%------------------------------------------------------------------------------
% File     : cocATP---0.2.0
% Problem  : ALG291^5 : TPTP v6.1.0. Released v4.0.0.
% Transfm  : none
% Format   : tptp:raw
% Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p

% Computer : n104.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 300.02s
% Output   : None 
% Verified : 
% SZS Type : None (Parsing solution fails)
% Syntax   : Number of formulae    : 0

% Comments : 
%------------------------------------------------------------------------------
%----NO SOLUTION OUTPUT BY SYSTEM
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% % Problem  : ALG291^5 : TPTP v6.1.0. Released v4.0.0.
% % Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n104.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 09:09:54 CDT 2014
% % CPUTime  : 300.02 
% 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 0x2226440>, <kernel.Type object at 0x2226488>) of role type named a_type
% Using role type
% Declaring a:Type
% FOF formula (forall (Z:a) (P:(a->(a->a))) (L:(a->a)) (R:(a->a)) (X:(a->Prop)), (((and ((and ((and ((and ((and (((eq a) (L Z)) Z)) (((eq a) (R Z)) Z))) (forall (Xx:a) (Xy:a), (((eq a) (L ((P Xx) Xy))) Xx)))) (forall (Xx:a) (Xy:a), (((eq a) (R ((P Xx) Xy))) Xy)))) (forall (Xt:a), ((iff (not (((eq a) Xt) Z))) (((eq a) Xt) ((P (L Xt)) (R Xt))))))) (forall (X0:(a->Prop)), (((ex a) (fun (Xt:a)=> ((and (X0 Xt)) (forall (Xu:a), ((X0 Xu)->(X0 (L Xu)))))))->(X0 Z))))->(forall (X_0:(a->Prop)) (Xz:a), ((iff ((ex a) (fun (Xx:a)=> ((and (forall (Xx_9:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_9))))))->(X Xx_9)))) (X_0 ((P Xx) Xz)))))) ((ex a) (fun (Xx:a)=> ((and (forall (Xx_10:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_10))))))->(X_0 Xx_10)))) ((ex a) (fun (Xx_12:a)=> ((and (forall (Xx_11:a), ((forall (X0:(a->Prop)), (((and (X0 Xx_12)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_11))))))->(X Xx_11)))) (forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) ((P Xx_12) Xz))))))))))))))))) of role conjecture named cPU_X238B_pme
% Conjecture to prove = (forall (Z:a) (P:(a->(a->a))) (L:(a->a)) (R:(a->a)) (X:(a->Prop)), (((and ((and ((and ((and ((and (((eq a) (L Z)) Z)) (((eq a) (R Z)) Z))) (forall (Xx:a) (Xy:a), (((eq a) (L ((P Xx) Xy))) Xx)))) (forall (Xx:a) (Xy:a), (((eq a) (R ((P Xx) Xy))) Xy)))) (forall (Xt:a), ((iff (not (((eq a) Xt) Z))) (((eq a) Xt) ((P (L Xt)) (R Xt))))))) (forall (X0:(a->Prop)), (((ex a) (fun (Xt:a)=> ((and (X0 Xt)) (forall (Xu:a), ((X0 Xu)->(X0 (L Xu)))))))->(X0 Z))))->(forall (X_0:(a->Prop)) (Xz:a), ((iff ((ex a) (fun (Xx:a)=> ((and (forall (Xx_9:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_9))))))->(X Xx_9)))) (X_0 ((P Xx) Xz)))))) ((ex a) (fun (Xx:a)=> ((and (forall (Xx_10:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_10))))))->(X_0 Xx_10)))) ((ex a) (fun (Xx_12:a)=> ((and (forall (Xx_11:a), ((forall (X0:(a->Prop)), (((and (X0 Xx_12)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_11))))))->(X Xx_11)))) (forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) ((P Xx_12) Xz))))))))))))))))):Prop
% Parameter a_DUMMY:a.
% We need to prove ['(forall (Z:a) (P:(a->(a->a))) (L:(a->a)) (R:(a->a)) (X:(a->Prop)), (((and ((and ((and ((and ((and (((eq a) (L Z)) Z)) (((eq a) (R Z)) Z))) (forall (Xx:a) (Xy:a), (((eq a) (L ((P Xx) Xy))) Xx)))) (forall (Xx:a) (Xy:a), (((eq a) (R ((P Xx) Xy))) Xy)))) (forall (Xt:a), ((iff (not (((eq a) Xt) Z))) (((eq a) Xt) ((P (L Xt)) (R Xt))))))) (forall (X0:(a->Prop)), (((ex a) (fun (Xt:a)=> ((and (X0 Xt)) (forall (Xu:a), ((X0 Xu)->(X0 (L Xu)))))))->(X0 Z))))->(forall (X_0:(a->Prop)) (Xz:a), ((iff ((ex a) (fun (Xx:a)=> ((and (forall (Xx_9:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_9))))))->(X Xx_9)))) (X_0 ((P Xx) Xz)))))) ((ex a) (fun (Xx:a)=> ((and (forall (Xx_10:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_10))))))->(X_0 Xx_10)))) ((ex a) (fun (Xx_12:a)=> ((and (forall (Xx_11:a), ((forall (X0:(a->Prop)), (((and (X0 Xx_12)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_11))))))->(X Xx_11)))) (forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) ((P Xx_12) Xz)))))))))))))))))']
% Parameter a:Type.
% Trying to prove (forall (Z:a) (P:(a->(a->a))) (L:(a->a)) (R:(a->a)) (X:(a->Prop)), (((and ((and ((and ((and ((and (((eq a) (L Z)) Z)) (((eq a) (R Z)) Z))) (forall (Xx:a) (Xy:a), (((eq a) (L ((P Xx) Xy))) Xx)))) (forall (Xx:a) (Xy:a), (((eq a) (R ((P Xx) Xy))) Xy)))) (forall (Xt:a), ((iff (not (((eq a) Xt) Z))) (((eq a) Xt) ((P (L Xt)) (R Xt))))))) (forall (X0:(a->Prop)), (((ex a) (fun (Xt:a)=> ((and (X0 Xt)) (forall (Xu:a), ((X0 Xu)->(X0 (L Xu)))))))->(X0 Z))))->(forall (X_0:(a->Prop)) (Xz:a), ((iff ((ex a) (fun (Xx:a)=> ((and (forall (Xx_9:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_9))))))->(X Xx_9)))) (X_0 ((P Xx) Xz)))))) ((ex a) (fun (Xx:a)=> ((and (forall (Xx_10:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_10))))))->(X_0 Xx_10)))) ((ex a) (fun (Xx_12:a)=> ((and (forall (Xx_11:a), ((forall (X0:(a->Prop)), (((and (X0 Xx_12)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_11))))))->(X Xx_11)))) (forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) ((P Xx_12) Xz)))))))))))))))))
% Found Z:a
% Found Z 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 (Xx:a)=> ((and (forall (Xx_9:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_9))))))->(X Xx_9)))) (X_0 ((P Xx) Xz)))))->((ex a) (fun (x:a)=> ((and (forall (Xx_9:a), ((forall (X0:(a->Prop)), (((and (X0 x)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_9))))))->(X Xx_9)))) (X_0 ((P x) Xz))))))
% Found (eta_expansion000 (ex a)) as proof of (P0 (fun (Xx:a)=> ((and (forall (Xx_9:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_9))))))->(X Xx_9)))) (X_0 ((P Xx) Xz)))))
% Found ((eta_expansion00 (fun (Xx:a)=> ((and (forall (Xx_9:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_9))))))->(X Xx_9)))) (X_0 ((P Xx) Xz))))) (ex a)) as proof of (P0 (fun (Xx:a)=> ((and (forall (Xx_9:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_9))))))->(X Xx_9)))) (X_0 ((P Xx) Xz)))))
% Found (((eta_expansion0 Prop) (fun (Xx:a)=> ((and (forall (Xx_9:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_9))))))->(X Xx_9)))) (X_0 ((P Xx) Xz))))) (ex a)) as proof of (P0 (fun (Xx:a)=> ((and (forall (Xx_9:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_9))))))->(X Xx_9)))) (X_0 ((P Xx) Xz)))))
% Found ((((eta_expansion a) Prop) (fun (Xx:a)=> ((and (forall (Xx_9:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_9))))))->(X Xx_9)))) (X_0 ((P Xx) Xz))))) (ex a)) as proof of (P0 (fun (Xx:a)=> ((and (forall (Xx_9:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_9))))))->(X Xx_9)))) (X_0 ((P Xx) Xz)))))
% Found ((((eta_expansion a) Prop) (fun (Xx:a)=> ((and (forall (Xx_9:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_9))))))->(X Xx_9)))) (X_0 ((P Xx) Xz))))) (ex a)) as proof of (P0 (fun (Xx:a)=> ((and (forall (Xx_9:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_9))))))->(X Xx_9)))) (X_0 ((P Xx) Xz)))))
% Found eta_expansion0000:=(eta_expansion000 (ex a)):(((ex a) (fun (Xx:a)=> ((and (forall (Xx_10:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_10))))))->(X_0 Xx_10)))) ((ex a) (fun (Xx_12:a)=> ((and (forall (Xx_11:a), ((forall (X0:(a->Prop)), (((and (X0 Xx_12)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_11))))))->(X Xx_11)))) (forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) ((P Xx_12) Xz)))))))))))))->((ex a) (fun (x:a)=> ((and (forall (Xx_10:a), ((forall (X0:(a->Prop)), (((and (X0 x)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_10))))))->(X_0 Xx_10)))) ((ex a) (fun (Xx_12:a)=> ((and (forall (Xx_11:a), ((forall (X0:(a->Prop)), (((and (X0 Xx_12)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_11))))))->(X Xx_11)))) (forall (X0:(a->Prop)), (((and (X0 x)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) ((P Xx_12) Xz))))))))))))))
% Found (eta_expansion000 (ex a)) as proof of (P0 (fun (Xx:a)=> ((and (forall (Xx_10:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_10))))))->(X_0 Xx_10)))) ((ex a) (fun (Xx_12:a)=> ((and (forall (Xx_11:a), ((forall (X0:(a->Prop)), (((and (X0 Xx_12)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_11))))))->(X Xx_11)))) (forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) ((P Xx_12) Xz)))))))))))))
