%------------------------------------------------------------------------------
% File     : cocATP---0.2.0
% Problem  : SEV102^5 : TPTP v6.1.0. Released v4.0.0.
% Transfm  : none
% Format   : tptp:raw
% Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p

% Computer : n100.star.cs.uiowa.edu
% Model    : x86_64 x86_64
% CPU      : Intel(R) Xeon(R) CPU E5-2609 0 2.40GHz
% Memory   : 32286.75MB
% OS       : Linux 2.6.32-431.20.3.el6.x86_64
% CPULimit : 300s
% DateTime : Thu Jul 17 13:33:44 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  : SEV102^5 : TPTP v6.1.0. Released v4.0.0.
% % Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n100.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 08:04:46 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 0x1b97998>, <kernel.Type object at 0x1b978c0>) of role type named a_type
% Using role type
% Declaring a:Type
% FOF formula (forall (Xx:(a->Prop)) (Xy:(a->Prop)) (Xz:(a->Prop)), (((and ((ex (a->a)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_52:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) ((fun (Xx:a) (Xy:a)=> (((eq a) Xx) Xy)) Xy_52)))))))))) ((ex (a->a)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) ((fun (Xx:a) (Xy:a)=> (((eq a) Xx) Xy)) Xy_53))))))))))->((ex (a->a)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) ((fun (Xx:a) (Xy:a)=> (((eq a) Xx) Xy)) Xy_55))))))))))) of role conjecture named cEQP_1C_pme
% Conjecture to prove = (forall (Xx:(a->Prop)) (Xy:(a->Prop)) (Xz:(a->Prop)), (((and ((ex (a->a)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_52:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) ((fun (Xx:a) (Xy:a)=> (((eq a) Xx) Xy)) Xy_52)))))))))) ((ex (a->a)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) ((fun (Xx:a) (Xy:a)=> (((eq a) Xx) Xy)) Xy_53))))))))))->((ex (a->a)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) ((fun (Xx:a) (Xy:a)=> (((eq a) Xx) Xy)) Xy_55))))))))))):Prop
% Parameter a_DUMMY:a.
% We need to prove ['(forall (Xx:(a->Prop)) (Xy:(a->Prop)) (Xz:(a->Prop)), (((and ((ex (a->a)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_52:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) ((fun (Xx:a) (Xy:a)=> (((eq a) Xx) Xy)) Xy_52)))))))))) ((ex (a->a)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) ((fun (Xx:a) (Xy:a)=> (((eq a) Xx) Xy)) Xy_53))))))))))->((ex (a->a)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) ((fun (Xx:a) (Xy:a)=> (((eq a) Xx) Xy)) Xy_55)))))))))))']
% Parameter a:Type.
% Trying to prove (forall (Xx:(a->Prop)) (Xy:(a->Prop)) (Xz:(a->Prop)), (((and ((ex (a->a)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_52:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) ((fun (Xx:a) (Xy:a)=> (((eq a) Xx) Xy)) Xy_52)))))))))) ((ex (a->a)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) ((fun (Xx:a) (Xy:a)=> (((eq a) Xx) Xy)) Xy_53))))))))))->((ex (a->a)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) ((fun (Xx:a) (Xy:a)=> (((eq a) Xx) Xy)) Xy_55)))))))))))
% Found eq_ref000:=(eq_ref00 (ex (a->a))):(((ex (a->a)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_53) Xy00))))))))))->((ex (a->a)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_53) Xy00)))))))))))
% Found (eq_ref00 (ex (a->a))) as proof of (P (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0))))))))))
% Found ((eq_ref0 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_53) Xy00)))))))))) (ex (a->a))) as proof of (P (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0))))))))))
% Found (((eq_ref ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_53) Xy00)))))))))) (ex (a->a))) as proof of (P (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0))))))))))
% Found (((eq_ref ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_53) Xy00)))))))))) (ex (a->a))) as proof of (P (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0))))))))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 (ex (a->a))):(((ex (a->a)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_53) Xy00))))))))))->((ex (a->a)) (fun (x:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (x Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_53) Xy00)))))))))))
% Found (eta_expansion_dep000 (ex (a->a))) as proof of (P (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0))))))))))
% Found ((eta_expansion_dep00 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_53) Xy00)))))))))) (ex (a->a))) as proof of (P (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0))))))))))
% Found (((eta_expansion_dep0 (fun (x2:(a->a))=> Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_53) Xy00)))))))))) (ex (a->a))) as proof of (P (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0))))))))))
% Found ((((eta_expansion_dep (a->a)) (fun (x2:(a->a))=> Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_53) Xy00)))))))))) (ex (a->a))) as proof of (P (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0))))))))))
% Found ((((eta_expansion_dep (a->a)) (fun (x2:(a->a))=> Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_53) Xy00)))))))))) (ex (a->a))) as proof of (P (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0))))))))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 (ex (a->a))):(((ex (a->a)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_53) Xy00))))))))))->((ex (a->a)) (fun (x:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (x Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_53) Xy00)))))))))))
% Found (eta_expansion_dep000 (ex (a->a))) as proof of (P (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0))))))))))
% Found ((eta_expansion_dep00 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_53) Xy00)))))))))) (ex (a->a))) as proof of (P (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0))))))))))
% Found (((eta_expansion_dep0 (fun (x2:(a->a))=> Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_53) Xy00)))))))))) (ex (a->a))) as proof of (P (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0))))))))))
% Found ((((eta_expansion_dep (a->a)) (fun (x2:(a->a))=> Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_53) Xy00)))))))))) (ex (a->a))) as proof of (P (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0))))))))))
% Found ((((eta_expansion_dep (a->a)) (fun (x2:(a->a))=> Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_53) Xy00)))))))))) (ex (a->a))) as proof of (P (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0))))))))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_55) Xy)))))))))):(((eq ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_55) Xy)))))))))) (fun (x:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (x Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy:a)=> (((eq a) Xy_55) Xy))))))))))
% Found (eta_expansion_dep00 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_55) Xy)))))))))) as proof of (((eq ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_55) Xy)))))))))) b)
% Found ((eta_expansion_dep0 (fun (x1:(a->a))=> Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_55) Xy)))))))))) as proof of (((eq ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_55) Xy)))))))))) b)
% Found (((eta_expansion_dep (a->a)) (fun (x1:(a->a))=> Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_55) Xy)))))))))) as proof of (((eq ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_55) Xy)))))))))) b)
% Found (((eta_expansion_dep (a->a)) (fun (x1:(a->a))=> Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_55) Xy)))))))))) as proof of (((eq ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_55) Xy)))))))))) b)
% Found (((eta_expansion_dep (a->a)) (fun (x1:(a->a))=> Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_55) Xy)))))))))) as proof of (((eq ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_55) Xy)))))))))) b)
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (x0 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x0 Xx0))))) (fun (Xy:a)=> (((eq a) Xy_55) Xy)))))))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (x0 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x0 Xx0))))) (fun (Xy:a)=> (((eq a) Xy_55) Xy)))))))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (x0 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x0 Xx0))))) (fun (Xy:a)=> (((eq a) Xy_55) Xy)))))))))
% Found (fun (x0:(a->a))=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (x0 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x0 Xx0))))) (fun (Xy:a)=> (((eq a) Xy_55) Xy)))))))))
