%------------------------------------------------------------------------------
% File     : cocATP---0.2.0
% Problem  : SEV027^5 : TPTP v6.1.0. Released v4.0.0.
% Transfm  : none
% Format   : tptp:raw
% Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p

% Computer : n093.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 13:33:36 EDT 2014

% Result   : Timeout 300.01s
% Output   : None 
% Verified : 
% SZS Type : None (Parsing solution fails)
% Syntax   : Number of formulae    : 0

% Comments : 
%------------------------------------------------------------------------------
%----NO SOLUTION OUTPUT BY SYSTEM
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% % Problem  : SEV027^5 : TPTP v6.1.0. Released v4.0.0.
% % Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n093.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 07:36:16 CDT 2014
% % CPUTime  : 300.01 
% 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 0xb36830>, <kernel.Type object at 0xb36f38>) of role type named a_type
% Using role type
% Declaring a:Type
% FOF formula ((forall (Xp:(a->Prop)) (Xa:(a->Prop)), ((forall (Xb:a), (((eq Prop) (Xp Xb)) (Xa Xb)))->(forall (P:((a->Prop)->Prop)), ((P Xp)->(P Xa)))))->(forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((Q Xx) Xx))) (forall (Xx:a) (Xy:a), (((Q Xx) Xy)->((Q Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Q Xx) Xy)) ((Q Xy) Xz))->((Q Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy)))))))) P))))))) of role conjecture named cTHM262_EXT_pme
% Conjecture to prove = ((forall (Xp:(a->Prop)) (Xa:(a->Prop)), ((forall (Xb:a), (((eq Prop) (Xp Xb)) (Xa Xb)))->(forall (P:((a->Prop)->Prop)), ((P Xp)->(P Xa)))))->(forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((Q Xx) Xx))) (forall (Xx:a) (Xy:a), (((Q Xx) Xy)->((Q Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Q Xx) Xy)) ((Q Xy) Xz))->((Q Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy)))))))) P))))))):Prop
% Parameter a_DUMMY:a.
% We need to prove ['((forall (Xp:(a->Prop)) (Xa:(a->Prop)), ((forall (Xb:a), (((eq Prop) (Xp Xb)) (Xa Xb)))->(forall (P:((a->Prop)->Prop)), ((P Xp)->(P Xa)))))->(forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((Q Xx) Xx))) (forall (Xx:a) (Xy:a), (((Q Xx) Xy)->((Q Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Q Xx) Xy)) ((Q Xy) Xz))->((Q Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy)))))))) P)))))))']
% Parameter a:Type.
% Trying to prove ((forall (Xp:(a->Prop)) (Xa:(a->Prop)), ((forall (Xb:a), (((eq Prop) (Xp Xb)) (Xa Xb)))->(forall (P:((a->Prop)->Prop)), ((P Xp)->(P Xa)))))->(forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((Q Xx) Xx))) (forall (Xx:a) (Xy:a), (((Q Xx) Xy)->((Q Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Q Xx) Xy)) ((Q Xy) Xz))->((Q Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy)))))))) P)))))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Q:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((Q Xx) Xx))) (forall (Xx:a) (Xy:a), (((Q Xx) Xy)->((Q Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Q Xx) Xy)) ((Q Xy) Xz))->((Q Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy)))))))) P)))):(((eq ((a->(a->Prop))->Prop)) (fun (Q:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((Q Xx) Xx))) (forall (Xx:a) (Xy:a), (((Q Xx) Xy)->((Q Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Q Xx) Xy)) ((Q Xy) Xz))->((Q Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy)))))))) P)))) (fun (x:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((x Xx) Xx))) (forall (Xx:a) (Xy:a), (((x Xx) Xy)->((x Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x Xx) Xy)))))))) P))))
% Found (eta_expansion_dep00 (fun (Q:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((Q Xx) Xx))) (forall (Xx:a) (Xy:a), (((Q Xx) Xy)->((Q Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Q Xx) Xy)) ((Q Xy) Xz))->((Q Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy)))))))) P)))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (Q:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((Q Xx) Xx))) (forall (Xx:a) (Xy:a), (((Q Xx) Xy)->((Q Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Q Xx) Xy)) ((Q Xy) Xz))->((Q Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy)))))))) P)))) b)
% Found ((eta_expansion_dep0 (fun (x2:(a->(a->Prop)))=> Prop)) (fun (Q:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((Q Xx) Xx))) (forall (Xx:a) (Xy:a), (((Q Xx) Xy)->((Q Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Q Xx) Xy)) ((Q Xy) Xz))->((Q Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy)))))))) P)))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (Q:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((Q Xx) Xx))) (forall (Xx:a) (Xy:a), (((Q Xx) Xy)->((Q Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Q Xx) Xy)) ((Q Xy) Xz))->((Q Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy)))))))) P)))) b)
% Found (((eta_expansion_dep (a->(a->Prop))) (fun (x2:(a->(a->Prop)))=> Prop)) (fun (Q:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((Q Xx) Xx))) (forall (Xx:a) (Xy:a), (((Q Xx) Xy)->((Q Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Q Xx) Xy)) ((Q Xy) Xz))->((Q Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy)))))))) P)))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (Q:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((Q Xx) Xx))) (forall (Xx:a) (Xy:a), (((Q Xx) Xy)->((Q Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Q Xx) Xy)) ((Q Xy) Xz))->((Q Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy)))))))) P)))) b)
% Found (((eta_expansion_dep (a->(a->Prop))) (fun (x2:(a->(a->Prop)))=> Prop)) (fun (Q:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((Q Xx) Xx))) (forall (Xx:a) (Xy:a), (((Q Xx) Xy)->((Q Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Q Xx) Xy)) ((Q Xy) Xz))->((Q Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy)))))))) P)))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (Q:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((Q Xx) Xx))) (forall (Xx:a) (Xy:a), (((Q Xx) Xy)->((Q Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Q Xx) Xy)) ((Q Xy) Xz))->((Q Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy)))))))) P)))) b)
