall (Xt:a), ((iff (not (((eq a) Xt) cZ))) (((eq a) Xt) ((cP (cL Xt)) (cR Xt))))))) (forall (X0:(a->Prop)), (((ex a) (fun (Xt:a)=> ((and (X0 Xt)) (forall (Xu:a), ((X0 Xu)->(X0 (cL Xu)))))))->(X0 cZ))))->(((eq (a->Prop)) (fun (Xy:a)=> ((ex a) (fun (Xx:a)=> ((and (forall (Xx_17:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz:a), ((X0 Xz)->(X0 (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_17))))))->(cX Xx_17)))) ((or (cF ((cP Xx) Xy))) (cG ((cP Xx) Xy)))))))) (fun (Xz:a)=> ((or ((ex a) (fun (Xx:a)=> ((and (forall (Xx_18:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_18))))))->(cX Xx_18)))) (cF ((cP Xx) Xz)))))) ((ex a) (fun (Xx:a)=> ((and (forall (Xx_19:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_19))))))->(cX Xx_19)))) (cG ((cP Xx) Xz))))))))):Prop
% We need to prove ['(((and ((and ((and ((and ((and (((eq a) (cL cZ)) cZ)) (((eq a) (cR cZ)) cZ))) (forall (Xx:a) (Xy:a), (((eq a) (cL ((cP Xx) Xy))) Xx)))) (forall (Xx:a) (Xy:a), (((eq a) (cR ((cP Xx) Xy))) Xy)))) (forall (Xt:a), ((iff (not (((eq a) Xt) cZ))) (((eq a) Xt) ((cP (cL Xt)) (cR Xt))))))) (forall (X0:(a->Prop)), (((ex a) (fun (Xt:a)=> ((and (X0 Xt)) (forall (Xu:a), ((X0 Xu)->(X0 (cL Xu)))))))->(X0 cZ))))->(((eq (a->Prop)) (fun (Xy:a)=> ((ex a) (fun (Xx:a)=> ((and (forall (Xx_17:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz:a), ((X0 Xz)->(X0 (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_17))))))->(cX Xx_17)))) ((or (cF ((cP Xx) Xy))) (cG ((cP Xx) Xy)))))))) (fun (Xz:a)=> ((or ((ex a) (fun (Xx:a)=> ((and (forall (Xx_18:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_18))))))->(cX Xx_18)))) (cF ((cP Xx) Xz)))))) ((ex a) (fun (Xx:a)=> ((and (forall (Xx_19:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_19))))))->(cX Xx_19)))) (cG ((cP Xx) Xz)))))))))']
% Parameter a:Type.
% Parameter cP:(a->(a->a)).
% Parameter cG:(a->Prop).
% Parameter cX:(a->Prop).
% Parameter cR:(a->a).
% Parameter cL:(a->a).
% Parameter cF:(a->Prop).
% Parameter cZ:a.
% Trying to prove (((and ((and ((and ((and ((and (((eq a) (cL cZ)) cZ)) (((eq a) (cR cZ)) cZ))) (forall (Xx:a) (Xy:a), (((eq a) (cL ((cP Xx) Xy))) Xx)))) (forall (Xx:a) (Xy:a), (((eq a) (cR ((cP Xx) Xy))) Xy)))) (forall (Xt:a), ((iff (not (((eq a) Xt) cZ))) (((eq a) Xt) ((cP (cL Xt)) (cR Xt))))))) (forall (X0:(a->Prop)), (((ex a) (fun (Xt:a)=> ((and (X0 Xt)) (forall (Xu:a), ((X0 Xu)->(X0 (cL Xu)))))))->(X0 cZ))))->(((eq (a->Prop)) (fun (Xy:a)=> ((ex a) (fun (Xx:a)=> ((and (forall (Xx_17:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz:a), ((X0 Xz)->(X0 (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_17))))))->(cX Xx_17)))) ((or (cF ((cP Xx) Xy))) (cG ((cP Xx) Xy)))))))) (fun (Xz:a)=> ((or ((ex a) (fun (Xx:a)=> ((and (forall (Xx_18:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_18))))))->(cX Xx_18)))) (cF ((cP Xx) Xz)))))) ((ex a) (fun (Xx:a)=> ((and (forall (Xx_19:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_19))))))->(cX Xx_19)))) (cG ((cP Xx) Xz)))))))))
% Found eq_ref000:=(eq_ref00 P):((P (fun (Xy:a)=> ((ex a) (fun (Xx:a)=> ((and (forall (Xx_17:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz:a), ((X0 Xz)->(X0 (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_17))))))->(cX Xx_17)))) ((or (cF ((cP Xx) Xy))) (cG ((cP Xx) Xy))))))))->(P (fun (Xy:a)=> ((ex a) (fun (Xx:a)=> ((and (forall (Xx_17:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz:a), ((X0 Xz)->(X0 (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_17))))))->(cX Xx_17)))) ((or (cF ((cP Xx) Xy))) (cG ((cP Xx) Xy)))))))))
% Found (eq_ref00 P) as proof of (P0 (fun (Xy:a)=> ((ex a) (fun (Xx:a)=> ((and (forall (Xx_17:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz:a), ((X0 Xz)->(X0 (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_17))))))->(cX Xx_17)))) ((or (cF ((cP Xx) Xy))) (cG ((cP Xx) Xy))))))))
% Found ((eq_ref0 (fun (Xy:a)=> ((ex a) (fun (Xx:a)=> ((and (forall (Xx_17:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz:a), ((X0 Xz)->(X0 (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_17))))))->(cX Xx_17)))) ((or (cF ((cP Xx) Xy))) (cG ((cP Xx) Xy)))))))) P) as proof of (P0 (fun (Xy:a)=> ((ex a) (fun (Xx:a)=> ((and (forall (Xx_17:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz:a), ((X0 Xz)->(X0 (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_17))))))->(cX Xx_17)))) ((or (cF ((cP Xx) Xy))) (cG ((cP Xx) Xy))))))))
% Found (((eq_ref (a->Prop)) (fun (Xy:a)=> ((ex a) (fun (Xx:a)=> ((and (forall (Xx_17:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz:a), ((X0 Xz)->(X0 (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_17))))))->(cX Xx_17)))) ((or (cF ((cP Xx) Xy))) (cG ((cP Xx) Xy)))))))) P) as proof of (P0 (fun (Xy:a)=> ((ex a) (fun (Xx:a)=> ((and (forall (Xx_17:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz:a), ((X0 Xz)->(X0 (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_17))))))->(cX Xx_17)))) ((or (cF ((cP Xx) Xy))) (cG ((cP Xx) Xy))))))))
% Found (((eq_ref (a->Prop)) (fun (Xy:a)=> ((ex a) (fun (Xx:a)=> ((and (forall (Xx_17:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz:a), ((X0 Xz)->(X0 (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_17))))))->(cX Xx_17)))) ((or (cF ((cP Xx) Xy))) (cG ((cP Xx) Xy)))))))) P) as proof of (P0 (fun (Xy:a)=> ((ex a) (fun (Xx:a)=> ((and (forall (Xx_17:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz:a), ((X0 Xz)->(X0 (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_17))))))->(cX Xx_17)))) ((or (cF ((cP Xx) Xy))) (cG ((cP Xx) Xy))))))))
% Found cZ:a
% Found cZ as proof of a
% Found cZ:a
% Found cZ as proof of a
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P (fun (Xy:a)=> ((ex a) (fun (Xx:a)=> ((and (forall (Xx_17:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz:a), ((X0 Xz)->(X0 (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_17))))))->(cX Xx_17)))) ((or (cF ((cP Xx) Xy))) (cG ((cP Xx) Xy))))))))->(P (fun (x:a)=> ((ex a) (fun (Xx:a)=> ((and (forall (Xx_17:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz:a), ((X0 Xz)->(X0 (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_17))))))->(cX Xx_17)))) ((or (cF ((cP Xx) x))) (cG ((cP Xx) x)))))))))
% Found (eta_expansion_dep000 P) as proof of (P0 (fun (Xy:a)=> ((ex a) (fun (Xx:a)=> ((and (forall (Xx_17:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz:a), ((X0 Xz)->(X0 (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_17))))))->(cX Xx_17)))) ((or (cF ((cP Xx) Xy))) (cG ((cP Xx) Xy))))))))
% Found ((eta_expansion_dep00 (fun (Xy:a)=> ((ex a) (fun (Xx:a)=> ((and (forall (Xx_17:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz:a), ((X0 Xz)->(X0 (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_17))))))->(cX Xx_17)))) ((or (cF ((cP Xx) Xy))) (cG ((cP Xx) Xy)))))))) P) as proof of (P0 (fun (Xy:a)=> ((ex a) (fun (Xx:a)=> ((and (forall (Xx_17:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz:a), ((X0 Xz)->(X0 (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_17))))))->(cX Xx_17)))) ((or (cF ((cP Xx) Xy))) (cG ((cP Xx) Xy))))))))
