TSTP Solution File: SEU957^5 by cocATP---0.2.0
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%------------------------------------------------------------------------------
% File : cocATP---0.2.0
% Problem : SEU957^5 : TPTP v6.1.0. Released v4.0.0.
% Transfm : none
% Format : tptp:raw
% Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% Computer : n187.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 13:33:26 EDT 2014
% Result : Timeout 300.07s
% Output : None
% Verified :
% SZS Type : None (Parsing solution fails)
% Syntax : Number of formulae : 0
% Comments :
%------------------------------------------------------------------------------
%----NO SOLUTION OUTPUT BY SYSTEM
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% % Problem : SEU957^5 : TPTP v6.1.0. Released v4.0.0.
% % Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n187.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 11:46:21 CDT 2014
% % CPUTime : 300.07
% Python 2.7.5
% Using paths ['/home/cristobal/cocATP/CASC/TPTP/', '/export/starexec/sandbox/benchmark/', '/export/starexec/sandbox/benchmark/']
% FOF formula (<kernel.Constant object at 0x2677b00>, <kernel.Type object at 0x2677a28>) of role type named a_type
% Using role type
% Declaring a:Type
% FOF formula (<kernel.Constant object at 0x2362170>, <kernel.Type object at 0x2677a70>) of role type named b_type
% Using role type
% Declaring b:Type
% FOF formula (((ex ((a->Prop)->a)) (fun (Xc:((a->Prop)->a))=> (forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (Xc X))))))->(((ex (a->b)) (fun (Xg:(a->b))=> (forall (Y:b), ((ex a) (fun (X:a)=> (((eq b) (Xg X)) Y))))))->((ex (b->a)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy))))))) of role conjecture named cTHM607_pme
% Conjecture to prove = (((ex ((a->Prop)->a)) (fun (Xc:((a->Prop)->a))=> (forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (Xc X))))))->(((ex (a->b)) (fun (Xg:(a->b))=> (forall (Y:b), ((ex a) (fun (X:a)=> (((eq b) (Xg X)) Y))))))->((ex (b->a)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy))))))):Prop
% Parameter a_DUMMY:a.
% Parameter b_DUMMY:b.
% We need to prove ['(((ex ((a->Prop)->a)) (fun (Xc:((a->Prop)->a))=> (forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (Xc X))))))->(((ex (a->b)) (fun (Xg:(a->b))=> (forall (Y:b), ((ex a) (fun (X:a)=> (((eq b) (Xg X)) Y))))))->((ex (b->a)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy)))))))']
% Parameter a:Type.
% Parameter b:Type.
% Trying to prove (((ex ((a->Prop)->a)) (fun (Xc:((a->Prop)->a))=> (forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (Xc X))))))->(((ex (a->b)) (fun (Xg:(a->b))=> (forall (Y:b), ((ex a) (fun (X:a)=> (((eq b) (Xg X)) Y))))))->((ex (b->a)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy)))))))
% Found eq_ref00:=(eq_ref0 (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy))))):(((eq ((b->a)->Prop)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy))))) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy)))))
% Found (eq_ref0 (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy))))) as proof of (((eq ((b->a)->Prop)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy))))) b0)
% Found ((eq_ref ((b->a)->Prop)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy))))) as proof of (((eq ((b->a)->Prop)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy))))) b0)
% Found ((eq_ref ((b->a)->Prop)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy))))) as proof of (((eq ((b->a)->Prop)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy))))) b0)
% Found ((eq_ref ((b->a)->Prop)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy))))) as proof of (((eq ((b->a)->Prop)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy))))) b0)
% Found choice000:=(choice00 (fun (x5:b) (x40:a)=> (((eq b) (x1 x40)) x5))):((forall (x:b), ((ex a) (fun (y:a)=> (((eq b) (x1 y)) x))))->((ex (b->a)) (fun (f:(b->a))=> (forall (x:b), (((eq b) (x1 (f x))) x)))))
% Found (choice00 (fun (x5:b) (x40:a)=> (((eq b) (x1 x40)) x5))) as proof of ((forall (Y:b), ((ex a) (fun (X:a)=> (((eq b) (x1 X)) Y))))->(P b0))
% Found ((choice0 a) (fun (x5:b) (x40:a)=> (((eq b) (x1 x40)) x5))) as proof of ((forall (Y:b), ((ex a) (fun (X:a)=> (((eq b) (x1 X)) Y))))->(P b0))
% Found (((choice b) a) (fun (x5:b) (x40:a)=> (((eq b) (x1 x40)) x5))) as proof of ((forall (Y:b), ((ex a) (fun (X:a)=> (((eq b) (x1 X)) Y))))->(P b0))
% Found (((choice b) a) (fun (x5:b) (x40:a)=> (((eq b) (x1 x40)) x5))) as proof of ((forall (Y:b), ((ex a) (fun (X:a)=> (((eq b) (x1 X)) Y))))->(P b0))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy))))):(((eq ((b->a)->Prop)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy))))) (fun (x:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (x Xx)) (x Xy))->(((eq b) Xx) Xy)))))
% Found (eta_expansion_dep00 (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy))))) as proof of (((eq ((b->a)->Prop)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy))))) b0)
% Found ((eta_expansion_dep0 (fun (x4:(b->a))=> Prop)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy))))) as proof of (((eq ((b->a)->Prop)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy))))) b0)
% Found (((eta_expansion_dep (b->a)) (fun (x4:(b->a))=> Prop)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy))))) as proof of (((eq ((b->a)->Prop)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy))))) b0)
% Found (((eta_expansion_dep (b->a)) (fun (x4:(b->a))=> Prop)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy))))) as proof of (((eq ((b->a)->Prop)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy))))) b0)
% Found (((eta_expansion_dep (b->a)) (fun (x4:(b->a))=> Prop)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy))))) as proof of (((eq ((b->a)->Prop)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy))))) b0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy))))):(((eq ((b->a)->Prop)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy))))) (fun (x:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (x Xx)) (x Xy))->(((eq b) Xx) Xy)))))
% Found (eta_expansion_dep00 (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy))))) as proof of (((eq ((b->a)->Prop)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy))))) b0)
% Found ((eta_expansion_dep0 (fun (x4:(b->a))=> Prop)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy))))) as proof of (((eq ((b->a)->Prop)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy))))) b0)
% Found (((eta_expansion_dep (b->a)) (fun (x4:(b->a))=> Prop)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy))))) as proof of (((eq ((b->a)->Prop)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy))))) b0)
% Found (((eta_expansion_dep (b->a)) (fun (x4:(b->a))=> Prop)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy))))) as proof of (((eq ((b->a)->Prop)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy))))) b0)
% Found (((eta_expansion_dep (b->a)) (fun (x4:(b->a))=> Prop)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy))))) as proof of (((eq ((b->a)->Prop)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy))))) b0)