% Found ((eta_expansion00 (fun (Xx:a)=> ((and (forall (Xx_10:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_10))))))->(X_0 Xx_10)))) ((ex a) (fun (Xx_12:a)=> ((and (forall (Xx_11:a), ((forall (X0:(a->Prop)), (((and (X0 Xx_12)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_11))))))->(X Xx_11)))) (forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) ((P Xx_12) Xz))))))))))))) (ex a)) as proof of (P0 (fun (Xx:a)=> ((and (forall (Xx_10:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_10))))))->(X_0 Xx_10)))) ((ex a) (fun (Xx_12:a)=> ((and (forall (Xx_11:a), ((forall (X0:(a->Prop)), (((and (X0 Xx_12)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_11))))))->(X Xx_11)))) (forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) ((P Xx_12) Xz)))))))))))))
% Found (((eta_expansion0 Prop) (fun (Xx:a)=> ((and (forall (Xx_10:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_10))))))->(X_0 Xx_10)))) ((ex a) (fun (Xx_12:a)=> ((and (forall (Xx_11:a), ((forall (X0:(a->Prop)), (((and (X0 Xx_12)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_11))))))->(X Xx_11)))) (forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) ((P Xx_12) Xz))))))))))))) (ex a)) as proof of (P0 (fun (Xx:a)=> ((and (forall (Xx_10:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_10))))))->(X_0 Xx_10)))) ((ex a) (fun (Xx_12:a)=> ((and (forall (Xx_11:a), ((forall (X0:(a->Prop)), (((and (X0 Xx_12)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_11))))))->(X Xx_11)))) (forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) ((P Xx_12) Xz)))))))))))))
% Found ((((eta_expansion a) Prop) (fun (Xx:a)=> ((and (forall (Xx_10:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_10))))))->(X_0 Xx_10)))) ((ex a) (fun (Xx_12:a)=> ((and (forall (Xx_11:a), ((forall (X0:(a->Prop)), (((and (X0 Xx_12)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_11))))))->(X Xx_11)))) (forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) ((P Xx_12) Xz))))))))))))) (ex a)) as proof of (P0 (fun (Xx:a)=> ((and (forall (Xx_10:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_10))))))->(X_0 Xx_10)))) ((ex a) (fun (Xx_12:a)=> ((and (forall (Xx_11:a), ((forall (X0:(a->Prop)), (((and (X0 Xx_12)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_11))))))->(X Xx_11)))) (forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) ((P Xx_12) Xz)))))))))))))
% Found ((((eta_expansion a) Prop) (fun (Xx:a)=> ((and (forall (Xx_10:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_10))))))->(X_0 Xx_10)))) ((ex a) (fun (Xx_12:a)=> ((and (forall (Xx_11:a), ((forall (X0:(a->Prop)), (((and (X0 Xx_12)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_11))))))->(X Xx_11)))) (forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) ((P Xx_12) Xz))))))))))))) (ex a)) as proof of (P0 (fun (Xx:a)=> ((and (forall (Xx_10:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_10))))))->(X_0 Xx_10)))) ((ex a) (fun (Xx_12:a)=> ((and (forall (Xx_11:a), ((forall (X0:(a->Prop)), (((and (X0 Xx_12)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_11))))))->(X Xx_11)))) (forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) ((P Xx_12) Xz)))))))))))))
% 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_9:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_9))))))->(X Xx_9)))) (X_0 ((P Xx) Xz)))))->((ex a) (fun (Xx:a)=> ((and (forall (Xx_9:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_9))))))->(X Xx_9)))) (X_0 ((P Xx) Xz))))))
% Found (eq_ref00 (ex a)) as proof of (P0 (fun (Xx:a)=> ((and (forall (Xx_9:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_9))))))->(X Xx_9)))) (X_0 ((P Xx) Xz)))))
% Found ((eq_ref0 (fun (Xx:a)=> ((and (forall (Xx_9:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_9))))))->(X Xx_9)))) (X_0 ((P Xx) Xz))))) (ex a)) as proof of (P0 (fun (Xx:a)=> ((and (forall (Xx_9:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_9))))))->(X Xx_9)))) (X_0 ((P Xx) Xz)))))
% Found (((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (forall (Xx_9:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_9))))))->(X Xx_9)))) (X_0 ((P Xx) Xz))))) (ex a)) as proof of (P0 (fun (Xx:a)=> ((and (forall (Xx_9:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_9))))))->(X Xx_9)))) (X_0 ((P Xx) Xz)))))
% Found (((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (forall (Xx_9:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_9))))))->(X Xx_9)))) (X_0 ((P Xx) Xz))))) (ex a)) as proof of (P0 (fun (Xx:a)=> ((and (forall (Xx_9:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_9))))))->(X Xx_9)))) (X_0 ((P Xx) Xz)))))
% Found eta_expansion0000:=(eta_expansion000 (ex a)):(((ex a) (fun (Xx:a)=> ((and (forall (Xx_10:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_10))))))->(X_0 Xx_10)))) ((ex a) (fun (Xx_12:a)=> ((and (forall (Xx_11:a), ((forall (X0:(a->Prop)), (((and (X0 Xx_12)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_11))))))->(X Xx_11)))) (forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) ((P Xx_12) Xz)))))))))))))->((ex a) (fun (x:a)=> ((and (forall (Xx_10:a), ((forall (X0:(a->Prop)), (((and (X0 x)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_10))))))->(X_0 Xx_10)))) ((ex a) (fun (Xx_12:a)=> ((and (forall (Xx_11:a), ((forall (X0:(a->Prop)), (((and (X0 Xx_12)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_11))))))->(X Xx_11)))) (forall (X0:(a->Prop)), (((and (X0 x)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) ((P Xx_12) Xz))))))))))))))
% Found (eta_expansion000 (ex a)) as proof of (P0 (fun (Xx:a)=> ((and (forall (Xx_10:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_10))))))->(X_0 Xx_10)))) ((ex a) (fun (Xx_12:a)=> ((and (forall (Xx_11:a), ((forall (X0:(a->Prop)), (((and (X0 Xx_12)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_11))))))->(X Xx_11)))) (forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) ((P Xx_12) Xz)))))))))))))
% Found ((eta_expansion00 (fun (Xx:a)=> ((and (forall (Xx_10:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_10))))))->(X_0 Xx_10)))) ((ex a) (fun (Xx_12:a)=> ((and (forall (Xx_11:a), ((forall (X0:(a->Prop)), (((and (X0 Xx_12)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_11))))))->(X Xx_11)))) (forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) ((P Xx_12) Xz))))))))))))) (ex a)) as proof of (P0 (fun (Xx:a)=> ((and (forall (Xx_10:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_10))))))->(X_0 Xx_10)))) ((ex a) (fun (Xx_12:a)=> ((and (forall (Xx_11:a), ((forall (X0:(a->Prop)), (((and (X0 Xx_12)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_11))))))->(X Xx_11)))) (forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) ((P Xx_12) Xz)))))))))))))
% Found (((eta_expansion0 Prop) (fun (Xx:a)=> ((and (forall (Xx_10:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_10))))))->(X_0 Xx_10)))) ((ex a) (fun (Xx_12:a)=> ((and (forall (Xx_11:a), ((forall (X0:(a->Prop)), (((and (X0 Xx_12)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_11))))))->(X Xx_11)))) (forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) ((P Xx_12) Xz))))))))))))) (ex a)) as proof of (P0 (fun (Xx:a)=> ((and (forall (Xx_10:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_10))))))->(X_0 Xx_10)))) ((ex a) (fun (Xx_12:a)=> ((and (forall (Xx_11:a), ((forall (X0:(a->Prop)), (((and (X0 Xx_12)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_11))))))->(X Xx_11)))) (forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) ((P Xx_12) Xz)))))))))))))
% Found ((((eta_expansion a) Prop) (fun (Xx:a)=> ((and (forall (Xx_10:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_10))))))->(X_0 Xx_10)))) ((ex a) (fun (Xx_12:a)=> ((and (forall (Xx_11:a), ((forall (X0:(a->Prop)), (((and (X0 Xx_12)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_11))))))->(X Xx_11)))) (forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) ((P Xx_12) Xz))))))))))))) (ex a)) as proof of (P0 (fun (Xx:a)=> ((and (forall (Xx_10:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_10))))))->(X_0 Xx_10)))) ((ex a) (fun (Xx_12:a)=> ((and (forall (Xx_11:a), ((forall (X0:(a->Prop)), (((and (X0 Xx_12)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_11))))))->(X Xx_11)))) (forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) ((P Xx_12) Xz)))))))))))))
% Found ((((eta_expansion a) Prop) (fun (Xx:a)=> ((and (forall (Xx_10:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_10))))))->(X_0 Xx_10)))) ((ex a) (fun (Xx_12:a)=> ((and (forall (Xx_11:a), ((forall (X0:(a->Prop)), (((and (X0 Xx_12)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_11))))))->(X Xx_11)))) (forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) ((P Xx_12) Xz))))))))))))) (ex a)) as proof of (P0 (fun (Xx:a)=> ((and (forall (Xx_10:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_10))))))->(X_0 Xx_10)))) ((ex a) (fun (Xx_12:a)=> ((and (forall (Xx_11:a), ((forall (X0:(a->Prop)), (((and (X0 Xx_12)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_11))))))->(X Xx_11)))) (forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) ((P Xx_12) Xz)))))))))))))
