%------------------------------------------------------------------------------
% File     : cocATP---0.2.0
% Problem  : SEV089^5 : TPTP v6.1.0. Released v4.0.0.
% Transfm  : none
% Format   : tptp:raw
% Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p

% Computer : n117.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 13:33:43 EDT 2014

% Result   : Timeout 300.10s
% Output   : None 
% Verified : 
% SZS Type : None (Parsing solution fails)
% Syntax   : Number of formulae    : 0

% Comments : 
%------------------------------------------------------------------------------
%----NO SOLUTION OUTPUT BY SYSTEM
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% % Problem  : SEV089^5 : TPTP v6.1.0. Released v4.0.0.
% % Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n117.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 07:59:56 CDT 2014
% % CPUTime  : 300.10 
% 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 0x18a1368>, <kernel.Type object at 0x18a1fc8>) of role type named a_type
% Using role type
% Declaring a:Type
% FOF formula (forall (Xx:(a->Prop)) (Xy:(a->Prop)), (((ex (a->a)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) ((fun (Xx:a) (Xy:a)=> (((eq a) Xx) Xy)) Xy_38)))))))))->((ex (a->a)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (Xs Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) ((fun (Xx:a) (Xy:a)=> (((eq a) Xx) Xy)) Xy_39))))))))))) of role conjecture named cEQP_1B_pme
% Conjecture to prove = (forall (Xx:(a->Prop)) (Xy:(a->Prop)), (((ex (a->a)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) ((fun (Xx:a) (Xy:a)=> (((eq a) Xx) Xy)) Xy_38)))))))))->((ex (a->a)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (Xs Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) ((fun (Xx:a) (Xy:a)=> (((eq a) Xx) Xy)) Xy_39))))))))))):Prop
% Parameter a_DUMMY:a.
% We need to prove ['(forall (Xx:(a->Prop)) (Xy:(a->Prop)), (((ex (a->a)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) ((fun (Xx:a) (Xy:a)=> (((eq a) Xx) Xy)) Xy_38)))))))))->((ex (a->a)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (Xs Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) ((fun (Xx:a) (Xy:a)=> (((eq a) Xx) Xy)) Xy_39)))))))))))']
% Parameter a:Type.
% Trying to prove (forall (Xx:(a->Prop)) (Xy:(a->Prop)), (((ex (a->a)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) ((fun (Xx:a) (Xy:a)=> (((eq a) Xx) Xy)) Xy_38)))))))))->((ex (a->a)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (Xs Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) ((fun (Xx:a) (Xy:a)=> (((eq a) Xx) Xy)) Xy_39)))))))))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 (ex (a->a))):(((ex (a->a)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0))))))))))->((ex (a->a)) (fun (x:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0)))))))))))
% Found (eta_expansion_dep000 (ex (a->a))) as proof of (P (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0))))))))))
% Found ((eta_expansion_dep00 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0)))))))))) (ex (a->a))) as proof of (P (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0))))))))))
% Found (((eta_expansion_dep0 (fun (x1:(a->a))=> Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0)))))))))) (ex (a->a))) as proof of (P (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0))))))))))
% Found ((((eta_expansion_dep (a->a)) (fun (x1:(a->a))=> Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0)))))))))) (ex (a->a))) as proof of (P (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0))))))))))
% Found ((((eta_expansion_dep (a->a)) (fun (x1:(a->a))=> Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0)))))))))) (ex (a->a))) as proof of (P (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0))))))))))
% Found eta_expansion0000:=(eta_expansion000 (ex (a->a))):(((ex (a->a)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0))))))))))->((ex (a->a)) (fun (x:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0)))))))))))
% Found (eta_expansion000 (ex (a->a))) as proof of (P (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0))))))))))
% Found ((eta_expansion00 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0)))))))))) (ex (a->a))) as proof of (P (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0))))))))))
% Found (((eta_expansion0 Prop) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0)))))))))) (ex (a->a))) as proof of (P (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0))))))))))
% Found ((((eta_expansion (a->a)) Prop) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0)))))))))) (ex (a->a))) as proof of (P (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0))))))))))
% Found ((((eta_expansion (a->a)) Prop) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0)))))))))) (ex (a->a))) as proof of (P (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0))))))))))
% Found eta_expansion0000:=(eta_expansion000 (ex (a->a))):(((ex (a->a)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0))))))))))->((ex (a->a)) (fun (x:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0)))))))))))
% Found (eta_expansion000 (ex (a->a))) as proof of (P (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0))))))))))
% Found ((eta_expansion00 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0)))))))))) (ex (a->a))) as proof of (P (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0))))))))))
% Found (((eta_expansion0 Prop) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0)))))))))) (ex (a->a))) as proof of (P (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0))))))))))
% Found ((((eta_expansion (a->a)) Prop) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0)))))))))) (ex (a->a))) as proof of (P (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0))))))))))
% Found ((((eta_expansion (a->a)) Prop) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0)))))))))) (ex (a->a))) as proof of (P (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0))))))))))
% Found x:((ex (a->a)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) ((fun (Xx:a) (Xy:a)=> (((eq a) Xx) Xy)) Xy_38)))))))))
% Instantiate: b:=(fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) ((fun (Xx:a) (Xy:a)=> (((eq a) Xx) Xy)) Xy_38)))))))):((a->a)->Prop)
% Found x as proof of (P b)
% Found eq_ref00:=(eq_ref0 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (Xs Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_39) Xy)))))))))):(((eq ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (Xs Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_39) Xy)))))))))) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (Xs Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_39) Xy))))))))))
% Found (eq_ref0 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (Xs Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_39) Xy)))))))))) as proof of (((eq ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (Xs Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_39) Xy)))))))))) b)
% Found ((eq_ref ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (Xs Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_39) Xy)))))))))) as proof of (((eq ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (Xs Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_39) Xy)))))))))) b)
% Found ((eq_ref ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (Xs Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_39) Xy)))))))))) as proof of (((eq ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (Xs Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_39) Xy)))))))))) b)
% Found ((eq_ref ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (Xs Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_39) Xy)))))))))) as proof of (((eq ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (Xs Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_39) Xy)))))))))) b)
% Found x:((ex (a->a)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) ((fun (Xx:a) (Xy:a)=> (((eq a) Xx) Xy)) Xy_38)))))))))
% Instantiate: f:=(fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) ((fun (Xx:a) (Xy:a)=> (((eq a) Xx) Xy)) Xy_38)))))))):((a->a)->Prop)
% Found x as proof of (P f)
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x0 Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x0 Xx0))))) (fun (Xy:a)=> (((eq a) Xy_39) Xy)))))))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x0 Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x0 Xx0))))) (fun (Xy:a)=> (((eq a) Xy_39) Xy)))))))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x0 Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x0 Xx0))))) (fun (Xy:a)=> (((eq a) Xy_39) Xy)))))))))
% Found (fun (x0:(a->a))=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x0 Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x0 Xx0))))) (fun (Xy:a)=> (((eq a) Xy_39) Xy)))))))))
% Found (fun (x0:(a->a))=> ((eq_ref Prop) (f x0))) as proof of (forall (x:(a->a)), (((eq Prop) (f x)) ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy:a)=> (((eq a) Xy_39) Xy))))))))))
% Found x:((ex (a->a)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) ((fun (Xx:a) (Xy:a)=> (((eq a) Xx) Xy)) Xy_38)))))))))
% Instantiate: f:=(fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) ((fun (Xx:a) (Xy:a)=> (((eq a) Xx) Xy)) Xy_38)))))))):((a->a)->Prop)
% Found x as proof of (P f)
