TSTP Solution File: SEV106^5 by cocATP---0.2.0
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- Process Solution
%------------------------------------------------------------------------------
% File : cocATP---0.2.0
% Problem : SEV106^5 : TPTP v6.1.0. Released v4.0.0.
% Transfm : none
% Format : tptp:raw
% Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% Computer : n116.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 300.07s
% Output : None
% Verified :
% SZS Type : None (Parsing solution fails)
% Syntax : Number of formulae : 0
% Comments :
%------------------------------------------------------------------------------
%----NO SOLUTION OUTPUT BY SYSTEM
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% % Problem : SEV106^5 : TPTP v6.1.0. Released v4.0.0.
% % Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n116.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 08:06:01 CDT 2014
% % CPUTime : 300.07
% 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 0x23f0908>, <kernel.Type object at 0x23f0ea8>) 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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0)))))))))))) ((ex (a->a)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))))->((ex (a->a)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))))) of role conjecture named cEQP1_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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0)))))))))))) ((ex (a->a)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))))->((ex (a->a)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))))):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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0)))))))))))) ((ex (a->a)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))))->((ex (a->a)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0)))))))))))))']
% 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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0)))))))))))) ((ex (a->a)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))))->((ex (a->a)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0)))))))))))))
% 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 (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0)))))))))))->((ex (a->a)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))))
% 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 (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0)))))))))))
% Found ((eq_ref0 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))) (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 (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0)))))))))))
% 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 (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))) (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 (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0)))))))))))
% 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 (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))) (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 (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0)))))))))))
% 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 (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0)))))))))))->((ex (a->a)) (fun (x:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (x Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (x Xz0)))->(((eq a) Xz0) Xx0))))))))))))
% 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 (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0)))))))))))
% 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 (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))) (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 (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0)))))))))))
% 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 (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))) (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 (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0)))))))))))
% 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 (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))) (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 (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0)))))))))))
% 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 (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))) (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 (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0)))))))))))
% 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 (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0)))))))))))->((ex (a->a)) (fun (x:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (x Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (x Xz0)))->(((eq a) Xz0) Xx0))))))))))))
% 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 (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0)))))))))))
% 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 (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))) (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 (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0)))))))))))
% 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 (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))) (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 (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0)))))))))))
% 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 (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))) (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 (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0)))))))))))
% 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 (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))) (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 (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0)))))))))))
% 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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))):(((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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))) (fun (x:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (x Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (x Xz0)))->(((eq a) Xz0) Xx0)))))))))))
% 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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))) 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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))) 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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))) 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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))) 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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))) 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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))) 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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))) 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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))) 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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))) 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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))) 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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0)))))))))))
% Instantiate: b:=(fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0)))))))))):((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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))):(((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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))) (fun (x:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (x Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (x Xz0)))->(((eq a) Xz0) Xx0)))))))))))
% Found (eta_expansion00 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))) 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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))) 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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))) 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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))) 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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))) 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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))) 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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))) 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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))) 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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))) 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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))) 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 (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0)))))))))))->((ex (a->a)) (fun (x:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (x Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (x Xz0)))->(((eq a) Xz0) Xx0))))))))))))