% Found x000:=(x00 (fun (x2:a)=> (P Xx))):((P Xx)->(P Xx))
% Found (x00 (fun (x2:a)=> (P Xx))) as proof of (P0 Xx)
% Found (x00 (fun (x2:a)=> (P Xx))) as proof of (P0 Xx)
% Found eq_ref00:=(eq_ref0 (fun (Xt:a)=> ((ex (b->a)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy))))))):(((eq (a->Prop)) (fun (Xt:a)=> ((ex (b->a)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy))))))) (fun (Xt:a)=> ((ex (b->a)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy)))))))
% Found (eq_ref0 (fun (Xt:a)=> ((ex (b->a)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((ex (b->a)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy))))))) b0)
% Found ((eq_ref (a->Prop)) (fun (Xt:a)=> ((ex (b->a)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((ex (b->a)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy))))))) b0)
% Found ((eq_ref (a->Prop)) (fun (Xt:a)=> ((ex (b->a)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((ex (b->a)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy))))))) b0)
% Found ((eq_ref (a->Prop)) (fun (Xt:a)=> ((ex (b->a)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((ex (b->a)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy))))))) b0)
% Found eq_ref00:=(eq_ref0 (f x3)):(((eq Prop) (f x3)) (f x3))
% Found (eq_ref0 (f x3)) as proof of (((eq Prop) (f x3)) ((ex (b->a)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy))))))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((ex (b->a)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy))))))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((ex (b->a)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy))))))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (((eq Prop) (f x3)) ((ex (b->a)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy))))))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (forall (x:a), (((eq Prop) (f x)) ((ex (b->a)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy)))))))
% Found eq_ref00:=(eq_ref0 (f x3)):(((eq Prop) (f x3)) (f x3))
% Found (eq_ref0 (f x3)) as proof of (((eq Prop) (f x3)) ((ex (b->a)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy))))))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((ex (b->a)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy))))))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((ex (b->a)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy))))))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (((eq Prop) (f x3)) ((ex (b->a)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy))))))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (forall (x:a), (((eq Prop) (f x)) ((ex (b->a)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy)))))))
% Found eq_ref00:=(eq_ref0 Xx):(((eq b) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq b) Xx) b0)
% Found ((eq_ref b) Xx) as proof of (((eq b) Xx) b0)
% Found ((eq_ref b) Xx) as proof of (((eq b) Xx) b0)
% Found ((eq_ref b) Xx) as proof of (((eq b) Xx) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq b) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq b) b0) Xy)
% Found ((eq_ref b) b0) as proof of (((eq b) b0) Xy)
% Found ((eq_ref b) b0) as proof of (((eq b) b0) Xy)
% Found ((eq_ref b) b0) as proof of (((eq b) b0) Xy)
% Found choice000:=(choice00 (fun (x7:b) (x60:a)=> (((eq b) (x3 x60)) x7))):((forall (x:b), ((ex a) (fun (y:a)=> (((eq b) (x3 y)) x))))->((ex (b->a)) (fun (f:(b->a))=> (forall (x:b), (((eq b) (x3 (f x))) x)))))
% Found (choice00 (fun (x7:b) (x60:a)=> (((eq b) (x3 x60)) x7))) as proof of ((forall (Y:b), ((ex a) (fun (X:a)=> (((eq b) (x3 X)) Y))))->(P b0))
% Found ((choice0 a) (fun (x7:b) (x60:a)=> (((eq b) (x3 x60)) x7))) as proof of ((forall (Y:b), ((ex a) (fun (X:a)=> (((eq b) (x3 X)) Y))))->(P b0))
% Found (((choice b) a) (fun (x7:b) (x60:a)=> (((eq b) (x3 x60)) x7))) as proof of ((forall (Y:b), ((ex a) (fun (X:a)=> (((eq b) (x3 X)) Y))))->(P b0))
% Found (((choice b) a) (fun (x7:b) (x60:a)=> (((eq b) (x3 x60)) x7))) as proof of ((forall (Y:b), ((ex a) (fun (X:a)=> (((eq b) (x3 X)) Y))))->(P b0))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy))))):(((eq ((b->a)->Prop)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy))))) (fun (x:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (x Xx)) (x Xy))->(((eq b) Xx) Xy)))))
% Found (eta_expansion_dep00 (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy))))) as proof of (((eq ((b->a)->Prop)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy))))) b0)
% Found ((eta_expansion_dep0 (fun (x6:(b->a))=> Prop)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy))))) as proof of (((eq ((b->a)->Prop)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy))))) b0)
% Found (((eta_expansion_dep (b->a)) (fun (x6:(b->a))=> Prop)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy))))) as proof of (((eq ((b->a)->Prop)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy))))) b0)
% Found (((eta_expansion_dep (b->a)) (fun (x6:(b->a))=> Prop)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy))))) as proof of (((eq ((b->a)->Prop)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy))))) b0)
% Found (((eta_expansion_dep (b->a)) (fun (x6:(b->a))=> Prop)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy))))) as proof of (((eq ((b->a)->Prop)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy))))) b0)
% Found eta_expansion000:=(eta_expansion00 (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy))))):(((eq ((b->a)->Prop)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy))))) (fun (x:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (x Xx)) (x Xy))->(((eq b) Xx) Xy)))))
% Found (eta_expansion00 (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy))))) as proof of (((eq ((b->a)->Prop)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy))))) b0)
% Found ((eta_expansion0 Prop) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy))))) as proof of (((eq ((b->a)->Prop)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy))))) b0)
% Found (((eta_expansion (b->a)) Prop) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy))))) as proof of (((eq ((b->a)->Prop)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy))))) b0)
% Found (((eta_expansion (b->a)) Prop) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy))))) as proof of (((eq ((b->a)->Prop)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy))))) b0)
% Found (((eta_expansion (b->a)) Prop) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy))))) as proof of (((eq ((b->a)->Prop)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy))))) b0)
% Found choice000:=(choice00 (fun (x5:b) (x40:a)=> (((eq b) (x1 x40)) x5))):((forall (x:b), ((ex a) (fun (y:a)=> (((eq b) (x1 y)) x))))->((ex (b->a)) (fun (f:(b->a))=> (forall (x:b), (((eq b) (x1 (f x))) x)))))
% Found (choice00 (fun (x5:b) (x40:a)=> (((eq b) (x1 x40)) x5))) as proof of ((forall (Y:b), ((ex a) (fun (X:a)=> (((eq b) (x1 X)) Y))))->(P f))