% Found (((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xy:a)=> ((ex a) (fun (Xx:a)=> ((and (forall (Xx_17:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz:a), ((X0 Xz)->(X0 (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_17))))))->(cX Xx_17)))) ((or (cF ((cP Xx) Xy))) (cG ((cP Xx) Xy)))))))) P) as proof of (P0 (fun (Xy:a)=> ((ex a) (fun (Xx:a)=> ((and (forall (Xx_17:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz:a), ((X0 Xz)->(X0 (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_17))))))->(cX Xx_17)))) ((or (cF ((cP Xx) Xy))) (cG ((cP Xx) Xy))))))))
% Found ((((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xy:a)=> ((ex a) (fun (Xx:a)=> ((and (forall (Xx_17:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz:a), ((X0 Xz)->(X0 (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_17))))))->(cX Xx_17)))) ((or (cF ((cP Xx) Xy))) (cG ((cP Xx) Xy)))))))) P) as proof of (P0 (fun (Xy:a)=> ((ex a) (fun (Xx:a)=> ((and (forall (Xx_17:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz:a), ((X0 Xz)->(X0 (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_17))))))->(cX Xx_17)))) ((or (cF ((cP Xx) Xy))) (cG ((cP Xx) Xy))))))))
% Found ((((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xy:a)=> ((ex a) (fun (Xx:a)=> ((and (forall (Xx_17:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz:a), ((X0 Xz)->(X0 (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_17))))))->(cX Xx_17)))) ((or (cF ((cP Xx) Xy))) (cG ((cP Xx) Xy)))))))) P) as proof of (P0 (fun (Xy:a)=> ((ex a) (fun (Xx:a)=> ((and (forall (Xx_17:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz:a), ((X0 Xz)->(X0 (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_17))))))->(cX Xx_17)))) ((or (cF ((cP Xx) Xy))) (cG ((cP Xx) Xy))))))))
% Found eta_expansion0000:=(eta_expansion000 P):((P (fun (Xy:a)=> ((ex a) (fun (Xx:a)=> ((and (forall (Xx_17:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz:a), ((X0 Xz)->(X0 (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_17))))))->(cX Xx_17)))) ((or (cF ((cP Xx) Xy))) (cG ((cP Xx) Xy))))))))->(P (fun (x:a)=> ((ex a) (fun (Xx:a)=> ((and (forall (Xx_17:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz:a), ((X0 Xz)->(X0 (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_17))))))->(cX Xx_17)))) ((or (cF ((cP Xx) x))) (cG ((cP Xx) x)))))))))
% Found (eta_expansion000 P) as proof of (P0 (fun (Xy:a)=> ((ex a) (fun (Xx:a)=> ((and (forall (Xx_17:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz:a), ((X0 Xz)->(X0 (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_17))))))->(cX Xx_17)))) ((or (cF ((cP Xx) Xy))) (cG ((cP Xx) Xy))))))))
% Found ((eta_expansion00 (fun (Xy:a)=> ((ex a) (fun (Xx:a)=> ((and (forall (Xx_17:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz:a), ((X0 Xz)->(X0 (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_17))))))->(cX Xx_17)))) ((or (cF ((cP Xx) Xy))) (cG ((cP Xx) Xy)))))))) P) as proof of (P0 (fun (Xy:a)=> ((ex a) (fun (Xx:a)=> ((and (forall (Xx_17:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz:a), ((X0 Xz)->(X0 (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_17))))))->(cX Xx_17)))) ((or (cF ((cP Xx) Xy))) (cG ((cP Xx) Xy))))))))
% Found (((eta_expansion0 Prop) (fun (Xy:a)=> ((ex a) (fun (Xx:a)=> ((and (forall (Xx_17:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz:a), ((X0 Xz)->(X0 (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_17))))))->(cX Xx_17)))) ((or (cF ((cP Xx) Xy))) (cG ((cP Xx) Xy)))))))) P) as proof of (P0 (fun (Xy:a)=> ((ex a) (fun (Xx:a)=> ((and (forall (Xx_17:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz:a), ((X0 Xz)->(X0 (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_17))))))->(cX Xx_17)))) ((or (cF ((cP Xx) Xy))) (cG ((cP Xx) Xy))))))))
% Found ((((eta_expansion a) Prop) (fun (Xy:a)=> ((ex a) (fun (Xx:a)=> ((and (forall (Xx_17:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz:a), ((X0 Xz)->(X0 (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_17))))))->(cX Xx_17)))) ((or (cF ((cP Xx) Xy))) (cG ((cP Xx) Xy)))))))) P) as proof of (P0 (fun (Xy:a)=> ((ex a) (fun (Xx:a)=> ((and (forall (Xx_17:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz:a), ((X0 Xz)->(X0 (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_17))))))->(cX Xx_17)))) ((or (cF ((cP Xx) Xy))) (cG ((cP Xx) Xy))))))))
% Found ((((eta_expansion a) Prop) (fun (Xy:a)=> ((ex a) (fun (Xx:a)=> ((and (forall (Xx_17:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz:a), ((X0 Xz)->(X0 (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_17))))))->(cX Xx_17)))) ((or (cF ((cP Xx) Xy))) (cG ((cP Xx) Xy)))))))) P) as proof of (P0 (fun (Xy:a)=> ((ex a) (fun (Xx:a)=> ((and (forall (Xx_17:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz:a), ((X0 Xz)->(X0 (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_17))))))->(cX Xx_17)))) ((or (cF ((cP Xx) Xy))) (cG ((cP Xx) Xy))))))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xy:a)=> ((ex a) (fun (Xx:a)=> ((and (forall (Xx_17:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz:a), ((X0 Xz)->(X0 (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_17))))))->(cX Xx_17)))) ((or (cF ((cP Xx) Xy))) (cG ((cP Xx) Xy)))))))):(((eq (a->Prop)) (fun (Xy:a)=> ((ex a) (fun (Xx:a)=> ((and (forall (Xx_17:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz:a), ((X0 Xz)->(X0 (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_17))))))->(cX Xx_17)))) ((or (cF ((cP Xx) Xy))) (cG ((cP Xx) Xy)))))))) (fun (x:a)=> ((ex a) (fun (Xx:a)=> ((and (forall (Xx_17:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz:a), ((X0 Xz)->(X0 (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_17))))))->(cX Xx_17)))) ((or (cF ((cP Xx) x))) (cG ((cP Xx) x))))))))
% Found (eta_expansion_dep00 (fun (Xy:a)=> ((ex a) (fun (Xx:a)=> ((and (forall (Xx_17:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz:a), ((X0 Xz)->(X0 (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_17))))))->(cX Xx_17)))) ((or (cF ((cP Xx) Xy))) (cG ((cP Xx) Xy)))))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> ((ex a) (fun (Xx:a)=> ((and (forall (Xx_17:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz:a), ((X0 Xz)->(X0 (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_17))))))->(cX Xx_17)))) ((or (cF ((cP Xx) Xy))) (cG ((cP Xx) Xy)))))))) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xy:a)=> ((ex a) (fun (Xx:a)=> ((and (forall (Xx_17:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz:a), ((X0 Xz)->(X0 (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_17))))))->(cX Xx_17)))) ((or (cF ((cP Xx) Xy))) (cG ((cP Xx) Xy)))))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> ((ex a) (fun (Xx:a)=> ((and (forall (Xx_17:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz:a), ((X0 Xz)->(X0 (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_17))))))->(cX Xx_17)))) ((or (cF ((cP Xx) Xy))) (cG ((cP Xx) Xy)))))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xy:a)=> ((ex a) (fun (Xx:a)=> ((and (forall (Xx_17:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz:a), ((X0 Xz)->(X0 (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_17))))))->(cX Xx_17)))) ((or (cF ((cP Xx) Xy))) (cG ((cP Xx) Xy)))))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> ((ex a) (fun (Xx:a)=> ((and (forall (Xx_17:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz:a), ((X0 Xz)->(X0 (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_17))))))->(cX Xx_17)))) ((or (cF ((cP Xx) Xy))) (cG ((cP Xx) Xy)))))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xy:a)=> ((ex a) (fun (Xx:a)=> ((and (forall (Xx_17:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz:a), ((X0 Xz)->(X0 (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_17))))))->(cX Xx_17)))) ((or (cF ((cP Xx) Xy))) (cG ((cP Xx) Xy)))))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> ((ex a) (fun (Xx:a)=> ((and (forall (Xx_17:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz:a), ((X0 Xz)->(X0 (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_17))))))->(cX Xx_17)))) ((or (cF ((cP Xx) Xy))) (cG ((cP Xx) Xy)))))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xy:a)=> ((ex a) (fun (Xx:a)=> ((and (forall (Xx_17:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz:a), ((X0 Xz)->(X0 (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_17))))))->(cX Xx_17)))) ((or (cF ((cP Xx) Xy))) (cG ((cP Xx) Xy)))))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> ((ex a) (fun (Xx:a)=> ((and (forall (Xx_17:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz:a), ((X0 Xz)->(X0 (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_17))))))->(cX Xx_17)))) ((or (cF ((cP Xx) Xy))) (cG ((cP Xx) Xy)))))))) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((ex a) (fun (Xx:a)=> ((and (forall (Xx_18:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_18))))))->(cX Xx_18)))) (cF ((cP Xx) Xz)))))) ((ex a) (fun (Xx:a)=> ((and (forall (Xx_19:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_19))))))->(cX Xx_19)))) (cG ((cP Xx) Xz))))))))
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((ex a) (fun (Xx:a)=> ((and (forall (Xx_18:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_18))))))->(cX Xx_18)))) (cF ((cP Xx) Xz)))))) ((ex a) (fun (Xx:a)=> ((and (forall (Xx_19:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_19))))))->(cX Xx_19)))) (cG ((cP Xx) Xz))))))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((ex a) (fun (Xx:a)=> ((and (forall (Xx_18:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_18))))))->(cX Xx_18)))) (cF ((cP Xx) Xz)))))) ((ex a) (fun (Xx:a)=> ((and (forall (Xx_19:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_19))))))->(cX Xx_19)))) (cG ((cP Xx) Xz))))))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((ex a) (fun (Xx:a)=> ((and (forall (Xx_18:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_18))))))->(cX Xx_18)))) (cF ((cP Xx) Xz)))))) ((ex a) (fun (Xx:a)=> ((and (forall (Xx_19:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_19))))))->(cX Xx_19)))) (cG ((cP Xx) Xz))))))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((ex a) (fun (Xx:a)=> ((and (forall (Xx_18:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_18))))))->(cX Xx_18)))) (cF ((cP Xx) Xz)))))) ((ex a) (fun (Xx:a)=> ((and (forall (Xx_19:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_19))))))->(cX Xx_19)))) (cG ((cP Xx) Xz))))))))
% Found cZ:a
% Found cZ as proof of a
% Found cZ:a
% Found cZ as proof of a
% Found eq_ref000:=(eq_ref00 P):((P (fun (Xy:a)=> ((ex a) (fun (Xx:a)=> ((and (forall (Xx_17:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz:a), ((X0 Xz)->(X0 (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_17))))))->(cX Xx_17)))) ((or (cF ((cP Xx) Xy))) (cG ((cP Xx) Xy))))))))->(P (fun (Xy:a)=> ((ex a) (fun (Xx:a)=> ((and (forall (Xx_17:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz:a), ((X0 Xz)->(X0 (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_17))))))->(cX Xx_17)))) ((or (cF ((cP Xx) Xy))) (cG ((cP Xx) Xy)))))))))
% Found (eq_ref00 P) as proof of (P0 (fun (Xy:a)=> ((ex a) (fun (Xx:a)=> ((and (forall (Xx_17:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz:a), ((X0 Xz)->(X0 (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_17))))))->(cX Xx_17)))) ((or (cF ((cP Xx) Xy))) (cG ((cP Xx) Xy))))))))
% Found ((eq_ref0 (fun (Xy:a)=> ((ex a) (fun (Xx:a)=> ((and (forall (Xx_17:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz:a), ((X0 Xz)->(X0 (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_17))))))->(cX Xx_17)))) ((or (cF ((cP Xx) Xy))) (cG ((cP Xx) Xy)))))))) P) as proof of (P0 (fun (Xy:a)=> ((ex a) (fun (Xx:a)=> ((and (forall (Xx_17:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz:a), ((X0 Xz)->(X0 (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_17))))))->(cX Xx_17)))) ((or (cF ((cP Xx) Xy))) (cG ((cP Xx) Xy))))))))
% Found (((eq_ref (a->Prop)) (fun (Xy:a)=> ((ex a) (fun (Xx:a)=> ((and (forall (Xx_17:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz:a), ((X0 Xz)->(X0 (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_17))))))->(cX Xx_17)))) ((or (cF ((cP Xx) Xy))) (cG ((cP Xx) Xy)))))))) P) as proof of (P0 (fun (Xy:a)=> ((ex a) (fun (Xx:a)=> ((and (forall (Xx_17:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz:a), ((X0 Xz)->(X0 (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_17))))))->(cX Xx_17)))) ((or (cF ((cP Xx) Xy))) (cG ((cP Xx) Xy))))))))
% Found (((eq_ref (a->Prop)) (fun (Xy:a)=> ((ex a) (fun (Xx:a)=> ((and (forall (Xx_17:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz:a), ((X0 Xz)->(X0 (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_17))))))->(cX Xx_17)))) ((or (cF ((cP Xx) Xy))) (cG ((cP Xx) Xy)))))))) P) as proof of (P0 (fun (Xy:a)=> ((ex a) (fun (Xx:a)=> ((and (forall (Xx_17:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz:a), ((X0 Xz)->(X0 (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_17))))))->(cX Xx_17)))) ((or (cF ((cP Xx) Xy))) (cG ((cP Xx) Xy))))))))
% Found eq_ref00:=(eq_ref0 (fun (Xy:a)=> ((ex a) (fun (Xx:a)=> ((and (forall (Xx_17:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz:a), ((X0 Xz)->(X0 (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_17))))))->(cX Xx_17)))) ((or (cF ((cP Xx) Xy))) (cG ((cP Xx) Xy)))))))):(((eq (a->Prop)) (fun (Xy:a)=> ((ex a) (fun (Xx:a)=> ((and (forall (Xx_17:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz:a), ((X0 Xz)->(X0 (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_17))))))->(cX Xx_17)))) ((or (cF ((cP Xx) Xy))) (cG ((cP Xx) Xy)))))))) (fun (Xy:a)=> ((ex a) (fun (Xx:a)=> ((and (forall (Xx_17:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz:a), ((X0 Xz)->(X0 (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_17))))))->(cX Xx_17)))) ((or (cF ((cP Xx) Xy))) (cG ((cP Xx) Xy))))))))
