%------------------------------------------------------------------------------
% File     : cocATP---0.2.0
% Problem  : ALG289^5 : TPTP v6.1.0. Released v4.0.0.
% Transfm  : none
% Format   : tptp:raw
% Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p

% Computer : n090.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 300.09s
% Output   : None 
% Verified : 
% SZS Type : None (Parsing solution fails)
% Syntax   : Number of formulae    : 0

% Comments : 
%------------------------------------------------------------------------------
%----NO SOLUTION OUTPUT BY SYSTEM
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% % Problem  : ALG289^5 : TPTP v6.1.0. Released v4.0.0.
% % Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n090.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 09:09:31 CDT 2014
% % CPUTime  : 300.09 
% 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 0x27b7fc8>, <kernel.Type object at 0x27b95f0>) of role type named a_type
% Using role type
% Declaring a:Type
% FOF formula (<kernel.Constant object at 0x27b78c0>, <kernel.DependentProduct object at 0x27b7f38>) of role type named cP
% Using role type
% Declaring cP:(a->(a->a))
% FOF formula (<kernel.Constant object at 0x27b7680>, <kernel.DependentProduct object at 0x2739cb0>) of role type named cR
% Using role type
% Declaring cR:(a->a)
% FOF formula (<kernel.Constant object at 0x27b9128>, <kernel.DependentProduct object at 0x273c200>) of role type named cL
% Using role type
% Declaring cL:(a->a)
% FOF formula (<kernel.Constant object at 0x27b90e0>, <kernel.Constant object at 0x27b78c0>) of role type named cZ
% Using role type
% Declaring cZ:a
% FOF formula (((and ((and ((and ((and ((and (((eq a) (cL cZ)) cZ)) (((eq a) (cR cZ)) cZ))) (forall (Xx:a) (Xy:a), (((eq a) (cL ((cP Xx) Xy))) Xx)))) (forall (Xx:a) (Xy:a), (((eq a) (cR ((cP Xx) Xy))) Xy)))) (forall (Xt:a), ((iff (not (((eq a) Xt) cZ))) (((eq a) Xt) ((cP (cL Xt)) (cR Xt))))))) (forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> ((and (X Xt)) (forall (Xu:a), ((X Xu)->(X (cL Xu)))))))->(X cZ))))->(forall (X:(a->Prop)) (Xz:a), ((iff ((ex a) (fun (Xz_0:a)=> (X ((cP Xz) Xz_0))))) ((ex a) (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1))))))))))))))) of role conjecture named cPU_SETL_CTS_pme
% Conjecture to prove = (((and ((and ((and ((and ((and (((eq a) (cL cZ)) cZ)) (((eq a) (cR cZ)) cZ))) (forall (Xx:a) (Xy:a), (((eq a) (cL ((cP Xx) Xy))) Xx)))) (forall (Xx:a) (Xy:a), (((eq a) (cR ((cP Xx) Xy))) Xy)))) (forall (Xt:a), ((iff (not (((eq a) Xt) cZ))) (((eq a) Xt) ((cP (cL Xt)) (cR Xt))))))) (forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> ((and (X Xt)) (forall (Xu:a), ((X Xu)->(X (cL Xu)))))))->(X cZ))))->(forall (X:(a->Prop)) (Xz:a), ((iff ((ex a) (fun (Xz_0:a)=> (X ((cP Xz) Xz_0))))) ((ex a) (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1))))))))))))))):Prop
% We need to prove ['(((and ((and ((and ((and ((and (((eq a) (cL cZ)) cZ)) (((eq a) (cR cZ)) cZ))) (forall (Xx:a) (Xy:a), (((eq a) (cL ((cP Xx) Xy))) Xx)))) (forall (Xx:a) (Xy:a), (((eq a) (cR ((cP Xx) Xy))) Xy)))) (forall (Xt:a), ((iff (not (((eq a) Xt) cZ))) (((eq a) Xt) ((cP (cL Xt)) (cR Xt))))))) (forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> ((and (X Xt)) (forall (Xu:a), ((X Xu)->(X (cL Xu)))))))->(X cZ))))->(forall (X:(a->Prop)) (Xz:a), ((iff ((ex a) (fun (Xz_0:a)=> (X ((cP Xz) Xz_0))))) ((ex a) (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1)))))))))))))))']
% Parameter a:Type.
% Parameter cP:(a->(a->a)).
% Parameter cR:(a->a).
% Parameter cL:(a->a).
% Parameter cZ:a.
% Trying to prove (((and ((and ((and ((and ((and (((eq a) (cL cZ)) cZ)) (((eq a) (cR cZ)) cZ))) (forall (Xx:a) (Xy:a), (((eq a) (cL ((cP Xx) Xy))) Xx)))) (forall (Xx:a) (Xy:a), (((eq a) (cR ((cP Xx) Xy))) Xy)))) (forall (Xt:a), ((iff (not (((eq a) Xt) cZ))) (((eq a) Xt) ((cP (cL Xt)) (cR Xt))))))) (forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> ((and (X Xt)) (forall (Xu:a), ((X Xu)->(X (cL Xu)))))))->(X cZ))))->(forall (X:(a->Prop)) (Xz:a), ((iff ((ex a) (fun (Xz_0:a)=> (X ((cP Xz) Xz_0))))) ((ex a) (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1)))))))))))))))
% Found cZ:a
% Found cZ as proof of a
% Found Xz:a
% Found Xz as proof of a
% Found cZ:a
% Found cZ as proof of a
% Found Xz:a
% Found Xz as proof of a
% Found cZ:a
% Found cZ as proof of a
% Found Xz:a
% Found Xz as proof of a
% Found Xz:a
% Found Xz as proof of a
% Found Xz:a
% Found Xz as proof of a
% Found eq_ref000:=(eq_ref00 (ex a)):(((ex a) (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1))))))))))))->((ex a) (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1)))))))))))))
% Found (eq_ref00 (ex a)) as proof of (P (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1))))))))))))
% Found ((eq_ref0 (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1)))))))))))) (ex a)) as proof of (P (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1))))))))))))
% Found (((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1)))))))))))) (ex a)) as proof of (P (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1))))))))))))
% Found (((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1)))))))))))) (ex a)) as proof of (P (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1))))))))))))
% Found eq_ref000:=(eq_ref00 (ex a)):(((ex a) (fun (Xz_0:a)=> (X ((cP Xz) Xz_0))))->((ex a) (fun (Xz_0:a)=> (X ((cP Xz) Xz_0)))))
% Found (eq_ref00 (ex a)) as proof of (P (fun (Xz_0:a)=> (X ((cP Xz) Xz_0))))
% Found ((eq_ref0 (fun (Xz_0:a)=> (X ((cP Xz) Xz_0)))) (ex a)) as proof of (P (fun (Xz_0:a)=> (X ((cP Xz) Xz_0))))
% Found (((eq_ref (a->Prop)) (fun (Xz_0:a)=> (X ((cP Xz) Xz_0)))) (ex a)) as proof of (P (fun (Xz_0:a)=> (X ((cP Xz) Xz_0))))
% Found (((eq_ref (a->Prop)) (fun (Xz_0:a)=> (X ((cP Xz) Xz_0)))) (ex a)) as proof of (P (fun (Xz_0:a)=> (X ((cP Xz) Xz_0))))
% Found Xz:a
% Found Xz as proof of a
% Found Xz:a
% Found Xz as proof of a
% Found Xz:a
% Found Xz as proof of a
% Found Xz:a
% Found Xz as proof of a
% Found Xz:a
% Found Xz as proof of a
% Found Xz:a
% Found Xz as proof of a
% Found eta_expansion0000:=(eta_expansion000 (ex a)):(((ex a) (fun (Xz_0:a)=> (X ((cP Xz) Xz_0))))->((ex a) (fun (x:a)=> (X ((cP Xz) x)))))
% Found (eta_expansion000 (ex a)) as proof of (P (fun (Xz_0:a)=> (X ((cP Xz) Xz_0))))
% Found ((eta_expansion00 (fun (Xz_0:a)=> (X ((cP Xz) Xz_0)))) (ex a)) as proof of (P (fun (Xz_0:a)=> (X ((cP Xz) Xz_0))))
% Found (((eta_expansion0 Prop) (fun (Xz_0:a)=> (X ((cP Xz) Xz_0)))) (ex a)) as proof of (P (fun (Xz_0:a)=> (X ((cP Xz) Xz_0))))
% Found ((((eta_expansion a) Prop) (fun (Xz_0:a)=> (X ((cP Xz) Xz_0)))) (ex a)) as proof of (P (fun (Xz_0:a)=> (X ((cP Xz) Xz_0))))
% Found ((((eta_expansion a) Prop) (fun (Xz_0:a)=> (X ((cP Xz) Xz_0)))) (ex a)) as proof of (P (fun (Xz_0:a)=> (X ((cP Xz) Xz_0))))
% Found eta_expansion0000:=(eta_expansion000 (ex a)):(((ex a) (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1))))))))))))->((ex a) (fun (x:a)=> ((and (forall (Xx_13:a), ((forall (X0:(a->Prop)), (((and (X0 x)) (forall (Xz0:a), ((X0 Xz0)->(X0 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1:a)=> (forall (X0:(a->Prop)), (((and (X0 x)) (forall (Xz0:a), ((X0 Xz0)->(X0 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) ((cP Xz) Xz_1)))))))))))))
% Found (eta_expansion000 (ex a)) as proof of (P (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1))))))))))))
% Found ((eta_expansion00 (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1)))))))))))) (ex a)) as proof of (P (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1))))))))))))