% Found (fun (x0:(a->a))=> ((eq_ref Prop) (f x0))) as proof of (forall (x:(a->a)), (((eq Prop) (f x)) ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (x Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy:a)=> (((eq a) Xy_55) Xy))))))))))
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (x0 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x0 Xx0))))) (fun (Xy:a)=> (((eq a) Xy_55) Xy)))))))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (x0 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x0 Xx0))))) (fun (Xy:a)=> (((eq a) Xy_55) Xy)))))))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (x0 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x0 Xx0))))) (fun (Xy:a)=> (((eq a) Xy_55) Xy)))))))))
% Found (fun (x0:(a->a))=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (x0 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x0 Xx0))))) (fun (Xy:a)=> (((eq a) Xy_55) Xy)))))))))
% Found (fun (x0:(a->a))=> ((eq_ref Prop) (f x0))) as proof of (forall (x:(a->a)), (((eq Prop) (f x)) ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (x Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy:a)=> (((eq a) Xy_55) Xy))))))))))
% Found x0:((ex (a->a)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_52:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_52) Xy))))))))))
% Instantiate: b:=(fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_52:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_52) Xy))))))))):((a->a)->Prop)
% Found x0 as proof of (P b)
% Found eta_expansion000:=(eta_expansion00 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_55) Xy)))))))))):(((eq ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_55) Xy)))))))))) (fun (x:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (x Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy:a)=> (((eq a) Xy_55) Xy))))))))))
% Found (eta_expansion00 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_55) Xy)))))))))) as proof of (((eq ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_55) Xy)))))))))) b)
% Found ((eta_expansion0 Prop) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_55) Xy)))))))))) as proof of (((eq ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_55) Xy)))))))))) b)
% Found (((eta_expansion (a->a)) Prop) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_55) Xy)))))))))) as proof of (((eq ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_55) Xy)))))))))) b)
% Found (((eta_expansion (a->a)) Prop) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_55) Xy)))))))))) as proof of (((eq ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_55) Xy)))))))))) b)
% Found (((eta_expansion (a->a)) Prop) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_55) Xy)))))))))) as proof of (((eq ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_55) Xy)))))))))) b)
% Found eta_expansion_dep0000:=(eta_expansion_dep000 (ex (a->a))):(((ex (a->a)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0))))))))))->((ex (a->a)) (fun (x:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (x Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0)))))))))))
% Found (eta_expansion_dep000 (ex (a->a))) as proof of (((ex (a->a)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0))))))))))->(P b))
% Found ((eta_expansion_dep00 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0)))))))))) (ex (a->a))) as proof of (((ex (a->a)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0))))))))))->(P b))
% Found (((eta_expansion_dep0 (fun (x2:(a->a))=> Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0)))))))))) (ex (a->a))) as proof of (((ex (a->a)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0))))))))))->(P b))
% Found ((((eta_expansion_dep (a->a)) (fun (x2:(a->a))=> Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0)))))))))) (ex (a->a))) as proof of (((ex (a->a)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0))))))))))->(P b))
% Found ((((eta_expansion_dep (a->a)) (fun (x2:(a->a))=> Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0)))))))))) (ex (a->a))) as proof of (((ex (a->a)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0))))))))))->(P b))
% Found (fun (x0:((ex (a->a)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_52:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_52) Xy)))))))))))=> ((((eta_expansion_dep (a->a)) (fun (x2:(a->a))=> Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0)))))))))) (ex (a->a)))) as proof of (((ex (a->a)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0))))))))))->(P b))
% Found (fun (x0:((ex (a->a)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_52:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_52) Xy)))))))))))=> ((((eta_expansion_dep (a->a)) (fun (x2:(a->a))=> Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0)))))))))) (ex (a->a)))) as proof of (((ex (a->a)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_52:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_52) Xy))))))))))->(((ex (a->a)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0))))))))))->(P b)))
% Found (and_rect00 (fun (x0:((ex (a->a)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_52:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_52) Xy)))))))))))=> ((((eta_expansion_dep (a->a)) (fun (x2:(a->a))=> Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0)))))))))) (ex (a->a))))) as proof of (P b)
% Found ((and_rect0 (P b)) (fun (x0:((ex (a->a)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_52:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_52) Xy)))))))))))=> ((((eta_expansion_dep (a->a)) (fun (x2:(a->a))=> Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0)))))))))) (ex (a->a))))) as proof of (P b)
% Found (((fun (P0:Type) (x0:(((ex (a->a)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_52:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_52) Xy))))))))))->(((ex (a->a)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0))))))))))->P0)))=> (((((and_rect ((ex (a->a)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_52:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_52) Xy0))))))))))) ((ex (a->a)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0))))))))))) P0) x0) x)) (P b)) (fun (x0:((ex (a->a)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_52:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_52) Xy0)))))))))))=> ((((eta_expansion_dep (a->a)) (fun (x2:(a->a))=> Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0)))))))))) (ex (a->a))))) as proof of (P b)
% Found (((fun (P0:Type) (x0:(((ex (a->a)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_52:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_52) Xy))))))))))->(((ex (a->a)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0))))))))))->P0)))=> (((((and_rect ((ex (a->a)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_52:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_52) Xy0))))))))))) ((ex (a->a)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0))))))))))) P0) x0) x)) (P b)) (fun (x0:((ex (a->a)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_52:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_52) Xy0)))))))))))=> ((((eta_expansion_dep (a->a)) (fun (x2:(a->a))=> Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0)))))))))) (ex (a->a))))) as proof of (P b)
% Found x0:((ex (a->a)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_52:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_52) Xy))))))))))
% Instantiate: f:=(fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_52:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_52) Xy))))))))):((a->a)->Prop)
% Found x0 as proof of (P f)
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (x2 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x2 Xx0))))) (fun (Xy:a)=> (((eq a) Xy_55) Xy)))))))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (x2 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x2 Xx0))))) (fun (Xy:a)=> (((eq a) Xy_55) Xy)))))))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (x2 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x2 Xx0))))) (fun (Xy:a)=> (((eq a) Xy_55) Xy)))))))))
% Found (fun (x2:(a->a))=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (x2 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x2 Xx0))))) (fun (Xy:a)=> (((eq a) Xy_55) Xy)))))))))
% Found (fun (x2:(a->a))=> ((eq_ref Prop) (f x2))) as proof of (forall (x:(a->a)), (((eq Prop) (f x)) ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (x Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy:a)=> (((eq a) Xy_55) Xy))))))))))