% 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 (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0)))))))))))->(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 (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))) (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 (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0)))))))))))->(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 (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))) (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 (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0)))))))))))->(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 (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))) (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 (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0)))))))))))->(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 (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))) (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 (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0)))))))))))->(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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))))=> ((((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 (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))) (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 (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0)))))))))))->(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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))))=> ((((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 (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))) (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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0)))))))))))->(((ex (a->a)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0)))))))))))->(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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))))=> ((((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 (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))) (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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))))=> ((((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 (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))) (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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0)))))))))))->(((ex (a->a)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0)))))))))))->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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0)))))))))))) ((ex (a->a)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0)))))))))))) 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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))))=> ((((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 (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))) (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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0)))))))))))->(((ex (a->a)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0)))))))))))->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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0)))))))))))) ((ex (a->a)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0)))))))))))) 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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))))=> ((((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 (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))) (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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0)))))))))))
% Instantiate: f:=(fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0)))))))))):((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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0)))))))))))
% Instantiate: f:=(fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0)))))))))):((a->a)->Prop)
% Found x0 as proof of (P f)
% 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), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0)))))))))))
% Found ((eq_ref ((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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0)))))))))))
% Found ((eq_ref ((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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0)))))))))))
% Found ((eq_ref ((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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0)))))))))))
% 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 (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))):(((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 (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))) (fun (x:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (x Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (x Xz0)))->(((eq a) Xz0) Xx0)))))))))))
% Found (eta_expansion00 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))) 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 (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))) 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 (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))) 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 (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))) 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 (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))) 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 (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))) 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 (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))) 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 (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))) 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 (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))) 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 (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))) b)
% 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 (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0)))))))))))->((ex (a->a)) (fun (x:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (x Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (x Xz0)))->(((eq a) Xz0) Xx0))))))))))))
% 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 (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0)))))))))))
% Found ((eta_expansion00 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))) (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 (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0)))))))))))
% 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 (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))) (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 (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0)))))))))))
% 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 (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))) (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 (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0)))))))))))
% 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 (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))) (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 (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0)))))))))))
% 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), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0)))))))))))
% Found ((eq_ref ((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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0)))))))))))
% Found ((eq_ref ((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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0)))))))))))
% Found ((eq_ref ((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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0)))))))))))
% 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 (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))):(((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 (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0)))))))))))
% Found (eq_ref0 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))) 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 (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))) 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 (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))) 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 (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))) 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 (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))) 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 (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))) 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 (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))) 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 (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))) 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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0)))))))))))
% Instantiate: b:=(fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0)))))))))):((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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))):(((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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))) (fun (x:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (x Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (x Xz0)))->(((eq a) Xz0) Xx0)))))))))))
% 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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))) 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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))) 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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))) 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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))) 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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))) 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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))) 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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))) 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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))) 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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))) 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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))) 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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0)))))))))))