% Found (((eta_expansion0 Prop) (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1)))))))))))) (ex a)) as proof of (P (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1))))))))))))
% Found ((((eta_expansion a) Prop) (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1)))))))))))) (ex a)) as proof of (P (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1))))))))))))
% Found ((((eta_expansion a) Prop) (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1)))))))))))) (ex a)) as proof of (P (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1))))))))))))
% Found eta_expansion0000:=(eta_expansion000 (ex a)):(((ex a) (fun (Xz_0:a)=> (X ((cP Xz) Xz_0))))->((ex a) (fun (x:a)=> (X ((cP Xz) x)))))
% Found (eta_expansion000 (ex a)) as proof of (P (fun (Xz_0:a)=> (X ((cP Xz) Xz_0))))
% Found ((eta_expansion00 (fun (Xz_0:a)=> (X ((cP Xz) Xz_0)))) (ex a)) as proof of (P (fun (Xz_0:a)=> (X ((cP Xz) Xz_0))))
% Found (((eta_expansion0 Prop) (fun (Xz_0:a)=> (X ((cP Xz) Xz_0)))) (ex a)) as proof of (P (fun (Xz_0:a)=> (X ((cP Xz) Xz_0))))
% Found ((((eta_expansion a) Prop) (fun (Xz_0:a)=> (X ((cP Xz) Xz_0)))) (ex a)) as proof of (P (fun (Xz_0:a)=> (X ((cP Xz) Xz_0))))
% Found ((((eta_expansion a) Prop) (fun (Xz_0:a)=> (X ((cP Xz) Xz_0)))) (ex a)) as proof of (P (fun (Xz_0:a)=> (X ((cP Xz) Xz_0))))
% Found eta_expansion0000:=(eta_expansion000 (ex a)):(((ex a) (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1))))))))))))->((ex a) (fun (x:a)=> ((and (forall (Xx_13:a), ((forall (X0:(a->Prop)), (((and (X0 x)) (forall (Xz0:a), ((X0 Xz0)->(X0 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1:a)=> (forall (X0:(a->Prop)), (((and (X0 x)) (forall (Xz0:a), ((X0 Xz0)->(X0 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) ((cP Xz) Xz_1)))))))))))))
% Found (eta_expansion000 (ex a)) as proof of (P (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1))))))))))))
% Found ((eta_expansion00 (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1)))))))))))) (ex a)) as proof of (P (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1))))))))))))
% Found (((eta_expansion0 Prop) (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1)))))))))))) (ex a)) as proof of (P (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1))))))))))))
% Found ((((eta_expansion a) Prop) (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1)))))))))))) (ex a)) as proof of (P (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1))))))))))))
% Found ((((eta_expansion a) Prop) (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1)))))))))))) (ex a)) as proof of (P (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1))))))))))))
% Found cZ:a
% Found cZ as proof of a
% Found cZ:a
% Found cZ as proof of a
% Found Xz:a
% Found Xz as proof of a
% Found Xz:a
% Found Xz as proof of a
% Found eta_expansion0000:=(eta_expansion000 (ex a)):(((ex a) (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1))))))))))))->((ex a) (fun (x:a)=> ((and (forall (Xx_13:a), ((forall (X0:(a->Prop)), (((and (X0 x)) (forall (Xz0:a), ((X0 Xz0)->(X0 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1:a)=> (forall (X0:(a->Prop)), (((and (X0 x)) (forall (Xz0:a), ((X0 Xz0)->(X0 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) ((cP Xz) Xz_1)))))))))))))
% Found (eta_expansion000 (ex a)) as proof of (P (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1))))))))))))
% Found ((eta_expansion00 (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1)))))))))))) (ex a)) as proof of (P (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1))))))))))))
% Found (((eta_expansion0 Prop) (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1)))))))))))) (ex a)) as proof of (P (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1))))))))))))
% Found ((((eta_expansion a) Prop) (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1)))))))))))) (ex a)) as proof of (P (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1))))))))))))
% Found ((((eta_expansion a) Prop) (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1)))))))))))) (ex a)) as proof of (P (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1))))))))))))
% Found eta_expansion0000:=(eta_expansion000 (ex a)):(((ex a) (fun (Xz_0:a)=> (X ((cP Xz) Xz_0))))->((ex a) (fun (x:a)=> (X ((cP Xz) x)))))
% Found (eta_expansion000 (ex a)) as proof of (P (fun (Xz_0:a)=> (X ((cP Xz) Xz_0))))
% Found ((eta_expansion00 (fun (Xz_0:a)=> (X ((cP Xz) Xz_0)))) (ex a)) as proof of (P (fun (Xz_0:a)=> (X ((cP Xz) Xz_0))))
% Found (((eta_expansion0 Prop) (fun (Xz_0:a)=> (X ((cP Xz) Xz_0)))) (ex a)) as proof of (P (fun (Xz_0:a)=> (X ((cP Xz) Xz_0))))
% Found ((((eta_expansion a) Prop) (fun (Xz_0:a)=> (X ((cP Xz) Xz_0)))) (ex a)) as proof of (P (fun (Xz_0:a)=> (X ((cP Xz) Xz_0))))
% Found ((((eta_expansion a) Prop) (fun (Xz_0:a)=> (X ((cP Xz) Xz_0)))) (ex a)) as proof of (P (fun (Xz_0:a)=> (X ((cP Xz) Xz_0))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 (ex a)):(((ex a) (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1))))))))))))->((ex a) (fun (x:a)=> ((and (forall (Xx_13:a), ((forall (X0:(a->Prop)), (((and (X0 x)) (forall (Xz0:a), ((X0 Xz0)->(X0 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1:a)=> (forall (X0:(a->Prop)), (((and (X0 x)) (forall (Xz0:a), ((X0 Xz0)->(X0 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) ((cP Xz) Xz_1)))))))))))))
% Found (eta_expansion_dep000 (ex a)) as proof of (P (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1))))))))))))
% Found ((eta_expansion_dep00 (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1)))))))))))) (ex a)) as proof of (P (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1))))))))))))
% Found (((eta_expansion_dep0 (fun (x3:a)=> Prop)) (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1)))))))))))) (ex a)) as proof of (P (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1))))))))))))
% Found ((((eta_expansion_dep a) (fun (x3:a)=> Prop)) (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1)))))))))))) (ex a)) as proof of (P (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1))))))))))))
% Found ((((eta_expansion_dep a) (fun (x3:a)=> Prop)) (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1)))))))))))) (ex a)) as proof of (P (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1))))))))))))
% Found eq_ref000:=(eq_ref00 (ex a)):(((ex a) (fun (Xz_0:a)=> (X ((cP Xz) Xz_0))))->((ex a) (fun (Xz_0:a)=> (X ((cP Xz) Xz_0)))))
% Found (eq_ref00 (ex a)) as proof of (P (fun (Xz_0:a)=> (X ((cP Xz) Xz_0))))
% Found ((eq_ref0 (fun (Xz_0:a)=> (X ((cP Xz) Xz_0)))) (ex a)) as proof of (P (fun (Xz_0:a)=> (X ((cP Xz) Xz_0))))
% Found (((eq_ref (a->Prop)) (fun (Xz_0:a)=> (X ((cP Xz) Xz_0)))) (ex a)) as proof of (P (fun (Xz_0:a)=> (X ((cP Xz) Xz_0))))
% Found (((eq_ref (a->Prop)) (fun (Xz_0:a)=> (X ((cP Xz) Xz_0)))) (ex a)) as proof of (P (fun (Xz_0:a)=> (X ((cP Xz) Xz_0))))
% Found cZ:a
% Found cZ as proof of a
% Found Xz:a
% Found Xz as proof of a
% Found cZ:a
% Found cZ as proof of a
% Found Xz:a
% Found Xz as proof of a
% Found cZ:a
% Found cZ as proof of a
% Found Xz:a
% Found Xz as proof of a
% Found eq_ref000:=(eq_ref00 (ex a)):(((ex a) (fun (Xz_0:a)=> (X ((cP Xz) Xz_0))))->((ex a) (fun (Xz_0:a)=> (X ((cP Xz) Xz_0)))))
% Found (eq_ref00 (ex a)) as proof of (P (fun (Xz_0:a)=> (X ((cP Xz) Xz_0))))
% Found ((eq_ref0 (fun (Xz_0:a)=> (X ((cP Xz) Xz_0)))) (ex a)) as proof of (P (fun (Xz_0:a)=> (X ((cP Xz) Xz_0))))
% Found (((eq_ref (a->Prop)) (fun (Xz_0:a)=> (X ((cP Xz) Xz_0)))) (ex a)) as proof of (P (fun (Xz_0:a)=> (X ((cP Xz) Xz_0))))