% Found x0:((ex (a->a)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_52:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_52) Xy))))))))))
% Instantiate: f:=(fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_52:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_52) Xy))))))))):((a->a)->Prop)
% Found x0 as proof of (P f)
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (x2 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x2 Xx0))))) (fun (Xy:a)=> (((eq a) Xy_55) Xy)))))))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (x2 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x2 Xx0))))) (fun (Xy:a)=> (((eq a) Xy_55) Xy)))))))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (x2 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x2 Xx0))))) (fun (Xy:a)=> (((eq a) Xy_55) Xy)))))))))
% Found (fun (x2:(a->a))=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (x2 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x2 Xx0))))) (fun (Xy:a)=> (((eq a) Xy_55) Xy)))))))))
% Found (fun (x2:(a->a))=> ((eq_ref Prop) (f x2))) as proof of (forall (x:(a->a)), (((eq Prop) (f x)) ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (x Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy:a)=> (((eq a) Xy_55) Xy))))))))))
% Found eq_ref000:=(eq_ref00 (ex (a->a))):(((ex (a->a)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0))))))))))->((ex (a->a)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0)))))))))))
% Found (eq_ref00 (ex (a->a))) as proof of (((ex (a->a)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0))))))))))->(P f))
% Found ((eq_ref0 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0)))))))))) (ex (a->a))) as proof of (((ex (a->a)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0))))))))))->(P f))
% Found (((eq_ref ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0)))))))))) (ex (a->a))) as proof of (((ex (a->a)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0))))))))))->(P f))
% Found (((eq_ref ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0)))))))))) (ex (a->a))) as proof of (((ex (a->a)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0))))))))))->(P f))
% Found (fun (x0:((ex (a->a)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_52:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_52) Xy)))))))))))=> (((eq_ref ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0)))))))))) (ex (a->a)))) as proof of (((ex (a->a)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0))))))))))->(P f))
% Found (fun (x0:((ex (a->a)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_52:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_52) Xy)))))))))))=> (((eq_ref ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0)))))))))) (ex (a->a)))) as proof of (((ex (a->a)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_52:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_52) Xy))))))))))->(((ex (a->a)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0))))))))))->(P f)))
% Found (and_rect00 (fun (x0:((ex (a->a)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_52:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_52) Xy)))))))))))=> (((eq_ref ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0)))))))))) (ex (a->a))))) as proof of (P f)
% Found ((and_rect0 (P f)) (fun (x0:((ex (a->a)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_52:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_52) Xy)))))))))))=> (((eq_ref ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0)))))))))) (ex (a->a))))) as proof of (P f)
% Found (((fun (P0:Type) (x0:(((ex (a->a)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_52:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_52) Xy))))))))))->(((ex (a->a)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0))))))))))->P0)))=> (((((and_rect ((ex (a->a)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_52:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_52) Xy0))))))))))) ((ex (a->a)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0))))))))))) P0) x0) x)) (P f)) (fun (x0:((ex (a->a)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_52:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_52) Xy0)))))))))))=> (((eq_ref ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0)))))))))) (ex (a->a))))) as proof of (P f)
% Found (((fun (P0:Type) (x0:(((ex (a->a)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_52:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_52) Xy))))))))))->(((ex (a->a)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0))))))))))->P0)))=> (((((and_rect ((ex (a->a)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_52:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_52) Xy0))))))))))) ((ex (a->a)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0))))))))))) P0) x0) x)) (P f)) (fun (x0:((ex (a->a)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_52:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_52) Xy0)))))))))))=> (((eq_ref ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0)))))))))) (ex (a->a))))) as proof of (P f)
% Found eq_ref000:=(eq_ref00 (ex (a->a))):(((ex (a->a)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0))))))))))->((ex (a->a)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0)))))))))))
% Found (eq_ref00 (ex (a->a))) as proof of (((ex (a->a)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0))))))))))->(P f))
% Found ((eq_ref0 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0)))))))))) (ex (a->a))) as proof of (((ex (a->a)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0))))))))))->(P f))
% Found (((eq_ref ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0)))))))))) (ex (a->a))) as proof of (((ex (a->a)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0))))))))))->(P f))
% Found (((eq_ref ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0)))))))))) (ex (a->a))) as proof of (((ex (a->a)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0))))))))))->(P f))
% Found (fun (x0:((ex (a->a)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_52:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_52) Xy)))))))))))=> (((eq_ref ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0)))))))))) (ex (a->a)))) as proof of (((ex (a->a)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0))))))))))->(P f))
% Found (fun (x0:((ex (a->a)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_52:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_52) Xy)))))))))))=> (((eq_ref ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0)))))))))) (ex (a->a)))) as proof of (((ex (a->a)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_52:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_52) Xy))))))))))->(((ex (a->a)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0))))))))))->(P f)))
% Found (and_rect00 (fun (x0:((ex (a->a)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_52:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_52) Xy)))))))))))=> (((eq_ref ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0)))))))))) (ex (a->a))))) as proof of (P f)
% Found ((and_rect0 (P f)) (fun (x0:((ex (a->a)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_52:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_52) Xy)))))))))))=> (((eq_ref ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0)))))))))) (ex (a->a))))) as proof of (P f)
% Found (((fun (P0:Type) (x0:(((ex (a->a)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_52:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_52) Xy))))))))))->(((ex (a->a)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0))))))))))->P0)))=> (((((and_rect ((ex (a->a)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_52:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_52) Xy0))))))))))) ((ex (a->a)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0))))))))))) P0) x0) x)) (P f)) (fun (x0:((ex (a->a)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_52:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_52) Xy0)))))))))))=> (((eq_ref ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0)))))))))) (ex (a->a))))) as proof of (P f)