% Instantiate: b:=(fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0)))))))))):((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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))):(((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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))) (fun (x:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (x Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (x Xz0)))->(((eq a) Xz0) Xx0)))))))))))
% 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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))) 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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))) 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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))) 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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))) 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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))) 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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))) 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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))) 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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))) 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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))) 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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))) 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 (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0)))))))))))
% Instantiate: f:=(fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0)))))))))):((a->a)->Prop)
% Found x1 as proof of (P f)
% 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 (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0)))))))))))
% Instantiate: f:=(fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0)))))))))):((a->a)->Prop)
% Found x1 as proof of (P f)
% 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 (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0)))))))))))
% Instantiate: f:=(fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0)))))))))):((a->a)->Prop)
% Found x1 as proof of (P f)
% 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 (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0)))))))))))
% Instantiate: f:=(fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0)))))))))):((a->a)->Prop)
% Found x1 as proof of (P f)
% 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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0)))))))))))->(P0 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))))
% 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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0)))))))))))
% Found ((eq_ref0 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))) 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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0)))))))))))
% 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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))) 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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0)))))))))))
% 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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))) 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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0)))))))))))
% 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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0)))))))))))->(P0 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))))
% 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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0)))))))))))
% Found ((eq_ref0 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))) 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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0)))))))))))
% 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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))) 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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0)))))))))))
% 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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))) 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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0)))))))))))
% 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 (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0)))))))))))
% 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 (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0)))))))))))
% 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 (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0)))))))))))
% 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 (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0)))))))))))
% 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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))):(((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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))) (fun (x:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (x Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (x Xz0)))->(((eq a) Xz0) Xx0)))))))))))
% 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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))) 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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))) b)
% 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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))) 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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))) b)
% 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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))) 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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))) b)
% 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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))) 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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))) b)
% 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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))) 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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))) b)
% 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 (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0)))))))))))
% 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 (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0)))))))))))
% 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 (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0)))))))))))
% 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 (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0)))))))))))
% 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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))):(((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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))) (fun (x:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (x Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (x Xz0)))->(((eq a) Xz0) Xx0)))))))))))
% 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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))) 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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))) b)
% 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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))) 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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))) b)
% 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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))) 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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))) b)
% 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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))) 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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))) b)
% 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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))) 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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))) b)
% 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 (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0)))))))))))->((ex (a->a)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))))