% Found (((eta_expansion_dep (a->(a->Prop))) (fun (x2:(a->(a->Prop)))=> Prop)) (fun (Q:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((Q Xx) Xx))) (forall (Xx:a) (Xy:a), (((Q Xx) Xy)->((Q Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Q Xx) Xy)) ((Q Xy) Xz))->((Q Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy)))))))) P)))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (Q:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((Q Xx) Xx))) (forall (Xx:a) (Xy:a), (((Q Xx) Xy)->((Q Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Q Xx) Xy)) ((Q Xy) Xz))->((Q Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy)))))))) P)))) b)
% Found eta_expansion000:=(eta_expansion00 (fun (Q:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((Q Xx) Xx))) (forall (Xx:a) (Xy:a), (((Q Xx) Xy)->((Q Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Q Xx) Xy)) ((Q Xy) Xz))->((Q Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy)))))))) P)))):(((eq ((a->(a->Prop))->Prop)) (fun (Q:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((Q Xx) Xx))) (forall (Xx:a) (Xy:a), (((Q Xx) Xy)->((Q Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Q Xx) Xy)) ((Q Xy) Xz))->((Q Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy)))))))) P)))) (fun (x:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((x Xx) Xx))) (forall (Xx:a) (Xy:a), (((x Xx) Xy)->((x Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x Xx) Xy)))))))) P))))
% Found (eta_expansion00 (fun (Q:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((Q Xx) Xx))) (forall (Xx:a) (Xy:a), (((Q Xx) Xy)->((Q Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Q Xx) Xy)) ((Q Xy) Xz))->((Q Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy)))))))) P)))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (Q:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((Q Xx) Xx))) (forall (Xx:a) (Xy:a), (((Q Xx) Xy)->((Q Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Q Xx) Xy)) ((Q Xy) Xz))->((Q Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy)))))))) P)))) b)
% Found ((eta_expansion0 Prop) (fun (Q:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((Q Xx) Xx))) (forall (Xx:a) (Xy:a), (((Q Xx) Xy)->((Q Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Q Xx) Xy)) ((Q Xy) Xz))->((Q Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy)))))))) P)))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (Q:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((Q Xx) Xx))) (forall (Xx:a) (Xy:a), (((Q Xx) Xy)->((Q Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Q Xx) Xy)) ((Q Xy) Xz))->((Q Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy)))))))) P)))) b)
% Found (((eta_expansion (a->(a->Prop))) Prop) (fun (Q:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((Q Xx) Xx))) (forall (Xx:a) (Xy:a), (((Q Xx) Xy)->((Q Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Q Xx) Xy)) ((Q Xy) Xz))->((Q Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy)))))))) P)))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (Q:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((Q Xx) Xx))) (forall (Xx:a) (Xy:a), (((Q Xx) Xy)->((Q Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Q Xx) Xy)) ((Q Xy) Xz))->((Q Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy)))))))) P)))) b)
% Found (((eta_expansion (a->(a->Prop))) Prop) (fun (Q:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((Q Xx) Xx))) (forall (Xx:a) (Xy:a), (((Q Xx) Xy)->((Q Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Q Xx) Xy)) ((Q Xy) Xz))->((Q Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy)))))))) P)))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (Q:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((Q Xx) Xx))) (forall (Xx:a) (Xy:a), (((Q Xx) Xy)->((Q Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Q Xx) Xy)) ((Q Xy) Xz))->((Q Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy)))))))) P)))) b)
% Found (((eta_expansion (a->(a->Prop))) Prop) (fun (Q:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((Q Xx) Xx))) (forall (Xx:a) (Xy:a), (((Q Xx) Xy)->((Q Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Q Xx) Xy)) ((Q Xy) Xz))->((Q Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy)))))))) P)))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (Q:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((Q Xx) Xx))) (forall (Xx:a) (Xy:a), (((Q Xx) Xy)->((Q Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Q Xx) Xy)) ((Q Xy) Xz))->((Q Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy)))))))) P)))) b)
% Found eta_expansion000:=(eta_expansion00 (fun (Q:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((Q Xx) Xx))) (forall (Xx:a) (Xy:a), (((Q Xx) Xy)->((Q Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Q Xx) Xy)) ((Q Xy) Xz))->((Q Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy)))))))) P)))):(((eq ((a->(a->Prop))->Prop)) (fun (Q:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((Q Xx) Xx))) (forall (Xx:a) (Xy:a), (((Q Xx) Xy)->((Q Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Q Xx) Xy)) ((Q Xy) Xz))->((Q Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy)))))))) P)))) (fun (x:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((x Xx) Xx))) (forall (Xx:a) (Xy:a), (((x Xx) Xy)->((x Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x Xx) Xy)))))))) P))))