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x0 Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x0 Xx0))))) (fun (Xy:a)=> (((eq a) Xy_39) Xy)))))))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x0 Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x0 Xx0))))) (fun (Xy:a)=> (((eq a) Xy_39) Xy)))))))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x0 Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x0 Xx0))))) (fun (Xy:a)=> (((eq a) Xy_39) Xy)))))))))
% Found (fun (x0:(a->a))=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x0 Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x0 Xx0))))) (fun (Xy:a)=> (((eq a) Xy_39) Xy)))))))))
% Found (fun (x0:(a->a))=> ((eq_ref Prop) (f x0))) as proof of (forall (x:(a->a)), (((eq Prop) (f x)) ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy:a)=> (((eq a) Xy_39) Xy))))))))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0)))))))))):(((eq ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0)))))))))) (fun (x:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0))))))))))
% Found (eta_expansion_dep00 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0)))))))))) as proof of (((eq ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0)))))))))) b)
% Found ((eta_expansion_dep0 (fun (x1:(a->a))=> Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0)))))))))) as proof of (((eq ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0)))))))))) b)
% Found (((eta_expansion_dep (a->a)) (fun (x1:(a->a))=> Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0)))))))))) as proof of (((eq ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0)))))))))) b)
% Found (((eta_expansion_dep (a->a)) (fun (x1:(a->a))=> Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0)))))))))) as proof of (((eq ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0)))))))))) b)
% Found (((eta_expansion_dep (a->a)) (fun (x1:(a->a))=> Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0)))))))))) as proof of (((eq ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0)))))))))) b)
% Found eta_expansion000:=(eta_expansion00 b):(((eq ((a->a)->Prop)) b) (fun (x:(a->a))=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq ((a->a)->Prop)) b) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (Xs Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_39) Xy))))))))))
% Found ((eta_expansion0 Prop) b) as proof of (((eq ((a->a)->Prop)) b) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (Xs Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_39) Xy))))))))))
% Found (((eta_expansion (a->a)) Prop) b) as proof of (((eq ((a->a)->Prop)) b) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (Xs Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_39) Xy))))))))))
% Found (((eta_expansion (a->a)) Prop) b) as proof of (((eq ((a->a)->Prop)) b) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (Xs Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_39) Xy))))))))))
% Found (((eta_expansion (a->a)) Prop) b) as proof of (((eq ((a->a)->Prop)) b) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (Xs Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_39) Xy))))))))))
% Found eta_expansion0000:=(eta_expansion000 (ex (a->a))):(((ex (a->a)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0))))))))))->((ex (a->a)) (fun (x:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0)))))))))))
% Found (eta_expansion000 (ex (a->a))) as proof of (P (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0))))))))))
% Found ((eta_expansion00 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0)))))))))) (ex (a->a))) as proof of (P (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0))))))))))
% Found (((eta_expansion0 Prop) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0)))))))))) (ex (a->a))) as proof of (P (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0))))))))))
% Found ((((eta_expansion (a->a)) Prop) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0)))))))))) (ex (a->a))) as proof of (P (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0))))))))))
% Found ((((eta_expansion (a->a)) Prop) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0)))))))))) (ex (a->a))) as proof of (P (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0))))))))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0)))))))))):(((eq ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0)))))))))) (fun (x:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0))))))))))
% Found (eta_expansion_dep00 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0)))))))))) as proof of (((eq ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0)))))))))) b)
% Found ((eta_expansion_dep0 (fun (x1:(a->a))=> Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0)))))))))) as proof of (((eq ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0)))))))))) b)
% Found (((eta_expansion_dep (a->a)) (fun (x1:(a->a))=> Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0)))))))))) as proof of (((eq ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0)))))))))) b)
% Found (((eta_expansion_dep (a->a)) (fun (x1:(a->a))=> Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0)))))))))) as proof of (((eq ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0)))))))))) b)
% Found (((eta_expansion_dep (a->a)) (fun (x1:(a->a))=> Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0)))))))))) as proof of (((eq ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0)))))))))) b)
% Found eta_expansion000:=(eta_expansion00 b):(((eq ((a->a)->Prop)) b) (fun (x:(a->a))=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq ((a->a)->Prop)) b) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (Xs Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_39) Xy))))))))))
% Found ((eta_expansion0 Prop) b) as proof of (((eq ((a->a)->Prop)) b) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (Xs Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_39) Xy))))))))))
% Found (((eta_expansion (a->a)) Prop) b) as proof of (((eq ((a->a)->Prop)) b) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (Xs Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_39) Xy))))))))))
% Found (((eta_expansion (a->a)) Prop) b) as proof of (((eq ((a->a)->Prop)) b) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (Xs Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_39) Xy))))))))))
% Found (((eta_expansion (a->a)) Prop) b) as proof of (((eq ((a->a)->Prop)) b) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (Xs Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_39) Xy))))))))))
% Found x:((ex (a->a)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) ((fun (Xx:a) (Xy:a)=> (((eq a) Xx) Xy)) Xy_38)))))))))
% Instantiate: b:=(fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) ((fun (Xx:a) (Xy:a)=> (((eq a) Xx) Xy)) Xy_38)))))))):((a->a)->Prop)
% Found x as proof of (P b)
% Found eta_expansion000:=(eta_expansion00 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (Xs Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_39) Xy)))))))))):(((eq ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (Xs Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_39) Xy)))))))))) (fun (x:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy:a)=> (((eq a) Xy_39) Xy))))))))))
% Found (eta_expansion00 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (Xs Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_39) Xy)))))))))) as proof of (((eq ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (Xs Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_39) Xy)))))))))) b)
% Found ((eta_expansion0 Prop) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (Xs Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_39) Xy)))))))))) as proof of (((eq ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (Xs Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_39) Xy)))))))))) b)
% Found (((eta_expansion (a->a)) Prop) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (Xs Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_39) Xy)))))))))) as proof of (((eq ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (Xs Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_39) Xy)))))))))) b)
% Found (((eta_expansion (a->a)) Prop) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (Xs Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_39) Xy)))))))))) as proof of (((eq ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (Xs Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_39) Xy)))))))))) b)
% Found (((eta_expansion (a->a)) Prop) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (Xs Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_39) Xy)))))))))) as proof of (((eq ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (Xs Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_39) Xy)))))))))) b)
% Found x:((ex (a->a)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) ((fun (Xx:a) (Xy:a)=> (((eq a) Xx) Xy)) Xy_38)))))))))
% Instantiate: f:=(fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) ((fun (Xx:a) (Xy:a)=> (((eq a) Xx) Xy)) Xy_38)))))))):((a->a)->Prop)
% Found x as proof of (P f)
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x2 Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x2 Xx0))))) (fun (Xy:a)=> (((eq a) Xy_39) Xy)))))))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x2 Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x2 Xx0))))) (fun (Xy:a)=> (((eq a) Xy_39) Xy)))))))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x2 Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x2 Xx0))))) (fun (Xy:a)=> (((eq a) Xy_39) Xy)))))))))