% 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 (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0)))))))))))->(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 (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))) (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 (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0)))))))))))->(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 (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))) (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 (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0)))))))))))->(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 (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))) (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 (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0)))))))))))->(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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))))=> (((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 (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))) (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 (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0)))))))))))->(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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))))=> (((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 (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))) (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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0)))))))))))->(((ex (a->a)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0)))))))))))->(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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))))=> (((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 (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))) (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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))))=> (((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 (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))) (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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0)))))))))))->(((ex (a->a)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0)))))))))))->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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0)))))))))))) ((ex (a->a)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0)))))))))))) 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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))))=> (((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 (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))) (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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0)))))))))))->(((ex (a->a)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0)))))))))))->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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0)))))))))))) ((ex (a->a)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0)))))))))))) 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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))))=> (((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 (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))) (ex (a->a))))) as proof of (P f)
% 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 (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0)))))))))))->((ex (a->a)) (fun (x:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (x Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (x Xz0)))->(((eq a) Xz0) Xx0))))))))))))
% Found (eta_expansion000 (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 (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0)))))))))))->(P f))
% Found ((eta_expansion00 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))) (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 (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0)))))))))))->(P f))
% 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 (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))) (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 (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0)))))))))))->(P f))
% 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 (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))) (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 (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0)))))))))))->(P f))
% 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 (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))) (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 (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0)))))))))))->(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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))))=> ((((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 (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))) (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 (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0)))))))))))->(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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))))=> ((((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 (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))) (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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0)))))))))))->(((ex (a->a)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0)))))))))))->(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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))))=> ((((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 (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))) (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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))))=> ((((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 (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))) (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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0)))))))))))->(((ex (a->a)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0)))))))))))->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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0)))))))))))) ((ex (a->a)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0)))))))))))) 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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))))=> ((((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 (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))) (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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0)))))))))))->(((ex (a->a)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0)))))))))))->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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0)))))))))))) ((ex (a->a)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0)))))))))))) 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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))))=> ((((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 (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))) (ex (a->a))))) as proof of (P f)
% 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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0)))))))))))->(P0 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))))
% 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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0)))))))))))
% Found ((eq_ref0 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))) 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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0)))))))))))
% 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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))) 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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0)))))))))))
% 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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))) 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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0)))))))))))
% 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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0)))))))))))->(P0 (fun (x:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (x Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (x Xz0)))->(((eq a) Xz0) Xx0))))))))))))
% 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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0)))))))))))
% 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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))) 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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0)))))))))))
% 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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))) 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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0)))))))))))
% 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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))) 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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0)))))))))))
% 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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))) 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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0)))))))))))
% 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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0)))))))))))->(P0 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))))
% 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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0)))))))))))
% Found ((eq_ref0 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (Xs Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))) 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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0)))))))))))
% 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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))) 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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0)))))))))))
% 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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))) 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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0)))))))))))
% 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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0)))))))))))->(P0 (fun (x:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (x Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (x Xz0)))->(((eq a) Xz0) Xx0))))))))))))
% 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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0)))))))))))
% 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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))) 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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0)))))))))))