% Found (((fun (P0:Type) (x0:(((ex (a->a)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_52:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_52) Xy))))))))))->(((ex (a->a)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0))))))))))->P0)))=> (((((and_rect ((ex (a->a)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_52:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_52) Xy0))))))))))) ((ex (a->a)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0))))))))))) P0) x0) x)) (P f)) (fun (x0:((ex (a->a)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_52:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_52) Xy0)))))))))))=> (((eq_ref ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0)))))))))) (ex (a->a))))) as proof of (P f)
% Found eq_ref00:=(eq_ref0 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0)))))))))):(((eq ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0)))))))))) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0))))))))))
% Found (eq_ref0 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0)))))))))) as proof of (((eq ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0)))))))))) b)
% Found ((eq_ref ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0)))))))))) as proof of (((eq ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0)))))))))) b)
% Found ((eq_ref ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0)))))))))) as proof of (((eq ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0)))))))))) b)
% Found ((eq_ref ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0)))))))))) as proof of (((eq ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0)))))))))) b)
% Found eta_expansion000:=(eta_expansion00 b):(((eq ((a->a)->Prop)) b) (fun (x:(a->a))=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq ((a->a)->Prop)) b) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_55) Xy))))))))))
% Found ((eta_expansion0 Prop) b) as proof of (((eq ((a->a)->Prop)) b) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_55) Xy))))))))))
% Found (((eta_expansion (a->a)) Prop) b) as proof of (((eq ((a->a)->Prop)) b) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_55) Xy))))))))))
% Found (((eta_expansion (a->a)) Prop) b) as proof of (((eq ((a->a)->Prop)) b) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_55) Xy))))))))))
% Found (((eta_expansion (a->a)) Prop) b) as proof of (((eq ((a->a)->Prop)) b) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_55) Xy))))))))))
% Found eta_expansion0000:=(eta_expansion000 (ex (a->a))):(((ex (a->a)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_53) Xy00))))))))))->((ex (a->a)) (fun (x:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (x Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_53) Xy00)))))))))))
% Found (eta_expansion000 (ex (a->a))) as proof of (P (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0))))))))))
% Found ((eta_expansion00 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_53) Xy00)))))))))) (ex (a->a))) as proof of (P (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0))))))))))
% Found (((eta_expansion0 Prop) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_53) Xy00)))))))))) (ex (a->a))) as proof of (P (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0))))))))))
% Found ((((eta_expansion (a->a)) Prop) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_53) Xy00)))))))))) (ex (a->a))) as proof of (P (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0))))))))))
% Found ((((eta_expansion (a->a)) Prop) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_53) Xy00)))))))))) (ex (a->a))) as proof of (P (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0))))))))))
% Found eta_expansion000:=(eta_expansion00 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0)))))))))):(((eq ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0)))))))))) (fun (x:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (x Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0))))))))))
% Found (eta_expansion00 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0)))))))))) as proof of (((eq ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0)))))))))) b)
% Found ((eta_expansion0 Prop) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0)))))))))) as proof of (((eq ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0)))))))))) b)
% Found (((eta_expansion (a->a)) Prop) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0)))))))))) as proof of (((eq ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0)))))))))) b)
% Found (((eta_expansion (a->a)) Prop) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0)))))))))) as proof of (((eq ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0)))))))))) b)
% Found (((eta_expansion (a->a)) Prop) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0)))))))))) as proof of (((eq ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0)))))))))) b)
% Found eta_expansion000:=(eta_expansion00 b):(((eq ((a->a)->Prop)) b) (fun (x:(a->a))=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq ((a->a)->Prop)) b) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_55) Xy))))))))))
% Found ((eta_expansion0 Prop) b) as proof of (((eq ((a->a)->Prop)) b) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_55) Xy))))))))))
% Found (((eta_expansion (a->a)) Prop) b) as proof of (((eq ((a->a)->Prop)) b) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_55) Xy))))))))))
% Found (((eta_expansion (a->a)) Prop) b) as proof of (((eq ((a->a)->Prop)) b) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_55) Xy))))))))))
% Found (((eta_expansion (a->a)) Prop) b) as proof of (((eq ((a->a)->Prop)) b) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_55) Xy))))))))))
% Found x0:((ex (a->a)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_52:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_52) Xy))))))))))
% Instantiate: b:=(fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_52:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_52) Xy))))))))):((a->a)->Prop)
% Found x0 as proof of (P b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_55) Xy)))))))))):(((eq ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_55) Xy)))))))))) (fun (x:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (x Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy:a)=> (((eq a) Xy_55) Xy))))))))))
% Found (eta_expansion_dep00 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_55) Xy)))))))))) as proof of (((eq ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_55) Xy)))))))))) b)
% Found ((eta_expansion_dep0 (fun (x5:(a->a))=> Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_55) Xy)))))))))) as proof of (((eq ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_55) Xy)))))))))) b)
% Found (((eta_expansion_dep (a->a)) (fun (x5:(a->a))=> Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_55) Xy)))))))))) as proof of (((eq ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_55) Xy)))))))))) b)
% Found (((eta_expansion_dep (a->a)) (fun (x5:(a->a))=> Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_55) Xy)))))))))) as proof of (((eq ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_55) Xy)))))))))) b)
% Found (((eta_expansion_dep (a->a)) (fun (x5:(a->a))=> Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_55) Xy)))))))))) as proof of (((eq ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_55) Xy)))))))))) b)
% Found x0:((ex (a->a)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_52:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_52) Xy))))))))))
% Instantiate: b:=(fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_52:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_52) Xy))))))))):((a->a)->Prop)
% Found x0 as proof of (P b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_55) Xy)))))))))):(((eq ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_55) Xy)))))))))) (fun (x:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (x Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy:a)=> (((eq a) Xy_55) Xy))))))))))
% Found (eta_expansion_dep00 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_55) Xy)))))))))) as proof of (((eq ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_55) Xy)))))))))) b)
% Found ((eta_expansion_dep0 (fun (x5:(a->a))=> Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_55) Xy)))))))))) as proof of (((eq ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_55) Xy)))))))))) b)
% Found (((eta_expansion_dep (a->a)) (fun (x5:(a->a))=> Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_55) Xy)))))))))) as proof of (((eq ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_55) Xy)))))))))) b)
% Found (((eta_expansion_dep (a->a)) (fun (x5:(a->a))=> Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_55) Xy)))))))))) as proof of (((eq ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_55) Xy)))))))))) b)