% Found (((eq_ref (a->Prop)) (fun (Xz_0:a)=> (X ((cP Xz) Xz_0)))) (ex a)) as proof of (P (fun (Xz_0:a)=> (X ((cP Xz) Xz_0))))
% Found eta_expansion0000:=(eta_expansion000 (ex a)):(((ex a) (fun (Xz_0:a)=> (X ((cP Xz) Xz_0))))->((ex a) (fun (x:a)=> (X ((cP Xz) x)))))
% Found (eta_expansion000 (ex a)) as proof of (P (fun (Xz_0:a)=> (X ((cP Xz) Xz_0))))
% Found ((eta_expansion00 (fun (Xz_0:a)=> (X ((cP Xz) Xz_0)))) (ex a)) as proof of (P (fun (Xz_0:a)=> (X ((cP Xz) Xz_0))))
% Found (((eta_expansion0 Prop) (fun (Xz_0:a)=> (X ((cP Xz) Xz_0)))) (ex a)) as proof of (P (fun (Xz_0:a)=> (X ((cP Xz) Xz_0))))
% Found ((((eta_expansion a) Prop) (fun (Xz_0:a)=> (X ((cP Xz) Xz_0)))) (ex a)) as proof of (P (fun (Xz_0:a)=> (X ((cP Xz) Xz_0))))
% Found ((((eta_expansion a) Prop) (fun (Xz_0:a)=> (X ((cP Xz) Xz_0)))) (ex a)) as proof of (P (fun (Xz_0:a)=> (X ((cP Xz) Xz_0))))
% Found eta_expansion0000:=(eta_expansion000 (ex a)):(((ex a) (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1))))))))))))->((ex a) (fun (x:a)=> ((and (forall (Xx_13:a), ((forall (X0:(a->Prop)), (((and (X0 x)) (forall (Xz0:a), ((X0 Xz0)->(X0 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1:a)=> (forall (X0:(a->Prop)), (((and (X0 x)) (forall (Xz0:a), ((X0 Xz0)->(X0 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) ((cP Xz) Xz_1)))))))))))))
% Found (eta_expansion000 (ex a)) as proof of (P (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1))))))))))))
% Found ((eta_expansion00 (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1)))))))))))) (ex a)) as proof of (P (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1))))))))))))
% Found (((eta_expansion0 Prop) (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1)))))))))))) (ex a)) as proof of (P (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1))))))))))))
% Found ((((eta_expansion a) Prop) (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1)))))))))))) (ex a)) as proof of (P (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1))))))))))))
% Found ((((eta_expansion a) Prop) (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1)))))))))))) (ex a)) as proof of (P (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1))))))))))))
% Found eq_ref000:=(eq_ref00 (ex a)):(((ex a) (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1))))))))))))->((ex a) (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1)))))))))))))
% Found (eq_ref00 (ex a)) as proof of (P (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1))))))))))))
% Found ((eq_ref0 (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1)))))))))))) (ex a)) as proof of (P (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1))))))))))))
% Found (((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1)))))))))))) (ex a)) as proof of (P (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1))))))))))))
% Found (((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1)))))))))))) (ex a)) as proof of (P (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1))))))))))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 (ex a)):(((ex a) (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1))))))))))))->((ex a) (fun (x:a)=> ((and (forall (Xx_13:a), ((forall (X0:(a->Prop)), (((and (X0 x)) (forall (Xz0:a), ((X0 Xz0)->(X0 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1:a)=> (forall (X0:(a->Prop)), (((and (X0 x)) (forall (Xz0:a), ((X0 Xz0)->(X0 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) ((cP Xz) Xz_1)))))))))))))
% Found (eta_expansion_dep000 (ex a)) as proof of (P (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1))))))))))))
% Found ((eta_expansion_dep00 (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1)))))))))))) (ex a)) as proof of (P (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1))))))))))))
% Found (((eta_expansion_dep0 (fun (x3:a)=> Prop)) (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1)))))))))))) (ex a)) as proof of (P (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1))))))))))))
% Found ((((eta_expansion_dep a) (fun (x3:a)=> Prop)) (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1)))))))))))) (ex a)) as proof of (P (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1))))))))))))
% Found ((((eta_expansion_dep a) (fun (x3:a)=> Prop)) (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1)))))))))))) (ex a)) as proof of (P (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1))))))))))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 (ex a)):(((ex a) (fun (Xz_0:a)=> (X ((cP Xz) Xz_0))))->((ex a) (fun (x:a)=> (X ((cP Xz) x)))))
% Found (eta_expansion_dep000 (ex a)) as proof of (P (fun (Xz_0:a)=> (X ((cP Xz) Xz_0))))
% Found ((eta_expansion_dep00 (fun (Xz_0:a)=> (X ((cP Xz) Xz_0)))) (ex a)) as proof of (P (fun (Xz_0:a)=> (X ((cP Xz) Xz_0))))
% Found (((eta_expansion_dep0 (fun (x3:a)=> Prop)) (fun (Xz_0:a)=> (X ((cP Xz) Xz_0)))) (ex a)) as proof of (P (fun (Xz_0:a)=> (X ((cP Xz) Xz_0))))
% Found ((((eta_expansion_dep a) (fun (x3:a)=> Prop)) (fun (Xz_0:a)=> (X ((cP Xz) Xz_0)))) (ex a)) as proof of (P (fun (Xz_0:a)=> (X ((cP Xz) Xz_0))))
% Found ((((eta_expansion_dep a) (fun (x3:a)=> Prop)) (fun (Xz_0:a)=> (X ((cP Xz) Xz_0)))) (ex a)) as proof of (P (fun (Xz_0:a)=> (X ((cP Xz) Xz_0))))
% Found eq_ref000:=(eq_ref00 (ex a)):(((ex a) (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1))))))))))))->((ex a) (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1)))))))))))))
% Found (eq_ref00 (ex a)) as proof of (P (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1))))))))))))
% Found ((eq_ref0 (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1)))))))))))) (ex a)) as proof of (P (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1))))))))))))
% Found (((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1)))))))))))) (ex a)) as proof of (P (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1))))))))))))
% Found (((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1)))))))))))) (ex a)) as proof of (P (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1))))))))))))
% Found eq_ref000:=(eq_ref00 (ex a)):(((ex a) (fun (Xz_0:a)=> (X ((cP Xz) Xz_0))))->((ex a) (fun (Xz_0:a)=> (X ((cP Xz) Xz_0)))))
% Found (eq_ref00 (ex a)) as proof of (P (fun (Xz_0:a)=> (X ((cP Xz) Xz_0))))
% Found ((eq_ref0 (fun (Xz_0:a)=> (X ((cP Xz) Xz_0)))) (ex a)) as proof of (P (fun (Xz_0:a)=> (X ((cP Xz) Xz_0))))
% Found (((eq_ref (a->Prop)) (fun (Xz_0:a)=> (X ((cP Xz) Xz_0)))) (ex a)) as proof of (P (fun (Xz_0:a)=> (X ((cP Xz) Xz_0))))
% Found (((eq_ref (a->Prop)) (fun (Xz_0:a)=> (X ((cP Xz) Xz_0)))) (ex a)) as proof of (P (fun (Xz_0:a)=> (X ((cP Xz) Xz_0))))
% 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 Xz:a
% Found Xz as proof of a
% Found Xz:a
% Found Xz as proof of a
% Found Xz:a
% Found Xz as proof of a
% Found Xz:a
% Found Xz 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 Xt:=??:a
% Found Xt 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 Xz:a
% Found Xz as proof of a
% Found Xz:a
% Found Xz as proof of a
% Found Xt:=??:a
% Found Xt as proof of a
% Found cZ:a
% Found cZ as proof of a
% Found cZ:a
% Found cZ as proof of a
% Found Xz:a
% Found Xz as proof of a
% Found Xz:a
% Found Xz as proof of a
% Found Xt:=??:a
% Found Xt as proof of a
% Found eta_expansion_dep0000:=(eta_expansion_dep000 (ex a)):(((ex a) (fun (Xz_0:a)=> (X ((cP Xz) Xz_0))))->((ex a) (fun (x:a)=> (X ((cP Xz) x)))))
% Found (eta_expansion_dep000 (ex a)) as proof of (P (fun (Xz_0:a)=> (X ((cP Xz) Xz_0))))
% Found ((eta_expansion_dep00 (fun (Xz_0:a)=> (X ((cP Xz) Xz_0)))) (ex a)) as proof of (P (fun (Xz_0:a)=> (X ((cP Xz) Xz_0))))