% 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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))) 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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0)))))))))))
% 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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))) 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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0)))))))))))
% 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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))) 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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0)))))))))))
% Found eq_ref000:=(eq_ref00 P0):((P0 ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (x1 Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (x1 Xz0)))->(((eq a) Xz0) Xx0))))))))))->(P0 ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (x1 Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (x1 Xz0)))->(((eq a) Xz0) Xx0)))))))))))
% 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 (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (x1 Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (x1 Xz0)))->(((eq a) Xz0) Xx0))))))))))
% Found ((eq_ref0 ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (x1 Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (x1 Xz0)))->(((eq a) Xz0) Xx0)))))))))) P0) as proof of (P1 ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (x1 Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (x1 Xz0)))->(((eq a) Xz0) Xx0))))))))))
% Found (((eq_ref Prop) ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (x1 Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (x1 Xz0)))->(((eq a) Xz0) Xx0)))))))))) P0) as proof of (P1 ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (x1 Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (x1 Xz0)))->(((eq a) Xz0) Xx0))))))))))
% Found (((eq_ref Prop) ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (x1 Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (x1 Xz0)))->(((eq a) Xz0) Xx0)))))))))) P0) as proof of (P1 ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (x1 Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (x1 Xz0)))->(((eq a) Xz0) Xx0))))))))))
% Found eq_ref000:=(eq_ref00 P0):((P0 ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (x1 Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (x1 Xz0)))->(((eq a) Xz0) Xx0))))))))))->(P0 ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (x1 Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (x1 Xz0)))->(((eq a) Xz0) Xx0)))))))))))
% 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 (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (x1 Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (x1 Xz0)))->(((eq a) Xz0) Xx0))))))))))
% Found ((eq_ref0 ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (x1 Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (x1 Xz0)))->(((eq a) Xz0) Xx0)))))))))) P0) as proof of (P1 ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (x1 Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (x1 Xz0)))->(((eq a) Xz0) Xx0))))))))))
% Found (((eq_ref Prop) ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (x1 Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (x1 Xz0)))->(((eq a) Xz0) Xx0)))))))))) P0) as proof of (P1 ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (x1 Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (x1 Xz0)))->(((eq a) Xz0) Xx0))))))))))
% Found (((eq_ref Prop) ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (x1 Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (x1 Xz0)))->(((eq a) Xz0) Xx0)))))))))) P0) as proof of (P1 ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (x1 Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (x1 Xz0)))->(((eq a) Xz0) Xx0))))))))))
% Found eq_ref000:=(eq_ref00 P0):((P0 ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (x1 Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (x1 Xz0)))->(((eq a) Xz0) Xx0))))))))))->(P0 ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (x1 Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (x1 Xz0)))->(((eq a) Xz0) Xx0)))))))))))
% 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 (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (x1 Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (x1 Xz0)))->(((eq a) Xz0) Xx0))))))))))
% Found ((eq_ref0 ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (x1 Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (x1 Xz0)))->(((eq a) Xz0) Xx0)))))))))) P0) as proof of (P1 ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (x1 Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (x1 Xz0)))->(((eq a) Xz0) Xx0))))))))))
% Found (((eq_ref Prop) ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (x1 Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (x1 Xz0)))->(((eq a) Xz0) Xx0)))))))))) P0) as proof of (P1 ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (x1 Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (x1 Xz0)))->(((eq a) Xz0) Xx0))))))))))
% Found (((eq_ref Prop) ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (x1 Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (x1 Xz0)))->(((eq a) Xz0) Xx0)))))))))) P0) as proof of (P1 ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (x1 Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (x1 Xz0)))->(((eq a) Xz0) Xx0))))))))))
% Found eq_ref000:=(eq_ref00 P0):((P0 ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (x1 Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (x1 Xz0)))->(((eq a) Xz0) Xx0))))))))))->(P0 ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (x1 Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (x1 Xz0)))->(((eq a) Xz0) Xx0)))))))))))
% 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 (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (x1 Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (x1 Xz0)))->(((eq a) Xz0) Xx0))))))))))
% Found ((eq_ref0 ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (x1 Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (x1 Xz0)))->(((eq a) Xz0) Xx0)))))))))) P0) as proof of (P1 ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (x1 Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (x1 Xz0)))->(((eq a) Xz0) Xx0))))))))))
% Found (((eq_ref Prop) ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (x1 Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (x1 Xz0)))->(((eq a) Xz0) Xx0)))))))))) P0) as proof of (P1 ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (x1 Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (x1 Xz0)))->(((eq a) Xz0) Xx0))))))))))
% Found (((eq_ref Prop) ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (x1 Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (x1 Xz0)))->(((eq a) Xz0) Xx0)))))))))) P0) as proof of (P1 ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (x1 Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (x1 Xz0)))->(((eq a) Xz0) Xx0))))))))))
% Found eq_ref00:=(eq_ref0 ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (x1 Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (x1 Xz0)))->(((eq a) Xz0) Xx0)))))))))):(((eq Prop) ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (x1 Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (x1 Xz0)))->(((eq a) Xz0) Xx0)))))))))) ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (x1 Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (x1 Xz0)))->(((eq a) Xz0) Xx0))))))))))
% Found (eq_ref0 ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (x1 Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (x1 Xz0)))->(((eq a) Xz0) Xx0)))))))))) as proof of (((eq Prop) ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (x1 Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (x1 Xz0)))->(((eq a) Xz0) Xx0)))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (x1 Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (x1 Xz0)))->(((eq a) Xz0) Xx0)))))))))) as proof of (((eq Prop) ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (x1 Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (x1 Xz0)))->(((eq a) Xz0) Xx0)))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (x1 Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (x1 Xz0)))->(((eq a) Xz0) Xx0)))))))))) as proof of (((eq Prop) ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (x1 Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (x1 Xz0)))->(((eq a) Xz0) Xx0)))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (x1 Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (x1 Xz0)))->(((eq a) Xz0) Xx0)))))))))) as proof of (((eq Prop) ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (x1 Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (x1 Xz0)))->(((eq a) Xz0) Xx0)))))))))) 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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (x1 Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (x1 Xz0)))->(((eq a) Xz0) Xx0))))))))))
% 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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (x1 Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (x1 Xz0)))->(((eq a) Xz0) Xx0))))))))))
% 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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (x1 Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (x1 Xz0)))->(((eq a) Xz0) Xx0))))))))))
% 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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (x1 Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (x1 Xz0)))->(((eq a) Xz0) Xx0))))))))))
% Found eq_ref00:=(eq_ref0 ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (x1 Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (x1 Xz0)))->(((eq a) Xz0) Xx0)))))))))):(((eq Prop) ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (x1 Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (x1 Xz0)))->(((eq a) Xz0) Xx0)))))))))) ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (x1 Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (x1 Xz0)))->(((eq a) Xz0) Xx0))))))))))