% Found (eq_ref0 (fun (Xy:a)=> ((ex a) (fun (Xx:a)=> ((and (forall (Xx_17:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz:a), ((X0 Xz)->(X0 (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_17))))))->(cX Xx_17)))) ((or (cF ((cP Xx) Xy))) (cG ((cP Xx) Xy)))))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> ((ex a) (fun (Xx:a)=> ((and (forall (Xx_17:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz:a), ((X0 Xz)->(X0 (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_17))))))->(cX Xx_17)))) ((or (cF ((cP Xx) Xy))) (cG ((cP Xx) Xy)))))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xy:a)=> ((ex a) (fun (Xx:a)=> ((and (forall (Xx_17:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz:a), ((X0 Xz)->(X0 (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_17))))))->(cX Xx_17)))) ((or (cF ((cP Xx) Xy))) (cG ((cP Xx) Xy)))))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> ((ex a) (fun (Xx:a)=> ((and (forall (Xx_17:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz:a), ((X0 Xz)->(X0 (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_17))))))->(cX Xx_17)))) ((or (cF ((cP Xx) Xy))) (cG ((cP Xx) Xy)))))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xy:a)=> ((ex a) (fun (Xx:a)=> ((and (forall (Xx_17:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz:a), ((X0 Xz)->(X0 (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_17))))))->(cX Xx_17)))) ((or (cF ((cP Xx) Xy))) (cG ((cP Xx) Xy)))))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> ((ex a) (fun (Xx:a)=> ((and (forall (Xx_17:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz:a), ((X0 Xz)->(X0 (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_17))))))->(cX Xx_17)))) ((or (cF ((cP Xx) Xy))) (cG ((cP Xx) Xy)))))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xy:a)=> ((ex a) (fun (Xx:a)=> ((and (forall (Xx_17:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz:a), ((X0 Xz)->(X0 (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_17))))))->(cX Xx_17)))) ((or (cF ((cP Xx) Xy))) (cG ((cP Xx) Xy)))))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> ((ex a) (fun (Xx:a)=> ((and (forall (Xx_17:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz:a), ((X0 Xz)->(X0 (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_17))))))->(cX Xx_17)))) ((or (cF ((cP Xx) Xy))) (cG ((cP Xx) Xy)))))))) b)
% Found eta_expansion000:=(eta_expansion00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((ex a) (fun (Xx:a)=> ((and (forall (Xx_18:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_18))))))->(cX Xx_18)))) (cF ((cP Xx) Xz)))))) ((ex a) (fun (Xx:a)=> ((and (forall (Xx_19:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_19))))))->(cX Xx_19)))) (cG ((cP Xx) Xz))))))))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((ex a) (fun (Xx:a)=> ((and (forall (Xx_18:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_18))))))->(cX Xx_18)))) (cF ((cP Xx) Xz)))))) ((ex a) (fun (Xx:a)=> ((and (forall (Xx_19:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_19))))))->(cX Xx_19)))) (cG ((cP Xx) Xz))))))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((ex a) (fun (Xx:a)=> ((and (forall (Xx_18:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_18))))))->(cX Xx_18)))) (cF ((cP Xx) Xz)))))) ((ex a) (fun (Xx:a)=> ((and (forall (Xx_19:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_19))))))->(cX Xx_19)))) (cG ((cP Xx) Xz))))))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((ex a) (fun (Xx:a)=> ((and (forall (Xx_18:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_18))))))->(cX Xx_18)))) (cF ((cP Xx) Xz)))))) ((ex a) (fun (Xx:a)=> ((and (forall (Xx_19:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_19))))))->(cX Xx_19)))) (cG ((cP Xx) Xz))))))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((ex a) (fun (Xx:a)=> ((and (forall (Xx_18:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_18))))))->(cX Xx_18)))) (cF ((cP Xx) Xz)))))) ((ex a) (fun (Xx:a)=> ((and (forall (Xx_19:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_19))))))->(cX Xx_19)))) (cG ((cP Xx) Xz))))))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P (fun (Xy:a)=> ((ex a) (fun (Xx:a)=> ((and (forall (Xx_17:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz:a), ((X0 Xz)->(X0 (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_17))))))->(cX Xx_17)))) ((or (cF ((cP Xx) Xy))) (cG ((cP Xx) Xy))))))))->(P (fun (x:a)=> ((ex a) (fun (Xx:a)=> ((and (forall (Xx_17:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz:a), ((X0 Xz)->(X0 (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_17))))))->(cX Xx_17)))) ((or (cF ((cP Xx) x))) (cG ((cP Xx) x)))))))))
% Found (eta_expansion_dep000 P) as proof of (P0 (fun (Xy:a)=> ((ex a) (fun (Xx:a)=> ((and (forall (Xx_17:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz:a), ((X0 Xz)->(X0 (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_17))))))->(cX Xx_17)))) ((or (cF ((cP Xx) Xy))) (cG ((cP Xx) Xy))))))))
% Found ((eta_expansion_dep00 (fun (Xy:a)=> ((ex a) (fun (Xx:a)=> ((and (forall (Xx_17:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz:a), ((X0 Xz)->(X0 (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_17))))))->(cX Xx_17)))) ((or (cF ((cP Xx) Xy))) (cG ((cP Xx) Xy)))))))) P) as proof of (P0 (fun (Xy:a)=> ((ex a) (fun (Xx:a)=> ((and (forall (Xx_17:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz:a), ((X0 Xz)->(X0 (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_17))))))->(cX Xx_17)))) ((or (cF ((cP Xx) Xy))) (cG ((cP Xx) Xy))))))))
% Found (((eta_expansion_dep0 (fun (x3:a)=> Prop)) (fun (Xy:a)=> ((ex a) (fun (Xx:a)=> ((and (forall (Xx_17:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz:a), ((X0 Xz)->(X0 (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_17))))))->(cX Xx_17)))) ((or (cF ((cP Xx) Xy))) (cG ((cP Xx) Xy)))))))) P) as proof of (P0 (fun (Xy:a)=> ((ex a) (fun (Xx:a)=> ((and (forall (Xx_17:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz:a), ((X0 Xz)->(X0 (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_17))))))->(cX Xx_17)))) ((or (cF ((cP Xx) Xy))) (cG ((cP Xx) Xy))))))))
% Found ((((eta_expansion_dep a) (fun (x3:a)=> Prop)) (fun (Xy:a)=> ((ex a) (fun (Xx:a)=> ((and (forall (Xx_17:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz:a), ((X0 Xz)->(X0 (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_17))))))->(cX Xx_17)))) ((or (cF ((cP Xx) Xy))) (cG ((cP Xx) Xy)))))))) P) as proof of (P0 (fun (Xy:a)=> ((ex a) (fun (Xx:a)=> ((and (forall (Xx_17:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz:a), ((X0 Xz)->(X0 (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_17))))))->(cX Xx_17)))) ((or (cF ((cP Xx) Xy))) (cG ((cP Xx) Xy))))))))
% Found ((((eta_expansion_dep a) (fun (x3:a)=> Prop)) (fun (Xy:a)=> ((ex a) (fun (Xx:a)=> ((and (forall (Xx_17:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz:a), ((X0 Xz)->(X0 (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_17))))))->(cX Xx_17)))) ((or (cF ((cP Xx) Xy))) (cG ((cP Xx) Xy)))))))) P) as proof of (P0 (fun (Xy:a)=> ((ex a) (fun (Xx:a)=> ((and (forall (Xx_17:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz:a), ((X0 Xz)->(X0 (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_17))))))->(cX Xx_17)))) ((or (cF ((cP Xx) Xy))) (cG ((cP Xx) Xy))))))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P (fun (Xy:a)=> ((ex a) (fun (Xx:a)=> ((and (forall (Xx_17:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz:a), ((X0 Xz)->(X0 (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_17))))))->(cX Xx_17)))) ((or (cF ((cP Xx) Xy))) (cG ((cP Xx) Xy))))))))->(P (fun (x:a)=> ((ex a) (fun (Xx:a)=> ((and (forall (Xx_17:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz:a), ((X0 Xz)->(X0 (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_17))))))->(cX Xx_17)))) ((or (cF ((cP Xx) x))) (cG ((cP Xx) x)))))))))
% Found (eta_expansion_dep000 P) as proof of (P0 (fun (Xy:a)=> ((ex a) (fun (Xx:a)=> ((and (forall (Xx_17:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz:a), ((X0 Xz)->(X0 (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_17))))))->(cX Xx_17)))) ((or (cF ((cP Xx) Xy))) (cG ((cP Xx) Xy))))))))