% Found (((eta_expansion_dep0 (fun (x3:a)=> Prop)) (fun (Xz_0:a)=> (X ((cP Xz) Xz_0)))) (ex a)) as proof of (P (fun (Xz_0:a)=> (X ((cP Xz) Xz_0))))
% Found ((((eta_expansion_dep a) (fun (x3:a)=> Prop)) (fun (Xz_0:a)=> (X ((cP Xz) Xz_0)))) (ex a)) as proof of (P (fun (Xz_0:a)=> (X ((cP Xz) Xz_0))))
% Found ((((eta_expansion_dep a) (fun (x3:a)=> Prop)) (fun (Xz_0:a)=> (X ((cP Xz) Xz_0)))) (ex a)) as proof of (P (fun (Xz_0:a)=> (X ((cP Xz) Xz_0))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 (ex a)):(((ex a) (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1))))))))))))->((ex a) (fun (x:a)=> ((and (forall (Xx_13:a), ((forall (X0:(a->Prop)), (((and (X0 x)) (forall (Xz0:a), ((X0 Xz0)->(X0 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1:a)=> (forall (X0:(a->Prop)), (((and (X0 x)) (forall (Xz0:a), ((X0 Xz0)->(X0 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) ((cP Xz) Xz_1)))))))))))))
% Found (eta_expansion_dep000 (ex a)) as proof of (P (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1))))))))))))
% Found ((eta_expansion_dep00 (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1)))))))))))) (ex a)) as proof of (P (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1))))))))))))
% Found (((eta_expansion_dep0 (fun (x3:a)=> Prop)) (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1)))))))))))) (ex a)) as proof of (P (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1))))))))))))
% Found ((((eta_expansion_dep a) (fun (x3:a)=> Prop)) (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1)))))))))))) (ex a)) as proof of (P (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1))))))))))))
% Found ((((eta_expansion_dep a) (fun (x3:a)=> Prop)) (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1)))))))))))) (ex a)) as proof of (P (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1))))))))))))
% Found eq_ref000:=(eq_ref00 (ex a)):(((ex a) (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1))))))))))))->((ex a) (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1)))))))))))))
% Found (eq_ref00 (ex a)) as proof of (P (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1))))))))))))
% Found ((eq_ref0 (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1)))))))))))) (ex a)) as proof of (P (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1))))))))))))
% Found (((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1)))))))))))) (ex a)) as proof of (P (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1))))))))))))
% Found (((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1)))))))))))) (ex a)) as proof of (P (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1))))))))))))
% Found eta_expansion0000:=(eta_expansion000 (ex a)):(((ex a) (fun (Xz_0:a)=> (X ((cP Xz) Xz_0))))->((ex a) (fun (x:a)=> (X ((cP Xz) x)))))
% Found (eta_expansion000 (ex a)) as proof of (P (fun (Xz_0:a)=> (X ((cP Xz) Xz_0))))
% Found ((eta_expansion00 (fun (Xz_0:a)=> (X ((cP Xz) Xz_0)))) (ex a)) as proof of (P (fun (Xz_0:a)=> (X ((cP Xz) Xz_0))))
% Found (((eta_expansion0 Prop) (fun (Xz_0:a)=> (X ((cP Xz) Xz_0)))) (ex a)) as proof of (P (fun (Xz_0:a)=> (X ((cP Xz) Xz_0))))
% Found ((((eta_expansion a) Prop) (fun (Xz_0:a)=> (X ((cP Xz) Xz_0)))) (ex a)) as proof of (P (fun (Xz_0:a)=> (X ((cP Xz) Xz_0))))
% Found ((((eta_expansion a) Prop) (fun (Xz_0:a)=> (X ((cP Xz) Xz_0)))) (ex a)) as proof of (P (fun (Xz_0:a)=> (X ((cP Xz) Xz_0))))
% Found cZ:a
% Found cZ as proof of a
% Found cZ:a
% Found cZ as proof of a
% Found Xz:a
% Found Xz as proof of a
% Found Xz:a
% Found Xz as proof of a
% 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 (ex a)):(((ex a) (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1))))))))))))->((ex a) (fun (x:a)=> ((and (forall (Xx_13:a), ((forall (X0:(a->Prop)), (((and (X0 x)) (forall (Xz0:a), ((X0 Xz0)->(X0 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1:a)=> (forall (X0:(a->Prop)), (((and (X0 x)) (forall (Xz0:a), ((X0 Xz0)->(X0 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) ((cP Xz) Xz_1)))))))))))))
% Found (eta_expansion_dep000 (ex a)) as proof of (P (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1))))))))))))
% Found ((eta_expansion_dep00 (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1)))))))))))) (ex a)) as proof of (P (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1))))))))))))
% Found (((eta_expansion_dep0 (fun (x3:a)=> Prop)) (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1)))))))))))) (ex a)) as proof of (P (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1))))))))))))
% Found ((((eta_expansion_dep a) (fun (x3:a)=> Prop)) (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1)))))))))))) (ex a)) as proof of (P (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1))))))))))))
% Found ((((eta_expansion_dep a) (fun (x3:a)=> Prop)) (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1)))))))))))) (ex a)) as proof of (P (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1))))))))))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 (ex a)):(((ex a) (fun (Xz_0:a)=> (X ((cP Xz) Xz_0))))->((ex a) (fun (x:a)=> (X ((cP Xz) x)))))
% Found (eta_expansion_dep000 (ex a)) as proof of (P (fun (Xz_0:a)=> (X ((cP Xz) Xz_0))))
% Found ((eta_expansion_dep00 (fun (Xz_0:a)=> (X ((cP Xz) Xz_0)))) (ex a)) as proof of (P (fun (Xz_0:a)=> (X ((cP Xz) Xz_0))))
% Found (((eta_expansion_dep0 (fun (x3:a)=> Prop)) (fun (Xz_0:a)=> (X ((cP Xz) Xz_0)))) (ex a)) as proof of (P (fun (Xz_0:a)=> (X ((cP Xz) Xz_0))))
% Found ((((eta_expansion_dep a) (fun (x3:a)=> Prop)) (fun (Xz_0:a)=> (X ((cP Xz) Xz_0)))) (ex a)) as proof of (P (fun (Xz_0:a)=> (X ((cP Xz) Xz_0))))
% Found ((((eta_expansion_dep a) (fun (x3:a)=> Prop)) (fun (Xz_0:a)=> (X ((cP Xz) Xz_0)))) (ex a)) as proof of (P (fun (Xz_0:a)=> (X ((cP Xz) Xz_0))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 (ex a)):(((ex a) (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1))))))))))))->((ex a) (fun (x:a)=> ((and (forall (Xx_13:a), ((forall (X0:(a->Prop)), (((and (X0 x)) (forall (Xz0:a), ((X0 Xz0)->(X0 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1:a)=> (forall (X0:(a->Prop)), (((and (X0 x)) (forall (Xz0:a), ((X0 Xz0)->(X0 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) ((cP Xz) Xz_1)))))))))))))
% Found (eta_expansion_dep000 (ex a)) as proof of (P (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1))))))))))))
% Found ((eta_expansion_dep00 (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1)))))))))))) (ex a)) as proof of (P (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1))))))))))))
% Found (((eta_expansion_dep0 (fun (x3:a)=> Prop)) (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1)))))))))))) (ex a)) as proof of (P (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1))))))))))))
% Found ((((eta_expansion_dep a) (fun (x3:a)=> Prop)) (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1)))))))))))) (ex a)) as proof of (P (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1))))))))))))
% Found ((((eta_expansion_dep a) (fun (x3:a)=> Prop)) (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1)))))))))))) (ex a)) as proof of (P (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1))))))))))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 (ex a)):(((ex a) (fun (Xz_0:a)=> (X ((cP Xz) Xz_0))))->((ex a) (fun (x:a)=> (X ((cP Xz) x)))))
% Found (eta_expansion_dep000 (ex a)) as proof of (P (fun (Xz_0:a)=> (X ((cP Xz) Xz_0))))
% Found ((eta_expansion_dep00 (fun (Xz_0:a)=> (X ((cP Xz) Xz_0)))) (ex a)) as proof of (P (fun (Xz_0:a)=> (X ((cP Xz) Xz_0))))