% Found (fun (x2:(a->a))=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x2 Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x2 Xx0))))) (fun (Xy:a)=> (((eq a) Xy_39) Xy)))))))))
% Found (fun (x2:(a->a))=> ((eq_ref Prop) (f x2))) as proof of (forall (x:(a->a)), (((eq Prop) (f x)) ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy:a)=> (((eq a) Xy_39) Xy))))))))))
% Found x:((ex (a->a)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) ((fun (Xx:a) (Xy:a)=> (((eq a) Xx) Xy)) Xy_38)))))))))
% Instantiate: f:=(fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) ((fun (Xx:a) (Xy:a)=> (((eq a) Xx) Xy)) Xy_38)))))))):((a->a)->Prop)
% Found x as proof of (P f)
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x2 Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x2 Xx0))))) (fun (Xy:a)=> (((eq a) Xy_39) Xy)))))))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x2 Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x2 Xx0))))) (fun (Xy:a)=> (((eq a) Xy_39) Xy)))))))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x2 Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x2 Xx0))))) (fun (Xy:a)=> (((eq a) Xy_39) Xy)))))))))
% Found (fun (x2:(a->a))=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x2 Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x2 Xx0))))) (fun (Xy:a)=> (((eq a) Xy_39) Xy)))))))))
% Found (fun (x2:(a->a))=> ((eq_ref Prop) (f x2))) as proof of (forall (x:(a->a)), (((eq Prop) (f x)) ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy:a)=> (((eq a) Xy_39) Xy))))))))))
% Found eq_ref000:=(eq_ref00 P0):((P0 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (Xs Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_39) Xy00))))))))))->(P0 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (Xs Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_39) Xy00)))))))))))
% Found (eq_ref00 P0) as proof of (P1 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (Xs Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0))))))))))
% Found ((eq_ref0 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (Xs Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_39) Xy00)))))))))) P0) as proof of (P1 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (Xs Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0))))))))))
% Found (((eq_ref ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (Xs Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_39) Xy00)))))))))) P0) as proof of (P1 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (Xs Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0))))))))))
% Found (((eq_ref ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (Xs Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_39) Xy00)))))))))) P0) as proof of (P1 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (Xs Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0))))))))))
% Found eq_ref000:=(eq_ref00 P0):((P0 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (Xs Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_39) Xy00))))))))))->(P0 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (Xs Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_39) Xy00)))))))))))
% Found (eq_ref00 P0) as proof of (P1 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (Xs Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0))))))))))
% Found ((eq_ref0 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (Xs Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_39) Xy00)))))))))) P0) as proof of (P1 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (Xs Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0))))))))))
% Found (((eq_ref ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (Xs Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_39) Xy00)))))))))) P0) as proof of (P1 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (Xs Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0))))))))))
% Found (((eq_ref ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (Xs Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_39) Xy00)))))))))) P0) as proof of (P1 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (Xs Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0))))))))))
% Found eta_expansion000:=(eta_expansion00 b):(((eq ((a->a)->Prop)) b) (fun (x:(a->a))=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq ((a->a)->Prop)) b) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_38) Xy))))))))))
% Found ((eta_expansion0 Prop) b) as proof of (((eq ((a->a)->Prop)) b) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_38) Xy))))))))))
% Found (((eta_expansion (a->a)) Prop) b) as proof of (((eq ((a->a)->Prop)) b) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_38) Xy))))))))))
% Found (((eta_expansion (a->a)) Prop) b) as proof of (((eq ((a->a)->Prop)) b) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_38) Xy))))))))))
% Found (((eta_expansion (a->a)) Prop) b) as proof of (((eq ((a->a)->Prop)) b) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_38) Xy))))))))))
% Found eq_ref00:=(eq_ref0 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (Xs Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0)))))))))):(((eq ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (Xs Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0)))))))))) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (Xs Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0))))))))))
% Found (eq_ref0 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (Xs Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0)))))))))) as proof of (((eq ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (Xs Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0)))))))))) b)
% Found ((eq_ref ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (Xs Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0)))))))))) as proof of (((eq ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (Xs Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0)))))))))) b)
% Found ((eq_ref ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (Xs Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0)))))))))) as proof of (((eq ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (Xs Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0)))))))))) b)
% Found ((eq_ref ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (Xs Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0)))))))))) as proof of (((eq ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (Xs Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0)))))))))) b)
% Found eq_ref00:=(eq_ref0 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (Xs Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0)))))))))):(((eq ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (Xs Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0)))))))))) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (Xs Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0))))))))))
% Found (eq_ref0 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (Xs Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0)))))))))) as proof of (((eq ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (Xs Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0)))))))))) b)
% Found ((eq_ref ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (Xs Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0)))))))))) as proof of (((eq ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (Xs Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0)))))))))) b)
% Found ((eq_ref ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (Xs Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0)))))))))) as proof of (((eq ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (Xs Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0)))))))))) b)
% Found ((eq_ref ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (Xs Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0)))))))))) as proof of (((eq ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (Xs Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0)))))))))) 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), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_38) Xy))))))))))
% 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)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_38) Xy))))))))))
% 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)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_38) Xy))))))))))
% 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)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_38) Xy))))))))))
% Found eta_expansion0000:=(eta_expansion000 P0):((P0 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (Xs Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_39) Xy00))))))))))->(P0 (fun (x:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_39) Xy00)))))))))))
% Found (eta_expansion000 P0) as proof of (P1 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (Xs Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0))))))))))
% Found ((eta_expansion00 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (Xs Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_39) Xy00)))))))))) P0) as proof of (P1 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (Xs Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0))))))))))
% Found (((eta_expansion0 Prop) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (Xs Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_39) Xy00)))))))))) P0) as proof of (P1 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (Xs Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0))))))))))