% Found eta_expansion0000:=(eta_expansion000 (ex a)):(((ex a) (fun (Xx:a)=> ((and (forall (Xx_9:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_9))))))->(X Xx_9)))) (X_0 ((P Xx) Xz)))))->((ex a) (fun (x:a)=> ((and (forall (Xx_9:a), ((forall (X0:(a->Prop)), (((and (X0 x)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_9))))))->(X Xx_9)))) (X_0 ((P x) Xz))))))
% Found (eta_expansion000 (ex a)) as proof of (P0 (fun (Xx:a)=> ((and (forall (Xx_9:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_9))))))->(X Xx_9)))) (X_0 ((P Xx) Xz)))))
% Found ((eta_expansion00 (fun (Xx:a)=> ((and (forall (Xx_9:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_9))))))->(X Xx_9)))) (X_0 ((P Xx) Xz))))) (ex a)) as proof of (P0 (fun (Xx:a)=> ((and (forall (Xx_9:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_9))))))->(X Xx_9)))) (X_0 ((P Xx) Xz)))))
% Found (((eta_expansion0 Prop) (fun (Xx:a)=> ((and (forall (Xx_9:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_9))))))->(X Xx_9)))) (X_0 ((P Xx) Xz))))) (ex a)) as proof of (P0 (fun (Xx:a)=> ((and (forall (Xx_9:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_9))))))->(X Xx_9)))) (X_0 ((P Xx) Xz)))))
% Found ((((eta_expansion a) Prop) (fun (Xx:a)=> ((and (forall (Xx_9:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_9))))))->(X Xx_9)))) (X_0 ((P Xx) Xz))))) (ex a)) as proof of (P0 (fun (Xx:a)=> ((and (forall (Xx_9:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_9))))))->(X Xx_9)))) (X_0 ((P Xx) Xz)))))
% Found ((((eta_expansion a) Prop) (fun (Xx:a)=> ((and (forall (Xx_9:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_9))))))->(X Xx_9)))) (X_0 ((P Xx) Xz))))) (ex a)) as proof of (P0 (fun (Xx:a)=> ((and (forall (Xx_9:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_9))))))->(X Xx_9)))) (X_0 ((P Xx) Xz)))))
% Found eq_ref000:=(eq_ref00 (ex a)):(((ex a) (fun (Xx:a)=> ((and (forall (Xx_10:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_10))))))->(X_0 Xx_10)))) ((ex a) (fun (Xx_12:a)=> ((and (forall (Xx_11:a), ((forall (X0:(a->Prop)), (((and (X0 Xx_12)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_11))))))->(X Xx_11)))) (forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) ((P Xx_12) Xz)))))))))))))->((ex a) (fun (Xx:a)=> ((and (forall (Xx_10:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_10))))))->(X_0 Xx_10)))) ((ex a) (fun (Xx_12:a)=> ((and (forall (Xx_11:a), ((forall (X0:(a->Prop)), (((and (X0 Xx_12)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_11))))))->(X Xx_11)))) (forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) ((P Xx_12) Xz))))))))))))))
% Found (eq_ref00 (ex a)) as proof of (P0 (fun (Xx:a)=> ((and (forall (Xx_10:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_10))))))->(X_0 Xx_10)))) ((ex a) (fun (Xx_12:a)=> ((and (forall (Xx_11:a), ((forall (X0:(a->Prop)), (((and (X0 Xx_12)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_11))))))->(X Xx_11)))) (forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) ((P Xx_12) Xz)))))))))))))
% Found ((eq_ref0 (fun (Xx:a)=> ((and (forall (Xx_10:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_10))))))->(X_0 Xx_10)))) ((ex a) (fun (Xx_12:a)=> ((and (forall (Xx_11:a), ((forall (X0:(a->Prop)), (((and (X0 Xx_12)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_11))))))->(X Xx_11)))) (forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) ((P Xx_12) Xz))))))))))))) (ex a)) as proof of (P0 (fun (Xx:a)=> ((and (forall (Xx_10:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_10))))))->(X_0 Xx_10)))) ((ex a) (fun (Xx_12:a)=> ((and (forall (Xx_11:a), ((forall (X0:(a->Prop)), (((and (X0 Xx_12)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_11))))))->(X Xx_11)))) (forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) ((P Xx_12) Xz)))))))))))))
% Found (((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (forall (Xx_10:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_10))))))->(X_0 Xx_10)))) ((ex a) (fun (Xx_12:a)=> ((and (forall (Xx_11:a), ((forall (X0:(a->Prop)), (((and (X0 Xx_12)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_11))))))->(X Xx_11)))) (forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) ((P Xx_12) Xz))))))))))))) (ex a)) as proof of (P0 (fun (Xx:a)=> ((and (forall (Xx_10:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_10))))))->(X_0 Xx_10)))) ((ex a) (fun (Xx_12:a)=> ((and (forall (Xx_11:a), ((forall (X0:(a->Prop)), (((and (X0 Xx_12)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_11))))))->(X Xx_11)))) (forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) ((P Xx_12) Xz)))))))))))))
% Found (((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (forall (Xx_10:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_10))))))->(X_0 Xx_10)))) ((ex a) (fun (Xx_12:a)=> ((and (forall (Xx_11:a), ((forall (X0:(a->Prop)), (((and (X0 Xx_12)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_11))))))->(X Xx_11)))) (forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) ((P Xx_12) Xz))))))))))))) (ex a)) as proof of (P0 (fun (Xx:a)=> ((and (forall (Xx_10:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_10))))))->(X_0 Xx_10)))) ((ex a) (fun (Xx_12:a)=> ((and (forall (Xx_11:a), ((forall (X0:(a->Prop)), (((and (X0 Xx_12)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_11))))))->(X Xx_11)))) (forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) ((P Xx_12) Xz)))))))))))))
% Found Z:a
% Found Z as proof of a
% Found Xz:a
% Found Xz as proof of a
% Found Xz:a
% Found Xz as proof of a
% Found Z:a
% Found Z as proof of a
% Found Z:a
% Found Z 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 a_DUMMY:a
% Found a_DUMMY as proof of a
% Found a_DUMMY:a
% Found a_DUMMY as proof of a
% Found eta_expansion0000:=(eta_expansion000 (ex a)):(((ex a) (fun (Xx:a)=> ((and (forall (Xx_10:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_10))))))->(X_0 Xx_10)))) ((ex a) (fun (Xx_12:a)=> ((and (forall (Xx_11:a), ((forall (X0:(a->Prop)), (((and (X0 Xx_12)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_11))))))->(X Xx_11)))) (forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) ((P Xx_12) Xz)))))))))))))->((ex a) (fun (x:a)=> ((and (forall (Xx_10:a), ((forall (X0:(a->Prop)), (((and (X0 x)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_10))))))->(X_0 Xx_10)))) ((ex a) (fun (Xx_12:a)=> ((and (forall (Xx_11:a), ((forall (X0:(a->Prop)), (((and (X0 Xx_12)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_11))))))->(X Xx_11)))) (forall (X0:(a->Prop)), (((and (X0 x)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) ((P Xx_12) Xz))))))))))))))
% Found (eta_expansion000 (ex a)) as proof of (P0 (fun (Xx:a)=> ((and (forall (Xx_10:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_10))))))->(X_0 Xx_10)))) ((ex a) (fun (Xx_12:a)=> ((and (forall (Xx_11:a), ((forall (X0:(a->Prop)), (((and (X0 Xx_12)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_11))))))->(X Xx_11)))) (forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) ((P Xx_12) Xz)))))))))))))
% Found ((eta_expansion00 (fun (Xx:a)=> ((and (forall (Xx_10:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_10))))))->(X_0 Xx_10)))) ((ex a) (fun (Xx_12:a)=> ((and (forall (Xx_11:a), ((forall (X0:(a->Prop)), (((and (X0 Xx_12)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_11))))))->(X Xx_11)))) (forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) ((P Xx_12) Xz))))))))))))) (ex a)) as proof of (P0 (fun (Xx:a)=> ((and (forall (Xx_10:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_10))))))->(X_0 Xx_10)))) ((ex a) (fun (Xx_12:a)=> ((and (forall (Xx_11:a), ((forall (X0:(a->Prop)), (((and (X0 Xx_12)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_11))))))->(X Xx_11)))) (forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) ((P Xx_12) Xz)))))))))))))
% Found (((eta_expansion0 Prop) (fun (Xx:a)=> ((and (forall (Xx_10:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_10))))))->(X_0 Xx_10)))) ((ex a) (fun (Xx_12:a)=> ((and (forall (Xx_11:a), ((forall (X0:(a->Prop)), (((and (X0 Xx_12)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_11))))))->(X Xx_11)))) (forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) ((P Xx_12) Xz))))))))))))) (ex a)) as proof of (P0 (fun (Xx:a)=> ((and (forall (Xx_10:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_10))))))->(X_0 Xx_10)))) ((ex a) (fun (Xx_12:a)=> ((and (forall (Xx_11:a), ((forall (X0:(a->Prop)), (((and (X0 Xx_12)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_11))))))->(X Xx_11)))) (forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) ((P Xx_12) Xz)))))))))))))