% Found ((choice0 a) (fun (x5:b) (x40:a)=> (((eq b) (x1 x40)) x5))) as proof of ((forall (Y:b), ((ex a) (fun (X:a)=> (((eq b) (x1 X)) Y))))->(P f))
% Found (((choice b) a) (fun (x5:b) (x40:a)=> (((eq b) (x1 x40)) x5))) as proof of ((forall (Y:b), ((ex a) (fun (X:a)=> (((eq b) (x1 X)) Y))))->(P f))
% Found (((choice b) a) (fun (x5:b) (x40:a)=> (((eq b) (x1 x40)) x5))) as proof of ((forall (Y:b), ((ex a) (fun (X:a)=> (((eq b) (x1 X)) Y))))->(P f))
% Found choice000:=(choice00 (fun (x5:b) (x40:a)=> (((eq b) (x1 x40)) x5))):((forall (x:b), ((ex a) (fun (y:a)=> (((eq b) (x1 y)) x))))->((ex (b->a)) (fun (f:(b->a))=> (forall (x:b), (((eq b) (x1 (f x))) x)))))
% Found (choice00 (fun (x5:b) (x40:a)=> (((eq b) (x1 x40)) x5))) as proof of ((forall (Y:b), ((ex a) (fun (X:a)=> (((eq b) (x1 X)) Y))))->(P f))
% Found ((choice0 a) (fun (x5:b) (x40:a)=> (((eq b) (x1 x40)) x5))) as proof of ((forall (Y:b), ((ex a) (fun (X:a)=> (((eq b) (x1 X)) Y))))->(P f))
% Found (((choice b) a) (fun (x5:b) (x40:a)=> (((eq b) (x1 x40)) x5))) as proof of ((forall (Y:b), ((ex a) (fun (X:a)=> (((eq b) (x1 X)) Y))))->(P f))
% Found (((choice b) a) (fun (x5:b) (x40:a)=> (((eq b) (x1 x40)) x5))) as proof of ((forall (Y:b), ((ex a) (fun (X:a)=> (((eq b) (x1 X)) Y))))->(P f))
% Found x000:=(x00 (fun (x4:a)=> (P Xx))):((P Xx)->(P Xx))
% Found (x00 (fun (x4:a)=> (P Xx))) as proof of (P0 Xx)
% Found (x00 (fun (x4:a)=> (P Xx))) as proof of (P0 Xx)
% Found choice000:=(choice00 (fun (x7:b) (x60:a)=> (((eq b) (x3 x60)) x7))):((forall (x:b), ((ex a) (fun (y:a)=> (((eq b) (x3 y)) x))))->((ex (b->a)) (fun (f:(b->a))=> (forall (x:b), (((eq b) (x3 (f x))) x)))))
% Found (choice00 (fun (x7:b) (x60:a)=> (((eq b) (x3 x60)) x7))) as proof of ((forall (Y:b), ((ex a) (fun (X:a)=> (((eq b) (x3 X)) Y))))->(P b0))
% Found ((choice0 a) (fun (x7:b) (x60:a)=> (((eq b) (x3 x60)) x7))) as proof of ((forall (Y:b), ((ex a) (fun (X:a)=> (((eq b) (x3 X)) Y))))->(P b0))
% Found (((choice b) a) (fun (x7:b) (x60:a)=> (((eq b) (x3 x60)) x7))) as proof of ((forall (Y:b), ((ex a) (fun (X:a)=> (((eq b) (x3 X)) Y))))->(P b0))
% Found (((choice b) a) (fun (x7:b) (x60:a)=> (((eq b) (x3 x60)) x7))) as proof of ((forall (Y:b), ((ex a) (fun (X:a)=> (((eq b) (x3 X)) Y))))->(P b0))
% Found x000:=(x00 (fun (x4:a)=> (P Xx))):((P Xx)->(P Xx))
% Found (x00 (fun (x4:a)=> (P Xx))) as proof of (P0 Xx)
% Found (x00 (fun (x4:a)=> (P Xx))) as proof of (P0 Xx)
% Found eq_ref00:=(eq_ref0 (fun (Xt:a)=> (forall (x30:(a->b)), ((forall (Y:b), ((ex a) (fun (X:a)=> (((eq b) (x30 X)) Y))))->((ex (b->a)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy))))))))):(((eq (a->Prop)) (fun (Xt:a)=> (forall (x30:(a->b)), ((forall (Y:b), ((ex a) (fun (X:a)=> (((eq b) (x30 X)) Y))))->((ex (b->a)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy))))))))) (fun (Xt:a)=> (forall (x30:(a->b)), ((forall (Y:b), ((ex a) (fun (X:a)=> (((eq b) (x30 X)) Y))))->((ex (b->a)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy)))))))))
% Found (eq_ref0 (fun (Xt:a)=> (forall (x30:(a->b)), ((forall (Y:b), ((ex a) (fun (X:a)=> (((eq b) (x30 X)) Y))))->((ex (b->a)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy))))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> (forall (x30:(a->b)), ((forall (Y:b), ((ex a) (fun (X:a)=> (((eq b) (x30 X)) Y))))->((ex (b->a)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy))))))))) b0)
% Found ((eq_ref (a->Prop)) (fun (Xt:a)=> (forall (x30:(a->b)), ((forall (Y:b), ((ex a) (fun (X:a)=> (((eq b) (x30 X)) Y))))->((ex (b->a)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy))))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> (forall (x30:(a->b)), ((forall (Y:b), ((ex a) (fun (X:a)=> (((eq b) (x30 X)) Y))))->((ex (b->a)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy))))))))) b0)
% Found ((eq_ref (a->Prop)) (fun (Xt:a)=> (forall (x30:(a->b)), ((forall (Y:b), ((ex a) (fun (X:a)=> (((eq b) (x30 X)) Y))))->((ex (b->a)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy))))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> (forall (x30:(a->b)), ((forall (Y:b), ((ex a) (fun (X:a)=> (((eq b) (x30 X)) Y))))->((ex (b->a)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy))))))))) b0)
% Found ((eq_ref (a->Prop)) (fun (Xt:a)=> (forall (x30:(a->b)), ((forall (Y:b), ((ex a) (fun (X:a)=> (((eq b) (x30 X)) Y))))->((ex (b->a)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy))))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> (forall (x30:(a->b)), ((forall (Y:b), ((ex a) (fun (X:a)=> (((eq b) (x30 X)) Y))))->((ex (b->a)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy))))))))) b0)
% Found eta_expansion000:=(eta_expansion00 (fun (Xt:a)=> ((forall (Y:b), ((ex a) (fun (X:a)=> (((eq b) (x3 X)) Y))))->((ex (b->a)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy)))))))):(((eq (a->Prop)) (fun (Xt:a)=> ((forall (Y:b), ((ex a) (fun (X:a)=> (((eq b) (x3 X)) Y))))->((ex (b->a)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy)))))))) (fun (x:a)=> ((forall (Y:b), ((ex a) (fun (X:a)=> (((eq b) (x3 X)) Y))))->((ex (b->a)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy))))))))
% Found (eta_expansion00 (fun (Xt:a)=> ((forall (Y:b), ((ex a) (fun (X:a)=> (((eq b) (x3 X)) Y))))->((ex (b->a)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy)))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((forall (Y:b), ((ex a) (fun (X:a)=> (((eq b) (x3 X)) Y))))->((ex (b->a)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy)))))))) b0)
% Found ((eta_expansion0 Prop) (fun (Xt:a)=> ((forall (Y:b), ((ex a) (fun (X:a)=> (((eq b) (x3 X)) Y))))->((ex (b->a)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy)))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((forall (Y:b), ((ex a) (fun (X:a)=> (((eq b) (x3 X)) Y))))->((ex (b->a)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy)))))))) b0)
% Found (((eta_expansion a) Prop) (fun (Xt:a)=> ((forall (Y:b), ((ex a) (fun (X:a)=> (((eq b) (x3 X)) Y))))->((ex (b->a)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy)))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((forall (Y:b), ((ex a) (fun (X:a)=> (((eq b) (x3 X)) Y))))->((ex (b->a)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy)))))))) b0)
% Found (((eta_expansion a) Prop) (fun (Xt:a)=> ((forall (Y:b), ((ex a) (fun (X:a)=> (((eq b) (x3 X)) Y))))->((ex (b->a)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy)))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((forall (Y:b), ((ex a) (fun (X:a)=> (((eq b) (x3 X)) Y))))->((ex (b->a)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy)))))))) b0)
% Found (((eta_expansion a) Prop) (fun (Xt:a)=> ((forall (Y:b), ((ex a) (fun (X:a)=> (((eq b) (x3 X)) Y))))->((ex (b->a)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy)))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((forall (Y:b), ((ex a) (fun (X:a)=> (((eq b) (x3 X)) Y))))->((ex (b->a)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy)))))))) b0)