% Found ((((eta_expansion (a->a)) Prop) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (Xs Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_39) Xy00)))))))))) P0) as proof of (P1 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (Xs Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0))))))))))
% Found ((((eta_expansion (a->a)) Prop) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (Xs Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_39) Xy00)))))))))) P0) as proof of (P1 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (Xs Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0))))))))))
% Found eta_expansion0000:=(eta_expansion000 P0):((P0 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (Xs Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_39) Xy00))))))))))->(P0 (fun (x:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_39) Xy00)))))))))))
% Found (eta_expansion000 P0) as proof of (P1 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (Xs Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0))))))))))
% Found ((eta_expansion00 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (Xs Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_39) Xy00)))))))))) P0) as proof of (P1 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (Xs Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0))))))))))
% Found (((eta_expansion0 Prop) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (Xs Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_39) Xy00)))))))))) P0) as proof of (P1 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (Xs Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0))))))))))
% Found ((((eta_expansion (a->a)) Prop) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (Xs Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_39) Xy00)))))))))) P0) as proof of (P1 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (Xs Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0))))))))))
% Found ((((eta_expansion (a->a)) Prop) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (Xs Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_39) Xy00)))))))))) P0) as proof of (P1 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (Xs Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0))))))))))
% Found eta_expansion0000:=(eta_expansion000 P0):((P0 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (Xs Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_39) Xy00))))))))))->(P0 (fun (x:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_39) Xy00)))))))))))
% Found (eta_expansion000 P0) as proof of (P1 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (Xs Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0))))))))))
% Found ((eta_expansion00 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (Xs Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_39) Xy00)))))))))) P0) as proof of (P1 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (Xs Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0))))))))))
% Found (((eta_expansion0 Prop) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (Xs Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_39) Xy00)))))))))) P0) as proof of (P1 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (Xs Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0))))))))))
% Found ((((eta_expansion (a->a)) Prop) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (Xs Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_39) Xy00)))))))))) P0) as proof of (P1 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (Xs Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0))))))))))
% Found ((((eta_expansion (a->a)) Prop) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (Xs Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_39) Xy00)))))))))) P0) as proof of (P1 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (Xs Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0))))))))))
% Found eta_expansion0000:=(eta_expansion000 P0):((P0 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (Xs Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_39) Xy00))))))))))->(P0 (fun (x:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_39) Xy00)))))))))))
% Found (eta_expansion000 P0) as proof of (P1 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (Xs Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0))))))))))
% Found ((eta_expansion00 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (Xs Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_39) Xy00)))))))))) P0) as proof of (P1 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (Xs Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0))))))))))
% Found (((eta_expansion0 Prop) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (Xs Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_39) Xy00)))))))))) P0) as proof of (P1 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (Xs Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0))))))))))
% Found ((((eta_expansion (a->a)) Prop) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (Xs Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_39) Xy00)))))))))) P0) as proof of (P1 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (Xs Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0))))))))))
% Found ((((eta_expansion (a->a)) Prop) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (Xs Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_39) Xy00)))))))))) P0) as proof of (P1 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (Xs Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0))))))))))
% Found x:((ex (a->a)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) ((fun (Xx:a) (Xy:a)=> (((eq a) Xx) Xy)) Xy_38)))))))))
% Instantiate: b:=(fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) ((fun (Xx:a) (Xy:a)=> (((eq a) Xx) Xy)) Xy_38)))))))):((a->a)->Prop)
% Found x as proof of (P b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (Xs Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_39) Xy)))))))))):(((eq ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (Xs Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_39) Xy)))))))))) (fun (x:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy:a)=> (((eq a) Xy_39) Xy))))))))))
% Found (eta_expansion_dep00 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (Xs Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_39) Xy)))))))))) as proof of (((eq ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (Xs Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_39) Xy)))))))))) b)
% Found ((eta_expansion_dep0 (fun (x5:(a->a))=> Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (Xs Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_39) Xy)))))))))) as proof of (((eq ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (Xs Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_39) Xy)))))))))) b)
% Found (((eta_expansion_dep (a->a)) (fun (x5:(a->a))=> Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (Xs Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_39) Xy)))))))))) as proof of (((eq ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (Xs Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_39) Xy)))))))))) b)
% Found (((eta_expansion_dep (a->a)) (fun (x5:(a->a))=> Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (Xs Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_39) Xy)))))))))) as proof of (((eq ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (Xs Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_39) Xy)))))))))) b)
% Found (((eta_expansion_dep (a->a)) (fun (x5:(a->a))=> Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (Xs Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_39) Xy)))))))))) as proof of (((eq ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (Xs Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy:a)=> (((eq a) Xy_39) Xy)))))))))) b)
% Found eq_ref000:=(eq_ref00 P0):((P0 ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_38) Xy00)))))))))->(P0 ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_38) Xy00))))))))))
% Found (eq_ref00 P0) as proof of (P1 ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0)))))))))
% Found ((eq_ref0 ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_38) Xy00))))))))) P0) as proof of (P1 ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0)))))))))
% Found (((eq_ref Prop) ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_38) Xy00))))))))) P0) as proof of (P1 ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0)))))))))
% Found (((eq_ref Prop) ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_38) Xy00))))))))) P0) as proof of (P1 ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0)))))))))
% Found eq_ref000:=(eq_ref00 P0):((P0 ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_38) Xy00)))))))))->(P0 ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_38) Xy00))))))))))
% Found (eq_ref00 P0) as proof of (P1 ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0)))))))))
% Found ((eq_ref0 ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_38) Xy00))))))))) P0) as proof of (P1 ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0)))))))))
% Found (((eq_ref Prop) ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_38) Xy00))))))))) P0) as proof of (P1 ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0)))))))))
% Found (((eq_ref Prop) ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_38) Xy00))))))))) P0) as proof of (P1 ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0)))))))))