% Found (eta_expansion00 (fun (Q:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((Q Xx) Xx))) (forall (Xx:a) (Xy:a), (((Q Xx) Xy)->((Q Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Q Xx) Xy)) ((Q Xy) Xz))->((Q Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy)))))))) P)))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (Q:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((Q Xx) Xx))) (forall (Xx:a) (Xy:a), (((Q Xx) Xy)->((Q Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Q Xx) Xy)) ((Q Xy) Xz))->((Q Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy)))))))) P)))) b)
% Found ((eta_expansion0 Prop) (fun (Q:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((Q Xx) Xx))) (forall (Xx:a) (Xy:a), (((Q Xx) Xy)->((Q Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Q Xx) Xy)) ((Q Xy) Xz))->((Q Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy)))))))) P)))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (Q:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((Q Xx) Xx))) (forall (Xx:a) (Xy:a), (((Q Xx) Xy)->((Q Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Q Xx) Xy)) ((Q Xy) Xz))->((Q Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy)))))))) P)))) b)
% Found (((eta_expansion (a->(a->Prop))) Prop) (fun (Q:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((Q Xx) Xx))) (forall (Xx:a) (Xy:a), (((Q Xx) Xy)->((Q Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Q Xx) Xy)) ((Q Xy) Xz))->((Q Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy)))))))) P)))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (Q:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((Q Xx) Xx))) (forall (Xx:a) (Xy:a), (((Q Xx) Xy)->((Q Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Q Xx) Xy)) ((Q Xy) Xz))->((Q Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy)))))))) P)))) b)
% Found (((eta_expansion (a->(a->Prop))) Prop) (fun (Q:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((Q Xx) Xx))) (forall (Xx:a) (Xy:a), (((Q Xx) Xy)->((Q Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Q Xx) Xy)) ((Q Xy) Xz))->((Q Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy)))))))) P)))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (Q:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((Q Xx) Xx))) (forall (Xx:a) (Xy:a), (((Q Xx) Xy)->((Q Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Q Xx) Xy)) ((Q Xy) Xz))->((Q Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy)))))))) P)))) b)
% Found (((eta_expansion (a->(a->Prop))) Prop) (fun (Q:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((Q Xx) Xx))) (forall (Xx:a) (Xy:a), (((Q Xx) Xy)->((Q Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Q Xx) Xy)) ((Q Xy) Xz))->((Q Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy)))))))) P)))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (Q:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((Q Xx) Xx))) (forall (Xx:a) (Xy:a), (((Q Xx) Xy)->((Q Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Q Xx) Xy)) ((Q Xy) Xz))->((Q Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy)))))))) P)))) b)
% Found eta_expansion000:=(eta_expansion00 (fun (Q:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((Q Xx) Xx))) (forall (Xx:a) (Xy:a), (((Q Xx) Xy)->((Q Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Q Xx) Xy)) ((Q Xy) Xz))->((Q Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy)))))))) P)))):(((eq ((a->(a->Prop))->Prop)) (fun (Q:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((Q Xx) Xx))) (forall (Xx:a) (Xy:a), (((Q Xx) Xy)->((Q Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Q Xx) Xy)) ((Q Xy) Xz))->((Q Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy)))))))) P)))) (fun (x:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((x Xx) Xx))) (forall (Xx:a) (Xy:a), (((x Xx) Xy)->((x Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x Xx) Xy)))))))) P))))
% Found (eta_expansion00 (fun (Q:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((Q Xx) Xx))) (forall (Xx:a) (Xy:a), (((Q Xx) Xy)->((Q Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Q Xx) Xy)) ((Q Xy) Xz))->((Q Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy)))))))) P)))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (Q:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((Q Xx) Xx))) (forall (Xx:a) (Xy:a), (((Q Xx) Xy)->((Q Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Q Xx) Xy)) ((Q Xy) Xz))->((Q Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy)))))))) P)))) b)
% Found ((eta_expansion0 Prop) (fun (Q:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((Q Xx) Xx))) (forall (Xx:a) (Xy:a), (((Q Xx) Xy)->((Q Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Q Xx) Xy)) ((Q Xy) Xz))->((Q Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy)))))))) P)))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (Q:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((Q Xx) Xx))) (forall (Xx:a) (Xy:a), (((Q Xx) Xy)->((Q Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Q Xx) Xy)) ((Q Xy) Xz))->((Q Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy)))))))) P)))) b)
% Found (((eta_expansion (a->(a->Prop))) Prop) (fun (Q:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((Q Xx) Xx))) (forall (Xx:a) (Xy:a), (((Q Xx) Xy)->((Q Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Q Xx) Xy)) ((Q Xy) Xz))->((Q Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy)))))))) P)))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (Q:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((Q Xx) Xx))) (forall (Xx:a) (Xy:a), (((Q Xx) Xy)->((Q Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Q Xx) Xy)) ((Q Xy) Xz))->((Q Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy)))))))) P)))) b)
% Found (((eta_expansion (a->(a->Prop))) Prop) (fun (Q:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((Q Xx) Xx))) (forall (Xx:a) (Xy:a), (((Q Xx) Xy)->((Q Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Q Xx) Xy)) ((Q Xy) Xz))->((Q Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy)))))))) P)))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (Q:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((Q Xx) Xx))) (forall (Xx:a) (Xy:a), (((Q Xx) Xy)->((Q Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Q Xx) Xy)) ((Q Xy) Xz))->((Q Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy)))))))) P)))) b)