% Found eq_ref00:=(eq_ref0 (fun (Xt:a)=> ((ex (b->a)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy))))))):(((eq (a->Prop)) (fun (Xt:a)=> ((ex (b->a)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy))))))) (fun (Xt:a)=> ((ex (b->a)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy)))))))
% Found (eq_ref0 (fun (Xt:a)=> ((ex (b->a)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((ex (b->a)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy))))))) b0)
% Found ((eq_ref (a->Prop)) (fun (Xt:a)=> ((ex (b->a)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((ex (b->a)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy))))))) b0)
% Found ((eq_ref (a->Prop)) (fun (Xt:a)=> ((ex (b->a)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((ex (b->a)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy))))))) b0)
% Found ((eq_ref (a->Prop)) (fun (Xt:a)=> ((ex (b->a)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((ex (b->a)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy))))))) b0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xt:a)=> ((ex (b->a)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy))))))):(((eq (a->Prop)) (fun (Xt:a)=> ((ex (b->a)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy))))))) (fun (x:a)=> ((ex (b->a)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy)))))))
% Found (eta_expansion_dep00 (fun (Xt:a)=> ((ex (b->a)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((ex (b->a)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy))))))) b0)
% Found ((eta_expansion_dep0 (fun (x6:a)=> Prop)) (fun (Xt:a)=> ((ex (b->a)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((ex (b->a)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy))))))) b0)
% Found (((eta_expansion_dep a) (fun (x6:a)=> Prop)) (fun (Xt:a)=> ((ex (b->a)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((ex (b->a)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy))))))) b0)
% Found (((eta_expansion_dep a) (fun (x6:a)=> Prop)) (fun (Xt:a)=> ((ex (b->a)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((ex (b->a)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy))))))) b0)
% Found (((eta_expansion_dep a) (fun (x6:a)=> Prop)) (fun (Xt:a)=> ((ex (b->a)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((ex (b->a)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy))))))) b0)
% Found x000:=(x00 (fun (x4:a)=> (P Xx))):((P Xx)->(P Xx))
% Found (x00 (fun (x4:a)=> (P Xx))) as proof of (P0 Xx)
% Found (x00 (fun (x4:a)=> (P Xx))) as proof of (P0 Xx)
% Found eq_ref00:=(eq_ref0 Xx):(((eq b) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq b) Xx) b0)
% Found ((eq_ref b) Xx) as proof of (((eq b) Xx) b0)
% Found ((eq_ref b) Xx) as proof of (((eq b) Xx) b0)
% Found ((eq_ref b) Xx) as proof of (((eq b) Xx) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq b) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq b) b0) Xy)
% Found ((eq_ref b) b0) as proof of (((eq b) b0) Xy)
% Found ((eq_ref b) b0) as proof of (((eq b) b0) Xy)
% Found ((eq_ref b) b0) as proof of (((eq b) b0) Xy)
% Found x000:=(x00 (fun (x4:a)=> (P Xx))):((P Xx)->(P Xx))
% Found (x00 (fun (x4:a)=> (P Xx))) as proof of (P0 Xx)
% Found (x00 (fun (x4:a)=> (P Xx))) as proof of (P0 Xx)
% Found eq_ref00:=(eq_ref0 Xx):(((eq b) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq b) Xx) b0)
% Found ((eq_ref b) Xx) as proof of (((eq b) Xx) b0)
% Found ((eq_ref b) Xx) as proof of (((eq b) Xx) b0)
% Found ((eq_ref b) Xx) as proof of (((eq b) Xx) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq b) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq b) b0) Xy)
% Found ((eq_ref b) b0) as proof of (((eq b) b0) Xy)
% Found ((eq_ref b) b0) as proof of (((eq b) b0) Xy)
% Found ((eq_ref b) b0) as proof of (((eq b) b0) Xy)
% Found eta_expansion000:=(eta_expansion00 (fun (Xt:a)=> ((ex (b->a)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy))))))):(((eq (a->Prop)) (fun (Xt:a)=> ((ex (b->a)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy))))))) (fun (x:a)=> ((ex (b->a)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy)))))))
% Found (eta_expansion00 (fun (Xt:a)=> ((ex (b->a)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((ex (b->a)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy))))))) b0)
% Found ((eta_expansion0 Prop) (fun (Xt:a)=> ((ex (b->a)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((ex (b->a)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy))))))) b0)
% Found (((eta_expansion a) Prop) (fun (Xt:a)=> ((ex (b->a)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((ex (b->a)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy))))))) b0)
% Found (((eta_expansion a) Prop) (fun (Xt:a)=> ((ex (b->a)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((ex (b->a)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy))))))) b0)
% Found (((eta_expansion a) Prop) (fun (Xt:a)=> ((ex (b->a)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((ex (b->a)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy))))))) b0)
% Found eq_ref00:=(eq_ref0 (f x3)):(((eq Prop) (f x3)) (f x3))
% Found (eq_ref0 (f x3)) as proof of (((eq Prop) (f x3)) (forall (x30:(a->b)), ((forall (Y:b), ((ex a) (fun (X:a)=> (((eq b) (x30 X)) Y))))->((ex (b->a)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy))))))))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) (forall (x30:(a->b)), ((forall (Y:b), ((ex a) (fun (X:a)=> (((eq b) (x30 X)) Y))))->((ex (b->a)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy))))))))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) (forall (x30:(a->b)), ((forall (Y:b), ((ex a) (fun (X:a)=> (((eq b) (x30 X)) Y))))->((ex (b->a)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy))))))))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (((eq Prop) (f x3)) (forall (x30:(a->b)), ((forall (Y:b), ((ex a) (fun (X:a)=> (((eq b) (x30 X)) Y))))->((ex (b->a)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy))))))))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (forall (x:a), (((eq Prop) (f x)) (forall (x30:(a->b)), ((forall (Y:b), ((ex a) (fun (X:a)=> (((eq b) (x30 X)) Y))))->((ex (b->a)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy)))))))))
% Found eq_ref00:=(eq_ref0 (f x3)):(((eq Prop) (f x3)) (f x3))
% Found (eq_ref0 (f x3)) as proof of (((eq Prop) (f x3)) (forall (x30:(a->b)), ((forall (Y:b), ((ex a) (fun (X:a)=> (((eq b) (x30 X)) Y))))->((ex (b->a)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy))))))))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) (forall (x30:(a->b)), ((forall (Y:b), ((ex a) (fun (X:a)=> (((eq b) (x30 X)) Y))))->((ex (b->a)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy))))))))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) (forall (x30:(a->b)), ((forall (Y:b), ((ex a) (fun (X:a)=> (((eq b) (x30 X)) Y))))->((ex (b->a)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy))))))))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (((eq Prop) (f x3)) (forall (x30:(a->b)), ((forall (Y:b), ((ex a) (fun (X:a)=> (((eq b) (x30 X)) Y))))->((ex (b->a)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy))))))))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (forall (x:a), (((eq Prop) (f x)) (forall (x30:(a->b)), ((forall (Y:b), ((ex a) (fun (X:a)=> (((eq b) (x30 X)) Y))))->((ex (b->a)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy)))))))))