% Found (((eta_expansion_dep0 (fun (x3:a)=> Prop)) (fun (Xz_0:a)=> (X ((cP Xz) Xz_0)))) (ex a)) as proof of (P (fun (Xz_0:a)=> (X ((cP Xz) Xz_0))))
% Found ((((eta_expansion_dep a) (fun (x3:a)=> Prop)) (fun (Xz_0:a)=> (X ((cP Xz) Xz_0)))) (ex a)) as proof of (P (fun (Xz_0:a)=> (X ((cP Xz) Xz_0))))
% Found ((((eta_expansion_dep a) (fun (x3:a)=> Prop)) (fun (Xz_0:a)=> (X ((cP Xz) Xz_0)))) (ex a)) as proof of (P (fun (Xz_0:a)=> (X ((cP Xz) Xz_0))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 (ex a)):(((ex a) (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1))))))))))))->((ex a) (fun (x:a)=> ((and (forall (Xx_13:a), ((forall (X0:(a->Prop)), (((and (X0 x)) (forall (Xz0:a), ((X0 Xz0)->(X0 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1:a)=> (forall (X0:(a->Prop)), (((and (X0 x)) (forall (Xz0:a), ((X0 Xz0)->(X0 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) ((cP Xz) Xz_1)))))))))))))
% Found (eta_expansion_dep000 (ex a)) as proof of (P (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1))))))))))))
% Found ((eta_expansion_dep00 (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1)))))))))))) (ex a)) as proof of (P (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1))))))))))))
% Found (((eta_expansion_dep0 (fun (x3:a)=> Prop)) (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1)))))))))))) (ex a)) as proof of (P (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1))))))))))))
% Found ((((eta_expansion_dep a) (fun (x3:a)=> Prop)) (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1)))))))))))) (ex a)) as proof of (P (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1))))))))))))
% Found ((((eta_expansion_dep a) (fun (x3:a)=> Prop)) (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1)))))))))))) (ex a)) as proof of (P (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1))))))))))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 (ex a)):(((ex a) (fun (Xz_0:a)=> (X ((cP Xz) Xz_0))))->((ex a) (fun (x:a)=> (X ((cP Xz) x)))))
% Found (eta_expansion_dep000 (ex a)) as proof of (P (fun (Xz_0:a)=> (X ((cP Xz) Xz_0))))
% Found ((eta_expansion_dep00 (fun (Xz_0:a)=> (X ((cP Xz) Xz_0)))) (ex a)) as proof of (P (fun (Xz_0:a)=> (X ((cP Xz) Xz_0))))
% Found (((eta_expansion_dep0 (fun (x3:a)=> Prop)) (fun (Xz_0:a)=> (X ((cP Xz) Xz_0)))) (ex a)) as proof of (P (fun (Xz_0:a)=> (X ((cP Xz) Xz_0))))
% Found ((((eta_expansion_dep a) (fun (x3:a)=> Prop)) (fun (Xz_0:a)=> (X ((cP Xz) Xz_0)))) (ex a)) as proof of (P (fun (Xz_0:a)=> (X ((cP Xz) Xz_0))))
% Found ((((eta_expansion_dep a) (fun (x3:a)=> Prop)) (fun (Xz_0:a)=> (X ((cP Xz) Xz_0)))) (ex a)) as proof of (P (fun (Xz_0:a)=> (X ((cP Xz) Xz_0))))
% Found eta_expansion0000:=(eta_expansion000 (ex a)):(((ex a) (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1))))))))))))->((ex a) (fun (x:a)=> ((and (forall (Xx_13:a), ((forall (X0:(a->Prop)), (((and (X0 x)) (forall (Xz0:a), ((X0 Xz0)->(X0 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1:a)=> (forall (X0:(a->Prop)), (((and (X0 x)) (forall (Xz0:a), ((X0 Xz0)->(X0 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) ((cP Xz) Xz_1)))))))))))))
% Found (eta_expansion000 (ex a)) as proof of (P (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1))))))))))))
% Found ((eta_expansion00 (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1)))))))))))) (ex a)) as proof of (P (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1))))))))))))
% Found (((eta_expansion0 Prop) (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1)))))))))))) (ex a)) as proof of (P (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1))))))))))))
% Found ((((eta_expansion a) Prop) (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1)))))))))))) (ex a)) as proof of (P (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1))))))))))))
% Found ((((eta_expansion a) Prop) (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1)))))))))))) (ex a)) as proof of (P (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1))))))))))))
% Found eta_expansion0000:=(eta_expansion000 (ex a)):(((ex a) (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1))))))))))))->((ex a) (fun (x:a)=> ((and (forall (Xx_13:a), ((forall (X0:(a->Prop)), (((and (X0 x)) (forall (Xz0:a), ((X0 Xz0)->(X0 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1:a)=> (forall (X0:(a->Prop)), (((and (X0 x)) (forall (Xz0:a), ((X0 Xz0)->(X0 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) ((cP Xz) Xz_1)))))))))))))
% Found (eta_expansion000 (ex a)) as proof of (P (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1))))))))))))
% Found ((eta_expansion00 (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1)))))))))))) (ex a)) as proof of (P (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1))))))))))))
% Found (((eta_expansion0 Prop) (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1)))))))))))) (ex a)) as proof of (P (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1))))))))))))
% Found ((((eta_expansion a) Prop) (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1)))))))))))) (ex a)) as proof of (P (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1))))))))))))
% Found ((((eta_expansion a) Prop) (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1)))))))))))) (ex a)) as proof of (P (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1))))))))))))
% Found eta_expansion0000:=(eta_expansion000 (ex a)):(((ex a) (fun (Xz_0:a)=> (X ((cP Xz) Xz_0))))->((ex a) (fun (x:a)=> (X ((cP Xz) x)))))
% Found (eta_expansion000 (ex a)) as proof of (P (fun (Xz_0:a)=> (X ((cP Xz) Xz_0))))
% Found ((eta_expansion00 (fun (Xz_0:a)=> (X ((cP Xz) Xz_0)))) (ex a)) as proof of (P (fun (Xz_0:a)=> (X ((cP Xz) Xz_0))))
% Found (((eta_expansion0 Prop) (fun (Xz_0:a)=> (X ((cP Xz) Xz_0)))) (ex a)) as proof of (P (fun (Xz_0:a)=> (X ((cP Xz) Xz_0))))
% Found ((((eta_expansion a) Prop) (fun (Xz_0:a)=> (X ((cP Xz) Xz_0)))) (ex a)) as proof of (P (fun (Xz_0:a)=> (X ((cP Xz) Xz_0))))
% Found ((((eta_expansion a) Prop) (fun (Xz_0:a)=> (X ((cP Xz) Xz_0)))) (ex a)) as proof of (P (fun (Xz_0:a)=> (X ((cP Xz) Xz_0))))
% Found eta_expansion0000:=(eta_expansion000 (ex a)):(((ex a) (fun (Xz_0:a)=> (X ((cP Xz) Xz_0))))->((ex a) (fun (x:a)=> (X ((cP Xz) x)))))
% Found (eta_expansion000 (ex a)) as proof of (P (fun (Xz_0:a)=> (X ((cP Xz) Xz_0))))
% Found ((eta_expansion00 (fun (Xz_0:a)=> (X ((cP Xz) Xz_0)))) (ex a)) as proof of (P (fun (Xz_0:a)=> (X ((cP Xz) Xz_0))))
% Found (((eta_expansion0 Prop) (fun (Xz_0:a)=> (X ((cP Xz) Xz_0)))) (ex a)) as proof of (P (fun (Xz_0:a)=> (X ((cP Xz) Xz_0))))
% Found ((((eta_expansion a) Prop) (fun (Xz_0:a)=> (X ((cP Xz) Xz_0)))) (ex a)) as proof of (P (fun (Xz_0:a)=> (X ((cP Xz) Xz_0))))
% Found ((((eta_expansion a) Prop) (fun (Xz_0:a)=> (X ((cP Xz) Xz_0)))) (ex a)) as proof of (P (fun (Xz_0:a)=> (X ((cP Xz) Xz_0))))
% Found Xz:a
% Found Xz as proof of a
% Found Xz:a
% Found Xz as proof of a