% Found (((eta_expansion (a->(a->Prop))) Prop) (fun (Q:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((Q Xx) Xx))) (forall (Xx:a) (Xy:a), (((Q Xx) Xy)->((Q Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Q Xx) Xy)) ((Q Xy) Xz))->((Q Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy)))))))) P)))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (Q:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((Q Xx) Xx))) (forall (Xx:a) (Xy:a), (((Q Xx) Xy)->((Q Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Q Xx) Xy)) ((Q Xy) Xz))->((Q Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy)))))))) P)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq ((a->Prop)->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq ((a->Prop)->Prop)) b) P)
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) P)
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) P)
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) P)
% Found eq_ref00:=(eq_ref0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))):(((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found (eq_ref0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) b)
% Found ((eq_ref ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) b)
% Found ((eq_ref ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) b)
% Found ((eq_ref ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) b)
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P0):((P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))->(P0 (fun (x:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (x Xz)))) (forall (Xx:a), ((x Xx)->(forall (Xy:a), ((iff (x Xy)) ((x1 Xx) Xy)))))))))
% Found (eta_expansion_dep000 P0) as proof of (P00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found ((eta_expansion_dep00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) P0) as proof of (P00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found (((eta_expansion_dep0 (fun (x3:(a->Prop))=> Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) P0) as proof of (P00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found ((((eta_expansion_dep (a->Prop)) (fun (x3:(a->Prop))=> Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) P0) as proof of (P00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found ((((eta_expansion_dep (a->Prop)) (fun (x3:(a->Prop))=> Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) P0) as proof of (P00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found eta_expansion0000:=(eta_expansion000 P0):((P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))->(P0 (fun (x:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (x Xz)))) (forall (Xx:a), ((x Xx)->(forall (Xy:a), ((iff (x Xy)) ((x1 Xx) Xy)))))))))
% Found (eta_expansion000 P0) as proof of (P00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found ((eta_expansion00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) P0) as proof of (P00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found (((eta_expansion0 Prop) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) P0) as proof of (P00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found ((((eta_expansion (a->Prop)) Prop) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) P0) as proof of (P00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found ((((eta_expansion (a->Prop)) Prop) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) P0) as proof of (P00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found eq_ref000:=(eq_ref00 P0):((P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))->(P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))))
% Found (eq_ref00 P0) as proof of (P00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found ((eq_ref0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) P0) as proof of (P00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found (((eq_ref ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) P0) as proof of (P00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found (((eq_ref ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) P0) as proof of (P00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found eq_ref00:=(eq_ref0 b):(((eq ((a->Prop)->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq ((a->Prop)->Prop)) b) P)
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) P)
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) P)
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) P)
% Found eq_ref00:=(eq_ref0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))):(((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found (eq_ref0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) b)
% Found ((eq_ref ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) b)
% Found ((eq_ref ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) b)
% Found ((eq_ref ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) b)
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P0):((P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))->(P0 (fun (x:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (x Xz)))) (forall (Xx:a), ((x Xx)->(forall (Xy:a), ((iff (x Xy)) ((x1 Xx) Xy)))))))))
% Found (eta_expansion_dep000 P0) as proof of (P00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found ((eta_expansion_dep00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) P0) as proof of (P00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found (((eta_expansion_dep0 (fun (x3:(a->Prop))=> Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) P0) as proof of (P00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found ((((eta_expansion_dep (a->Prop)) (fun (x3:(a->Prop))=> Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) P0) as proof of (P00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found ((((eta_expansion_dep (a->Prop)) (fun (x3:(a->Prop))=> Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) P0) as proof of (P00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found eq_ref00:=(eq_ref0 b):(((eq ((a->Prop)->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq ((a->Prop)->Prop)) b) P)