% Found (((eta_expansion_dep (a->a)) (fun (x5:(a->a))=> Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_55) Xy)))))))))) as proof of (((eq ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_55) Xy)))))))))) b)
% Found x1:((ex (a->a)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0))))))))))
% Instantiate: f:=(fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0))))))))):((a->a)->Prop)
% Found x1 as proof of (P f)
% Found eq_ref00:=(eq_ref0 (f x4)):(((eq Prop) (f x4)) (f x4))
% Found (eq_ref0 (f x4)) as proof of (((eq Prop) (f x4)) ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (x4 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x4 Xx0))))) (fun (Xy:a)=> (((eq a) Xy_55) Xy)))))))))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (x4 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x4 Xx0))))) (fun (Xy:a)=> (((eq a) Xy_55) Xy)))))))))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (x4 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x4 Xx0))))) (fun (Xy:a)=> (((eq a) Xy_55) Xy)))))))))
% Found (fun (x4:(a->a))=> ((eq_ref Prop) (f x4))) as proof of (((eq Prop) (f x4)) ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (x4 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x4 Xx0))))) (fun (Xy:a)=> (((eq a) Xy_55) Xy)))))))))
% Found (fun (x4:(a->a))=> ((eq_ref Prop) (f x4))) as proof of (forall (x:(a->a)), (((eq Prop) (f x)) ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (x Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy:a)=> (((eq a) Xy_55) Xy))))))))))
% Found x1:((ex (a->a)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0))))))))))
% Instantiate: f:=(fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0))))))))):((a->a)->Prop)
% Found x1 as proof of (P f)
% Found eq_ref00:=(eq_ref0 (f x4)):(((eq Prop) (f x4)) (f x4))
% Found (eq_ref0 (f x4)) as proof of (((eq Prop) (f x4)) ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (x4 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x4 Xx0))))) (fun (Xy:a)=> (((eq a) Xy_55) Xy)))))))))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (x4 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x4 Xx0))))) (fun (Xy:a)=> (((eq a) Xy_55) Xy)))))))))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (x4 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x4 Xx0))))) (fun (Xy:a)=> (((eq a) Xy_55) Xy)))))))))
% Found (fun (x4:(a->a))=> ((eq_ref Prop) (f x4))) as proof of (((eq Prop) (f x4)) ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (x4 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x4 Xx0))))) (fun (Xy:a)=> (((eq a) Xy_55) Xy)))))))))
% Found (fun (x4:(a->a))=> ((eq_ref Prop) (f x4))) as proof of (forall (x:(a->a)), (((eq Prop) (f x)) ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (x Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy:a)=> (((eq a) Xy_55) Xy))))))))))
% Found x1:((ex (a->a)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0))))))))))
% Instantiate: f:=(fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0))))))))):((a->a)->Prop)
% Found x1 as proof of (P f)
% Found eq_ref00:=(eq_ref0 (f x4)):(((eq Prop) (f x4)) (f x4))
% Found (eq_ref0 (f x4)) as proof of (((eq Prop) (f x4)) ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (x4 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x4 Xx0))))) (fun (Xy:a)=> (((eq a) Xy_55) Xy)))))))))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (x4 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x4 Xx0))))) (fun (Xy:a)=> (((eq a) Xy_55) Xy)))))))))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (x4 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x4 Xx0))))) (fun (Xy:a)=> (((eq a) Xy_55) Xy)))))))))
% Found (fun (x4:(a->a))=> ((eq_ref Prop) (f x4))) as proof of (((eq Prop) (f x4)) ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (x4 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x4 Xx0))))) (fun (Xy:a)=> (((eq a) Xy_55) Xy)))))))))
% Found (fun (x4:(a->a))=> ((eq_ref Prop) (f x4))) as proof of (forall (x:(a->a)), (((eq Prop) (f x)) ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (x Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy:a)=> (((eq a) Xy_55) Xy))))))))))
% Found x1:((ex (a->a)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0))))))))))
% Instantiate: f:=(fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0))))))))):((a->a)->Prop)
% Found x1 as proof of (P f)
% Found eq_ref00:=(eq_ref0 (f x4)):(((eq Prop) (f x4)) (f x4))
% Found (eq_ref0 (f x4)) as proof of (((eq Prop) (f x4)) ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (x4 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x4 Xx0))))) (fun (Xy:a)=> (((eq a) Xy_55) Xy)))))))))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (x4 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x4 Xx0))))) (fun (Xy:a)=> (((eq a) Xy_55) Xy)))))))))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (x4 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x4 Xx0))))) (fun (Xy:a)=> (((eq a) Xy_55) Xy)))))))))
% Found (fun (x4:(a->a))=> ((eq_ref Prop) (f x4))) as proof of (((eq Prop) (f x4)) ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (x4 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x4 Xx0))))) (fun (Xy:a)=> (((eq a) Xy_55) Xy)))))))))
% Found (fun (x4:(a->a))=> ((eq_ref Prop) (f x4))) as proof of (forall (x:(a->a)), (((eq Prop) (f x)) ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (x Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy:a)=> (((eq a) Xy_55) Xy))))))))))
% Found eq_ref000:=(eq_ref00 P0):((P0 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_55) Xy00))))))))))->(P0 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_55) Xy00)))))))))))
% Found (eq_ref00 P0) as proof of (P1 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_55) Xy0))))))))))
% Found ((eq_ref0 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_55) Xy00)))))))))) P0) as proof of (P1 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_55) Xy0))))))))))
% Found (((eq_ref ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_55) Xy00)))))))))) P0) as proof of (P1 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_55) Xy0))))))))))
% Found (((eq_ref ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_55) Xy00)))))))))) P0) as proof of (P1 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_55) Xy0))))))))))
% Found eq_ref000:=(eq_ref00 P0):((P0 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_55) Xy00))))))))))->(P0 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_55) Xy00)))))))))))
% Found (eq_ref00 P0) as proof of (P1 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_55) Xy0))))))))))
% Found ((eq_ref0 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_55) Xy00)))))))))) P0) as proof of (P1 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_55) Xy0))))))))))
% Found (((eq_ref ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_55) Xy00)))))))))) P0) as proof of (P1 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_55) Xy0))))))))))
% Found (((eq_ref ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_55) Xy00)))))))))) P0) as proof of (P1 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_55) Xy0))))))))))
% Found eta_expansion000:=(eta_expansion00 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_55) Xy0)))))))))):(((eq ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_55) Xy0)))))))))) (fun (x:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (x Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_55) Xy0))))))))))
% Found (eta_expansion00 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_55) Xy0)))))))))) as proof of (((eq ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_55) Xy0)))))))))) b)
% Found ((eta_expansion0 Prop) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_55) Xy0)))))))))) as proof of (((eq ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_55) Xy0)))))))))) b)
% Found (((eta_expansion (a->a)) Prop) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_55) Xy0)))))))))) as proof of (((eq ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_55) Xy0)))))))))) b)
% Found (((eta_expansion (a->a)) Prop) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_55) Xy0)))))))))) as proof of (((eq ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_55) Xy0)))))))))) b)