% Found ((((eta_expansion a) Prop) (fun (Xx:a)=> ((and (forall (Xx_10:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_10))))))->(X_0 Xx_10)))) ((ex a) (fun (Xx_12:a)=> ((and (forall (Xx_11:a), ((forall (X0:(a->Prop)), (((and (X0 Xx_12)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_11))))))->(X Xx_11)))) (forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) ((P Xx_12) Xz))))))))))))) (ex a)) as proof of (P0 (fun (Xx:a)=> ((and (forall (Xx_10:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_10))))))->(X_0 Xx_10)))) ((ex a) (fun (Xx_12:a)=> ((and (forall (Xx_11:a), ((forall (X0:(a->Prop)), (((and (X0 Xx_12)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_11))))))->(X Xx_11)))) (forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) ((P Xx_12) Xz)))))))))))))
% Found ((((eta_expansion a) Prop) (fun (Xx:a)=> ((and (forall (Xx_10:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_10))))))->(X_0 Xx_10)))) ((ex a) (fun (Xx_12:a)=> ((and (forall (Xx_11:a), ((forall (X0:(a->Prop)), (((and (X0 Xx_12)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_11))))))->(X Xx_11)))) (forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) ((P Xx_12) Xz))))))))))))) (ex a)) as proof of (P0 (fun (Xx:a)=> ((and (forall (Xx_10:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_10))))))->(X_0 Xx_10)))) ((ex a) (fun (Xx_12:a)=> ((and (forall (Xx_11:a), ((forall (X0:(a->Prop)), (((and (X0 Xx_12)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_11))))))->(X Xx_11)))) (forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) ((P Xx_12) Xz)))))))))))))
% Found eta_expansion0000:=(eta_expansion000 (ex a)):(((ex a) (fun (Xx:a)=> ((and (forall (Xx_9:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_9))))))->(X Xx_9)))) (X_0 ((P Xx) Xz)))))->((ex a) (fun (x:a)=> ((and (forall (Xx_9:a), ((forall (X0:(a->Prop)), (((and (X0 x)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_9))))))->(X Xx_9)))) (X_0 ((P x) Xz))))))
% Found (eta_expansion000 (ex a)) as proof of (P0 (fun (Xx:a)=> ((and (forall (Xx_9:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_9))))))->(X Xx_9)))) (X_0 ((P Xx) Xz)))))
% Found ((eta_expansion00 (fun (Xx:a)=> ((and (forall (Xx_9:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_9))))))->(X Xx_9)))) (X_0 ((P Xx) Xz))))) (ex a)) as proof of (P0 (fun (Xx:a)=> ((and (forall (Xx_9:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_9))))))->(X Xx_9)))) (X_0 ((P Xx) Xz)))))
% Found (((eta_expansion0 Prop) (fun (Xx:a)=> ((and (forall (Xx_9:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_9))))))->(X Xx_9)))) (X_0 ((P Xx) Xz))))) (ex a)) as proof of (P0 (fun (Xx:a)=> ((and (forall (Xx_9:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_9))))))->(X Xx_9)))) (X_0 ((P Xx) Xz)))))
% Found ((((eta_expansion a) Prop) (fun (Xx:a)=> ((and (forall (Xx_9:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_9))))))->(X Xx_9)))) (X_0 ((P Xx) Xz))))) (ex a)) as proof of (P0 (fun (Xx:a)=> ((and (forall (Xx_9:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_9))))))->(X Xx_9)))) (X_0 ((P Xx) Xz)))))
% Found ((((eta_expansion a) Prop) (fun (Xx:a)=> ((and (forall (Xx_9:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_9))))))->(X Xx_9)))) (X_0 ((P Xx) Xz))))) (ex a)) as proof of (P0 (fun (Xx:a)=> ((and (forall (Xx_9:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_9))))))->(X Xx_9)))) (X_0 ((P Xx) Xz)))))
% Found eta_expansion0000:=(eta_expansion000 (ex a)):(((ex a) (fun (Xx:a)=> ((and (forall (Xx_10:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_10))))))->(X_0 Xx_10)))) ((ex a) (fun (Xx_12:a)=> ((and (forall (Xx_11:a), ((forall (X0:(a->Prop)), (((and (X0 Xx_12)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_11))))))->(X Xx_11)))) (forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) ((P Xx_12) Xz)))))))))))))->((ex a) (fun (x:a)=> ((and (forall (Xx_10:a), ((forall (X0:(a->Prop)), (((and (X0 x)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_10))))))->(X_0 Xx_10)))) ((ex a) (fun (Xx_12:a)=> ((and (forall (Xx_11:a), ((forall (X0:(a->Prop)), (((and (X0 Xx_12)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_11))))))->(X Xx_11)))) (forall (X0:(a->Prop)), (((and (X0 x)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) ((P Xx_12) Xz))))))))))))))
% Found (eta_expansion000 (ex a)) as proof of (P0 (fun (Xx:a)=> ((and (forall (Xx_10:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_10))))))->(X_0 Xx_10)))) ((ex a) (fun (Xx_12:a)=> ((and (forall (Xx_11:a), ((forall (X0:(a->Prop)), (((and (X0 Xx_12)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_11))))))->(X Xx_11)))) (forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) ((P Xx_12) Xz)))))))))))))
% Found ((eta_expansion00 (fun (Xx:a)=> ((and (forall (Xx_10:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_10))))))->(X_0 Xx_10)))) ((ex a) (fun (Xx_12:a)=> ((and (forall (Xx_11:a), ((forall (X0:(a->Prop)), (((and (X0 Xx_12)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_11))))))->(X Xx_11)))) (forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) ((P Xx_12) Xz))))))))))))) (ex a)) as proof of (P0 (fun (Xx:a)=> ((and (forall (Xx_10:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_10))))))->(X_0 Xx_10)))) ((ex a) (fun (Xx_12:a)=> ((and (forall (Xx_11:a), ((forall (X0:(a->Prop)), (((and (X0 Xx_12)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_11))))))->(X Xx_11)))) (forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) ((P Xx_12) Xz)))))))))))))
% Found (((eta_expansion0 Prop) (fun (Xx:a)=> ((and (forall (Xx_10:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_10))))))->(X_0 Xx_10)))) ((ex a) (fun (Xx_12:a)=> ((and (forall (Xx_11:a), ((forall (X0:(a->Prop)), (((and (X0 Xx_12)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_11))))))->(X Xx_11)))) (forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) ((P Xx_12) Xz))))))))))))) (ex a)) as proof of (P0 (fun (Xx:a)=> ((and (forall (Xx_10:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_10))))))->(X_0 Xx_10)))) ((ex a) (fun (Xx_12:a)=> ((and (forall (Xx_11:a), ((forall (X0:(a->Prop)), (((and (X0 Xx_12)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_11))))))->(X Xx_11)))) (forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) ((P Xx_12) Xz)))))))))))))
% Found ((((eta_expansion a) Prop) (fun (Xx:a)=> ((and (forall (Xx_10:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_10))))))->(X_0 Xx_10)))) ((ex a) (fun (Xx_12:a)=> ((and (forall (Xx_11:a), ((forall (X0:(a->Prop)), (((and (X0 Xx_12)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_11))))))->(X Xx_11)))) (forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) ((P Xx_12) Xz))))))))))))) (ex a)) as proof of (P0 (fun (Xx:a)=> ((and (forall (Xx_10:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_10))))))->(X_0 Xx_10)))) ((ex a) (fun (Xx_12:a)=> ((and (forall (Xx_11:a), ((forall (X0:(a->Prop)), (((and (X0 Xx_12)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_11))))))->(X Xx_11)))) (forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) ((P Xx_12) Xz)))))))))))))
% Found ((((eta_expansion a) Prop) (fun (Xx:a)=> ((and (forall (Xx_10:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_10))))))->(X_0 Xx_10)))) ((ex a) (fun (Xx_12:a)=> ((and (forall (Xx_11:a), ((forall (X0:(a->Prop)), (((and (X0 Xx_12)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_11))))))->(X Xx_11)))) (forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) ((P Xx_12) Xz))))))))))))) (ex a)) as proof of (P0 (fun (Xx:a)=> ((and (forall (Xx_10:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_10))))))->(X_0 Xx_10)))) ((ex a) (fun (Xx_12:a)=> ((and (forall (Xx_11:a), ((forall (X0:(a->Prop)), (((and (X0 Xx_12)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_11))))))->(X Xx_11)))) (forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) ((P Xx_12) Xz)))))))))))))
% Found eta_expansion0000:=(eta_expansion000 (ex a)):(((ex a) (fun (Xx:a)=> ((and (forall (Xx_9:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_9))))))->(X Xx_9)))) (X_0 ((P Xx) Xz)))))->((ex a) (fun (x:a)=> ((and (forall (Xx_9:a), ((forall (X0:(a->Prop)), (((and (X0 x)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_9))))))->(X Xx_9)))) (X_0 ((P x) Xz))))))