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) P)
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) P)
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) P)
% Found eta_expansion000:=(eta_expansion00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy)))))))):(((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy)))))))) (fun (x:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (x Xz)))) (forall (Xx:a), ((x Xx)->(forall (Xy:a), ((iff (x Xy)) ((x3 Xx) Xy))))))))
% Found (eta_expansion00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy)))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy)))))))) b)
% Found ((eta_expansion0 Prop) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy)))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy)))))))) b)
% Found (((eta_expansion (a->Prop)) Prop) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy)))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy)))))))) b)
% Found (((eta_expansion (a->Prop)) Prop) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy)))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy)))))))) b)
% Found (((eta_expansion (a->Prop)) Prop) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy)))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy)))))))) b)
% Found eq_ref000:=(eq_ref00 P0):((P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))->(P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))))
% Found (eq_ref00 P0) as proof of (P00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found ((eq_ref0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) P0) as proof of (P00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found (((eq_ref ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) P0) as proof of (P00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found (((eq_ref ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) P0) as proof of (P00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found eta_expansion0000:=(eta_expansion000 P0):((P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))->(P0 (fun (x:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (x Xz)))) (forall (Xx:a), ((x Xx)->(forall (Xy:a), ((iff (x Xy)) ((x1 Xx) Xy)))))))))
% Found (eta_expansion000 P0) as proof of (P00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found ((eta_expansion00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) P0) as proof of (P00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found (((eta_expansion0 Prop) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) P0) as proof of (P00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found ((((eta_expansion (a->Prop)) Prop) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) P0) as proof of (P00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found ((((eta_expansion (a->Prop)) Prop) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) P0) as proof of (P00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P0):((P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy))))))))->(P0 (fun (x:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (x Xz)))) (forall (Xx:a), ((x Xx)->(forall (Xy:a), ((iff (x Xy)) ((x3 Xx) Xy)))))))))
% Found (eta_expansion_dep000 P0) as proof of (P00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy))))))))
% Found ((eta_expansion_dep00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy)))))))) P0) as proof of (P00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy))))))))
% Found (((eta_expansion_dep0 (fun (x5:(a->Prop))=> Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy)))))))) P0) as proof of (P00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy))))))))
% Found ((((eta_expansion_dep (a->Prop)) (fun (x5:(a->Prop))=> Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy)))))))) P0) as proof of (P00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy))))))))
% Found ((((eta_expansion_dep (a->Prop)) (fun (x5:(a->Prop))=> Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy)))))))) P0) as proof of (P00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy))))))))
% Found eq_ref00:=(eq_ref0 b):(((eq ((a->Prop)->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq ((a->Prop)->Prop)) b) P)
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) P)
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) P)
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) P)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))):(((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) (fun (x:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (x Xz)))) (forall (Xx:a), ((x Xx)->(forall (Xy:a), ((iff (x Xy)) ((x1 Xx) Xy))))))))
% Found (eta_expansion_dep00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) b)
% Found ((eta_expansion_dep0 (fun (x5:(a->Prop))=> Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x5:(a->Prop))=> Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x5:(a->Prop))=> Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x5:(a->Prop))=> Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) b)
% Found eq_ref000:=(eq_ref00 P0):((P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))->(P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))))