% Found ((eta_expansion_dep00 (fun (Xy:a)=> ((ex a) (fun (Xx:a)=> ((and (forall (Xx_17:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz:a), ((X0 Xz)->(X0 (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_17))))))->(cX Xx_17)))) ((or (cF ((cP Xx) Xy))) (cG ((cP Xx) Xy)))))))) P) as proof of (P0 (fun (Xy:a)=> ((ex a) (fun (Xx:a)=> ((and (forall (Xx_17:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz:a), ((X0 Xz)->(X0 (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_17))))))->(cX Xx_17)))) ((or (cF ((cP Xx) Xy))) (cG ((cP Xx) Xy))))))))
% Found (((eta_expansion_dep0 (fun (x3:a)=> Prop)) (fun (Xy:a)=> ((ex a) (fun (Xx:a)=> ((and (forall (Xx_17:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz:a), ((X0 Xz)->(X0 (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_17))))))->(cX Xx_17)))) ((or (cF ((cP Xx) Xy))) (cG ((cP Xx) Xy)))))))) P) as proof of (P0 (fun (Xy:a)=> ((ex a) (fun (Xx:a)=> ((and (forall (Xx_17:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz:a), ((X0 Xz)->(X0 (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_17))))))->(cX Xx_17)))) ((or (cF ((cP Xx) Xy))) (cG ((cP Xx) Xy))))))))
% Found ((((eta_expansion_dep a) (fun (x3:a)=> Prop)) (fun (Xy:a)=> ((ex a) (fun (Xx:a)=> ((and (forall (Xx_17:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz:a), ((X0 Xz)->(X0 (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_17))))))->(cX Xx_17)))) ((or (cF ((cP Xx) Xy))) (cG ((cP Xx) Xy)))))))) P) as proof of (P0 (fun (Xy:a)=> ((ex a) (fun (Xx:a)=> ((and (forall (Xx_17:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz:a), ((X0 Xz)->(X0 (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_17))))))->(cX Xx_17)))) ((or (cF ((cP Xx) Xy))) (cG ((cP Xx) Xy))))))))
% Found ((((eta_expansion_dep a) (fun (x3:a)=> Prop)) (fun (Xy:a)=> ((ex a) (fun (Xx:a)=> ((and (forall (Xx_17:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz:a), ((X0 Xz)->(X0 (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_17))))))->(cX Xx_17)))) ((or (cF ((cP Xx) Xy))) (cG ((cP Xx) Xy)))))))) P) as proof of (P0 (fun (Xy:a)=> ((ex a) (fun (Xx:a)=> ((and (forall (Xx_17:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz:a), ((X0 Xz)->(X0 (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_17))))))->(cX Xx_17)))) ((or (cF ((cP Xx) Xy))) (cG ((cP Xx) Xy))))))))
% Found cZ:a
% Found cZ as proof of a
% Found cZ:a
% Found cZ as proof of a
% Found cZ:a
% Found cZ as proof of a
% Found eta_expansion000:=(eta_expansion00 (fun (Xz:a)=> ((or ((ex a) (fun (Xx:a)=> ((and (forall (Xx_18:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_18))))))->(cX Xx_18)))) (cF ((cP Xx) Xz)))))) ((ex a) (fun (Xx:a)=> ((and (forall (Xx_19:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_19))))))->(cX Xx_19)))) (cG ((cP Xx) Xz)))))))):(((eq (a->Prop)) (fun (Xz:a)=> ((or ((ex a) (fun (Xx:a)=> ((and (forall (Xx_18:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_18))))))->(cX Xx_18)))) (cF ((cP Xx) Xz)))))) ((ex a) (fun (Xx:a)=> ((and (forall (Xx_19:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_19))))))->(cX Xx_19)))) (cG ((cP Xx) Xz)))))))) (fun (x:a)=> ((or ((ex a) (fun (Xx:a)=> ((and (forall (Xx_18:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_18))))))->(cX Xx_18)))) (cF ((cP Xx) x)))))) ((ex a) (fun (Xx:a)=> ((and (forall (Xx_19:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_19))))))->(cX Xx_19)))) (cG ((cP Xx) x))))))))
% Found (eta_expansion00 (fun (Xz:a)=> ((or ((ex a) (fun (Xx:a)=> ((and (forall (Xx_18:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_18))))))->(cX Xx_18)))) (cF ((cP Xx) Xz)))))) ((ex a) (fun (Xx:a)=> ((and (forall (Xx_19:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_19))))))->(cX Xx_19)))) (cG ((cP Xx) Xz)))))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((ex a) (fun (Xx:a)=> ((and (forall (Xx_18:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_18))))))->(cX Xx_18)))) (cF ((cP Xx) Xz)))))) ((ex a) (fun (Xx:a)=> ((and (forall (Xx_19:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_19))))))->(cX Xx_19)))) (cG ((cP Xx) Xz)))))))) b)
% Found ((eta_expansion0 Prop) (fun (Xz:a)=> ((or ((ex a) (fun (Xx:a)=> ((and (forall (Xx_18:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_18))))))->(cX Xx_18)))) (cF ((cP Xx) Xz)))))) ((ex a) (fun (Xx:a)=> ((and (forall (Xx_19:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_19))))))->(cX Xx_19)))) (cG ((cP Xx) Xz)))))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((ex a) (fun (Xx:a)=> ((and (forall (Xx_18:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_18))))))->(cX Xx_18)))) (cF ((cP Xx) Xz)))))) ((ex a) (fun (Xx:a)=> ((and (forall (Xx_19:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_19))))))->(cX Xx_19)))) (cG ((cP Xx) Xz)))))))) b)
% Found (((eta_expansion a) Prop) (fun (Xz:a)=> ((or ((ex a) (fun (Xx:a)=> ((and (forall (Xx_18:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_18))))))->(cX Xx_18)))) (cF ((cP Xx) Xz)))))) ((ex a) (fun (Xx:a)=> ((and (forall (Xx_19:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_19))))))->(cX Xx_19)))) (cG ((cP Xx) Xz)))))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((ex a) (fun (Xx:a)=> ((and (forall (Xx_18:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_18))))))->(cX Xx_18)))) (cF ((cP Xx) Xz)))))) ((ex a) (fun (Xx:a)=> ((and (forall (Xx_19:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_19))))))->(cX Xx_19)))) (cG ((cP Xx) Xz)))))))) b)
% Found (((eta_expansion a) Prop) (fun (Xz:a)=> ((or ((ex a) (fun (Xx:a)=> ((and (forall (Xx_18:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_18))))))->(cX Xx_18)))) (cF ((cP Xx) Xz)))))) ((ex a) (fun (Xx:a)=> ((and (forall (Xx_19:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_19))))))->(cX Xx_19)))) (cG ((cP Xx) Xz)))))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((ex a) (fun (Xx:a)=> ((and (forall (Xx_18:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_18))))))->(cX Xx_18)))) (cF ((cP Xx) Xz)))))) ((ex a) (fun (Xx:a)=> ((and (forall (Xx_19:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_19))))))->(cX Xx_19)))) (cG ((cP Xx) Xz)))))))) b)
% Found (((eta_expansion a) Prop) (fun (Xz:a)=> ((or ((ex a) (fun (Xx:a)=> ((and (forall (Xx_18:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_18))))))->(cX Xx_18)))) (cF ((cP Xx) Xz)))))) ((ex a) (fun (Xx:a)=> ((and (forall (Xx_19:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_19))))))->(cX Xx_19)))) (cG ((cP Xx) Xz)))))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((ex a) (fun (Xx:a)=> ((and (forall (Xx_18:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_18))))))->(cX Xx_18)))) (cF ((cP Xx) Xz)))))) ((ex a) (fun (Xx:a)=> ((and (forall (Xx_19:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_19))))))->(cX Xx_19)))) (cG ((cP Xx) Xz)))))))) b)