% Found (eta_expansion000 (ex a)) as proof of (P0 (fun (Xx:a)=> ((and (forall (Xx_9:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_9))))))->(X Xx_9)))) (X_0 ((P Xx) Xz)))))
% Found ((eta_expansion00 (fun (Xx:a)=> ((and (forall (Xx_9:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_9))))))->(X Xx_9)))) (X_0 ((P Xx) Xz))))) (ex a)) as proof of (P0 (fun (Xx:a)=> ((and (forall (Xx_9:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_9))))))->(X Xx_9)))) (X_0 ((P Xx) Xz)))))
% Found (((eta_expansion0 Prop) (fun (Xx:a)=> ((and (forall (Xx_9:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_9))))))->(X Xx_9)))) (X_0 ((P Xx) Xz))))) (ex a)) as proof of (P0 (fun (Xx:a)=> ((and (forall (Xx_9:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_9))))))->(X Xx_9)))) (X_0 ((P Xx) Xz)))))
% Found ((((eta_expansion a) Prop) (fun (Xx:a)=> ((and (forall (Xx_9:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_9))))))->(X Xx_9)))) (X_0 ((P Xx) Xz))))) (ex a)) as proof of (P0 (fun (Xx:a)=> ((and (forall (Xx_9:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_9))))))->(X Xx_9)))) (X_0 ((P Xx) Xz)))))
% Found ((((eta_expansion a) Prop) (fun (Xx:a)=> ((and (forall (Xx_9:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_9))))))->(X Xx_9)))) (X_0 ((P Xx) Xz))))) (ex a)) as proof of (P0 (fun (Xx:a)=> ((and (forall (Xx_9:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_9))))))->(X Xx_9)))) (X_0 ((P Xx) Xz)))))
% Found eta_expansion0000:=(eta_expansion000 (ex a)):(((ex a) (fun (Xx:a)=> ((and (forall (Xx_10:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_10))))))->(X_0 Xx_10)))) ((ex a) (fun (Xx_12:a)=> ((and (forall (Xx_11:a), ((forall (X0:(a->Prop)), (((and (X0 Xx_12)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_11))))))->(X Xx_11)))) (forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) ((P Xx_12) Xz)))))))))))))->((ex a) (fun (x:a)=> ((and (forall (Xx_10:a), ((forall (X0:(a->Prop)), (((and (X0 x)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_10))))))->(X_0 Xx_10)))) ((ex a) (fun (Xx_12:a)=> ((and (forall (Xx_11:a), ((forall (X0:(a->Prop)), (((and (X0 Xx_12)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_11))))))->(X Xx_11)))) (forall (X0:(a->Prop)), (((and (X0 x)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) ((P Xx_12) Xz))))))))))))))
% Found (eta_expansion000 (ex a)) as proof of (P0 (fun (Xx:a)=> ((and (forall (Xx_10:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_10))))))->(X_0 Xx_10)))) ((ex a) (fun (Xx_12:a)=> ((and (forall (Xx_11:a), ((forall (X0:(a->Prop)), (((and (X0 Xx_12)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_11))))))->(X Xx_11)))) (forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) ((P Xx_12) Xz)))))))))))))
% Found ((eta_expansion00 (fun (Xx:a)=> ((and (forall (Xx_10:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_10))))))->(X_0 Xx_10)))) ((ex a) (fun (Xx_12:a)=> ((and (forall (Xx_11:a), ((forall (X0:(a->Prop)), (((and (X0 Xx_12)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_11))))))->(X Xx_11)))) (forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) ((P Xx_12) Xz))))))))))))) (ex a)) as proof of (P0 (fun (Xx:a)=> ((and (forall (Xx_10:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_10))))))->(X_0 Xx_10)))) ((ex a) (fun (Xx_12:a)=> ((and (forall (Xx_11:a), ((forall (X0:(a->Prop)), (((and (X0 Xx_12)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_11))))))->(X Xx_11)))) (forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) ((P Xx_12) Xz)))))))))))))
% Found (((eta_expansion0 Prop) (fun (Xx:a)=> ((and (forall (Xx_10:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_10))))))->(X_0 Xx_10)))) ((ex a) (fun (Xx_12:a)=> ((and (forall (Xx_11:a), ((forall (X0:(a->Prop)), (((and (X0 Xx_12)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_11))))))->(X Xx_11)))) (forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) ((P Xx_12) Xz))))))))))))) (ex a)) as proof of (P0 (fun (Xx:a)=> ((and (forall (Xx_10:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_10))))))->(X_0 Xx_10)))) ((ex a) (fun (Xx_12:a)=> ((and (forall (Xx_11:a), ((forall (X0:(a->Prop)), (((and (X0 Xx_12)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_11))))))->(X Xx_11)))) (forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) ((P Xx_12) Xz)))))))))))))
% Found ((((eta_expansion a) Prop) (fun (Xx:a)=> ((and (forall (Xx_10:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_10))))))->(X_0 Xx_10)))) ((ex a) (fun (Xx_12:a)=> ((and (forall (Xx_11:a), ((forall (X0:(a->Prop)), (((and (X0 Xx_12)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_11))))))->(X Xx_11)))) (forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) ((P Xx_12) Xz))))))))))))) (ex a)) as proof of (P0 (fun (Xx:a)=> ((and (forall (Xx_10:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_10))))))->(X_0 Xx_10)))) ((ex a) (fun (Xx_12:a)=> ((and (forall (Xx_11:a), ((forall (X0:(a->Prop)), (((and (X0 Xx_12)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_11))))))->(X Xx_11)))) (forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) ((P Xx_12) Xz)))))))))))))
% Found ((((eta_expansion a) Prop) (fun (Xx:a)=> ((and (forall (Xx_10:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_10))))))->(X_0 Xx_10)))) ((ex a) (fun (Xx_12:a)=> ((and (forall (Xx_11:a), ((forall (X0:(a->Prop)), (((and (X0 Xx_12)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_11))))))->(X Xx_11)))) (forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) ((P Xx_12) Xz))))))))))))) (ex a)) as proof of (P0 (fun (Xx:a)=> ((and (forall (Xx_10:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_10))))))->(X_0 Xx_10)))) ((ex a) (fun (Xx_12:a)=> ((and (forall (Xx_11:a), ((forall (X0:(a->Prop)), (((and (X0 Xx_12)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_11))))))->(X Xx_11)))) (forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) ((P Xx_12) Xz)))))))))))))
% Found eta_expansion0000:=(eta_expansion000 (ex a)):(((ex a) (fun (Xx:a)=> ((and (forall (Xx_10:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_10))))))->(X_0 Xx_10)))) ((ex a) (fun (Xx_12:a)=> ((and (forall (Xx_11:a), ((forall (X0:(a->Prop)), (((and (X0 Xx_12)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_11))))))->(X Xx_11)))) (forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) ((P Xx_12) Xz)))))))))))))->((ex a) (fun (x:a)=> ((and (forall (Xx_10:a), ((forall (X0:(a->Prop)), (((and (X0 x)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_10))))))->(X_0 Xx_10)))) ((ex a) (fun (Xx_12:a)=> ((and (forall (Xx_11:a), ((forall (X0:(a->Prop)), (((and (X0 Xx_12)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_11))))))->(X Xx_11)))) (forall (X0:(a->Prop)), (((and (X0 x)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) ((P Xx_12) Xz))))))))))))))
% Found (eta_expansion000 (ex a)) as proof of (P0 (fun (Xx:a)=> ((and (forall (Xx_10:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_10))))))->(X_0 Xx_10)))) ((ex a) (fun (Xx_12:a)=> ((and (forall (Xx_11:a), ((forall (X0:(a->Prop)), (((and (X0 Xx_12)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_11))))))->(X Xx_11)))) (forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) ((P Xx_12) Xz)))))))))))))
% Found ((eta_expansion00 (fun (Xx:a)=> ((and (forall (Xx_10:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_10))))))->(X_0 Xx_10)))) ((ex a) (fun (Xx_12:a)=> ((and (forall (Xx_11:a), ((forall (X0:(a->Prop)), (((and (X0 Xx_12)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_11))))))->(X Xx_11)))) (forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) ((P Xx_12) Xz))))))))))))) (ex a)) as proof of (P0 (fun (Xx:a)=> ((and (forall (Xx_10:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_10))))))->(X_0 Xx_10)))) ((ex a) (fun (Xx_12:a)=> ((and (forall (Xx_11:a), ((forall (X0:(a->Prop)), (((and (X0 Xx_12)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_11))))))->(X Xx_11)))) (forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) ((P Xx_12) Xz)))))))))))))
% Found (((eta_expansion0 Prop) (fun (Xx:a)=> ((and (forall (Xx_10:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_10))))))->(X_0 Xx_10)))) ((ex a) (fun (Xx_12:a)=> ((and (forall (Xx_11:a), ((forall (X0:(a->Prop)), (((and (X0 Xx_12)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_11))))))->(X Xx_11)))) (forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) ((P Xx_12) Xz))))))))))))) (ex a)) as proof of (P0 (fun (Xx:a)=> ((and (forall (Xx_10:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_10))))))->(X_0 Xx_10)))) ((ex a) (fun (Xx_12:a)=> ((and (forall (Xx_11:a), ((forall (X0:(a->Prop)), (((and (X0 Xx_12)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_11))))))->(X Xx_11)))) (forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) ((P Xx_12) Xz)))))))))))))