% 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)) ((forall (Y:b), ((ex a) (fun (X:a)=> (((eq b) (x3 X)) Y))))->((ex (b->a)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy)))))))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((forall (Y:b), ((ex a) (fun (X:a)=> (((eq b) (x3 X)) Y))))->((ex (b->a)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy)))))))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((forall (Y:b), ((ex a) (fun (X:a)=> (((eq b) (x3 X)) Y))))->((ex (b->a)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy)))))))
% Found (fun (x4:a)=> ((eq_ref Prop) (f x4))) as proof of (((eq Prop) (f x4)) ((forall (Y:b), ((ex a) (fun (X:a)=> (((eq b) (x3 X)) Y))))->((ex (b->a)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy)))))))
% Found (fun (x4:a)=> ((eq_ref Prop) (f x4))) as proof of (forall (x:a), (((eq Prop) (f x)) ((forall (Y:b), ((ex a) (fun (X:a)=> (((eq b) (x3 X)) Y))))->((ex (b->a)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy))))))))
% 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)) ((forall (Y:b), ((ex a) (fun (X:a)=> (((eq b) (x3 X)) Y))))->((ex (b->a)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy)))))))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((forall (Y:b), ((ex a) (fun (X:a)=> (((eq b) (x3 X)) Y))))->((ex (b->a)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy)))))))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((forall (Y:b), ((ex a) (fun (X:a)=> (((eq b) (x3 X)) Y))))->((ex (b->a)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy)))))))
% Found (fun (x4:a)=> ((eq_ref Prop) (f x4))) as proof of (((eq Prop) (f x4)) ((forall (Y:b), ((ex a) (fun (X:a)=> (((eq b) (x3 X)) Y))))->((ex (b->a)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy)))))))
% Found (fun (x4:a)=> ((eq_ref Prop) (f x4))) as proof of (forall (x:a), (((eq Prop) (f x)) ((forall (Y:b), ((ex a) (fun (X:a)=> (((eq b) (x3 X)) Y))))->((ex (b->a)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy))))))))
% Found eq_ref00:=(eq_ref0 b0):(((eq b) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq b) b0) Xx)
% Found ((eq_ref b) b0) as proof of (((eq b) b0) Xx)
% Found ((eq_ref b) b0) as proof of (((eq b) b0) Xx)
% Found ((eq_ref b) b0) as proof of (((eq b) b0) Xx)
% Found eq_ref00:=(eq_ref0 Xy):(((eq b) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq b) Xy) b0)
% Found ((eq_ref b) Xy) as proof of (((eq b) Xy) b0)
% Found ((eq_ref b) Xy) as proof of (((eq b) Xy) b0)
% Found ((eq_ref b) Xy) as proof of (((eq b) Xy) b0)
% Found eq_ref00:=(eq_ref0 (f x5)):(((eq Prop) (f x5)) (f x5))
% Found (eq_ref0 (f x5)) as proof of (((eq Prop) (f x5)) ((ex (b->a)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy))))))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) ((ex (b->a)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy))))))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) ((ex (b->a)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy))))))
% Found (fun (x5:a)=> ((eq_ref Prop) (f x5))) as proof of (((eq Prop) (f x5)) ((ex (b->a)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy))))))
% Found (fun (x5:a)=> ((eq_ref Prop) (f x5))) as proof of (forall (x:a), (((eq Prop) (f x)) ((ex (b->a)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy)))))))
% Found eq_ref00:=(eq_ref0 (f x5)):(((eq Prop) (f x5)) (f x5))
% Found (eq_ref0 (f x5)) as proof of (((eq Prop) (f x5)) ((ex (b->a)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy))))))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) ((ex (b->a)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy))))))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) ((ex (b->a)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy))))))
% Found (fun (x5:a)=> ((eq_ref Prop) (f x5))) as proof of (((eq Prop) (f x5)) ((ex (b->a)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy))))))
% Found (fun (x5:a)=> ((eq_ref Prop) (f x5))) as proof of (forall (x:a), (((eq Prop) (f x)) ((ex (b->a)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy)))))))
% Found eq_ref00:=(eq_ref0 (f x5)):(((eq Prop) (f x5)) (f x5))
% Found (eq_ref0 (f x5)) as proof of (((eq Prop) (f x5)) ((ex (b->a)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy))))))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) ((ex (b->a)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy))))))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) ((ex (b->a)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy))))))
% Found (fun (x5:a)=> ((eq_ref Prop) (f x5))) as proof of (((eq Prop) (f x5)) ((ex (b->a)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy))))))
% Found (fun (x5:a)=> ((eq_ref Prop) (f x5))) as proof of (forall (x:a), (((eq Prop) (f x)) ((ex (b->a)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy)))))))
% Found eq_ref00:=(eq_ref0 (f x5)):(((eq Prop) (f x5)) (f x5))
% Found (eq_ref0 (f x5)) as proof of (((eq Prop) (f x5)) ((ex (b->a)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy))))))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) ((ex (b->a)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy))))))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) ((ex (b->a)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy))))))
% Found (fun (x5:a)=> ((eq_ref Prop) (f x5))) as proof of (((eq Prop) (f x5)) ((ex (b->a)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy))))))
% Found (fun (x5:a)=> ((eq_ref Prop) (f x5))) as proof of (forall (x:a), (((eq Prop) (f x)) ((ex (b->a)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy)))))))
% Found eq_ref00:=(eq_ref0 Xx):(((eq b) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq b) Xx) b0)
% Found ((eq_ref b) Xx) as proof of (((eq b) Xx) b0)
% Found ((eq_ref b) Xx) as proof of (((eq b) Xx) b0)
% Found ((eq_ref b) Xx) as proof of (((eq b) Xx) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq b) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq b) b0) Xy)
% Found ((eq_ref b) b0) as proof of (((eq b) b0) Xy)
% Found ((eq_ref b) b0) as proof of (((eq b) b0) Xy)
% Found ((eq_ref b) b0) as proof of (((eq b) b0) Xy)
% Found eq_ref00:=(eq_ref0 Xx):(((eq b) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq b) Xx) b0)
% Found ((eq_ref b) Xx) as proof of (((eq b) Xx) b0)
% Found ((eq_ref b) Xx) as proof of (((eq b) Xx) b0)
% Found ((eq_ref b) Xx) as proof of (((eq b) Xx) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq b) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq b) b0) Xy)
% Found ((eq_ref b) b0) as proof of (((eq b) b0) Xy)
% Found ((eq_ref b) b0) as proof of (((eq b) b0) Xy)
% Found ((eq_ref b) b0) as proof of (((eq b) b0) Xy)
% Found eq_ref00:=(eq_ref0 (f x5)):(((eq Prop) (f x5)) (f x5))