% Found eta_expansion000:=(eta_expansion00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> ((ex a) (fun (Xx:a)=> ((and (forall (Xx_17:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz:a), ((X0 Xz)->(X0 (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_17))))))->(cX Xx_17)))) ((or (cF ((cP Xx) Xy))) (cG ((cP Xx) Xy))))))))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> ((ex a) (fun (Xx:a)=> ((and (forall (Xx_17:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz:a), ((X0 Xz)->(X0 (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_17))))))->(cX Xx_17)))) ((or (cF ((cP Xx) Xy))) (cG ((cP Xx) Xy))))))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> ((ex a) (fun (Xx:a)=> ((and (forall (Xx_17:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz:a), ((X0 Xz)->(X0 (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_17))))))->(cX Xx_17)))) ((or (cF ((cP Xx) Xy))) (cG ((cP Xx) Xy))))))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> ((ex a) (fun (Xx:a)=> ((and (forall (Xx_17:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz:a), ((X0 Xz)->(X0 (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_17))))))->(cX Xx_17)))) ((or (cF ((cP Xx) Xy))) (cG ((cP Xx) Xy))))))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> ((ex a) (fun (Xx:a)=> ((and (forall (Xx_17:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz:a), ((X0 Xz)->(X0 (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_17))))))->(cX Xx_17)))) ((or (cF ((cP Xx) Xy))) (cG ((cP Xx) Xy))))))))
% Found eq_ref000:=(eq_ref00 P):((P (fun (Xz:a)=> ((or ((ex a) (fun (Xx:a)=> ((and (forall (Xx_18:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_18))))))->(cX Xx_18)))) (cF ((cP Xx) Xz)))))) ((ex a) (fun (Xx:a)=> ((and (forall (Xx_19:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_19))))))->(cX Xx_19)))) (cG ((cP Xx) Xz))))))))->(P (fun (Xz:a)=> ((or ((ex a) (fun (Xx:a)=> ((and (forall (Xx_18:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_18))))))->(cX Xx_18)))) (cF ((cP Xx) Xz)))))) ((ex a) (fun (Xx:a)=> ((and (forall (Xx_19:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_19))))))->(cX Xx_19)))) (cG ((cP Xx) Xz)))))))))
% Found (eq_ref00 P) as proof of (P0 (fun (Xz:a)=> ((or ((ex a) (fun (Xx:a)=> ((and (forall (Xx_18:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_18))))))->(cX Xx_18)))) (cF ((cP Xx) Xz)))))) ((ex a) (fun (Xx:a)=> ((and (forall (Xx_19:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_19))))))->(cX Xx_19)))) (cG ((cP Xx) Xz))))))))
% Found ((eq_ref0 (fun (Xz:a)=> ((or ((ex a) (fun (Xx:a)=> ((and (forall (Xx_18:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_18))))))->(cX Xx_18)))) (cF ((cP Xx) Xz)))))) ((ex a) (fun (Xx:a)=> ((and (forall (Xx_19:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_19))))))->(cX Xx_19)))) (cG ((cP Xx) Xz)))))))) P) as proof of (P0 (fun (Xz:a)=> ((or ((ex a) (fun (Xx:a)=> ((and (forall (Xx_18:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_18))))))->(cX Xx_18)))) (cF ((cP Xx) Xz)))))) ((ex a) (fun (Xx:a)=> ((and (forall (Xx_19:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_19))))))->(cX Xx_19)))) (cG ((cP Xx) Xz))))))))
% Found (((eq_ref (a->Prop)) (fun (Xz:a)=> ((or ((ex a) (fun (Xx:a)=> ((and (forall (Xx_18:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_18))))))->(cX Xx_18)))) (cF ((cP Xx) Xz)))))) ((ex a) (fun (Xx:a)=> ((and (forall (Xx_19:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_19))))))->(cX Xx_19)))) (cG ((cP Xx) Xz)))))))) P) as proof of (P0 (fun (Xz:a)=> ((or ((ex a) (fun (Xx:a)=> ((and (forall (Xx_18:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_18))))))->(cX Xx_18)))) (cF ((cP Xx) Xz)))))) ((ex a) (fun (Xx:a)=> ((and (forall (Xx_19:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_19))))))->(cX Xx_19)))) (cG ((cP Xx) Xz))))))))
% Found (((eq_ref (a->Prop)) (fun (Xz:a)=> ((or ((ex a) (fun (Xx:a)=> ((and (forall (Xx_18:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_18))))))->(cX Xx_18)))) (cF ((cP Xx) Xz)))))) ((ex a) (fun (Xx:a)=> ((and (forall (Xx_19:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_19))))))->(cX Xx_19)))) (cG ((cP Xx) Xz)))))))) P) as proof of (P0 (fun (Xz:a)=> ((or ((ex a) (fun (Xx:a)=> ((and (forall (Xx_18:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_18))))))->(cX Xx_18)))) (cF ((cP Xx) Xz)))))) ((ex a) (fun (Xx:a)=> ((and (forall (Xx_19:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_19))))))->(cX Xx_19)))) (cG ((cP Xx) Xz))))))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P (fun (Xz:a)=> ((or ((ex a) (fun (Xx:a)=> ((and (forall (Xx_18:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_18))))))->(cX Xx_18)))) (cF ((cP Xx) Xz)))))) ((ex a) (fun (Xx:a)=> ((and (forall (Xx_19:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_19))))))->(cX Xx_19)))) (cG ((cP Xx) Xz))))))))->(P (fun (x:a)=> ((or ((ex a) (fun (Xx:a)=> ((and (forall (Xx_18:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_18))))))->(cX Xx_18)))) (cF ((cP Xx) x)))))) ((ex a) (fun (Xx:a)=> ((and (forall (Xx_19:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_19))))))->(cX Xx_19)))) (cG ((cP Xx) x)))))))))
% Found (eta_expansion_dep000 P) as proof of (P0 (fun (Xz:a)=> ((or ((ex a) (fun (Xx:a)=> ((and (forall (Xx_18:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_18))))))->(cX Xx_18)))) (cF ((cP Xx) Xz)))))) ((ex a) (fun (Xx:a)=> ((and (forall (Xx_19:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_19))))))->(cX Xx_19)))) (cG ((cP Xx) Xz))))))))
% Found ((eta_expansion_dep00 (fun (Xz:a)=> ((or ((ex a) (fun (Xx:a)=> ((and (forall (Xx_18:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_18))))))->(cX Xx_18)))) (cF ((cP Xx) Xz)))))) ((ex a) (fun (Xx:a)=> ((and (forall (Xx_19:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_19))))))->(cX Xx_19)))) (cG ((cP Xx) Xz)))))))) P) as proof of (P0 (fun (Xz:a)=> ((or ((ex a) (fun (Xx:a)=> ((and (forall (Xx_18:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_18))))))->(cX Xx_18)))) (cF ((cP Xx) Xz)))))) ((ex a) (fun (Xx:a)=> ((and (forall (Xx_19:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_19))))))->(cX Xx_19)))) (cG ((cP Xx) Xz))))))))
% Found (((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((or ((ex a) (fun (Xx:a)=> ((and (forall (Xx_18:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_18))))))->(cX Xx_18)))) (cF ((cP Xx) Xz)))))) ((ex a) (fun (Xx:a)=> ((and (forall (Xx_19:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_19))))))->(cX Xx_19)))) (cG ((cP Xx) Xz)))))))) P) as proof of (P0 (fun (Xz:a)=> ((or ((ex a) (fun (Xx:a)=> ((and (forall (Xx_18:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_18))))))->(cX Xx_18)))) (cF ((cP Xx) Xz)))))) ((ex a) (fun (Xx:a)=> ((and (forall (Xx_19:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_19))))))->(cX Xx_19)))) (cG ((cP Xx) Xz))))))))
% Found ((((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((or ((ex a) (fun (Xx:a)=> ((and (forall (Xx_18:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_18))))))->(cX Xx_18)))) (cF ((cP Xx) Xz)))))) ((ex a) (fun (Xx:a)=> ((and (forall (Xx_19:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_19))))))->(cX Xx_19)))) (cG ((cP Xx) Xz)))))))) P) as proof of (P0 (fun (Xz:a)=> ((or ((ex a) (fun (Xx:a)=> ((and (forall (Xx_18:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_18))))))->(cX Xx_18)))) (cF ((cP Xx) Xz)))))) ((ex a) (fun (Xx:a)=> ((and (forall (Xx_19:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_19))))))->(cX Xx_19)))) (cG ((cP Xx) Xz))))))))