% Found (((eta_expansion (a->a)) Prop) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_55) Xy0)))))))))) as proof of (((eq ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_55) Xy0)))))))))) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq ((a->a)->Prop)) b) (fun (x:(a->a))=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq ((a->a)->Prop)) b) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0))))))))))
% Found ((eta_expansion_dep0 (fun (x2:(a->a))=> Prop)) b) as proof of (((eq ((a->a)->Prop)) b) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0))))))))))
% Found (((eta_expansion_dep (a->a)) (fun (x2:(a->a))=> Prop)) b) as proof of (((eq ((a->a)->Prop)) b) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0))))))))))
% Found (((eta_expansion_dep (a->a)) (fun (x2:(a->a))=> Prop)) b) as proof of (((eq ((a->a)->Prop)) b) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0))))))))))
% Found (((eta_expansion_dep (a->a)) (fun (x2:(a->a))=> Prop)) b) as proof of (((eq ((a->a)->Prop)) b) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0))))))))))
% Found eq_ref00:=(eq_ref0 b):(((eq ((a->a)->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq ((a->a)->Prop)) b) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0))))))))))
% Found ((eq_ref ((a->a)->Prop)) b) as proof of (((eq ((a->a)->Prop)) b) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0))))))))))
% Found ((eq_ref ((a->a)->Prop)) b) as proof of (((eq ((a->a)->Prop)) b) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0))))))))))
% Found ((eq_ref ((a->a)->Prop)) b) as proof of (((eq ((a->a)->Prop)) b) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0))))))))))
% Found eq_ref00:=(eq_ref0 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_55) Xy0)))))))))):(((eq ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_55) Xy0)))))))))) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_55) Xy0))))))))))
% Found (eq_ref0 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_55) Xy0)))))))))) as proof of (((eq ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_55) Xy0)))))))))) b)
% Found ((eq_ref ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_55) Xy0)))))))))) as proof of (((eq ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_55) Xy0)))))))))) b)
% Found ((eq_ref ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_55) Xy0)))))))))) as proof of (((eq ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_55) Xy0)))))))))) b)
% Found ((eq_ref ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_55) Xy0)))))))))) as proof of (((eq ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_55) Xy0)))))))))) b)
% Found eq_ref000:=(eq_ref00 P0):((P0 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_55) Xy00))))))))))->(P0 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_55) Xy00)))))))))))
% Found (eq_ref00 P0) as proof of (P1 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_55) Xy0))))))))))
% Found ((eq_ref0 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_55) Xy00)))))))))) P0) as proof of (P1 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_55) Xy0))))))))))
% Found (((eq_ref ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_55) Xy00)))))))))) P0) as proof of (P1 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_55) Xy0))))))))))
% Found (((eq_ref ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_55) Xy00)))))))))) P0) as proof of (P1 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_55) Xy0))))))))))
% Found eq_ref000:=(eq_ref00 P0):((P0 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_55) Xy00))))))))))->(P0 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_55) Xy00)))))))))))
% Found (eq_ref00 P0) as proof of (P1 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_55) Xy0))))))))))
% Found ((eq_ref0 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_55) Xy00)))))))))) P0) as proof of (P1 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_55) Xy0))))))))))
% Found (((eq_ref ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_55) Xy00)))))))))) P0) as proof of (P1 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_55) Xy0))))))))))
% Found (((eq_ref ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_55) Xy00)))))))))) P0) as proof of (P1 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_55) Xy0))))))))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P0):((P0 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_55) Xy00))))))))))->(P0 (fun (x:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (x Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_55) Xy00)))))))))))
% Found (eta_expansion_dep000 P0) as proof of (P1 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_55) Xy0))))))))))
% Found ((eta_expansion_dep00 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_55) Xy00)))))))))) P0) as proof of (P1 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_55) Xy0))))))))))
% Found (((eta_expansion_dep0 (fun (x2:(a->a))=> Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_55) Xy00)))))))))) P0) as proof of (P1 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_55) Xy0))))))))))
% Found ((((eta_expansion_dep (a->a)) (fun (x2:(a->a))=> Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_55) Xy00)))))))))) P0) as proof of (P1 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_55) Xy0))))))))))
% Found ((((eta_expansion_dep (a->a)) (fun (x2:(a->a))=> Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_55) Xy00)))))))))) P0) as proof of (P1 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_55) Xy0))))))))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P0):((P0 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_55) Xy00))))))))))->(P0 (fun (x:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (x Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_55) Xy00)))))))))))
% Found (eta_expansion_dep000 P0) as proof of (P1 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_55) Xy0))))))))))
% Found ((eta_expansion_dep00 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_55) Xy00)))))))))) P0) as proof of (P1 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_55) Xy0))))))))))
% Found (((eta_expansion_dep0 (fun (x2:(a->a))=> Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_55) Xy00)))))))))) P0) as proof of (P1 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_55) Xy0))))))))))
% Found ((((eta_expansion_dep (a->a)) (fun (x2:(a->a))=> Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_55) Xy00)))))))))) P0) as proof of (P1 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_55) Xy0))))))))))
% Found ((((eta_expansion_dep (a->a)) (fun (x2:(a->a))=> Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_55) Xy00)))))))))) P0) as proof of (P1 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_55) Xy0))))))))))
% Found eq_ref000:=(eq_ref00 P0):((P0 ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x1 Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_53) Xy00)))))))))->(P0 ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x1 Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_53) Xy00))))))))))
% Found (eq_ref00 P0) as proof of (P1 ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x1 Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0)))))))))
% Found ((eq_ref0 ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x1 Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_53) Xy00))))))))) P0) as proof of (P1 ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x1 Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0)))))))))
% Found (((eq_ref Prop) ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x1 Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_53) Xy00))))))))) P0) as proof of (P1 ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x1 Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0)))))))))
% Found (((eq_ref Prop) ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x1 Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_53) Xy00))))))))) P0) as proof of (P1 ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x1 Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0)))))))))
% Found eq_ref000:=(eq_ref00 P0):((P0 ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x1 Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_53) Xy00)))))))))->(P0 ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x1 Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_53) Xy00))))))))))
% Found (eq_ref00 P0) as proof of (P1 ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x1 Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0)))))))))