% Found (eq_ref0 (f x5)) as proof of (((eq Prop) (f x5)) ((ex (b->a)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy))))))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) ((ex (b->a)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy))))))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) ((ex (b->a)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy))))))
% Found (fun (x5:a)=> ((eq_ref Prop) (f x5))) as proof of (((eq Prop) (f x5)) ((ex (b->a)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy))))))
% Found (fun (x5:a)=> ((eq_ref Prop) (f x5))) as proof of (forall (x:a), (((eq Prop) (f x)) ((ex (b->a)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy)))))))
% Found eq_ref00:=(eq_ref0 (f x5)):(((eq Prop) (f x5)) (f x5))
% Found (eq_ref0 (f x5)) as proof of (((eq Prop) (f x5)) ((ex (b->a)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy))))))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) ((ex (b->a)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy))))))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) ((ex (b->a)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy))))))
% Found (fun (x5:a)=> ((eq_ref Prop) (f x5))) as proof of (((eq Prop) (f x5)) ((ex (b->a)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy))))))
% Found (fun (x5:a)=> ((eq_ref Prop) (f x5))) as proof of (forall (x:a), (((eq Prop) (f x)) ((ex (b->a)) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy)))))))
% Found choice000:=(choice00 (fun (x7:b) (x60:a)=> (((eq b) (x3 x60)) x7))):((forall (x:b), ((ex a) (fun (y:a)=> (((eq b) (x3 y)) x))))->((ex (b->a)) (fun (f:(b->a))=> (forall (x:b), (((eq b) (x3 (f x))) x)))))
% Found (choice00 (fun (x7:b) (x60:a)=> (((eq b) (x3 x60)) x7))) as proof of ((forall (Y:b), ((ex a) (fun (X:a)=> (((eq b) (x3 X)) Y))))->(P f))
% Found ((choice0 a) (fun (x7:b) (x60:a)=> (((eq b) (x3 x60)) x7))) as proof of ((forall (Y:b), ((ex a) (fun (X:a)=> (((eq b) (x3 X)) Y))))->(P f))
% Found (((choice b) a) (fun (x7:b) (x60:a)=> (((eq b) (x3 x60)) x7))) as proof of ((forall (Y:b), ((ex a) (fun (X:a)=> (((eq b) (x3 X)) Y))))->(P f))
% Found (((choice b) a) (fun (x7:b) (x60:a)=> (((eq b) (x3 x60)) x7))) as proof of ((forall (Y:b), ((ex a) (fun (X:a)=> (((eq b) (x3 X)) Y))))->(P f))
% Found choice000:=(choice00 (fun (x7:b) (x60:a)=> (((eq b) (x3 x60)) x7))):((forall (x:b), ((ex a) (fun (y:a)=> (((eq b) (x3 y)) x))))->((ex (b->a)) (fun (f:(b->a))=> (forall (x:b), (((eq b) (x3 (f x))) x)))))
% Found (choice00 (fun (x7:b) (x60:a)=> (((eq b) (x3 x60)) x7))) as proof of ((forall (Y:b), ((ex a) (fun (X:a)=> (((eq b) (x3 X)) Y))))->(P f))
% Found ((choice0 a) (fun (x7:b) (x60:a)=> (((eq b) (x3 x60)) x7))) as proof of ((forall (Y:b), ((ex a) (fun (X:a)=> (((eq b) (x3 X)) Y))))->(P f))
% Found (((choice b) a) (fun (x7:b) (x60:a)=> (((eq b) (x3 x60)) x7))) as proof of ((forall (Y:b), ((ex a) (fun (X:a)=> (((eq b) (x3 X)) Y))))->(P f))
% Found (((choice b) a) (fun (x7:b) (x60:a)=> (((eq b) (x3 x60)) x7))) as proof of ((forall (Y:b), ((ex a) (fun (X:a)=> (((eq b) (x3 X)) Y))))->(P f))
% Found x000:=(x00 (fun (x6:a)=> (P Xx))):((P Xx)->(P Xx))
% Found (x00 (fun (x6:a)=> (P Xx))) as proof of (P0 Xx)
% Found (x00 (fun (x6:a)=> (P Xx))) as proof of (P0 Xx)
% Found x000:=(x00 (fun (x6:a)=> (P Xx))):((P Xx)->(P Xx))
% Found (x00 (fun (x6:a)=> (P Xx))) as proof of (P0 Xx)
% Found (x00 (fun (x6:a)=> (P Xx))) as proof of (P0 Xx)
% Found eq_ref00:=(eq_ref0 Xy):(((eq b) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq b) Xy) b0)
% Found ((eq_ref b) Xy) as proof of (((eq b) Xy) b0)
% Found ((eq_ref b) Xy) as proof of (((eq b) Xy) b0)
% Found ((eq_ref b) Xy) as proof of (((eq b) Xy) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq b) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq b) b0) Xx)
% Found ((eq_ref b) b0) as proof of (((eq b) b0) Xx)
% Found ((eq_ref b) b0) as proof of (((eq b) b0) Xx)
% Found ((eq_ref b) b0) as proof of (((eq b) b0) Xx)
% Found eq_ref00:=(eq_ref0 b0):(((eq b) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq b) b0) Xx)
% Found ((eq_ref b) b0) as proof of (((eq b) b0) Xx)
% Found ((eq_ref b) b0) as proof of (((eq b) b0) Xx)
% Found ((eq_ref b) b0) as proof of (((eq b) b0) Xx)
% Found eq_ref00:=(eq_ref0 Xy):(((eq b) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq b) Xy) b0)
% Found ((eq_ref b) Xy) as proof of (((eq b) Xy) b0)
% Found ((eq_ref b) Xy) as proof of (((eq b) Xy) b0)
% Found ((eq_ref b) Xy) as proof of (((eq b) Xy) b0)
% Found x000:=(x00 (fun (x6:a)=> (P Xx))):((P Xx)->(P Xx))
% Found (x00 (fun (x6:a)=> (P Xx))) as proof of (P0 Xx)
% Found (x00 (fun (x6:a)=> (P Xx))) as proof of (P0 Xx)
% Found eq_ref00:=(eq_ref0 Xx):(((eq b) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq b) Xx) b0)
% Found ((eq_ref b) Xx) as proof of (((eq b) Xx) b0)
% Found ((eq_ref b) Xx) as proof of (((eq b) Xx) b0)
% Found ((eq_ref b) Xx) as proof of (((eq b) Xx) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq b) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq b) b0) Xy)
% Found ((eq_ref b) b0) as proof of (((eq b) b0) Xy)
% Found ((eq_ref b) b0) as proof of (((eq b) b0) Xy)
% Found ((eq_ref b) b0) as proof of (((eq b) b0) Xy)
% Found x000:=(x00 (fun (x6:a)=> (P Xx))):((P Xx)->(P Xx))
% Found (x00 (fun (x6:a)=> (P Xx))) as proof of (P0 Xx)
% Found (x00 (fun (x6:a)=> (P Xx))) as proof of (P0 Xx)
% Found eq_ref00:=(eq_ref0 Xx):(((eq b) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq b) Xx) b0)
% Found ((eq_ref b) Xx) as proof of (((eq b) Xx) b0)
% Found ((eq_ref b) Xx) as proof of (((eq b) Xx) b0)
% Found ((eq_ref b) Xx) as proof of (((eq b) Xx) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq b) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq b) b0) Xy)
% Found ((eq_ref b) b0) as proof of (((eq b) b0) Xy)
% Found ((eq_ref b) b0) as proof of (((eq b) b0) Xy)
% Found ((eq_ref b) b0) as proof of (((eq b) b0) Xy)
% Found eq_ref00:=(eq_ref0 b0):(((eq b) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq b) b0) Xx)
% Found ((eq_ref b) b0) as proof of (((eq b) b0) Xx)
% Found ((eq_ref b) b0) as proof of (((eq b) b0) Xx)
% Found ((eq_ref b) b0) as proof of (((eq b) b0) Xx)
% Found eq_ref00:=(eq_ref0 Xy):(((eq b) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq b) Xy) b0)
% Found ((eq_ref b) Xy) as proof of (((eq b) Xy) b0)
% Found ((eq_ref b) Xy) as proof of (((eq b) Xy) b0)
% Found ((eq_ref b) Xy) as proof of (((eq b) Xy) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq b) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq b) b0) Xx)
% Found ((eq_ref b) b0) as proof of (((eq b) b0) Xx)
% Found ((eq_ref b) b0) as proof of (((eq b) b0) Xx)
% Found ((eq_ref b) b0) as proof of (((eq b) b0) Xx)
% Found eq_ref00:=(eq_ref0 Xy):(((eq b) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq b) Xy) b0)
% Found ((eq_ref b) Xy) as proof of (((eq b) Xy) b0)
% Found ((eq_ref b) Xy) as proof of (((eq b) Xy) b0)
% Found ((eq_ref b) Xy) as proof of (((eq b) Xy) b0)
% Found x000:=(x00 (fun (x6:a)=> (P Xx))):((P Xx)->(P Xx))