% Found eq_ref000:=(eq_ref00 P0):((P0 ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_38) Xy00)))))))))->(P0 ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_38) Xy00))))))))))
% Found (eq_ref00 P0) as proof of (P1 ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0)))))))))
% Found ((eq_ref0 ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_38) Xy00))))))))) P0) as proof of (P1 ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0)))))))))
% Found (((eq_ref Prop) ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_38) Xy00))))))))) P0) as proof of (P1 ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0)))))))))
% Found (((eq_ref Prop) ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_38) Xy00))))))))) P0) as proof of (P1 ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0)))))))))
% Found eq_ref000:=(eq_ref00 P0):((P0 ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_38) Xy00)))))))))->(P0 ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_38) Xy00))))))))))
% Found (eq_ref00 P0) as proof of (P1 ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0)))))))))
% Found ((eq_ref0 ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_38) Xy00))))))))) P0) as proof of (P1 ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0)))))))))
% Found (((eq_ref Prop) ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_38) Xy00))))))))) P0) as proof of (P1 ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0)))))))))
% Found (((eq_ref Prop) ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_38) Xy00))))))))) P0) as proof of (P1 ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0)))))))))
% Found eq_ref00:=(eq_ref0 ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0))))))))):(((eq Prop) ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0))))))))) ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0)))))))))
% Found (eq_ref0 ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0))))))))) as proof of (((eq Prop) ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0))))))))) as proof of (((eq Prop) ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0))))))))) as proof of (((eq Prop) ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0))))))))) as proof of (((eq Prop) ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0))))))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy:a)=> (((eq a) Xy_39) Xy)))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy:a)=> (((eq a) Xy_39) Xy)))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy:a)=> (((eq a) Xy_39) Xy)))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy:a)=> (((eq a) Xy_39) Xy)))))))))
% Found eq_ref00:=(eq_ref0 ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0))))))))):(((eq Prop) ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0))))))))) ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0)))))))))
% Found (eq_ref0 ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0))))))))) as proof of (((eq Prop) ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0))))))))) as proof of (((eq Prop) ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0))))))))) as proof of (((eq Prop) ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0))))))))) as proof of (((eq Prop) ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0))))))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy:a)=> (((eq a) Xy_39) Xy)))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy:a)=> (((eq a) Xy_39) Xy)))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy:a)=> (((eq a) Xy_39) Xy)))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy:a)=> (((eq a) Xy_39) Xy)))))))))
% Found eq_ref00:=(eq_ref0 ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0))))))))):(((eq Prop) ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0))))))))) ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0)))))))))
% Found (eq_ref0 ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0))))))))) as proof of (((eq Prop) ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0))))))))) as proof of (((eq Prop) ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0))))))))) as proof of (((eq Prop) ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0))))))))) as proof of (((eq Prop) ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0))))))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy:a)=> (((eq a) Xy_39) Xy)))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy:a)=> (((eq a) Xy_39) Xy)))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy:a)=> (((eq a) Xy_39) Xy)))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy:a)=> (((eq a) Xy_39) Xy)))))))))
% Found eq_ref00:=(eq_ref0 ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0))))))))):(((eq Prop) ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0))))))))) ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0)))))))))
% Found (eq_ref0 ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0))))))))) as proof of (((eq Prop) ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0))))))))) as proof of (((eq Prop) ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0))))))))) as proof of (((eq Prop) ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0))))))))) as proof of (((eq Prop) ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0))))))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy:a)=> (((eq a) Xy_39) Xy)))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy:a)=> (((eq a) Xy_39) Xy)))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy:a)=> (((eq a) Xy_39) Xy)))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy:a)=> (((eq a) Xy_39) Xy)))))))))
% Found x:((ex (a->a)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) ((fun (Xx:a) (Xy:a)=> (((eq a) Xx) Xy)) Xy_38)))))))))
% Instantiate: f:=(fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) ((fun (Xx:a) (Xy:a)=> (((eq a) Xx) Xy)) Xy_38)))))))):((a->a)->Prop)
% Found x as proof of (P f)
% Found eq_ref00:=(eq_ref0 (f x4)):(((eq Prop) (f x4)) (f x4))
% Found (eq_ref0 (f x4)) as proof of (((eq Prop) (f x4)) ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x4 Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x4 Xx0))))) (fun (Xy:a)=> (((eq a) Xy_39) Xy)))))))))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x4 Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x4 Xx0))))) (fun (Xy:a)=> (((eq a) Xy_39) Xy)))))))))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x4 Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x4 Xx0))))) (fun (Xy:a)=> (((eq a) Xy_39) Xy)))))))))
% Found (fun (x4:(a->a))=> ((eq_ref Prop) (f x4))) as proof of (((eq Prop) (f x4)) ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x4 Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x4 Xx0))))) (fun (Xy:a)=> (((eq a) Xy_39) Xy)))))))))
% Found (fun (x4:(a->a))=> ((eq_ref Prop) (f x4))) as proof of (forall (x:(a->a)), (((eq Prop) (f x)) ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy:a)=> (((eq a) Xy_39) Xy))))))))))
% Found x:((ex (a->a)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) ((fun (Xx:a) (Xy:a)=> (((eq a) Xx) Xy)) Xy_38)))))))))
% Instantiate: f:=(fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) ((fun (Xx:a) (Xy:a)=> (((eq a) Xx) Xy)) Xy_38)))))))):((a->a)->Prop)
% Found x as proof of (P f)
% Found eq_ref00:=(eq_ref0 (f x4)):(((eq Prop) (f x4)) (f x4))
% Found (eq_ref0 (f x4)) as proof of (((eq Prop) (f x4)) ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x4 Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x4 Xx0))))) (fun (Xy:a)=> (((eq a) Xy_39) Xy)))))))))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x4 Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x4 Xx0))))) (fun (Xy:a)=> (((eq a) Xy_39) Xy)))))))))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x4 Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x4 Xx0))))) (fun (Xy:a)=> (((eq a) Xy_39) Xy)))))))))
% Found (fun (x4:(a->a))=> ((eq_ref Prop) (f x4))) as proof of (((eq Prop) (f x4)) ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x4 Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x4 Xx0))))) (fun (Xy:a)=> (((eq a) Xy_39) Xy)))))))))
% Found (fun (x4:(a->a))=> ((eq_ref Prop) (f x4))) as proof of (forall (x:(a->a)), (((eq Prop) (f x)) ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy:a)=> (((eq a) Xy_39) Xy))))))))))
% 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)->(Xx (Xs Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0))))))))))
% Found ((eq_ref ((a->a)->Prop)) b) as proof of (((eq ((a->a)->Prop)) b) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (Xs Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0))))))))))
% Found ((eq_ref ((a->a)->Prop)) b) as proof of (((eq ((a->a)->Prop)) b) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (Xs Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0))))))))))
% Found ((eq_ref ((a->a)->Prop)) b) as proof of (((eq ((a->a)->Prop)) b) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (Xs Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0))))))))))
% Found eta_expansion000:=(eta_expansion00 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0)))))))))):(((eq ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0)))))))))) (fun (x:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0))))))))))
% Found (eta_expansion00 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0)))))))))) as proof of (((eq ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0)))))))))) b)