% Found (eq_ref00 P0) as proof of (P00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found ((eq_ref0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) P0) as proof of (P00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found (((eq_ref ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) P0) as proof of (P00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found (((eq_ref ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) P0) as proof of (P00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P0):((P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy))))))))->(P0 (fun (x:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (x Xz)))) (forall (Xx:a), ((x Xx)->(forall (Xy:a), ((iff (x Xy)) ((x3 Xx) Xy)))))))))
% Found (eta_expansion_dep000 P0) as proof of (P00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy))))))))
% Found ((eta_expansion_dep00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy)))))))) P0) as proof of (P00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy))))))))
% Found (((eta_expansion_dep0 (fun (x5:(a->Prop))=> Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy)))))))) P0) as proof of (P00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy))))))))
% Found ((((eta_expansion_dep (a->Prop)) (fun (x5:(a->Prop))=> Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy)))))))) P0) as proof of (P00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy))))))))
% Found ((((eta_expansion_dep (a->Prop)) (fun (x5:(a->Prop))=> Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy)))))))) P0) as proof of (P00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy))))))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P0):((P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy))))))))->(P0 (fun (x:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (x Xz)))) (forall (Xx:a), ((x Xx)->(forall (Xy:a), ((iff (x Xy)) ((x3 Xx) Xy)))))))))
% Found (eta_expansion_dep000 P0) as proof of (P00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy))))))))
% Found ((eta_expansion_dep00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy)))))))) P0) as proof of (P00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy))))))))
% Found (((eta_expansion_dep0 (fun (x5:(a->Prop))=> Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy)))))))) P0) as proof of (P00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy))))))))
% Found ((((eta_expansion_dep (a->Prop)) (fun (x5:(a->Prop))=> Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy)))))))) P0) as proof of (P00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy))))))))
% Found ((((eta_expansion_dep (a->Prop)) (fun (x5:(a->Prop))=> Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy)))))))) P0) as proof of (P00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy))))))))
% Found eq_ref00:=(eq_ref0 b):(((eq ((a->Prop)->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq ((a->Prop)->Prop)) b) P)
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) P)
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) P)
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) P)
% Found eta_expansion000:=(eta_expansion00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy)))))))):(((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy)))))))) (fun (x:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (x Xz)))) (forall (Xx:a), ((x Xx)->(forall (Xy:a), ((iff (x Xy)) ((x3 Xx) Xy))))))))
% Found (eta_expansion00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy)))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy)))))))) b)
% Found ((eta_expansion0 Prop) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy)))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy)))))))) b)
% Found (((eta_expansion (a->Prop)) Prop) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy)))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy)))))))) b)
% Found (((eta_expansion (a->Prop)) Prop) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy)))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy)))))))) b)
% Found (((eta_expansion (a->Prop)) Prop) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy)))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy)))))))) b)
% Found eq_ref000:=(eq_ref00 P0):((P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy))))))))->(P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy)))))))))
% Found (eq_ref00 P0) as proof of (P00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy))))))))
% Found ((eq_ref0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy)))))))) P0) as proof of (P00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy))))))))
% Found (((eq_ref ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy)))))))) P0) as proof of (P00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy))))))))
% Found (((eq_ref ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy)))))))) P0) as proof of (P00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy))))))))
% Found eq_ref000:=(eq_ref00 P0):((P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))->(P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))))
% Found (eq_ref00 P0) as proof of (P00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found ((eq_ref0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) P0) as proof of (P00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found (((eq_ref ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) P0) as proof of (P00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found (((eq_ref ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) P0) as proof of (P00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found eta_expansion0000:=(eta_expansion000 P0):((P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))->(P0 (fun (x:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (x Xz)))) (forall (Xx:a), ((x Xx)->(forall (Xy:a), ((iff (x Xy)) ((x1 Xx) Xy)))))))))