% Found ((((eta_expansion a) Prop) (fun (Xx:a)=> ((and (forall (Xx_10:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_10))))))->(X_0 Xx_10)))) ((ex a) (fun (Xx_12:a)=> ((and (forall (Xx_11:a), ((forall (X0:(a->Prop)), (((and (X0 Xx_12)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_11))))))->(X Xx_11)))) (forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) ((P Xx_12) Xz))))))))))))) (ex a)) as proof of (P0 (fun (Xx:a)=> ((and (forall (Xx_10:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_10))))))->(X_0 Xx_10)))) ((ex a) (fun (Xx_12:a)=> ((and (forall (Xx_11:a), ((forall (X0:(a->Prop)), (((and (X0 Xx_12)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_11))))))->(X Xx_11)))) (forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) ((P Xx_12) Xz)))))))))))))
% Found ((((eta_expansion a) Prop) (fun (Xx:a)=> ((and (forall (Xx_10:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_10))))))->(X_0 Xx_10)))) ((ex a) (fun (Xx_12:a)=> ((and (forall (Xx_11:a), ((forall (X0:(a->Prop)), (((and (X0 Xx_12)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_11))))))->(X Xx_11)))) (forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) ((P Xx_12) Xz))))))))))))) (ex a)) as proof of (P0 (fun (Xx:a)=> ((and (forall (Xx_10:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_10))))))->(X_0 Xx_10)))) ((ex a) (fun (Xx_12:a)=> ((and (forall (Xx_11:a), ((forall (X0:(a->Prop)), (((and (X0 Xx_12)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_11))))))->(X Xx_11)))) (forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) ((P Xx_12) Xz)))))))))))))
% Found eta_expansion0000:=(eta_expansion000 (ex a)):(((ex a) (fun (Xx:a)=> ((and (forall (Xx_9:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_9))))))->(X Xx_9)))) (X_0 ((P Xx) Xz)))))->((ex a) (fun (x:a)=> ((and (forall (Xx_9:a), ((forall (X0:(a->Prop)), (((and (X0 x)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_9))))))->(X Xx_9)))) (X_0 ((P x) Xz))))))
% Found (eta_expansion000 (ex a)) as proof of (P0 (fun (Xx:a)=> ((and (forall (Xx_9:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_9))))))->(X Xx_9)))) (X_0 ((P Xx) Xz)))))
% Found ((eta_expansion00 (fun (Xx:a)=> ((and (forall (Xx_9:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_9))))))->(X Xx_9)))) (X_0 ((P Xx) Xz))))) (ex a)) as proof of (P0 (fun (Xx:a)=> ((and (forall (Xx_9:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_9))))))->(X Xx_9)))) (X_0 ((P Xx) Xz)))))
% Found (((eta_expansion0 Prop) (fun (Xx:a)=> ((and (forall (Xx_9:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_9))))))->(X Xx_9)))) (X_0 ((P Xx) Xz))))) (ex a)) as proof of (P0 (fun (Xx:a)=> ((and (forall (Xx_9:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_9))))))->(X Xx_9)))) (X_0 ((P Xx) Xz)))))
% Found ((((eta_expansion a) Prop) (fun (Xx:a)=> ((and (forall (Xx_9:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_9))))))->(X Xx_9)))) (X_0 ((P Xx) Xz))))) (ex a)) as proof of (P0 (fun (Xx:a)=> ((and (forall (Xx_9:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_9))))))->(X Xx_9)))) (X_0 ((P Xx) Xz)))))
% Found ((((eta_expansion a) Prop) (fun (Xx:a)=> ((and (forall (Xx_9:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_9))))))->(X Xx_9)))) (X_0 ((P Xx) Xz))))) (ex a)) as proof of (P0 (fun (Xx:a)=> ((and (forall (Xx_9:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_9))))))->(X Xx_9)))) (X_0 ((P Xx) Xz)))))
% Found eta_expansion0000:=(eta_expansion000 (ex a)):(((ex a) (fun (Xx:a)=> ((and (forall (Xx_9:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_9))))))->(X Xx_9)))) (X_0 ((P Xx) Xz)))))->((ex a) (fun (x:a)=> ((and (forall (Xx_9:a), ((forall (X0:(a->Prop)), (((and (X0 x)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_9))))))->(X Xx_9)))) (X_0 ((P x) Xz))))))
% Found (eta_expansion000 (ex a)) as proof of (P0 (fun (Xx:a)=> ((and (forall (Xx_9:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_9))))))->(X Xx_9)))) (X_0 ((P Xx) Xz)))))
% Found ((eta_expansion00 (fun (Xx:a)=> ((and (forall (Xx_9:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_9))))))->(X Xx_9)))) (X_0 ((P Xx) Xz))))) (ex a)) as proof of (P0 (fun (Xx:a)=> ((and (forall (Xx_9:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_9))))))->(X Xx_9)))) (X_0 ((P Xx) Xz)))))
% Found (((eta_expansion0 Prop) (fun (Xx:a)=> ((and (forall (Xx_9:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_9))))))->(X Xx_9)))) (X_0 ((P Xx) Xz))))) (ex a)) as proof of (P0 (fun (Xx:a)=> ((and (forall (Xx_9:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_9))))))->(X Xx_9)))) (X_0 ((P Xx) Xz)))))
% Found ((((eta_expansion a) Prop) (fun (Xx:a)=> ((and (forall (Xx_9:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_9))))))->(X Xx_9)))) (X_0 ((P Xx) Xz))))) (ex a)) as proof of (P0 (fun (Xx:a)=> ((and (forall (Xx_9:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_9))))))->(X Xx_9)))) (X_0 ((P Xx) Xz)))))
% Found ((((eta_expansion a) Prop) (fun (Xx:a)=> ((and (forall (Xx_9:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_9))))))->(X Xx_9)))) (X_0 ((P Xx) Xz))))) (ex a)) as proof of (P0 (fun (Xx:a)=> ((and (forall (Xx_9:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_9))))))->(X Xx_9)))) (X_0 ((P Xx) Xz)))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 (ex a)):(((ex a) (fun (Xx:a)=> ((and (forall (Xx_9:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_9))))))->(X Xx_9)))) (X_0 ((P Xx) Xz)))))->((ex a) (fun (x:a)=> ((and (forall (Xx_9:a), ((forall (X0:(a->Prop)), (((and (X0 x)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_9))))))->(X Xx_9)))) (X_0 ((P x) Xz))))))
% Found (eta_expansion_dep000 (ex a)) as proof of (P0 (fun (Xx:a)=> ((and (forall (Xx_9:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_9))))))->(X Xx_9)))) (X_0 ((P Xx) Xz)))))
% Found ((eta_expansion_dep00 (fun (Xx:a)=> ((and (forall (Xx_9:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_9))))))->(X Xx_9)))) (X_0 ((P Xx) Xz))))) (ex a)) as proof of (P0 (fun (Xx:a)=> ((and (forall (Xx_9:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_9))))))->(X Xx_9)))) (X_0 ((P Xx) Xz)))))
% Found (((eta_expansion_dep0 (fun (x3:a)=> Prop)) (fun (Xx:a)=> ((and (forall (Xx_9:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_9))))))->(X Xx_9)))) (X_0 ((P Xx) Xz))))) (ex a)) as proof of (P0 (fun (Xx:a)=> ((and (forall (Xx_9:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_9))))))->(X Xx_9)))) (X_0 ((P Xx) Xz)))))
% Found ((((eta_expansion_dep a) (fun (x3:a)=> Prop)) (fun (Xx:a)=> ((and (forall (Xx_9:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_9))))))->(X Xx_9)))) (X_0 ((P Xx) Xz))))) (ex a)) as proof of (P0 (fun (Xx:a)=> ((and (forall (Xx_9:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_9))))))->(X Xx_9)))) (X_0 ((P Xx) Xz)))))
% Found ((((eta_expansion_dep a) (fun (x3:a)=> Prop)) (fun (Xx:a)=> ((and (forall (Xx_9:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_9))))))->(X Xx_9)))) (X_0 ((P Xx) Xz))))) (ex a)) as proof of (P0 (fun (Xx:a)=> ((and (forall (Xx_9:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_9))))))->(X Xx_9)))) (X_0 ((P Xx) Xz)))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 (ex a)):(((ex a) (fun (Xx:a)=> ((and (forall (Xx_10:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_10))))))->(X_0 Xx_10)))) ((ex a) (fun (Xx_12:a)=> ((and (forall (Xx_11:a), ((forall (X0:(a->Prop)), (((and (X0 Xx_12)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_11))))))->(X Xx_11)))) (forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) ((P Xx_12) Xz)))))))))))))->((ex a) (fun (x:a)=> ((and (forall (Xx_10:a), ((forall (X0:(a->Prop)), (((and (X0 x)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_10))))))->(X_0 Xx_10)))) ((ex a) (fun (Xx_12:a)=> ((and (forall (Xx_11:a), ((forall (X0:(a->Prop)), (((and (X0 Xx_12)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_11))))))->(X Xx_11)))) (forall (X0:(a->Prop)), (((and (X0 x)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) ((P Xx_12) Xz))))))))))))))
% Found (eta_expansion_dep000 (ex a)) as proof of (P0 (fun (Xx:a)=> ((and (forall (Xx_10:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_10))))))->(X_0 Xx_10)))) ((ex a) (fun (Xx_12:a)=> ((and (forall (Xx_11:a), ((forall (X0:(a->Prop)), (((and (X0 Xx_12)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_11))))))->(X Xx_11)))) (forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) ((P Xx_12) Xz)))))))))))))