% Found ((((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((or ((ex a) (fun (Xx:a)=> ((and (forall (Xx_18:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_18))))))->(cX Xx_18)))) (cF ((cP Xx) Xz)))))) ((ex a) (fun (Xx:a)=> ((and (forall (Xx_19:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_19))))))->(cX Xx_19)))) (cG ((cP Xx) Xz)))))))) P) as proof of (P0 (fun (Xz:a)=> ((or ((ex a) (fun (Xx:a)=> ((and (forall (Xx_18:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_18))))))->(cX Xx_18)))) (cF ((cP Xx) Xz)))))) ((ex a) (fun (Xx:a)=> ((and (forall (Xx_19:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_19))))))->(cX Xx_19)))) (cG ((cP Xx) Xz))))))))
% Found cZ:a
% Found cZ as proof of a
% Found eq_ref000:=(eq_ref00 P):((P ((ex a) (fun (Xx:a)=> ((and (forall (Xx_17:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz:a), ((X0 Xz)->(X0 (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_17))))))->(cX Xx_17)))) ((or (cF ((cP Xx) x0))) (cG ((cP Xx) x0)))))))->(P ((ex a) (fun (Xx:a)=> ((and (forall (Xx_17:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz:a), ((X0 Xz)->(X0 (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_17))))))->(cX Xx_17)))) ((or (cF ((cP Xx) x0))) (cG ((cP Xx) x0))))))))
% Found (eq_ref00 P) as proof of (P0 ((ex a) (fun (Xx:a)=> ((and (forall (Xx_17:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz:a), ((X0 Xz)->(X0 (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_17))))))->(cX Xx_17)))) ((or (cF ((cP Xx) x0))) (cG ((cP Xx) x0)))))))
% Found ((eq_ref0 ((ex a) (fun (Xx:a)=> ((and (forall (Xx_17:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz:a), ((X0 Xz)->(X0 (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_17))))))->(cX Xx_17)))) ((or (cF ((cP Xx) x0))) (cG ((cP Xx) x0))))))) P) as proof of (P0 ((ex a) (fun (Xx:a)=> ((and (forall (Xx_17:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz:a), ((X0 Xz)->(X0 (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_17))))))->(cX Xx_17)))) ((or (cF ((cP Xx) x0))) (cG ((cP Xx) x0)))))))
% Found (((eq_ref Prop) ((ex a) (fun (Xx:a)=> ((and (forall (Xx_17:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz:a), ((X0 Xz)->(X0 (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_17))))))->(cX Xx_17)))) ((or (cF ((cP Xx) x0))) (cG ((cP Xx) x0))))))) P) as proof of (P0 ((ex a) (fun (Xx:a)=> ((and (forall (Xx_17:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz:a), ((X0 Xz)->(X0 (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_17))))))->(cX Xx_17)))) ((or (cF ((cP Xx) x0))) (cG ((cP Xx) x0)))))))
% Found (((eq_ref Prop) ((ex a) (fun (Xx:a)=> ((and (forall (Xx_17:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz:a), ((X0 Xz)->(X0 (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_17))))))->(cX Xx_17)))) ((or (cF ((cP Xx) x0))) (cG ((cP Xx) x0))))))) P) as proof of (P0 ((ex a) (fun (Xx:a)=> ((and (forall (Xx_17:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz:a), ((X0 Xz)->(X0 (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_17))))))->(cX Xx_17)))) ((or (cF ((cP Xx) x0))) (cG ((cP Xx) x0)))))))
% Found eq_ref000:=(eq_ref00 P):((P ((ex a) (fun (Xx:a)=> ((and (forall (Xx_17:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz:a), ((X0 Xz)->(X0 (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_17))))))->(cX Xx_17)))) ((or (cF ((cP Xx) x0))) (cG ((cP Xx) x0)))))))->(P ((ex a) (fun (Xx:a)=> ((and (forall (Xx_17:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz:a), ((X0 Xz)->(X0 (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_17))))))->(cX Xx_17)))) ((or (cF ((cP Xx) x0))) (cG ((cP Xx) x0))))))))
% Found (eq_ref00 P) as proof of (P0 ((ex a) (fun (Xx:a)=> ((and (forall (Xx_17:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz:a), ((X0 Xz)->(X0 (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_17))))))->(cX Xx_17)))) ((or (cF ((cP Xx) x0))) (cG ((cP Xx) x0)))))))
% Found ((eq_ref0 ((ex a) (fun (Xx:a)=> ((and (forall (Xx_17:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz:a), ((X0 Xz)->(X0 (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_17))))))->(cX Xx_17)))) ((or (cF ((cP Xx) x0))) (cG ((cP Xx) x0))))))) P) as proof of (P0 ((ex a) (fun (Xx:a)=> ((and (forall (Xx_17:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz:a), ((X0 Xz)->(X0 (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_17))))))->(cX Xx_17)))) ((or (cF ((cP Xx) x0))) (cG ((cP Xx) x0)))))))
% Found (((eq_ref Prop) ((ex a) (fun (Xx:a)=> ((and (forall (Xx_17:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz:a), ((X0 Xz)->(X0 (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_17))))))->(cX Xx_17)))) ((or (cF ((cP Xx) x0))) (cG ((cP Xx) x0))))))) P) as proof of (P0 ((ex a) (fun (Xx:a)=> ((and (forall (Xx_17:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz:a), ((X0 Xz)->(X0 (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_17))))))->(cX Xx_17)))) ((or (cF ((cP Xx) x0))) (cG ((cP Xx) x0)))))))
% Found (((eq_ref Prop) ((ex a) (fun (Xx:a)=> ((and (forall (Xx_17:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz:a), ((X0 Xz)->(X0 (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_17))))))->(cX Xx_17)))) ((or (cF ((cP Xx) x0))) (cG ((cP Xx) x0))))))) P) as proof of (P0 ((ex a) (fun (Xx:a)=> ((and (forall (Xx_17:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz:a), ((X0 Xz)->(X0 (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_17))))))->(cX Xx_17)))) ((or (cF ((cP Xx) x0))) (cG ((cP Xx) x0)))))))
% Found cZ:a
% Found cZ as proof of a
% Found cZ:a
% Found cZ as proof of a
% Found cZ:a
% Found cZ as proof of a
% Found cZ:a
% Found cZ as proof of a
% Found cZ:a
% Found cZ as proof of a
% Found cZ:a
% Found cZ as proof of a
% Found eq_ref00:=(eq_ref0 ((ex a) (fun (Xx:a)=> ((and (forall (Xx_17:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz:a), ((X0 Xz)->(X0 (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_17))))))->(cX Xx_17)))) ((or (cF ((cP Xx) x0))) (cG ((cP Xx) x0))))))):(((eq Prop) ((ex a) (fun (Xx:a)=> ((and (forall (Xx_17:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz:a), ((X0 Xz)->(X0 (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_17))))))->(cX Xx_17)))) ((or (cF ((cP Xx) x0))) (cG ((cP Xx) x0))))))) ((ex a) (fun (Xx:a)=> ((and (forall (Xx_17:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz:a), ((X0 Xz)->(X0 (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_17))))))->(cX Xx_17)))) ((or (cF ((cP Xx) x0))) (cG ((cP Xx) x0)))))))
% Found (eq_ref0 ((ex a) (fun (Xx:a)=> ((and (forall (Xx_17:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz:a), ((X0 Xz)->(X0 (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_17))))))->(cX Xx_17)))) ((or (cF ((cP Xx) x0))) (cG ((cP Xx) x0))))))) as proof of (((eq Prop) ((ex a) (fun (Xx:a)=> ((and (forall (Xx_17:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz:a), ((X0 Xz)->(X0 (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_17))))))->(cX Xx_17)))) ((or (cF ((cP Xx) x0))) (cG ((cP Xx) x0))))))) b)