% Found ((eta_expansion0 Prop) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0)))))))))) as proof of (((eq ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0)))))))))) b)
% Found (((eta_expansion (a->a)) Prop) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0)))))))))) as proof of (((eq ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0)))))))))) b)
% Found (((eta_expansion (a->a)) Prop) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0)))))))))) as proof of (((eq ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0)))))))))) b)
% Found (((eta_expansion (a->a)) Prop) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0)))))))))) as proof of (((eq ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0)))))))))) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq ((a->a)->Prop)) b) (fun (x:(a->a))=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq ((a->a)->Prop)) b) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (Xs Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0))))))))))
% Found ((eta_expansion_dep0 (fun (x1:(a->a))=> Prop)) b) as proof of (((eq ((a->a)->Prop)) b) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (Xs Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0))))))))))
% Found (((eta_expansion_dep (a->a)) (fun (x1:(a->a))=> Prop)) b) as proof of (((eq ((a->a)->Prop)) b) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (Xs Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0))))))))))
% Found (((eta_expansion_dep (a->a)) (fun (x1:(a->a))=> Prop)) b) as proof of (((eq ((a->a)->Prop)) b) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (Xs Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0))))))))))
% Found (((eta_expansion_dep (a->a)) (fun (x1:(a->a))=> Prop)) b) as proof of (((eq ((a->a)->Prop)) b) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (Xs Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0))))))))))
% Found eta_expansion000:=(eta_expansion00 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0)))))))))):(((eq ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0)))))))))) (fun (x:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0))))))))))
% Found (eta_expansion00 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0)))))))))) as proof of (((eq ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0)))))))))) b)
% Found ((eta_expansion0 Prop) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0)))))))))) as proof of (((eq ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0)))))))))) b)
% Found (((eta_expansion (a->a)) Prop) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0)))))))))) as proof of (((eq ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0)))))))))) b)
% Found (((eta_expansion (a->a)) Prop) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0)))))))))) as proof of (((eq ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0)))))))))) b)
% Found (((eta_expansion (a->a)) Prop) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0)))))))))) as proof of (((eq ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0)))))))))) b)
% Found eta_expansion0000:=(eta_expansion000 P0):((P0 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_38) Xy00))))))))))->(P0 (fun (x:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_38) Xy00)))))))))))
% Found (eta_expansion000 P0) as proof of (P1 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0))))))))))
% Found ((eta_expansion00 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_38) Xy00)))))))))) P0) as proof of (P1 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0))))))))))
% Found (((eta_expansion0 Prop) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_38) Xy00)))))))))) P0) as proof of (P1 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0))))))))))
% Found ((((eta_expansion (a->a)) Prop) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_38) Xy00)))))))))) P0) as proof of (P1 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0))))))))))
% Found ((((eta_expansion (a->a)) Prop) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_38) Xy00)))))))))) P0) as proof of (P1 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0))))))))))
% Found eta_expansion0000:=(eta_expansion000 P0):((P0 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_38) Xy00))))))))))->(P0 (fun (x:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_38) Xy00)))))))))))
% Found (eta_expansion000 P0) as proof of (P1 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0))))))))))
% Found ((eta_expansion00 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_38) Xy00)))))))))) P0) as proof of (P1 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0))))))))))
% Found (((eta_expansion0 Prop) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_38) Xy00)))))))))) P0) as proof of (P1 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0))))))))))
% Found ((((eta_expansion (a->a)) Prop) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_38) Xy00)))))))))) P0) as proof of (P1 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0))))))))))
% Found ((((eta_expansion (a->a)) Prop) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_38) Xy00)))))))))) P0) as proof of (P1 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0))))))))))
% Found eq_ref000:=(eq_ref00 P0):((P0 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_38) Xy00))))))))))->(P0 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_38) Xy00)))))))))))
% Found (eq_ref00 P0) as proof of (P1 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0))))))))))
% Found ((eq_ref0 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_38) Xy00)))))))))) P0) as proof of (P1 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0))))))))))
% Found (((eq_ref ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_38) Xy00)))))))))) P0) as proof of (P1 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0))))))))))
% Found (((eq_ref ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_38) Xy00)))))))))) P0) as proof of (P1 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0))))))))))
% Found eq_ref000:=(eq_ref00 P0):((P0 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_38) Xy00))))))))))->(P0 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_38) Xy00)))))))))))
% Found (eq_ref00 P0) as proof of (P1 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0))))))))))
% Found ((eq_ref0 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_38) Xy00)))))))))) P0) as proof of (P1 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0))))))))))
% Found (((eq_ref ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_38) Xy00)))))))))) P0) as proof of (P1 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0))))))))))
% Found (((eq_ref ((a->a)->Prop)) (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_38) Xy00)))))))))) P0) as proof of (P1 (fun (Xs:(a->a))=> ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (Xs Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (Xs Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0))))))))))
% Found eq_ref000:=(eq_ref00 P0):((P0 ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_39) Xy00)))))))))->(P0 ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_39) Xy00))))))))))
% Found (eq_ref00 P0) as proof of (P1 ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0)))))))))
% Found ((eq_ref0 ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_39) Xy00))))))))) P0) as proof of (P1 ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0)))))))))
% Found (((eq_ref Prop) ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_39) Xy00))))))))) P0) as proof of (P1 ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0)))))))))
% Found (((eq_ref Prop) ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_39) Xy00))))))))) P0) as proof of (P1 ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0)))))))))
% Found eq_ref000:=(eq_ref00 P0):((P0 ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_39) Xy00)))))))))->(P0 ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_39) Xy00))))))))))
% Found (eq_ref00 P0) as proof of (P1 ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0)))))))))
% Found ((eq_ref0 ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_39) Xy00))))))))) P0) as proof of (P1 ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0)))))))))
% Found (((eq_ref Prop) ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_39) Xy00))))))))) P0) as proof of (P1 ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0)))))))))
% Found (((eq_ref Prop) ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_39) Xy00))))))))) P0) as proof of (P1 ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0)))))))))
% Found eq_ref000:=(eq_ref00 P0):((P0 ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_39) Xy00)))))))))->(P0 ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_39) Xy00))))))))))
% Found (eq_ref00 P0) as proof of (P1 ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0)))))))))
% Found ((eq_ref0 ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_39) Xy00))))))))) P0) as proof of (P1 ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0)))))))))
% Found (((eq_ref Prop) ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_39) Xy00))))))))) P0) as proof of (P1 ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0)))))))))
% Found (((eq_ref Prop) ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_39) Xy00))))))))) P0) as proof of (P1 ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0)))))))))