% Found ((eq_ref0 ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x1 Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_53) Xy00))))))))) P0) as proof of (P1 ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x1 Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0)))))))))
% Found (((eq_ref Prop) ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x1 Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_53) Xy00))))))))) P0) as proof of (P1 ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x1 Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0)))))))))
% Found (((eq_ref Prop) ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x1 Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_53) Xy00))))))))) P0) as proof of (P1 ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x1 Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0)))))))))
% Found eq_ref000:=(eq_ref00 P0):((P0 ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x1 Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_53) Xy00)))))))))->(P0 ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x1 Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_53) Xy00))))))))))
% Found (eq_ref00 P0) as proof of (P1 ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x1 Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0)))))))))
% Found ((eq_ref0 ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x1 Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_53) Xy00))))))))) P0) as proof of (P1 ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x1 Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0)))))))))
% Found (((eq_ref Prop) ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x1 Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_53) Xy00))))))))) P0) as proof of (P1 ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x1 Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0)))))))))
% Found (((eq_ref Prop) ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x1 Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_53) Xy00))))))))) P0) as proof of (P1 ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x1 Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0)))))))))
% Found eq_ref000:=(eq_ref00 P0):((P0 ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x1 Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_53) Xy00)))))))))->(P0 ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x1 Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_53) Xy00))))))))))
% Found (eq_ref00 P0) as proof of (P1 ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x1 Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0)))))))))
% Found ((eq_ref0 ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x1 Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_53) Xy00))))))))) P0) as proof of (P1 ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x1 Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0)))))))))
% Found (((eq_ref Prop) ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x1 Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_53) Xy00))))))))) P0) as proof of (P1 ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x1 Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0)))))))))
% Found (((eq_ref Prop) ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x1 Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_53) Xy00))))))))) P0) as proof of (P1 ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x1 Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0)))))))))
% Found eq_ref00:=(eq_ref0 ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x1 Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0))))))))):(((eq Prop) ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x1 Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0))))))))) ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x1 Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0)))))))))
% Found (eq_ref0 ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x1 Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0))))))))) as proof of (((eq Prop) ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x1 Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x1 Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0))))))))) as proof of (((eq Prop) ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x1 Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x1 Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0))))))))) as proof of (((eq Prop) ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x1 Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x1 Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0))))))))) as proof of (((eq Prop) ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x1 Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0))))))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x1 Xx0))))) (fun (Xy:a)=> (((eq a) Xy_55) Xy)))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x1 Xx0))))) (fun (Xy:a)=> (((eq a) Xy_55) Xy)))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x1 Xx0))))) (fun (Xy:a)=> (((eq a) Xy_55) Xy)))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x1 Xx0))))) (fun (Xy:a)=> (((eq a) Xy_55) Xy)))))))))
% Found eq_ref00:=(eq_ref0 ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x1 Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0))))))))):(((eq Prop) ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x1 Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0))))))))) ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x1 Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0)))))))))
% Found (eq_ref0 ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x1 Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0))))))))) as proof of (((eq Prop) ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x1 Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x1 Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0))))))))) as proof of (((eq Prop) ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x1 Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x1 Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0))))))))) as proof of (((eq Prop) ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x1 Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x1 Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0))))))))) as proof of (((eq Prop) ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x1 Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0))))))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x1 Xx0))))) (fun (Xy:a)=> (((eq a) Xy_55) Xy)))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x1 Xx0))))) (fun (Xy:a)=> (((eq a) Xy_55) Xy)))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x1 Xx0))))) (fun (Xy:a)=> (((eq a) Xy_55) Xy)))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x1 Xx0))))) (fun (Xy:a)=> (((eq a) Xy_55) Xy)))))))))
% Found eq_ref00:=(eq_ref0 ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x1 Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0))))))))):(((eq Prop) ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x1 Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0))))))))) ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x1 Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0)))))))))
% Found (eq_ref0 ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x1 Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0))))))))) as proof of (((eq Prop) ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x1 Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x1 Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0))))))))) as proof of (((eq Prop) ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x1 Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x1 Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0))))))))) as proof of (((eq Prop) ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x1 Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x1 Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0))))))))) as proof of (((eq Prop) ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x1 Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0))))))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x1 Xx0))))) (fun (Xy:a)=> (((eq a) Xy_55) Xy)))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x1 Xx0))))) (fun (Xy:a)=> (((eq a) Xy_55) Xy)))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x1 Xx0))))) (fun (Xy:a)=> (((eq a) Xy_55) Xy)))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x1 Xx0))))) (fun (Xy:a)=> (((eq a) Xy_55) Xy)))))))))