% Found ((eta_expansion_dep00 (fun (Xx:a)=> ((and (forall (Xx_10:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_10))))))->(X_0 Xx_10)))) ((ex a) (fun (Xx_12:a)=> ((and (forall (Xx_11:a), ((forall (X0:(a->Prop)), (((and (X0 Xx_12)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_11))))))->(X Xx_11)))) (forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) ((P Xx_12) Xz))))))))))))) (ex a)) as proof of (P0 (fun (Xx:a)=> ((and (forall (Xx_10:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_10))))))->(X_0 Xx_10)))) ((ex a) (fun (Xx_12:a)=> ((and (forall (Xx_11:a), ((forall (X0:(a->Prop)), (((and (X0 Xx_12)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_11))))))->(X Xx_11)))) (forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) ((P Xx_12) Xz)))))))))))))
% Found (((eta_expansion_dep0 (fun (x3:a)=> Prop)) (fun (Xx:a)=> ((and (forall (Xx_10:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_10))))))->(X_0 Xx_10)))) ((ex a) (fun (Xx_12:a)=> ((and (forall (Xx_11:a), ((forall (X0:(a->Prop)), (((and (X0 Xx_12)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_11))))))->(X Xx_11)))) (forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) ((P Xx_12) Xz))))))))))))) (ex a)) as proof of (P0 (fun (Xx:a)=> ((and (forall (Xx_10:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_10))))))->(X_0 Xx_10)))) ((ex a) (fun (Xx_12:a)=> ((and (forall (Xx_11:a), ((forall (X0:(a->Prop)), (((and (X0 Xx_12)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_11))))))->(X Xx_11)))) (forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) ((P Xx_12) Xz)))))))))))))
% Found ((((eta_expansion_dep a) (fun (x3:a)=> Prop)) (fun (Xx:a)=> ((and (forall (Xx_10:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_10))))))->(X_0 Xx_10)))) ((ex a) (fun (Xx_12:a)=> ((and (forall (Xx_11:a), ((forall (X0:(a->Prop)), (((and (X0 Xx_12)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_11))))))->(X Xx_11)))) (forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) ((P Xx_12) Xz))))))))))))) (ex a)) as proof of (P0 (fun (Xx:a)=> ((and (forall (Xx_10:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_10))))))->(X_0 Xx_10)))) ((ex a) (fun (Xx_12:a)=> ((and (forall (Xx_11:a), ((forall (X0:(a->Prop)), (((and (X0 Xx_12)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_11))))))->(X Xx_11)))) (forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) ((P Xx_12) Xz)))))))))))))
% Found ((((eta_expansion_dep a) (fun (x3:a)=> Prop)) (fun (Xx:a)=> ((and (forall (Xx_10:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_10))))))->(X_0 Xx_10)))) ((ex a) (fun (Xx_12:a)=> ((and (forall (Xx_11:a), ((forall (X0:(a->Prop)), (((and (X0 Xx_12)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_11))))))->(X Xx_11)))) (forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) ((P Xx_12) Xz))))))))))))) (ex a)) as proof of (P0 (fun (Xx:a)=> ((and (forall (Xx_10:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_10))))))->(X_0 Xx_10)))) ((ex a) (fun (Xx_12:a)=> ((and (forall (Xx_11:a), ((forall (X0:(a->Prop)), (((and (X0 Xx_12)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_11))))))->(X Xx_11)))) (forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) ((P Xx_12) Xz)))))))))))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 (ex a)):(((ex a) (fun (Xx:a)=> ((and (forall (Xx_9:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_9))))))->(X Xx_9)))) (X_0 ((P Xx) Xz)))))->((ex a) (fun (x:a)=> ((and (forall (Xx_9:a), ((forall (X0:(a->Prop)), (((and (X0 x)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_9))))))->(X Xx_9)))) (X_0 ((P x) Xz))))))
% Found (eta_expansion_dep000 (ex a)) as proof of (P0 (fun (Xx:a)=> ((and (forall (Xx_9:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_9))))))->(X Xx_9)))) (X_0 ((P Xx) Xz)))))
% Found ((eta_expansion_dep00 (fun (Xx:a)=> ((and (forall (Xx_9:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_9))))))->(X Xx_9)))) (X_0 ((P Xx) Xz))))) (ex a)) as proof of (P0 (fun (Xx:a)=> ((and (forall (Xx_9:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_9))))))->(X Xx_9)))) (X_0 ((P Xx) Xz)))))
% Found (((eta_expansion_dep0 (fun (x3:a)=> Prop)) (fun (Xx:a)=> ((and (forall (Xx_9:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_9))))))->(X Xx_9)))) (X_0 ((P Xx) Xz))))) (ex a)) as proof of (P0 (fun (Xx:a)=> ((and (forall (Xx_9:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_9))))))->(X Xx_9)))) (X_0 ((P Xx) Xz)))))
% Found ((((eta_expansion_dep a) (fun (x3:a)=> Prop)) (fun (Xx:a)=> ((and (forall (Xx_9:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_9))))))->(X Xx_9)))) (X_0 ((P Xx) Xz))))) (ex a)) as proof of (P0 (fun (Xx:a)=> ((and (forall (Xx_9:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_9))))))->(X Xx_9)))) (X_0 ((P Xx) Xz)))))
% Found ((((eta_expansion_dep a) (fun (x3:a)=> Prop)) (fun (Xx:a)=> ((and (forall (Xx_9:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_9))))))->(X Xx_9)))) (X_0 ((P Xx) Xz))))) (ex a)) as proof of (P0 (fun (Xx:a)=> ((and (forall (Xx_9:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_9))))))->(X Xx_9)))) (X_0 ((P Xx) Xz)))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 (ex a)):(((ex a) (fun (Xx:a)=> ((and (forall (Xx_10:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_10))))))->(X_0 Xx_10)))) ((ex a) (fun (Xx_12:a)=> ((and (forall (Xx_11:a), ((forall (X0:(a->Prop)), (((and (X0 Xx_12)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_11))))))->(X Xx_11)))) (forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) ((P Xx_12) Xz)))))))))))))->((ex a) (fun (x:a)=> ((and (forall (Xx_10:a), ((forall (X0:(a->Prop)), (((and (X0 x)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_10))))))->(X_0 Xx_10)))) ((ex a) (fun (Xx_12:a)=> ((and (forall (Xx_11:a), ((forall (X0:(a->Prop)), (((and (X0 Xx_12)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_11))))))->(X Xx_11)))) (forall (X0:(a->Prop)), (((and (X0 x)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) ((P Xx_12) Xz))))))))))))))
% Found (eta_expansion_dep000 (ex a)) as proof of (P0 (fun (Xx:a)=> ((and (forall (Xx_10:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_10))))))->(X_0 Xx_10)))) ((ex a) (fun (Xx_12:a)=> ((and (forall (Xx_11:a), ((forall (X0:(a->Prop)), (((and (X0 Xx_12)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_11))))))->(X Xx_11)))) (forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) ((P Xx_12) Xz)))))))))))))
% Found ((eta_expansion_dep00 (fun (Xx:a)=> ((and (forall (Xx_10:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_10))))))->(X_0 Xx_10)))) ((ex a) (fun (Xx_12:a)=> ((and (forall (Xx_11:a), ((forall (X0:(a->Prop)), (((and (X0 Xx_12)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_11))))))->(X Xx_11)))) (forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) ((P Xx_12) Xz))))))))))))) (ex a)) as proof of (P0 (fun (Xx:a)=> ((and (forall (Xx_10:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_10))))))->(X_0 Xx_10)))) ((ex a) (fun (Xx_12:a)=> ((and (forall (Xx_11:a), ((forall (X0:(a->Prop)), (((and (X0 Xx_12)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_11))))))->(X Xx_11)))) (forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) ((P Xx_12) Xz)))))))))))))
% Found (((eta_expansion_dep0 (fun (x3:a)=> Prop)) (fun (Xx:a)=> ((and (forall (Xx_10:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_10))))))->(X_0 Xx_10)))) ((ex a) (fun (Xx_12:a)=> ((and (forall (Xx_11:a), ((forall (X0:(a->Prop)), (((and (X0 Xx_12)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_11))))))->(X Xx_11)))) (forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) ((P Xx_12) Xz))))))))))))) (ex a)) as proof of (P0 (fun (Xx:a)=> ((and (forall (Xx_10:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_10))))))->(X_0 Xx_10)))) ((ex a) (fun (Xx_12:a)=> ((and (forall (Xx_11:a), ((forall (X0:(a->Prop)), (((and (X0 Xx_12)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_11))))))->(X Xx_11)))) (forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) ((P Xx_12) Xz)))))))))))))
% Found ((((eta_expansion_dep a) (fun (x3:a)=> Prop)) (fun (Xx:a)=> ((and (forall (Xx_10:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_10))))))->(X_0 Xx_10)))) ((ex a) (fun (Xx_12:a)=> ((and (forall (Xx_11:a), ((forall (X0:(a->Prop)), (((and (X0 Xx_12)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_11))))))->(X Xx_11)))) (forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) ((P Xx_12) Xz))))))))))))) (ex a)) as proof of (P0 (fun (Xx:a)=> ((and (forall (Xx_10:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_10))))))->(X_0 Xx_10)))) ((ex a) (fun (Xx_12:a)=> ((and (forall (Xx_11:a), ((forall (X0:(a->Prop)), (((and (X0 Xx_12)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_11))))))->(X Xx_11)))) (forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) ((P Xx_12) Xz)))))))))))))