% Found (eq_ref0 ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (x1 Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (x1 Xz0)))->(((eq a) Xz0) Xx0)))))))))) as proof of (((eq Prop) ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (x1 Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (x1 Xz0)))->(((eq a) Xz0) Xx0)))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (x1 Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (x1 Xz0)))->(((eq a) Xz0) Xx0)))))))))) as proof of (((eq Prop) ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (x1 Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (x1 Xz0)))->(((eq a) Xz0) Xx0)))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (x1 Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (x1 Xz0)))->(((eq a) Xz0) Xx0)))))))))) as proof of (((eq Prop) ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (x1 Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (x1 Xz0)))->(((eq a) Xz0) Xx0)))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (x1 Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (x1 Xz0)))->(((eq a) Xz0) Xx0)))))))))) as proof of (((eq Prop) ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (x1 Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (x1 Xz0)))->(((eq a) Xz0) Xx0)))))))))) 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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (x1 Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (x1 Xz0)))->(((eq a) Xz0) Xx0))))))))))
% 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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (x1 Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (x1 Xz0)))->(((eq a) Xz0) Xx0))))))))))
% 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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (x1 Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (x1 Xz0)))->(((eq a) Xz0) Xx0))))))))))
% 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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (x1 Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (x1 Xz0)))->(((eq a) Xz0) Xx0))))))))))
% Found eq_ref00:=(eq_ref0 ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (x1 Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (x1 Xz0)))->(((eq a) Xz0) Xx0)))))))))):(((eq Prop) ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (x1 Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (x1 Xz0)))->(((eq a) Xz0) Xx0)))))))))) ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (x1 Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (x1 Xz0)))->(((eq a) Xz0) Xx0))))))))))
% Found (eq_ref0 ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (x1 Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (x1 Xz0)))->(((eq a) Xz0) Xx0)))))))))) as proof of (((eq Prop) ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (x1 Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (x1 Xz0)))->(((eq a) Xz0) Xx0)))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (x1 Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (x1 Xz0)))->(((eq a) Xz0) Xx0)))))))))) as proof of (((eq Prop) ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (x1 Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (x1 Xz0)))->(((eq a) Xz0) Xx0)))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (x1 Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (x1 Xz0)))->(((eq a) Xz0) Xx0)))))))))) as proof of (((eq Prop) ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (x1 Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (x1 Xz0)))->(((eq a) Xz0) Xx0)))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (x1 Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (x1 Xz0)))->(((eq a) Xz0) Xx0)))))))))) as proof of (((eq Prop) ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (x1 Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (x1 Xz0)))->(((eq a) Xz0) Xx0)))))))))) 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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (x1 Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (x1 Xz0)))->(((eq a) Xz0) Xx0))))))))))
% 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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (x1 Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (x1 Xz0)))->(((eq a) Xz0) Xx0))))))))))
% 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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (x1 Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (x1 Xz0)))->(((eq a) Xz0) Xx0))))))))))
% 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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (x1 Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (x1 Xz0)))->(((eq a) Xz0) Xx0))))))))))
% Found eq_ref00:=(eq_ref0 ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (x1 Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (x1 Xz0)))->(((eq a) Xz0) Xx0)))))))))):(((eq Prop) ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (x1 Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (x1 Xz0)))->(((eq a) Xz0) Xx0)))))))))) ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (x1 Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (x1 Xz0)))->(((eq a) Xz0) Xx0))))))))))
% Found (eq_ref0 ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (x1 Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (x1 Xz0)))->(((eq a) Xz0) Xx0)))))))))) as proof of (((eq Prop) ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (x1 Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (x1 Xz0)))->(((eq a) Xz0) Xx0)))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (x1 Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (x1 Xz0)))->(((eq a) Xz0) Xx0)))))))))) as proof of (((eq Prop) ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (x1 Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (x1 Xz0)))->(((eq a) Xz0) Xx0)))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (x1 Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (x1 Xz0)))->(((eq a) Xz0) Xx0)))))))))) as proof of (((eq Prop) ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (x1 Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (x1 Xz0)))->(((eq a) Xz0) Xx0)))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (x1 Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (x1 Xz0)))->(((eq a) Xz0) Xx0)))))))))) as proof of (((eq Prop) ((and (forall (Xx0:a), ((Xy Xx0)->(Xz (x1 Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xx0:a)=> ((and ((and (Xy Xx0)) (((eq a) Xy0) (x1 Xx0)))) (forall (Xz0:a), (((and (Xy Xz0)) (((eq a) Xy0) (x1 Xz0)))->(((eq a) Xz0) Xx0)))))))))) 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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (x1 Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (x1 Xz0)))->(((eq a) Xz0) Xx0))))))))))
% 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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (x1 Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (x1 Xz0)))->(((eq a) Xz0) Xx0))))))))))
% 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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (x1 Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (x1 Xz0)))->(((eq a) Xz0) Xx0))))))))))
% 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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (x1 Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (x1 Xz0)))->(((eq a) Xz0) Xx0))))))))))
% 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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0)))))))))))
% Instantiate: b:=(fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0)))))))))):((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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))):(((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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))) (fun (x:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (x Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (x Xz0)))->(((eq a) Xz0) Xx0)))))))))))
% 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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))) 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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))) 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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))) 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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))) 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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))) 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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))) 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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))) 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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))) 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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))) 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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))) 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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0)))))))))))
% Instantiate: b:=(fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0)))))))))):((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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))):(((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 (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (Xs Xz0)))->(((eq a) Xz0) Xx0))))))))))) (fun (x:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xz (x Xx0))))) (forall (Xy0:a), ((Xz Xy0)->((ex a) (fun (Xx0:a)=> ((and ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0)))) (forall (Xz0:a), (((and (Xx Xz0)) (((eq a) Xy0) (x Xz0)))->(((eq a) Xz0) Xx0)))))))))))
% Found (eta_expansion
% EOF
%------------------------------------------------------------------------------