% Found eta_expansion000:=(eta_expansion00 (fun (Xt:a)=> ((and ((iff ((ex a) (fun (Xz_0:a)=> (X ((cP Xz) Xz_0))))) ((ex a) (fun (Xx:a)=> ((and (forall (Xx_13:a), ((forall (X00:(a->Prop)), (((and (X00 Xx)) (forall (Xz0:a), ((X00 Xz0)->(X00 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X00 Xv)) (((eq a) (cR Xv)) Xx_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1:a)=> (forall (X00:(a->Prop)), (((and (X00 Xx)) (forall (Xz0:a), ((X00 Xz0)->(X00 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X00 Xv)) (((eq a) (cR Xv)) ((cP Xz) Xz_1)))))))))))))) (a->(((iff ((ex a) (fun (Xz_0:a)=> (X ((cP Xz) Xz_0))))) ((ex a) (fun (Xx:a)=> ((and (forall (Xx_13:a), ((forall (X00:(a->Prop)), (((and (X00 Xx)) (forall (Xz0:a), ((X00 Xz0)->(X00 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X00 Xv)) (((eq a) (cR Xv)) Xx_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1:a)=> (forall (X00:(a->Prop)), (((and (X00 Xx)) (forall (Xz0:a), ((X00 Xz0)->(X00 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X00 Xv)) (((eq a) (cR Xv)) ((cP Xz) Xz_1)))))))))))))->((iff ((ex a) (fun (Xz_0:a)=> (X ((cP Xz) Xz_0))))) ((ex a) (fun (Xx:a)=> ((and (forall (Xx_13:a), ((forall (X00:(a->Prop)), (((and (X00 Xx)) (forall (Xz0:a), ((X00 Xz0)->(X00 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X00 Xv)) (((eq a) (cR Xv)) Xx_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1:a)=> (forall (X00:(a->Prop)), (((and (X00 Xx)) (forall (Xz0:a), ((X00 Xz0)->(X00 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X00 Xv)) (((eq a) (cR Xv)) ((cP Xz) Xz_1)))))))))))))))))):(((eq (a->Prop)) (fun (Xt:a)=> ((and ((iff ((ex a) (fun (Xz_0:a)=> (X ((cP Xz) Xz_0))))) ((ex a) (fun (Xx:a)=> ((and (forall (Xx_13:a), ((forall (X00:(a->Prop)), (((and (X00 Xx)) (forall (Xz0:a), ((X00 Xz0)->(X00 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X00 Xv)) (((eq a) (cR Xv)) Xx_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1:a)=> (forall (X00:(a->Prop)), (((and (X00 Xx)) (forall (Xz0:a), ((X00 Xz0)->(X00 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X00 Xv)) (((eq a) (cR Xv)) ((cP Xz) Xz_1)))))))))))))) (a->(((iff ((ex a) (fun (Xz_0:a)=> (X ((cP Xz) Xz_0))))) ((ex a) (fun (Xx:a)=> ((and (forall (Xx_13:a), ((forall (X00:(a->Prop)), (((and (X00 Xx)) (forall (Xz0:a), ((X00 Xz0)->(X00 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X00 Xv)) (((eq a) (cR Xv)) Xx_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1:a)=> (forall (X00:(a->Prop)), (((and (X00 Xx)) (forall (Xz0:a), ((X00 Xz0)->(X00 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X00 Xv)) (((eq a) (cR Xv)) ((cP Xz) Xz_1)))))))))))))->((iff ((ex a) (fun (Xz_0:a)=> (X ((cP Xz) Xz_0))))) ((ex a) (fun (Xx:a)=> ((and (forall (Xx_13:a), ((forall (X00:(a->Prop)), (((and (X00 Xx)) (forall (Xz0:a), ((X00 Xz0)->(X00 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X00 Xv)) (((eq a) (cR Xv)) Xx_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1:a)=> (forall (X00:(a->Prop)), (((and (X00 Xx)) (forall (Xz0:a), ((X00 Xz0)->(X00 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X00 Xv)) (((eq a) (cR Xv)) ((cP Xz) Xz_1)))))))))))))))))) (fun (x:a)=> ((and ((iff ((ex a) (fun (Xz_0:a)=> (X ((cP Xz) Xz_0))))) ((ex a) (fun (Xx:a)=> ((and (forall (Xx_13:a), ((forall (X00:(a->Prop)), (((and (X00 Xx)) (forall (Xz0:a), ((X00 Xz0)->(X00 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X00 Xv)) (((eq a) (cR Xv)) Xx_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1:a)=> (forall (X00:(a->Prop)), (((and (X00 Xx)) (forall (Xz0:a), ((X00 Xz0)->(X00 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X00 Xv)) (((eq a) (cR Xv)) ((cP Xz) Xz_1)))))))))))))) (a->(((iff ((ex a) (fun (Xz_0:a)=> (X ((cP Xz) Xz_0))))) ((ex a) (fun (Xx:a)=> ((and (forall (Xx_13:a), ((forall (X00:(a->Prop)), (((and (X00 Xx)) (forall (Xz0:a), ((X00 Xz0)->(X00 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X00 Xv)) (((eq a) (cR Xv)) Xx_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1:a)=> (forall (X00:(a->Prop)), (((and (X00 Xx)) (forall (Xz0:a), ((X00 Xz0)->(X00 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X00 Xv)) (((eq a) (cR Xv)) ((cP Xz) Xz_1)))))))))))))->((iff ((ex a) (fun (Xz_0:a)=> (X ((cP Xz) Xz_0))))) ((ex a) (fun (Xx:a)=> ((and (forall (Xx_13:a), ((forall (X00:(a->Prop)), (((and (X00 Xx)) (forall (Xz0:a), ((X00 Xz0)->(X00 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X00 Xv)) (((eq a) (cR Xv)) Xx_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1:a)=> (forall (X00:(a->Prop)), (((and (X00 Xx)) (forall (Xz0:a), ((X00 Xz0)->(X00 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X00 Xv)) (((eq a) (cR Xv)) ((cP Xz) Xz_1))))))))))))))))))
% Found (eta_expansion00 (fun (Xt:a)=> ((and ((iff ((ex a) (fun (Xz_0:a)=> (X ((cP Xz) Xz_0))))) ((ex a) (fun (Xx:a)=> ((and (forall (Xx_13:a), ((forall (X00:(a->Prop)), (((and (X00 Xx)) (forall (Xz0:a), ((X00 Xz0)->(X00 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X00 Xv)) (((eq a) (cR Xv)) Xx_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1:a)=> (forall (X00:(a->Prop)), (((and (X00 Xx)) (forall (Xz0:a), ((X00 Xz0)->(X00 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X00 Xv)) (((eq a) (cR Xv)) ((cP Xz) Xz_1)))))))))))))) (a->(((iff ((ex a) (fun (Xz_0:a)=> (X ((cP Xz) Xz_0))))) ((ex a) (fun (Xx:a)=> ((and (forall (Xx_13:a), ((forall (X00:(a->Prop)), (((and (X00 Xx)) (forall (Xz0:a), ((X00 Xz0)->(X00 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X00 Xv)) (((eq a) (cR Xv)) Xx_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1:a)=> (forall (X00:(a->Prop)), (((and (X00 Xx)) (forall (Xz0:a), ((X00 Xz0)->(X00 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X00 Xv)) (((eq a) (cR Xv)) ((cP Xz) Xz_1)))))))))))))->((iff ((ex a) (fun (Xz_0:a)=> (X ((cP Xz) Xz_0))))) ((ex a) (fun (Xx:a)=> ((and (forall (Xx_13:a), ((forall (X00:(a->Prop)), (((and (X00 Xx)) (forall (Xz0:a), ((X00 Xz0)->(X00 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X00 Xv)) (((eq a) (cR Xv)) Xx_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1:a)=> (forall (X00:(a->Prop)), (((and (X00 Xx)) (forall (Xz0:a), ((X00 Xz0)->(X00 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X00 Xv)) (((eq a) (cR Xv)) ((cP Xz) Xz_1)))))))))))))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((and ((iff ((ex a) (fun (Xz_0:a)=> (X ((cP Xz) Xz_0))))) ((ex a) (fun (Xx:a)=> ((and (forall (Xx_13:a), ((forall (X00:(a->Prop)), (((and (X00 Xx)) (forall (Xz0:a), ((X00 Xz0)->(X00 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X00 Xv)) (((eq a) (cR Xv)) Xx_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1:a)=> (forall (X00:(a->Prop)), (((and (X00 Xx)) (forall (Xz0:a), ((X00 Xz0)->(X00 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X00 Xv)) (((eq a) (cR Xv)) ((cP Xz) Xz_1)))))))))))))) (a->(((iff ((ex a) (fun (Xz_0:a)=> (X ((cP Xz) Xz_0))))) ((ex a) (fun (Xx:a)=> ((and (forall (Xx_13:a), ((forall (X00:(a->Prop)), (((and (X00 Xx)) (forall (Xz0:a), ((X00 Xz0)->(X00 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X00 Xv)) (((eq a) (cR Xv)) Xx_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1:a)=> (forall (X00:(a->Prop)), (((and (X00 Xx)) (forall (Xz0:a), ((X00 Xz0)->(X00 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X00 Xv)) (((eq a) (cR Xv)) ((cP Xz) Xz_1)))))))))))))->((iff ((ex a) (fun (Xz_0:a)=> (X ((cP Xz) Xz_0))))) ((ex a) (fun (Xx:a)=> ((and (forall (Xx_13:a), ((forall (X00:(a->Prop)), (((and (X00 Xx)) (forall (Xz0:a), ((X00 Xz0)->(X00 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X00 Xv)) (((eq a) (cR Xv)) Xx_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1:a)=> (forall (X00:(a->Prop)), (((and (X00 Xx)) (forall (Xz0:a), ((X00 Xz0)->(X00 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X00 Xv)) (((eq a) (cR Xv)) ((cP Xz) Xz_1)))))))))))))))))) b)