% Found (eta_expansion000 P0) as proof of (P00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found ((eta_expansion00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) P0) as proof of (P00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found (((eta_expansion0 Prop) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) P0) as proof of (P00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found ((((eta_expansion (a->Prop)) Prop) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) P0) as proof of (P00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found ((((eta_expansion (a->Prop)) Prop) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) P0) as proof of (P00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found eq_ref00:=(eq_ref0 b):(((eq ((a->Prop)->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq ((a->Prop)->Prop)) b) P)
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) P)
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) P)
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) P)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))):(((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) (fun (x:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (x Xz)))) (forall (Xx:a), ((x Xx)->(forall (Xy:a), ((iff (x Xy)) ((x1 Xx) Xy))))))))
% Found (eta_expansion_dep00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) b)
% Found ((eta_expansion_dep0 (fun (x5:(a->Prop))=> Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x5:(a->Prop))=> Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x5:(a->Prop))=> Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x5:(a->Prop))=> Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) b)
% Found eq_ref000:=(eq_ref00 P0):((P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))->(P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))))
% Found (eq_ref00 P0) as proof of (P00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found ((eq_ref0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) P0) as proof of (P00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found (((eq_ref ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) P0) as proof of (P00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found (((eq_ref ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) P0) as proof of (P00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P0):((P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy))))))))->(P0 (fun (x:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (x Xz)))) (forall (Xx:a), ((x Xx)->(forall (Xy:a), ((iff (x Xy)) ((x3 Xx) Xy)))))))))
% Found (eta_expansion_dep000 P0) as proof of (P00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy))))))))
% Found ((eta_expansion_dep00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy)))))))) P0) as proof of (P00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy))))))))
% Found (((eta_expansion_dep0 (fun (x5:(a->Prop))=> Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy)))))))) P0) as proof of (P00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy))))))))
% Found ((((eta_expansion_dep (a->Prop)) (fun (x5:(a->Prop))=> Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy)))))))) P0) as proof of (P00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy))))))))
% Found ((((eta_expansion_dep (a->Prop)) (fun (x5:(a->Prop))=> Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy)))))))) P0) as proof of (P00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy))))))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P0):((P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy))))))))->(P0 (fun (x:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (x Xz)))) (forall (Xx:a), ((x Xx)->(forall (Xy:a), ((iff (x Xy)) ((x3 Xx) Xy)))))))))
% Found (eta_expansion_dep000 P0) as proof of (P00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy))))))))
% Found ((eta_expansion_dep00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy)))))))) P0) as proof of (P00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy))))))))
% Found (((eta_expansion_dep0 (fun (x5:(a->Prop))=> Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy)))))))) P0) as proof of (P00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy))))))))
% Found ((((eta_expansion_dep (a->Prop)) (fun (x5:(a->Prop))=> Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy)))))))) P0) as proof of (P00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy))))))))
% Found ((((eta_expansion_dep (a->Prop)) (fun (x5:(a->Prop))=> Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy)))))))) P0) as proof of (P00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy))))))))
% Found eq_ref00:=(eq_ref0 b):(((eq ((a->Prop)->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq ((a->Prop)->Prop)) b) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found eq_ref00:=(eq_ref0 P):(((eq ((a->Prop)->Prop)) P) P)
% Found (eq_ref0 P) as proof of (((eq ((a->Prop)->Prop)) P) b)
% Found ((eq_ref ((a->Prop)->Prop)) P) as proof of (((eq ((a->Prop)->Prop)) P) b)
% Found ((eq_ref ((a->Prop)->Prop)) P) as proof of (((eq ((a->Prop)->Prop)) P) b)
% Found ((eq_ref ((a->Prop)->Prop)) P) as proof of (((eq ((a->Prop)->Prop)) P) b)
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P0):((P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))->(P0 (fun (x:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (x Xz)))) (forall (Xx:a), ((x Xx)->(forall (Xy:a), ((iff (x Xy)) ((x1 Xx) Xy)))))))))
% Found (eta_expansion_dep000 P0) as proof of (P00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found ((eta_expansion_dep00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) P0) as proof of (P00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found (((eta_expansion_dep0 (fun (x5:(a->Prop))=> Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) P0) as proof of (P00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found ((((eta_expansion_dep (a->Prop)) (fun (x5:(a->Prop))=> Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) P0) as proof of (P00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found ((((eta_expansion_dep (a->Prop)) (fun (x5:(a->Prop))=> Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) P0) as proof of (P00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P0):((P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))->(P0 (fun (x:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (x Xz)))) (forall (Xx:a), ((x Xx)->(forall (Xy:a), ((iff (x Xy)) ((x1 Xx) Xy)))))))))