% Found (x00 (fun (x6:a)=> (P Xx))) as proof of (P0 Xx)
% Found (x00 (fun (x6:a)=> (P Xx))) as proof of (P0 Xx)
% Found eq_ref00:=(eq_ref0 Xx):(((eq b) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq b) Xx) b0)
% Found ((eq_ref b) Xx) as proof of (((eq b) Xx) b0)
% Found ((eq_ref b) Xx) as proof of (((eq b) Xx) b0)
% Found ((eq_ref b) Xx) as proof of (((eq b) Xx) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq b) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq b) b0) Xy)
% Found ((eq_ref b) b0) as proof of (((eq b) b0) Xy)
% Found ((eq_ref b) b0) as proof of (((eq b) b0) Xy)
% Found ((eq_ref b) b0) as proof of (((eq b) b0) Xy)
% Found x000:=(x00 (fun (x6:a)=> (P Xx))):((P Xx)->(P Xx))
% Found (x00 (fun (x6:a)=> (P Xx))) as proof of (P0 Xx)
% Found (x00 (fun (x6:a)=> (P Xx))) as proof of (P0 Xx)
% Found eq_ref00:=(eq_ref0 Xx):(((eq b) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq b) Xx) b0)
% Found ((eq_ref b) Xx) as proof of (((eq b) Xx) b0)
% Found ((eq_ref b) Xx) as proof of (((eq b) Xx) b0)
% Found ((eq_ref b) Xx) as proof of (((eq b) Xx) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq b) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq b) b0) Xy)
% Found ((eq_ref b) b0) as proof of (((eq b) b0) Xy)
% Found ((eq_ref b) b0) as proof of (((eq b) b0) Xy)
% Found ((eq_ref b) b0) as proof of (((eq b) b0) Xy)
% Found eq_ref00:=(eq_ref0 b0):(((eq b) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq b) b0) Xx)
% Found ((eq_ref b) b0) as proof of (((eq b) b0) Xx)
% Found ((eq_ref b) b0) as proof of (((eq b) b0) Xx)
% Found ((eq_ref b) b0) as proof of (((eq b) b0) Xx)
% Found eq_ref00:=(eq_ref0 Xy):(((eq b) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq b) Xy) b0)
% Found ((eq_ref b) Xy) as proof of (((eq b) Xy) b0)
% Found ((eq_ref b) Xy) as proof of (((eq b) Xy) b0)
% Found ((eq_ref b) Xy) as proof of (((eq b) Xy) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq b) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq b) b0) Xx)
% Found ((eq_ref b) b0) as proof of (((eq b) b0) Xx)
% Found ((eq_ref b) b0) as proof of (((eq b) b0) Xx)
% Found ((eq_ref b) b0) as proof of (((eq b) b0) Xx)
% Found eq_ref00:=(eq_ref0 Xy):(((eq b) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq b) Xy) b0)
% Found ((eq_ref b) Xy) as proof of (((eq b) Xy) b0)
% Found ((eq_ref b) Xy) as proof of (((eq b) Xy) b0)
% Found ((eq_ref b) Xy) as proof of (((eq b) Xy) b0)
% Found choice000:=(choice00 (fun (x7:b) (x60:a)=> (((eq b) (x3 x60)) x7))):((forall (x:b), ((ex a) (fun (y:a)=> (((eq b) (x3 y)) x))))->((ex (b->a)) (fun (f:(b->a))=> (forall (x:b), (((eq b) (x3 (f x))) x)))))
% Found (choice00 (fun (x7:b) (x60:a)=> (((eq b) (x3 x60)) x7))) as proof of ((forall (Y:b), ((ex a) (fun (X:a)=> (((eq b) (x3 X)) Y))))->(P f))
% Found ((choice0 a) (fun (x7:b) (x60:a)=> (((eq b) (x3 x60)) x7))) as proof of ((forall (Y:b), ((ex a) (fun (X:a)=> (((eq b) (x3 X)) Y))))->(P f))
% Found (((choice b) a) (fun (x7:b) (x60:a)=> (((eq b) (x3 x60)) x7))) as proof of ((forall (Y:b), ((ex a) (fun (X:a)=> (((eq b) (x3 X)) Y))))->(P f))
% Found (((choice b) a) (fun (x7:b) (x60:a)=> (((eq b) (x3 x60)) x7))) as proof of ((forall (Y:b), ((ex a) (fun (X:a)=> (((eq b) (x3 X)) Y))))->(P f))
% Found choice000:=(choice00 (fun (x7:b) (x60:a)=> (((eq b) (x3 x60)) x7))):((forall (x:b), ((ex a) (fun (y:a)=> (((eq b) (x3 y)) x))))->((ex (b->a)) (fun (f:(b->a))=> (forall (x:b), (((eq b) (x3 (f x))) x)))))
% Found (choice00 (fun (x7:b) (x60:a)=> (((eq b) (x3 x60)) x7))) as proof of ((forall (Y:b), ((ex a) (fun (X:a)=> (((eq b) (x3 X)) Y))))->(P f))
% Found ((choice0 a) (fun (x7:b) (x60:a)=> (((eq b) (x3 x60)) x7))) as proof of ((forall (Y:b), ((ex a) (fun (X:a)=> (((eq b) (x3 X)) Y))))->(P f))
% Found (((choice b) a) (fun (x7:b) (x60:a)=> (((eq b) (x3 x60)) x7))) as proof of ((forall (Y:b), ((ex a) (fun (X:a)=> (((eq b) (x3 X)) Y))))->(P f))
% Found (((choice b) a) (fun (x7:b) (x60:a)=> (((eq b) (x3 x60)) x7))) as proof of ((forall (Y:b), ((ex a) (fun (X:a)=> (((eq b) (x3 X)) Y))))->(P f))
% Found eq_ref00:=(eq_ref0 Xx):(((eq b) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq b) Xx) b0)
% Found ((eq_ref b) Xx) as proof of (((eq b) Xx) b0)
% Found ((eq_ref b) Xx) as proof of (((eq b) Xx) b0)
% Found ((eq_ref b) Xx) as proof of (((eq b) Xx) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq b) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq b) b0) Xy)
% Found ((eq_ref b) b0) as proof of (((eq b) b0) Xy)
% Found ((eq_ref b) b0) as proof of (((eq b) b0) Xy)
% Found ((eq_ref b) b0) as proof of (((eq b) b0) Xy)
% Found eq_ref00:=(eq_ref0 Xx):(((eq b) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq b) Xx) b0)
% Found ((eq_ref b) Xx) as proof of (((eq b) Xx) b0)
% Found ((eq_ref b) Xx) as proof of (((eq b) Xx) b0)
% Found ((eq_ref b) Xx) as proof of (((eq b) Xx) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq b) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq b) b0) Xy)
% Found ((eq_ref b) b0) as proof of (((eq b) b0) Xy)
% Found ((eq_ref b) b0) as proof of (((eq b) b0) Xy)
% Found ((eq_ref b) b0) as proof of (((eq b) b0) Xy)
% Found eq_ref00:=(eq_ref0 Xy):(((eq b) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq b) Xy) b0)
% Found ((eq_ref b) Xy) as proof of (((eq b) Xy) b0)
% Found ((eq_ref b) Xy) as proof of (((eq b) Xy) b0)
% Found ((eq_ref b) Xy) as proof of (((eq b) Xy) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq b) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq b) b0) Xx)
% Found ((eq_ref b) b0) as proof of (((eq b) b0) Xx)
% Found ((eq_ref b) b0) as proof of (((eq b) b0) Xx)
% Found ((eq_ref b) b0) as proof of (((eq b) b0) Xx)
% Found eq_ref00:=(eq_ref0 b0):(((eq b) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq b) b0) Xx)
% Found ((eq_ref b) b0) as proof of (((eq b) b0) Xx)
% Found ((eq_ref b) b0) as proof of (((eq b) b0) Xx)
% Found ((eq_ref b) b0) as proof of (((eq b) b0) Xx)
% Found eq_ref00:=(eq_ref0 Xy):(((eq b) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq b) Xy) b0)
% Found ((eq_ref b) Xy) as proof of (((eq b) Xy) b0)
% Found ((eq_ref b) Xy) as proof of (((eq b) Xy) b0)
% Found ((eq_ref b) Xy) as proof of (((eq b) Xy) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq b) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq b) b0) Xx)
% Found ((eq_ref b) b0) as proof of (((eq b) b0) Xx)
% Found ((eq_ref b) b0) as proof of (((eq b) b0) Xx)
% Found ((eq_ref b) b0) as proof of (((eq b) b0) Xx)
% Found eq_ref00:=(eq_ref0 Xy):(((eq b) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq b) Xy) b0)
% Found ((eq_ref b) Xy) as proof of (((eq b) Xy) b0)
% Found ((eq_ref b) Xy) as proof of (((eq b) Xy) b0)
% Found ((eq_ref b) Xy) as proof of (((eq b) Xy) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq b) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq b) b0) Xx)
% Found ((eq_ref b) b0) as proof of (((eq b) b0) Xx)
% Found ((eq_ref b) b0) as proof of (((eq b) b0) Xx)
% Found ((eq_ref b) b0) as proof of (((eq b) b0) Xx)