% Found eq_ref00:=(eq_ref0 ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x1 Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0))))))))):(((eq Prop) ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x1 Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0))))))))) ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x1 Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0)))))))))
% Found (eq_ref0 ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x1 Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0))))))))) as proof of (((eq Prop) ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x1 Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x1 Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0))))))))) as proof of (((eq Prop) ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x1 Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x1 Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0))))))))) as proof of (((eq Prop) ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x1 Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x1 Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0))))))))) as proof of (((eq Prop) ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_53:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x1 Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_53) Xy0))))))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x1 Xx0))))) (fun (Xy:a)=> (((eq a) Xy_55) Xy)))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x1 Xx0))))) (fun (Xy:a)=> (((eq a) Xy_55) Xy)))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x1 Xx0))))) (fun (Xy:a)=> (((eq a) Xy_55) Xy)))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x1 Xx0))))) (fun (Xy:a)=> (((eq a) Xy_55) Xy)))))))))
% Found x0:((ex (a->a)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_52:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_52) Xy))))))))))
% Instantiate: b:=(fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_52:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_52) Xy))))))))):((a->a)->Prop)
% Found x0 as proof of (P b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_55) Xy)))))))))):(((eq ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_55) Xy)))))))))) (fun (x:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (x Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy:a)=> (((eq a) Xy_55) Xy))))))))))
% Found (eta_expansion_dep00 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_55) Xy)))))))))) as proof of (((eq ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_55) Xy)))))))))) b)
% Found ((eta_expansion_dep0 (fun (x7:(a->a))=> Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_55) Xy)))))))))) as proof of (((eq ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_55) Xy)))))))))) b)
% Found (((eta_expansion_dep (a->a)) (fun (x7:(a->a))=> Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_55) Xy)))))))))) as proof of (((eq ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_55) Xy)))))))))) b)
% Found (((eta_expansion_dep (a->a)) (fun (x7:(a->a))=> Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_55) Xy)))))))))) as proof of (((eq ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_55) Xy)))))))))) b)
% Found (((eta_expansion_dep (a->a)) (fun (x7:(a->a))=> Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_55) Xy)))))))))) as proof of (((eq ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_55) Xy)))))))))) b)
% Found x0:((ex (a->a)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_52:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_52) Xy))))))))))
% Instantiate: b:=(fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_52:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_52) Xy))))))))):((a->a)->Prop)
% Found x0 as proof of (P b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_55) Xy)))))))))):(((eq ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_55) Xy)))))))))) (fun (x:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (x Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy:a)=> (((eq a) Xy_55) Xy))))))))))
% Found (eta_expansion_dep00 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_55) Xy)))))))))) as proof of (((eq ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_55) Xy)))))))))) b)
% Found ((eta_expansion_dep0 (fun (x7:(a->a))=> Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_55) Xy)))))))))) as proof of (((eq ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_55) Xy)))))))))) b)
% Found (((eta_expansion_dep (a->a)) (fun (x7:(a->a))=> Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_55) Xy)))))))))) as proof of (((eq ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_55) Xy)))))))))) b)
% Found (((eta_expansion_dep (a->a)) (fun (x7:(a->a))=> Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_55) Xy)))))))))) as proof of (((eq ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_55) Xy)))))))))) b)
% Found (((eta_expansion_dep (a->a)) (fun (x7:(a->a))=> Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_55) Xy)))))))))) as proof of (((eq ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_55) Xy)))))))))) b)
% Found x0:((ex (a->a)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_52:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_52) Xy))))))))))
% Instantiate: f:=(fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_52:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_52) Xy))))))))):((a->a)->Prop)
% Found x0 as proof of (P f)
% Found x0:((ex (a->a)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_52:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_52) Xy))))))))))
% Instantiate: f:=(fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_52:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_52) Xy))))))))):((a->a)->Prop)
% Found x0 as proof of (P f)
% Found x0:((ex (a->a)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_52:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_52) Xy))))))))))
% Instantiate: f:=(fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_52:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_52) Xy))))))))):((a->a)->Prop)
% Found x0 as proof of (P f)
% Found x0:((ex (a->a)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_52:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_52) Xy))))))))))
% Instantiate: f:=(fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_52:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_52) Xy))))))))):((a->a)->Prop)
% Found x0 as proof of (P f)
% Found eq_ref00:=(eq_ref0 (f x6)):(((eq Prop) (f x6)) (f x6))
% Found (eq_ref0 (f x6)) as proof of (((eq Prop) (f x6)) ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (x6 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x6 Xx0))))) (fun (Xy:a)=> (((eq a) Xy_55) Xy)))))))))
% Found ((eq_ref Prop) (f x6)) as proof of (((eq Prop) (f x6)) ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (x6 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x6 Xx0))))) (fun (Xy:a)=> (((eq a) Xy_55) Xy)))))))))
% Found ((eq_ref Prop) (f x6)) as proof of (((eq Prop) (f x6)) ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (x6 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x6 Xx0))))) (fun (Xy:a)=> (((eq a) Xy_55) Xy)))))))))
% Found (fun (x6:(a->a))=> ((eq_ref Prop) (f x6))) as proof of (((eq Prop) (f x6)) ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (x6 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x6 Xx0))))) (fun (Xy:a)=> (((eq a) Xy_55) Xy)))))))))
% Found (fun (x6:(a->a))=> ((eq_ref Prop) (f x6))) as proof of (forall (x:(a->a)), (((eq Prop) (f x)) ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (x Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy:a)=> (((eq a) Xy_55) Xy))))))))))
% Found eq_ref00:=(eq_ref0 (f x6)):(((eq Prop) (f x6)) (f x6))
% Found (eq_ref0 (f x6)) as proof of (((eq Prop) (f x6)) ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (x6 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x6 Xx0))))) (fun (Xy:a)=> (((eq a) Xy_55) Xy)))))))))
% Found ((eq_ref Prop) (f x6)) as proof of (((eq Prop) (f x6)) ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (x6 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xy_55:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x6 Xx0))))) (fun (Xy:a)=> (((eq a) Xy_55) Xy)))))))))
% Found ((eq_ref Prop) 
% EOF
%------------------------------------------------------------------------------