% Found ((((eta_expansion_dep a) (fun (x3:a)=> Prop)) (fun (Xx:a)=> ((and (forall (Xx_10:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_10))))))->(X_0 Xx_10)))) ((ex a) (fun (Xx_12:a)=> ((and (forall (Xx_11:a), ((forall (X0:(a->Prop)), (((and (X0 Xx_12)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_11))))))->(X Xx_11)))) (forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) ((P Xx_12) Xz))))))))))))) (ex a)) as proof of (P0 (fun (Xx:a)=> ((and (forall (Xx_10:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_10))))))->(X_0 Xx_10)))) ((ex a) (fun (Xx_12:a)=> ((and (forall (Xx_11:a), ((forall (X0:(a->Prop)), (((and (X0 Xx_12)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_11))))))->(X Xx_11)))) (forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) ((P Xx_12) Xz)))))))))))))
% Found a_DUMMY:a
% Found a_DUMMY 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 a_DUMMY:a
% Found a_DUMMY as proof of a
% Found Xz:a
% Found Xz as proof of a
% Found Xz:a
% Found Xz as proof of a
% Found a_DUMMY:a
% Found a_DUMMY 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 a_DUMMY:a
% Found a_DUMMY as proof of a
% Found a_DUMMY:a
% Found a_DUMMY as proof of a
% Found a_DUMMY:a
% Found a_DUMMY 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 a_DUMMY:a
% Found a_DUMMY as proof of a
% Found a_DUMMY:a
% Found a_DUMMY as proof of a
% Found a_DUMMY:a
% Found a_DUMMY as proof of a
% Found a_DUMMY:a
% Found a_DUMMY as proof of a
% Found a_DUMMY:a
% Found a_DUMMY as proof of a
% Found Z:a
% Found Z as proof of a
% Found Z:a
% Found Z as proof of a
% Found Xz:a
% Found Xz as proof of a
% Found Xz:a
% Found Xz as proof of a
% Found a_DUMMY:a
% Found a_DUMMY as proof of a
% Found a_DUMMY:a
% Found a_DUMMY 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 (Xx:a)=> ((and (forall (Xx_10:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_10))))))->(X_0 Xx_10)))) ((ex a) (fun (Xx_12:a)=> ((and (forall (Xx_11:a), ((forall (X0:(a->Prop)), (((and (X0 Xx_12)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_11))))))->(X Xx_11)))) (forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) ((P Xx_12) Xz)))))))))))))->((ex a) (fun (x:a)=> ((and (forall (Xx_10:a), ((forall (X0:(a->Prop)), (((and (X0 x)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_10))))))->(X_0 Xx_10)))) ((ex a) (fun (Xx_12:a)=> ((and (forall (Xx_11:a), ((forall (X0:(a->Prop)), (((and (X0 Xx_12)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_11))))))->(X Xx_11)))) (forall (X0:(a->Prop)), (((and (X0 x)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) ((P Xx_12) Xz))))))))))))))
% Found (eta_expansion_dep000 (ex a)) as proof of (P0 (fun (Xx:a)=> ((and (forall (Xx_10:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_10))))))->(X_0 Xx_10)))) ((ex a) (fun (Xx_12:a)=> ((and (forall (Xx_11:a), ((forall (X0:(a->Prop)), (((and (X0 Xx_12)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_11))))))->(X Xx_11)))) (forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) ((P Xx_12) Xz)))))))))))))
% Found ((eta_expansion_dep00 (fun (Xx:a)=> ((and (forall (Xx_10:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_10))))))->(X_0 Xx_10)))) ((ex a) (fun (Xx_12:a)=> ((and (forall (Xx_11:a), ((forall (X0:(a->Prop)), (((and (X0 Xx_12)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_11))))))->(X Xx_11)))) (forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) ((P Xx_12) Xz))))))))))))) (ex a)) as proof of (P0 (fun (Xx:a)=> ((and (forall (Xx_10:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_10))))))->(X_0 Xx_10)))) ((ex a) (fun (Xx_12:a)=> ((and (forall (Xx_11:a), ((forall (X0:(a->Prop)), (((and (X0 Xx_12)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_11))))))->(X Xx_11)))) (forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) ((P Xx_12) Xz)))))))))))))
% Found (((eta_expansion_dep0 (fun (x3:a)=> Prop)) (fun (Xx:a)=> ((and (forall (Xx_10:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_10))))))->(X_0 Xx_10)))) ((ex a) (fun (Xx_12:a)=> ((and (forall (Xx_11:a), ((forall (X0:(a->Prop)), (((and (X0 Xx_12)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_11))))))->(X Xx_11)))) (forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) ((P Xx_12) Xz))))))))))))) (ex a)) as proof of (P0 (fun (Xx:a)=> ((and (forall (Xx_10:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_10))))))->(X_0 Xx_10)))) ((ex a) (fun (Xx_12:a)=> ((and (forall (Xx_11:a), ((forall (X0:(a->Prop)), (((and (X0 Xx_12)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_11))))))->(X Xx_11)))) (forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) ((P Xx_12) Xz)))))))))))))
% Found ((((eta_expansion_dep a) (fun (x3:a)=> Prop)) (fun (Xx:a)=> ((and (forall (Xx_10:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_10))))))->(X_0 Xx_10)))) ((ex a) (fun (Xx_12:a)=> ((and (forall (Xx_11:a), ((forall (X0:(a->Prop)), (((and (X0 Xx_12)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_11))))))->(X Xx_11)))) (forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) ((P Xx_12) Xz))))))))))))) (ex a)) as proof of (P0 (fun (Xx:a)=> ((and (forall (Xx_10:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_10))))))->(X_0 Xx_10)))) ((ex a) (fun (Xx_12:a)=> ((and (forall (Xx_11:a), ((forall (X0:(a->Prop)), (((and (X0 Xx_12)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_11))))))->(X Xx_11)))) (forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) ((P Xx_12) Xz)))))))))))))
% Found ((((eta_expansion_dep a) (fun (x3:a)=> Prop)) (fun (Xx:a)=> ((and (forall (Xx_10:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_10))))))->(X_0 Xx_10)))) ((ex a) (fun (Xx_12:a)=> ((and (forall (Xx_11:a), ((forall (X0:(a->Prop)), (((and (X0 Xx_12)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_11))))))->(X Xx_11)))) (forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) ((P Xx_12) Xz))))))))))))) (ex a)) as proof of (P0 (fun (Xx:a)=> ((and (forall (Xx_10:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_10))))))->(X_0 Xx_10)))) ((ex a) (fun (Xx_12:a)=> ((and (forall (Xx_11:a), ((forall (X0:(a->Prop)), (((and (X0 Xx_12)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_11))))))->(X Xx_11)))) (forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) ((P Xx_12) Xz)))))))))))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 (ex a)):(((ex a) (fun (Xx:a)=> ((and (forall (Xx_9:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_9))))))->(X Xx_9)))) (X_0 ((P Xx) Xz)))))->((ex a) (fun (x:a)=> ((and (forall (Xx_9:a), ((forall (X0:(a->Prop)), (((and (X0 x)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_9))))))->(X Xx_9)))) (X_0 ((P x) Xz))))))
% Found (eta_expansion_dep000 (ex a)) as proof of (P0 (fun (Xx:a)=> ((and (forall (Xx_9:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_9))))))->(X Xx_9)))) (X_0 ((P Xx) Xz)))))
% Found ((eta_expansion_dep00 (fun (Xx:a)=> ((and (forall (Xx_9:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_9))))))->(X Xx_9)))) (X_0 ((P Xx) Xz))))) (ex a)) as proof of (P0 (fun (Xx:a)=> ((and (forall (Xx_9:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_9))))))->(X Xx_9)))) (X_0 ((P Xx) Xz)))))
% Found (((eta_expansion_dep0 (fun (x3:a)=> Prop)) (fun (Xx:a)=> ((and (forall (Xx_9:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_9))))))->(X Xx_9)))) (X_0 ((P Xx) Xz))))) (ex a)) as proof of (P0 (fun (Xx:a)=> ((and (forall (Xx_9:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_9))))))->(X Xx_9)))) (X_0 ((P Xx) Xz)))))
% Found ((((eta_expansion_dep a) (fun (x3:a)=> Prop)) (fun (Xx:a)=> ((and (forall (Xx_9:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_9))))))->(X Xx_9)))) (X_0 ((P Xx) Xz))))) (ex a)) as proof of (P0 (fun (Xx:a)=> ((and (forall (Xx_9:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_9))))))->(X Xx_9)))) (X_0 ((P Xx) Xz)))))
% Found ((((eta_expansion_dep a) (fun (x3:a)=> Prop)) (fun (Xx:a)=> ((and (forall (Xx_9:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_9))))))->(X Xx_9)))) (X_0 ((P Xx) Xz))))) (ex a)) as proof of (P0 (fun (Xx:a)=> ((and (forall (Xx_9:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (L Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (R Xv)) Xx_9))))))->(X Xx_9))
% EOF
%------------------------------------------------------------------------------