% Found eq_ref00:=(eq_ref0 Xy):(((eq b) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq b) Xy) b0)
% Found ((eq_ref b) Xy) as proof of (((eq b) Xy) b0)
% Found ((eq_ref b) Xy) as proof of (((eq b) Xy) b0)
% Found ((eq_ref b) Xy) as proof of (((eq b) Xy) b0)
% Found eq_ref00:=(eq_ref0 Xy):(((eq b) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq b) Xy) b0)
% Found ((eq_ref b) Xy) as proof of (((eq b) Xy) b0)
% Found ((eq_ref b) Xy) as proof of (((eq b) Xy) b0)
% Found ((eq_ref b) Xy) as proof of (((eq b) Xy) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq b) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq b) b0) Xx)
% Found ((eq_ref b) b0) as proof of (((eq b) b0) Xx)
% Found ((eq_ref b) b0) as proof of (((eq b) b0) Xx)
% Found ((eq_ref b) b0) as proof of (((eq b) b0) Xx)
% Found eq_ref00:=(eq_ref0 b0):(((eq b) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq b) b0) Xx)
% Found ((eq_ref b) b0) as proof of (((eq b) b0) Xx)
% Found ((eq_ref b) b0) as proof of (((eq b) b0) Xx)
% Found ((eq_ref b) b0) as proof of (((eq b) b0) Xx)
% Found eq_ref00:=(eq_ref0 Xy):(((eq b) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq b) Xy) b0)
% Found ((eq_ref b) Xy) as proof of (((eq b) Xy) b0)
% Found ((eq_ref b) Xy) as proof of (((eq b) Xy) b0)
% Found ((eq_ref b) Xy) as proof of (((eq b) Xy) b0)
% Found choice000:=(choice00 (fun (x5:b) (x40:a)=> (((eq b) (x1 x40)) x5))):((forall (x:b), ((ex a) (fun (y:a)=> (((eq b) (x1 y)) x))))->((ex (b->a)) (fun (f:(b->a))=> (forall (x:b), (((eq b) (x1 (f x))) x)))))
% Found (choice00 (fun (x5:b) (x40:a)=> (((eq b) (x1 x40)) x5))) as proof of ((forall (Y:b), ((ex a) (fun (X:a)=> (((eq b) (x1 X)) Y))))->((ex (b->a)) b0))
% Found ((choice0 a) (fun (x5:b) (x40:a)=> (((eq b) (x1 x40)) x5))) as proof of ((forall (Y:b), ((ex a) (fun (X:a)=> (((eq b) (x1 X)) Y))))->((ex (b->a)) b0))
% Found (((choice b) a) (fun (x5:b) (x40:a)=> (((eq b) (x1 x40)) x5))) as proof of ((forall (Y:b), ((ex a) (fun (X:a)=> (((eq b) (x1 X)) Y))))->((ex (b->a)) b0))
% Found (((choice b) a) (fun (x5:b) (x40:a)=> (((eq b) (x1 x40)) x5))) as proof of ((forall (Y:b), ((ex a) (fun (X:a)=> (((eq b) (x1 X)) Y))))->((ex (b->a)) b0))
% Found eq_ref00:=(eq_ref0 Xy):(((eq b) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq b) Xy) b0)
% Found ((eq_ref b) Xy) as proof of (((eq b) Xy) b0)
% Found ((eq_ref b) Xy) as proof of (((eq b) Xy) b0)
% Found ((eq_ref b) Xy) as proof of (((eq b) Xy) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq b) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq b) b0) Xx)
% Found ((eq_ref b) b0) as proof of (((eq b) b0) Xx)
% Found ((eq_ref b) b0) as proof of (((eq b) b0) Xx)
% Found ((eq_ref b) b0) as proof of (((eq b) b0) Xx)
% Found eq_ref00:=(eq_ref0 Xy):(((eq b) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq b) Xy) b0)
% Found ((eq_ref b) Xy) as proof of (((eq b) Xy) b0)
% Found ((eq_ref b) Xy) as proof of (((eq b) Xy) b0)
% Found ((eq_ref b) Xy) as proof of (((eq b) Xy) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq b) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq b) b0) Xx)
% Found ((eq_ref b) b0) as proof of (((eq b) b0) Xx)
% Found ((eq_ref b) b0) as proof of (((eq b) b0) Xx)
% Found ((eq_ref b) b0) as proof of (((eq b) b0) Xx)
% Found eq_ref00:=(eq_ref0 b0):(((eq ((b->a)->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq ((b->a)->Prop)) b0) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy)))))
% Found ((eq_ref ((b->a)->Prop)) b0) as proof of (((eq ((b->a)->Prop)) b0) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy)))))
% Found ((eq_ref ((b->a)->Prop)) b0) as proof of (((eq ((b->a)->Prop)) b0) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy)))))
% Found ((eq_ref ((b->a)->Prop)) b0) as proof of (((eq ((b->a)->Prop)) b0) (fun (Xf:(b->a))=> (forall (Xx:b) (Xy:b), ((((eq a) (Xf Xx)) (Xf Xy))->(((eq b) Xx) Xy)))))
% Found eta_expansion000:=(eta_expansion00 a0):(((eq ((b->a)->Prop)) a0) (fun (x:(b->a))=> (a0 x)))
% Found (eta_expansion00 a0) as proof of (((eq ((b->a)->Prop)) a0) b0)
% Found ((eta_expansion0 Prop) a0) as proof of (((eq ((b->a)->Prop)) a0) b0)
% Found (((eta_expansion (b->a)) Prop) a0) as proof of (((eq ((b->a)->Prop)) a0) b0)
% Found (((eta_expansion (b->a)) Prop) a0) as proof of (((eq ((b->a)->Prop)) a0) b0)
% Found (((eta_expansion (b->a)) Prop) a0) as proof of (((eq ((b->a)->Prop)) a0) b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq ((b->a)->Prop)) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq ((b->a)->Prop)) a0) b0)
% Found ((eq_ref ((b->a)->Prop)) a0) as proof of (((eq ((b->a)->Prop)) a0) b0)
% Found ((eq_ref ((b->a)->Prop)) a0) as proof of (((eq ((b->a)->Prop)) a0) b0)
% Found ((eq_ref ((b->a)->Prop)) a0) as proof of (((eq ((b->a)->Prop)) a0) b0)
% Found choice000:=(choice00 (fun (x5:b) (x40:a)=> (((eq b) (x1 x40)) x5))):((forall (x:b), ((ex a) (fun (y:a)=> (((eq b) (x1 y)) x))))->((ex (b->a)) (fun (f:(b->a))=> (forall (x:b), (((eq b) (x1 (f x))) x)))))
% Found (choice00 (fun (x5:b) (x40:a)=> (((eq b) (x1 x40)) x5))) as proof of ((forall (Y:b), ((ex a) (fun (X:a)=> (((eq b) (x1 X)) Y))))->(P a0))
% Found ((choice0 a) (fun (x5:b) (x40:a)=> (((eq b) (x1 x40)) x5))) as proof of ((forall (Y:b), ((ex a) (fun (X:a)=> (((eq b) (x1 X)) Y))))->(P a0))
% Found (((choice b) a) (fun (x5:b) (x40:a)=> (((eq b) (x1 x40)) x5))) as proof of ((forall (Y:b), ((ex a) (fun (X:a)=> (((eq b) (x1 X)) Y))))->(P a0))
% Found (((choice b) a) (fun (x5:b) (x40:a)=> (((eq b) (x1 x40)) x5))) as proof of ((forall (Y:b), ((ex a) (fun (X:a)=> (((eq b) (x1 X)) Y))))->(P a0))
% Found eq_ref00:=(eq_ref0 b0):(((eq b) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq b) b0) Xx)
% Found ((eq_ref b) b0) as proof of (((eq b) b0) Xx)
% Found ((eq_ref b) b0) as proof of (((eq b) b0) Xx)
% Found ((eq_ref b) b0) as proof of (((eq b) b0) Xx)
% Found eq_ref00:=(eq_ref0 Xy):(((eq b) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq b) Xy) b0)
% Found ((eq_ref b) Xy) as proof of (((eq b) Xy) b0)
% Found ((eq_ref b) Xy) as proof of (((eq b) Xy) b0)
% Found ((eq_ref b) Xy) as proof of (((eq b) Xy) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq b) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq b) b0) Xx)
% Found ((eq_ref b) b0) as proof of (((eq b) b0) Xx)
% Found ((eq_ref b) b0) as proof of (((eq b) b0) Xx)
% Found ((eq_ref b) b0) as proof of (((eq b) b0) Xx)
% Found eq_ref00:=(eq_ref0 Xy):(((eq b) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq b) Xy) b0)
% Found ((eq_ref b) Xy) as proof of (((eq b) Xy) b0)
% Found ((eq_ref b) Xy) as proof of (((eq b) Xy) b0)
% Found ((eq_ref b) Xy) as proof of (((eq b) Xy) b0)
% Found x000:=(x00 (fun (x2:a)=> (P Xx))):((P Xx)->(P Xx))
% Found (x00 (fun (x2:a)=> (P Xx))) as proof of (P0 Xx)
% Found (x00 (fun (x2:a)=> (P Xx))) as proof of (P0 Xx)
% Found choice000:=(choice00 (fun (x5:b) (x40:a)=> (((eq b) (x1 x40)) x5))):((forall (x:b), ((ex a) (fun (y:a)=> (((eq b) (x
% EOF
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