% Found eq_ref000:=(eq_ref00 P0):((P0 ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_39) Xy00)))))))))->(P0 ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_39) Xy00))))))))))
% Found (eq_ref00 P0) as proof of (P1 ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0)))))))))
% Found ((eq_ref0 ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_39) Xy00))))))))) P0) as proof of (P1 ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0)))))))))
% Found (((eq_ref Prop) ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_39) Xy00))))))))) P0) as proof of (P1 ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0)))))))))
% Found (((eq_ref Prop) ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_39) Xy00))))))))) P0) as proof of (P1 ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0)))))))))
% 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)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0)))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0)))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0)))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0)))))))))
% Found eq_ref00:=(eq_ref0 ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0))))))))):(((eq Prop) ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0))))))))) ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0)))))))))
% Found (eq_ref0 ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0))))))))) as proof of (((eq Prop) ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0))))))))) as proof of (((eq Prop) ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0))))))))) as proof of (((eq Prop) ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0))))))))) as proof of (((eq Prop) ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0))))))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0)))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0)))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0)))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0)))))))))
% Found eq_ref00:=(eq_ref0 ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0))))))))):(((eq Prop) ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0))))))))) ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0)))))))))
% Found (eq_ref0 ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0))))))))) as proof of (((eq Prop) ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0))))))))) as proof of (((eq Prop) ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0))))))))) as proof of (((eq Prop) ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0))))))))) as proof of (((eq Prop) ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0))))))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0)))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0)))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0)))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0)))))))))
% Found eq_ref00:=(eq_ref0 ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0))))))))):(((eq Prop) ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0))))))))) ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0)))))))))
% Found (eq_ref0 ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0))))))))) as proof of (((eq Prop) ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0))))))))) as proof of (((eq Prop) ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0))))))))) as proof of (((eq Prop) ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0))))))))) as proof of (((eq Prop) ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0))))))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0)))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0)))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0)))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0)))))))))
% Found eq_ref00:=(eq_ref0 ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0))))))))):(((eq Prop) ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0))))))))) ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0)))))))))
% Found (eq_ref0 ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0))))))))) as proof of (((eq Prop) ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0))))))))) as proof of (((eq Prop) ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0))))))))) as proof of (((eq Prop) ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0))))))))) as proof of (((eq Prop) ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0))))))))) b)
% Found eq_ref00:=(eq_ref0 ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0))))))))):(((eq Prop) ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0))))))))) ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0)))))))))
% Found (eq_ref0 ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0))))))))) as proof of (((eq Prop) ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0))))))))) as proof of (((eq Prop) ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0))))))))) as proof of (((eq Prop) ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0))))))))) as proof of (((eq Prop) ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0))))))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy:a)=> (((eq a) Xy_38) Xy)))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy:a)=> (((eq a) Xy_38) Xy)))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy:a)=> (((eq a) Xy_38) Xy)))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy:a)=> (((eq a) Xy_38) Xy)))))))))
% Found eq_ref00:=(eq_ref0 ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0))))))))):(((eq Prop) ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0))))))))) ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0)))))))))
% Found (eq_ref0 ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0))))))))) as proof of (((eq Prop) ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0))))))))) as proof of (((eq Prop) ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0))))))))) as proof of (((eq Prop) ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0))))))))) as proof of (((eq Prop) ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0))))))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy:a)=> (((eq a) Xy_38) Xy)))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy:a)=> (((eq a) Xy_38) Xy)))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy:a)=> (((eq a) Xy_38) Xy)))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy:a)=> (((eq a) Xy_38) Xy)))))))))
% Found eq_ref00:=(eq_ref0 ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0))))))))):(((eq Prop) ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0))))))))) ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0)))))))))
% Found (eq_ref0 ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0))))))))) as proof of (((eq Prop) ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0))))))))) as proof of (((eq Prop) ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0))))))))) as proof of (((eq Prop) ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0))))))))) as proof of (((eq Prop) ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0))))))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy:a)=> (((eq a) Xy_38) Xy)))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy:a)=> (((eq a) Xy_38) Xy)))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy:a)=> (((eq a) Xy_38) Xy)))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy:a)=> (((eq a) Xy_38) Xy)))))))))
% Found eq_ref00:=(eq_ref0 ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0))))))))):(((eq Prop) ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0))))))))) ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0)))))))))
% Found (eq_ref0 ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0))))))))) as proof of (((eq Prop) ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0))))))))) as proof of (((eq Prop) ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0))))))))) as proof of (((eq Prop) ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0))))))))) as proof of (((eq Prop) ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0))))))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy:a)=> (((eq a) Xy_38) Xy)))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy:a)=> (((eq a) Xy_38) Xy)))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy:a)=> (((eq a) Xy_38) Xy)))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy:a)=> (((eq a) Xy_38) Xy)))))))))
% Found eq_ref000:=(eq_ref00 P0):((P0 ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_38) Xy00)))))))))->(P0 ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_38) Xy00))))))))))
% Found (eq_ref00 P0) as proof of (P1 ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0)))))))))
% Found ((eq_ref0 ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_38) Xy00))))))))) P0) as proof of (P1 ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0)))))))))
% Found (((eq_ref Prop) ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_38) Xy00))))))))) P0) as proof of (P1 ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0)))))))))
% Found (((eq_ref Prop) ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_38) Xy00))))))))) P0) as proof of (P1 ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0)))))))))
% Found eq_ref000:=(eq_ref00 P0):((P0 ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_38) Xy00)))))))))->(P0 ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_38) Xy00))))))))))