% Found ((eta_expansion0 Prop) (fun (Xt:a)=> ((and ((iff ((ex a) (fun (Xz_0:a)=> (X ((cP Xz) Xz_0))))) ((ex a) (fun (Xx:a)=> ((and (forall (Xx_13:a), ((forall (X00:(a->Prop)), (((and (X00 Xx)) (forall (Xz0:a), ((X00 Xz0)->(X00 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X00 Xv)) (((eq a) (cR Xv)) Xx_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1:a)=> (forall (X00:(a->Prop)), (((and (X00 Xx)) (forall (Xz0:a), ((X00 Xz0)->(X00 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X00 Xv)) (((eq a) (cR Xv)) ((cP Xz) Xz_1)))))))))))))) (a->(((iff ((ex a) (fun (Xz_0:a)=> (X ((cP Xz) Xz_0))))) ((ex a) (fun (Xx:a)=> ((and (forall (Xx_13:a), ((forall (X00:(a->Prop)), (((and (X00 Xx)) (forall (Xz0:a), ((X00 Xz0)->(X00 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X00 Xv)) (((eq a) (cR Xv)) Xx_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1:a)=> (forall (X00:(a->Prop)), (((and (X00 Xx)) (forall (Xz0:a), ((X00 Xz0)->(X00 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X00 Xv)) (((eq a) (cR Xv)) ((cP Xz) Xz_1)))))))))))))->((iff ((ex a) (fun (Xz_0:a)=> (X ((cP Xz) Xz_0))))) ((ex a) (fun (Xx:a)=> ((and (forall (Xx_13:a), ((forall (X00:(a->Prop)), (((and (X00 Xx)) (forall (Xz0:a), ((X00 Xz0)->(X00 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X00 Xv)) (((eq a) (cR Xv)) Xx_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1:a)=> (forall (X00:(a->Prop)), (((and (X00 Xx)) (forall (Xz0:a), ((X00 Xz0)->(X00 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X00 Xv)) (((eq a) (cR Xv)) ((cP Xz) Xz_1)))))))))))))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((and ((iff ((ex a) (fun (Xz_0:a)=> (X ((cP Xz) Xz_0))))) ((ex a) (fun (Xx:a)=> ((and (forall (Xx_13:a), ((forall (X00:(a->Prop)), (((and (X00 Xx)) (forall (Xz0:a), ((X00 Xz0)->(X00 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X00 Xv)) (((eq a) (cR Xv)) Xx_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1:a)=> (forall (X00:(a->Prop)), (((and (X00 Xx)) (forall (Xz0:a), ((X00 Xz0)->(X00 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X00 Xv)) (((eq a) (cR Xv)) ((cP Xz) Xz_1)))))))))))))) (a->(((iff ((ex a) (fun (Xz_0:a)=> (X ((cP Xz) Xz_0))))) ((ex a) (fun (Xx:a)=> ((and (forall (Xx_13:a), ((forall (X00:(a->Prop)), (((and (X00 Xx)) (forall (Xz0:a), ((X00 Xz0)->(X00 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X00 Xv)) (((eq a) (cR Xv)) Xx_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1:a)=> (forall (X00:(a->Prop)), (((and (X00 Xx)) (forall (Xz0:a), ((X00 Xz0)->(X00 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X00 Xv)) (((eq a) (cR Xv)) ((cP Xz) Xz_1)))))))))))))->((iff ((ex a) (fun (Xz_0:a)=> (X ((cP Xz) Xz_0))))) ((ex a) (fun (Xx:a)=> ((and (forall (Xx_13:a), ((forall (X00:(a->Prop)), (((and (X00 Xx)) (forall (Xz0:a), ((X00 Xz0)->(X00 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X00 Xv)) (((eq a) (cR Xv)) Xx_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1:a)=> (forall (X00:(a->Prop)), (((and (X00 Xx)) (forall (Xz0:a), ((X00 Xz0)->(X00 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X00 Xv)) (((eq a) (cR Xv)) ((cP Xz) Xz_1)))))))))))))))))) b)
% Found (((eta_expansion a) Prop) (fun (Xt:a)=> ((and ((iff ((ex a) (fun (Xz_0:a)=> (X ((cP Xz) Xz_0))))) ((ex a) (fun (Xx:a)=> ((and (forall (Xx_13:a), ((forall (X00:(a->Prop)), (((and (X00 Xx)) (forall (Xz0:a), ((X00 Xz0)->(X00 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X00 Xv)) (((eq a) (cR Xv)) Xx_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1:a)=> (forall (X00:(a->Prop)), (((and (X00 Xx)) (forall (Xz0:a), ((X00 Xz0)->(X00 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X00 Xv)) (((eq a) (cR Xv)) ((cP Xz) Xz_1)))))))))))))) (a->(((iff ((ex a) (fun (Xz_0:a)=> (X ((cP Xz) Xz_0))))) ((ex a) (fun (Xx:a)=> ((and (forall (Xx_13:a), ((forall (X00:(a->Prop)), (((and (X00 Xx)) (forall (Xz0:a), ((X00 Xz0)->(X00 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X00 Xv)) (((eq a) (cR Xv)) Xx_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1:a)=> (forall (X00:(a->Prop)), (((and (X00 Xx)) (forall (Xz0:a), ((X00 Xz0)->(X00 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X00 Xv)) (((eq a) (cR Xv)) ((cP Xz) Xz_1)))))))))))))->((iff ((ex a) (fun (Xz_0:a)=> (X ((cP Xz) Xz_0))))) ((ex a) (fun (Xx:a)=> ((and (forall (Xx_13:a), ((forall (X00:(a->Prop)), (((and (X00 Xx)) (forall (Xz0:a), ((X00 Xz0)->(X00 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X00 Xv)) (((eq a) (cR Xv)) Xx_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1:a)=> (forall (X00:(a->Prop)), (((and (X00 Xx)) (forall (Xz0:a), ((X00 Xz0)->(X00 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X00 Xv)) (((eq a) (cR Xv)) ((cP Xz) Xz_1)))))))))))))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((and ((iff ((ex a) (fun (Xz_0:a)=> (X ((cP Xz) Xz_0))))) ((ex a) (fun (Xx:a)=> ((and (forall (Xx_13:a), ((forall (X00:(a->Prop)), (((and (X00 Xx)) (forall (Xz0:a), ((X00 Xz0)->(X00 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X00 Xv)) (((eq a) (cR Xv)) Xx_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1:a)=> (forall (X00:(a->Prop)), (((and (X00 Xx)) (forall (Xz0:a), ((X00 Xz0)->(X00 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X00 Xv)) (((eq a) (cR Xv)) ((cP Xz) Xz_1)))))))))))))) (a->(((iff ((ex a) (fun (Xz_0:a)=> (X ((cP Xz) Xz_0))))) ((ex a) (fun (Xx:a)=> ((and (forall (Xx_13:a), ((forall (X00:(a->Prop)), (((and (X00 Xx)) (forall (Xz0:a), ((X00 Xz0)->(X00 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X00 Xv)) (((eq a) (cR Xv)) Xx_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1:a)=> (forall (X00:(a->Prop)), (((and (X00 Xx)) (forall (Xz0:a), ((X00 Xz0)->(X00 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X00 Xv)) (((eq a) (cR Xv)) ((cP Xz) Xz_1)))))))))))))->((iff ((ex a) (fun (Xz_0:a)=> (X ((cP Xz) Xz_0))))) ((ex a) (fun (Xx:a)=> ((and (forall (Xx_13:a), ((forall (X00:(a->Prop)), (((and (X00 Xx)) (forall (Xz0:a), ((X00 Xz0)->(X00 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X00 Xv)) (((eq a) (cR Xv)) Xx_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1:a)=> (forall (X00:(a->Prop)), (((and (X00 Xx)) (forall (Xz0:a), ((X00 Xz0)->(X00 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X00 Xv)) (((eq a) (cR Xv)) ((cP Xz) Xz_1)))))))))))))))))) b)
% Found (((eta_expansion a) Prop) (fun (Xt:a)=> ((and ((iff ((ex a) (fun (Xz_0:a)=> (X ((cP Xz) Xz_0))))) ((ex a) (fun (Xx:a)=> ((and (forall (Xx_13:a), ((forall (X00:(a->Prop)), (((and (X00 Xx)) (forall (Xz0:a), ((X00 Xz0)->(X00 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X00 Xv)) (((eq a) (cR Xv)) Xx_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1:a)=> (forall (X00:(a->Prop)), (((and (X00 Xx)) (forall (Xz0:a), ((X00 Xz0)->(X00 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X00 Xv)) (((eq a) (cR Xv)) ((cP Xz) Xz_1)))))))))))))) (a->(((iff ((ex a) (fun (Xz_0:a)=> (X ((cP Xz) Xz_0))))) ((ex a) (fun (Xx:a)=> ((and (forall (Xx_13:a), ((forall (X00:(a->Prop)), (((and (X00 Xx)) (forall (Xz0:a), ((X00 Xz0)->(X00 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X00 Xv)) (((eq a) (cR Xv)) Xx_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1:a)=> (forall (X00:(a->Prop)), (((and (X00 Xx)) (forall (Xz0:a), ((X00 Xz0)->(X00 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X00 Xv)) (((eq a) (cR Xv)) ((cP Xz) Xz_1)))))))))))))->((iff ((ex a) (fun (Xz_0:a)=> (X ((cP Xz) Xz_0))))) ((ex a) (fun (Xx:a)=> ((and (forall (Xx_13:a), ((forall (X00:(a->Prop)), (((and (X00 Xx)) (forall (Xz0:a), ((X00 Xz0)->(X00 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X00 Xv)) (((eq a) (cR Xv)) Xx_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1:a)=> (forall (X00:(a->Prop)), (((and (X00 Xx)) (forall (Xz0:a), ((X00 Xz0)->(X00 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X00 Xv)) (((eq a) (cR Xv)) ((cP Xz) Xz_1)))))))))))))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((and ((iff ((ex a) (fun (Xz_0:a)=> (X ((cP Xz) Xz_0))))) ((ex a) (fun (Xx:a)=> ((and (forall (Xx_13:a), ((forall (X00:(a->Prop)), (((and (X00 Xx)) (forall (Xz0:a), ((X00 Xz0)->(X00 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X00 Xv)) (((eq a) (cR Xv)) Xx_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1:a)=> (forall (X00:(a->Prop)), (((and (X00 Xx)) (forall (Xz0:a), ((X00 Xz0)->(X00 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X00 Xv)) (((eq a) (cR Xv)) ((cP Xz) Xz_1)))))))))))))) (a->(((iff ((ex a) (fun (Xz_0:a)=> (X ((cP Xz) Xz_0))))) ((ex a) (fun (Xx:a)=> ((and (forall (Xx_13:a), ((forall (X00:(a->Prop)), (((and (X00 Xx)) (forall (Xz0:a), ((X00 Xz0)->(X00 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X00 Xv)) (((eq a) (cR Xv)) Xx_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1:a)=> (forall (X00:(a->Prop)), (((and (X00 Xx)) (forall (Xz0:a), ((X00 Xz0)->(X00 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X00 Xv)) (((eq a) (cR Xv)) ((cP Xz) Xz_1)))))))))))))->((iff ((ex a) (fun (Xz_0:a)=> (X ((cP Xz) Xz_0))))) ((ex a) (fun (Xx:a)=> ((and (forall (Xx_13:a), ((forall (X00:(a->Prop)), (((and (X00 Xx)) (forall (Xz0:a), ((X00 Xz0)->(X00 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X00 Xv)) (((eq a) (cR Xv)) Xx_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1:a)=> (forall (X00:(a->Prop)), (((and (X00 Xx)) (forall (Xz0:a), ((X00 Xz0)->(X00 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X00 Xv)) (((eq a) (cR Xv)) ((cP Xz) Xz_1)))))))))))))))))) b)
% Found (((eta_expansion a) Prop) (fun (Xt:a)=> ((and ((iff ((ex a) (fun (Xz_0:a)=> (X ((cP Xz) Xz_0))))) ((ex a) (fun (Xx:a)=> ((and (forall (Xx_13:a), ((forall (X00:(a->Prop)), (((and (X00 Xx)) (forall (Xz0:a), ((X00 Xz0)->(X00 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X00 Xv)) (((eq a) (cR Xv)) Xx_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1:a)=> (forall (X00:(a->Prop)), (((and (X00 Xx)) (forall (Xz0:a), ((X00 Xz0)->(X00 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X00 Xv)) (((eq a) (cR Xv)) ((cP Xz) Xz_1)))))))))))))) (a->(((iff ((ex a) (fun (Xz_0:a)=> (X ((cP Xz) Xz_0))))) ((ex a) (fun (Xx:a)=> ((and (forall (Xx_13:a), ((forall (X00:(a->Prop)), (((and (X00 Xx)) (forall (Xz0:a), ((X00 Xz0)->(X00 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X00 Xv)) (((eq a) (cR Xv)) Xx_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1:a)=> (forall (X00:(a->Prop)), (((and (X00 Xx)) (forall (Xz0:a), ((X00 Xz0)->(X00 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X00 Xv)) (((eq a) (cR Xv)) ((cP Xz) Xz_1)))))))))))))->((iff ((ex a) (fun (Xz_0:a)=> (X ((cP Xz) Xz_0))))) ((ex a) (fun (Xx:a)=> ((and (forall (Xx_13:a), ((forall (X00:(a->Prop)), (((and (X00 Xx)) (forall (Xz0:a), ((X00 Xz0)->(X00 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X00 Xv)) (((eq a) (cR Xv)) Xx_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1:a)=> (forall (X00:(a->Prop)), (((and (X00 Xx)) (forall (Xz0:a), ((X00 Xz0)->(X00 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X00 Xv)) (((eq a) (cR Xv)) ((cP Xz) Xz_1)))))))))))))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((and ((iff ((ex a) (fun (Xz_0:a)=> (X ((cP Xz) Xz_0))))) ((ex a) (fun (Xx:a)=> ((and (forall (Xx_13:a), ((forall (X00:(a->Prop)), (((and (X00 Xx)) (forall (Xz0:a), ((X00 Xz0)->(X00 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X00 Xv)) (((eq a) (cR Xv)) Xx_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1:a)=> (forall (X00:(a->Prop)), (((and (X00 Xx)) (forall (Xz0:a), ((X00 Xz0)->(X00 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X00 Xv)) (((eq a) (cR Xv)) ((cP Xz) Xz_1)))))))))))))) (a->(((iff ((ex a) (fun (Xz_0:a)=> (X ((cP Xz) Xz_0))))) ((ex a) (fun (Xx:a)=> ((and (forall (Xx_13:a), ((forall (X00:(a->Prop)), (((and (X00 Xx)) (forall (Xz0:a), ((X00 Xz0)->(X00 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X00 Xv)) (((eq a) (cR Xv)) Xx_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1:a)=> (forall (X00:(a->Prop)), (((and (X00 Xx)) (forall (Xz0:a), ((X00 Xz0)->(X00 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X00 Xv)) (((eq a) (cR Xv)) ((cP Xz) Xz_1)))))))))))))->((iff ((ex a) (fun (Xz_0:a)=> (X ((cP Xz) Xz_0))))) ((ex a) (fun (Xx:a)=> ((and (forall (Xx_13:a), ((forall (X00:(a->Prop)), (((and (X00 Xx)) (forall (Xz0:a), ((X00 Xz0)->(X00 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X00 Xv)) (((eq a) (cR Xv)) Xx_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1:a)=> (forall (X00:(a->Prop)), (((and (X00 Xx)) (forall (Xz0:a), ((X00 Xz0)->(X00 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X00 Xv)) (((eq a) (cR Xv)) ((cP Xz) Xz_1)))))))))))))))))) b)
% Found eta_expansion_dep0000:=(eta_expansion_dep000 (ex a)):(((ex a) (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1))))))))))))->((ex a) (fun (x:a)=> ((and (forall (Xx_13:a), ((forall (X0:(a->Prop)), (((and (X0 x)) (forall (Xz0:a), ((X0 Xz0)->(X0 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1:a)=> (forall (X0:(a->Prop)), (((and (X0 x)) (forall (Xz0:a), ((X0 Xz0)->(X0 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) ((cP Xz) Xz_1)))))))))))))
% Found (eta_expansion_dep000 (ex a)) as proof of (P (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1))))))))))))
% Found ((eta_expansion_dep00 (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1)))))))))))) (ex a)) as proof of (P (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1))))))))))))
% Found (((eta_expansion_dep0 (fun (x3:a)=> Prop)) (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1)))))))))))) (ex a)) as proof of (P (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1))))))))))))
% Found ((((eta_expansion_dep a) (fun (x3:a)=> Prop)) (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1)))))))))))) (ex a)) as proof of (P (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1))))))))))))
% Found ((((eta_expansion_dep a) (fun (x3:a)=> Prop)) (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1)))))))))))) (ex a)) as proof of (P (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1))))))))))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 (ex a)):(((ex a) (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1))))))))))))->((ex a) (fun (x:a)=> ((and (forall (Xx_13:a), ((forall (X0:(a->Prop)), (((and (X0 x)) (forall (Xz0:a), ((X0 Xz0)->(X0 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1:a)=> (forall (X0:(a->Prop)), (((and (X0 x)) (forall (Xz0:a), ((X0 Xz0)->(X0 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) ((cP Xz) Xz_1)))))))))))))
% Found (eta_expansion_dep000 (ex a)) as proof of (P (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1))))))))))))
% Found ((eta_expansion_dep00 (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1)))))))))))) (ex a)) as proof of (P (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1))))))))))))
% Found (((eta_expansion_dep0 (fun (x3:a)=> Prop)) (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1)))))))))))) (ex a)) as proof of (P (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1))))))))))))
% Found ((((eta_expansion_dep a) (fun (x3:a)=> Prop)) (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1)))))))))))) (ex a)) as proof of (P (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1))))))))))))
% Found ((((eta_expansion_dep a) (fun (x3:a)=> Prop)) (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1)))))))))))) (ex a)) as proof of (P (fun (Xx:a)=> ((and (forall (Xx_13: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_13))))))->(X Xx_13)))) ((ex a) (fun (Xz_1: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)) ((cP Xz) Xz_1))))))))))))
% Found eta_expansion_dep
% EOF
%------------------------------------------------------------------------------