% Found ((eq_ref Prop) ((ex a) (fun (Xx:a)=> ((and (forall (Xx_17:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz:a), ((X0 Xz)->(X0 (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_17))))))->(cX Xx_17)))) ((or (cF ((cP Xx) x0))) (cG ((cP Xx) x0))))))) as proof of (((eq Prop) ((ex a) (fun (Xx:a)=> ((and (forall (Xx_17:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz:a), ((X0 Xz)->(X0 (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_17))))))->(cX Xx_17)))) ((or (cF ((cP Xx) x0))) (cG ((cP Xx) x0))))))) b)
% Found ((eq_ref Prop) ((ex a) (fun (Xx:a)=> ((and (forall (Xx_17:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz:a), ((X0 Xz)->(X0 (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_17))))))->(cX Xx_17)))) ((or (cF ((cP Xx) x0))) (cG ((cP Xx) x0))))))) as proof of (((eq Prop) ((ex a) (fun (Xx:a)=> ((and (forall (Xx_17:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz:a), ((X0 Xz)->(X0 (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_17))))))->(cX Xx_17)))) ((or (cF ((cP Xx) x0))) (cG ((cP Xx) x0))))))) b)
% Found ((eq_ref Prop) ((ex a) (fun (Xx:a)=> ((and (forall (Xx_17:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz:a), ((X0 Xz)->(X0 (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_17))))))->(cX Xx_17)))) ((or (cF ((cP Xx) x0))) (cG ((cP Xx) x0))))))) as proof of (((eq Prop) ((ex a) (fun (Xx:a)=> ((and (forall (Xx_17:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz:a), ((X0 Xz)->(X0 (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_17))))))->(cX Xx_17)))) ((or (cF ((cP Xx) x0))) (cG ((cP Xx) x0))))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or ((ex a) (fun (Xx:a)=> ((and (forall (Xx_18:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_18))))))->(cX Xx_18)))) (cF ((cP Xx) x0)))))) ((ex a) (fun (Xx:a)=> ((and (forall (Xx_19:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_19))))))->(cX Xx_19)))) (cG ((cP Xx) x0)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((ex a) (fun (Xx:a)=> ((and (forall (Xx_18:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_18))))))->(cX Xx_18)))) (cF ((cP Xx) x0)))))) ((ex a) (fun (Xx:a)=> ((and (forall (Xx_19:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_19))))))->(cX Xx_19)))) (cG ((cP Xx) x0)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((ex a) (fun (Xx:a)=> ((and (forall (Xx_18:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_18))))))->(cX Xx_18)))) (cF ((cP Xx) x0)))))) ((ex a) (fun (Xx:a)=> ((and (forall (Xx_19:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_19))))))->(cX Xx_19)))) (cG ((cP Xx) x0)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((ex a) (fun (Xx:a)=> ((and (forall (Xx_18:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_18))))))->(cX Xx_18)))) (cF ((cP Xx) x0)))))) ((ex a) (fun (Xx:a)=> ((and (forall (Xx_19:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_19))))))->(cX Xx_19)))) (cG ((cP Xx) x0)))))))
% Found eq_ref00:=(eq_ref0 ((ex a) (fun (Xx:a)=> ((and (forall (Xx_17:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz:a), ((X0 Xz)->(X0 (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_17))))))->(cX Xx_17)))) ((or (cF ((cP Xx) x0))) (cG ((cP Xx) x0))))))):(((eq Prop) ((ex a) (fun (Xx:a)=> ((and (forall (Xx_17:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz:a), ((X0 Xz)->(X0 (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_17))))))->(cX Xx_17)))) ((or (cF ((cP Xx) x0))) (cG ((cP Xx) x0))))))) ((ex a) (fun (Xx:a)=> ((and (forall (Xx_17:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz:a), ((X0 Xz)->(X0 (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_17))))))->(cX Xx_17)))) ((or (cF ((cP Xx) x0))) (cG ((cP Xx) x0)))))))
% Found (eq_ref0 ((ex a) (fun (Xx:a)=> ((and (forall (Xx_17:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz:a), ((X0 Xz)->(X0 (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_17))))))->(cX Xx_17)))) ((or (cF ((cP Xx) x0))) (cG ((cP Xx) x0))))))) as proof of (((eq Prop) ((ex a) (fun (Xx:a)=> ((and (forall (Xx_17:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz:a), ((X0 Xz)->(X0 (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_17))))))->(cX Xx_17)))) ((or (cF ((cP Xx) x0))) (cG ((cP Xx) x0))))))) b)
% Found ((eq_ref Prop) ((ex a) (fun (Xx:a)=> ((and (forall (Xx_17:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz:a), ((X0 Xz)->(X0 (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_17))))))->(cX Xx_17)))) ((or (cF ((cP Xx) x0))) (cG ((cP Xx) x0))))))) as proof of (((eq Prop) ((ex a) (fun (Xx:a)=> ((and (forall (Xx_17:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz:a), ((X0 Xz)->(X0 (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_17))))))->(cX Xx_17)))) ((or (cF ((cP Xx) x0))) (cG ((cP Xx) x0))))))) b)
% Found ((eq_ref Prop) ((ex a) (fun (Xx:a)=> ((and (forall (Xx_17:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz:a), ((X0 Xz)->(X0 (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_17))))))->(cX Xx_17)))) ((or (cF ((cP Xx) x0))) (cG ((cP Xx) x0))))))) as proof of (((eq Prop) ((ex a) (fun (Xx:a)=> ((and (forall (Xx_17:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz:a), ((X0 Xz)->(X0 (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_17))))))->(cX Xx_17)))) ((or (cF ((cP Xx) x0))) (cG ((cP Xx) x0))))))) b)
% Found ((eq_ref Prop) ((ex a) (fun (Xx:a)=> ((and (forall (Xx_17:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz:a), ((X0 Xz)->(X0 (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_17))))))->(cX Xx_17)))) ((or (cF ((cP Xx) x0))) (cG ((cP Xx) x0))))))) as proof of (((eq Prop) ((ex a) (fun (Xx:a)=> ((and (forall (Xx_17:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz:a), ((X0 Xz)->(X0 (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_17))))))->(cX Xx_17)))) ((or (cF ((cP Xx) x0))) (cG ((cP Xx) x0))))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or ((ex a) (fun (Xx:a)=> ((and (forall (Xx_18:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_18))))))->(cX Xx_18)))) (cF ((cP Xx) x0)))))) ((ex a) (fun (Xx:a)=> ((and (forall (Xx_19:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_19))))))->(cX Xx_19)))) (cG ((cP Xx) x0)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((ex a) (fun (Xx:a)=> ((and (forall (Xx_18:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_18))))))->(cX Xx_18)))) (cF ((cP Xx) x0)))))) ((ex a) (fun (Xx:a)=> ((and (forall (Xx_19:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_19))))))->(cX Xx_19)))) (cG ((cP Xx) x0)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((ex a) (fun (Xx:a)=> ((and (forall (Xx_18:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_18))))))->(cX Xx_18)))) (cF ((cP Xx) x0)))))) ((ex a) (fun (Xx:a)=> ((and (forall (Xx_19:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_19))))))->(cX Xx_19)))) (cG ((cP Xx) x0)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((ex a) (fun (Xx:a)=> ((and (forall (Xx_18:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_18))))))->(cX Xx_18)))) (cF ((cP Xx) x0)))))) ((ex a) (fun (Xx:a)=> ((and (forall (Xx_19:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_19))))))->(cX Xx_19)))) (cG ((cP Xx) x0)))))))
% Found cZ:a
% Found cZ as proof of a
% Found cZ:a
% Found cZ as proof of a
% Found x2:a
% Found x2 as proof of a
% Found x2:a
% Found x2
% EOF
%------------------------------------------------------------------------------