% Found (eq_ref00 P0) as proof of (P1 ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0)))))))))
% Found ((eq_ref0 ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_38) Xy00))))))))) P0) as proof of (P1 ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0)))))))))
% Found (((eq_ref Prop) ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_38) Xy00))))))))) P0) as proof of (P1 ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0)))))))))
% Found (((eq_ref Prop) ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_38) Xy00))))))))) P0) as proof of (P1 ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0)))))))))
% Found eq_ref000:=(eq_ref00 P0):((P0 ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_38) Xy00)))))))))->(P0 ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_38) Xy00))))))))))
% Found (eq_ref00 P0) as proof of (P1 ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0)))))))))
% Found ((eq_ref0 ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_38) Xy00))))))))) P0) as proof of (P1 ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0)))))))))
% Found (((eq_ref Prop) ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_38) Xy00))))))))) P0) as proof of (P1 ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0)))))))))
% Found (((eq_ref Prop) ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_38) Xy00))))))))) P0) as proof of (P1 ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0)))))))))
% Found eq_ref000:=(eq_ref00 P0):((P0 ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_38) Xy00)))))))))->(P0 ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_38) Xy00))))))))))
% Found (eq_ref00 P0) as proof of (P1 ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0)))))))))
% Found ((eq_ref0 ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_38) Xy00))))))))) P0) as proof of (P1 ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0)))))))))
% Found (((eq_ref Prop) ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_38) Xy00))))))))) P0) as proof of (P1 ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0)))))))))
% Found (((eq_ref Prop) ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy00:a)=> (((eq a) Xy_38) Xy00))))))))) P0) as proof of (P1 ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0)))))))))
% Found eq_ref00:=(eq_ref0 ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0))))))))):(((eq Prop) ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0))))))))) ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0)))))))))
% Found (eq_ref0 ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0))))))))) as proof of (((eq Prop) ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0))))))))) as proof of (((eq Prop) ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0))))))))) as proof of (((eq Prop) ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0))))))))) as proof of (((eq Prop) ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0))))))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0)))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0)))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0)))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0)))))))))
% Found eq_ref00:=(eq_ref0 ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0))))))))):(((eq Prop) ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0))))))))) ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0)))))))))
% Found (eq_ref0 ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0))))))))) as proof of (((eq Prop) ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0))))))))) as proof of (((eq Prop) ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0))))))))) as proof of (((eq Prop) ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0))))))))) as proof of (((eq Prop) ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0))))))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0)))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0)))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0)))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0)))))))))
% Found eq_ref00:=(eq_ref0 ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0))))))))):(((eq Prop) ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0))))))))) ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0)))))))))
% Found (eq_ref0 ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0))))))))) as proof of (((eq Prop) ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0))))))))) as proof of (((eq Prop) ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0))))))))) as proof of (((eq Prop) ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0))))))))) as proof of (((eq Prop) ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0))))))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0)))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0)))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0)))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0)))))))))
% Found eq_ref00:=(eq_ref0 ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0))))))))):(((eq Prop) ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0))))))))) ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0)))))))))
% Found (eq_ref0 ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0))))))))) as proof of (((eq Prop) ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0))))))))) as proof of (((eq Prop) ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0))))))))) as proof of (((eq Prop) ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0))))))))) as proof of (((eq Prop) ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0))))))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0)))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0)))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0)))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0)))))))))
% Found eq_ref00:=(eq_ref0 ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0))))))))):(((eq Prop) ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0))))))))) ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0)))))))))
% Found (eq_ref0 ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0))))))))) as proof of (((eq Prop) ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0))))))))) as proof of (((eq Prop) ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0))))))))) as proof of (((eq Prop) ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0))))))))) as proof of (((eq Prop) ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0))))))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0)))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0)))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0)))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0)))))))))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0)))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0)))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0)))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0)))))))))
% Found eq_ref00:=(eq_ref0 ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0))))))))):(((eq Prop) ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0))))))))) ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0)))))))))
% Found (eq_ref0 ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0))))))))) as proof of (((eq Prop) ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0))))))))) as proof of (((eq Prop) ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0))))))))) as proof of (((eq Prop) ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0))))))))) as proof of (((eq Prop) ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0))))))))) b)
% Found eq_ref00:=(eq_ref0 ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0))))))))):(((eq Prop) ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0))))))))) ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0)))))))))
% Found (eq_ref0 ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0))))))))) as proof of (((eq Prop) ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0))))))))) as proof of (((eq Prop) ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0))))))))) as proof of (((eq Prop) ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0))))))))) as proof of (((eq Prop) ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0))))))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0)))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0)))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0)))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx0:a), ((Xy Xx0)->(Xx (x Xx0))))) (forall (Xy0:a), ((Xx Xy0)->((ex a) (fun (Xy_39:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xy Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_39) Xy0)))))))))
% Found eq_ref00:=(eq_ref0 ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0))))))))):(((eq Prop) ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0))))))))) ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0)))))))))
% Found (eq_ref0 ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0))))))))) as proof of (((eq Prop) ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0))))) (forall (Xy0:a), ((Xy Xy0)->((ex a) (fun (Xy_38:a)=> (((eq (a->Prop)) (fun (Xx0:a)=> ((and (Xx Xx0)) (((eq a) Xy0) (x Xx0))))) (fun (Xy0:a)=> (((eq a) Xy_38) Xy0))))))))) as proof of (((eq Prop) ((and (forall (Xx0:a), ((Xx Xx0)->(Xy (x Xx0)))
% EOF
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