% Found (eta_expansion_dep000 P0) as proof of (P00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found ((eta_expansion_dep00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) P0) as proof of (P00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found (((eta_expansion_dep0 (fun (x5:(a->Prop))=> Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) P0) as proof of (P00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found ((((eta_expansion_dep (a->Prop)) (fun (x5:(a->Prop))=> Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) P0) as proof of (P00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found ((((eta_expansion_dep (a->Prop)) (fun (x5:(a->Prop))=> Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) P0) as proof of (P00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found eq_ref00:=(eq_ref0 b):(((eq ((a->Prop)->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq ((a->Prop)->Prop)) b) P)
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) P)
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) P)
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) P)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))):(((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) (fun (x:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (x Xz)))) (forall (Xx:a), ((x Xx)->(forall (Xy:a), ((iff (x Xy)) ((x1 Xx) Xy))))))))
% Found (eta_expansion_dep00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) b)
% Found ((eta_expansion_dep0 (fun (x5:(a->Prop))=> Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x5:(a->Prop))=> Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x5:(a->Prop))=> Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x5:(a->Prop))=> Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) b)
% Found eq_ref000:=(eq_ref00 P0):((P0 P)->(P0 P))
% Found (eq_ref00 P0) as proof of (P00 P)
% Found ((eq_ref0 P) P0) as proof of (P00 P)
% Found (((eq_ref ((a->Prop)->Prop)) P) P0) as proof of (P00 P)
% Found (((eq_ref ((a->Prop)->Prop)) P) P0) as proof of (P00 P)
% Found eq_ref000:=(eq_ref00 P0):((P0 P)->(P0 P))
% Found (eq_ref00 P0) as proof of (P00 P)
% Found ((eq_ref0 P) P0) as proof of (P00 P)
% Found (((eq_ref ((a->Prop)->Prop)) P) P0) as proof of (P00 P)
% Found (((eq_ref ((a->Prop)->Prop)) P) P0) as proof of (P00 P)
% Found eq_ref000:=(eq_ref00 P0):((P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))->(P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))))
% Found (eq_ref00 P0) as proof of (P00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found ((eq_ref0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) P0) as proof of (P00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found (((eq_ref ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) P0) as proof of (P00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found (((eq_ref ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) P0) as proof of (P00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found eq_ref00:=(eq_ref0 b):(((eq ((a->Prop)->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq ((a->Prop)->Prop)) b) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found eq_ref00:=(eq_ref0 P):(((eq ((a->Prop)->Prop)) P) P)
% Found (eq_ref0 P) as proof of (((eq ((a->Prop)->Prop)) P) b)
% Found ((eq_ref ((a->Prop)->Prop)) P) as proof of (((eq ((a->Prop)->Prop)) P) b)
% Found ((eq_ref ((a->Prop)->Prop)) P) as proof of (((eq ((a->Prop)->Prop)) P) b)
% Found ((eq_ref ((a->Prop)->Prop)) P) as proof of (((eq ((a->Prop)->Prop)) P) b)
% Found eq_ref00:=(eq_ref0 ((and ((ex a) (fun (Xz:a)=> (x2 Xz)))) (forall (Xx:a), ((x2 Xx)->(forall (Xy:a), ((iff (x2 Xy)) ((x1 Xx) Xy))))))):(((eq Prop) ((and ((ex a) (fun (Xz:a)=> (x2 Xz)))) (forall (Xx:a), ((x2 Xx)->(forall (Xy:a), ((iff (x2 Xy)) ((x1 Xx) Xy))))))) ((and ((ex a) (fun (Xz:a)=> (x2 Xz)))) (forall (Xx:a), ((x2 Xx)->(forall (Xy:a), ((iff (x2 Xy)) ((x1 Xx) Xy)))))))
% Found (eq_ref0 ((and ((ex a) (fun (Xz:a)=> (x2 Xz)))) (forall (Xx:a), ((x2 Xx)->(forall (Xy:a), ((iff (x2 Xy)) ((x1 Xx) Xy))))))) as proof of (((eq Prop) ((and ((ex a) (fun (Xz:a)=> (x2 Xz)))) (forall (Xx:a), ((x2 Xx)->(forall (Xy:a), ((iff (x2 Xy)) ((x1 Xx) Xy))))))) b)
% Found ((eq_ref Prop) ((and ((ex a) (fun (Xz:a)=> (x2 Xz)))) (forall (Xx:a), ((x2 Xx)->(forall (Xy:a), ((iff (x2 Xy)) ((x1 Xx) Xy))))))) as proof of (((eq Prop) ((and ((ex a) (fun (Xz:a)=> (x2 Xz)))) (forall (Xx:a), ((x2 Xx)->(forall (Xy:a), ((iff (x2 Xy)) ((x1 Xx) Xy))))))) b)
% Found ((eq_ref Prop) ((and ((ex a) (fun (Xz:a)=> (x2 Xz)))) (forall (Xx:a), ((x2 Xx)->(forall (Xy:a), ((iff (x2 Xy)) ((x1 Xx) Xy))))))) as proof of (((eq Prop) ((and ((ex a) (fun (Xz:a)=> (x2 Xz)))) (forall (Xx:a), ((x2 Xx)->(forall (Xy:a), ((iff (x2 Xy)) ((x1 Xx) Xy))))))) b)
% Found ((eq_ref Prop) ((and ((ex a) (fun (Xz:a)=> (x2 Xz)))) (forall (Xx:a), ((x2 Xx)->(forall (Xy:a), ((iff (x2 Xy)) ((x1 Xx) Xy))))))) as proof of (((eq Prop) ((and ((ex a) (fun (Xz:a)=> (x2 Xz)))) (forall (Xx:a), ((x2 Xx)->(forall (Xy:a), ((iff (x2 Xy)) ((x1 Xx) Xy))))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (P x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (P x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (P x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (P x2))
% Found eq_ref00:=(eq_ref0 ((and ((ex a) (fun (Xz:a)=> (x2 Xz)))) (forall (Xx:a), ((x2 Xx)->(forall (Xy:a), ((iff (x2 Xy)) ((x1 Xx) Xy))))))):(((eq Prop) ((and ((ex a) (fun (Xz:a)=> (x2 Xz)))) (forall (Xx:a), ((x2 Xx)->(forall (Xy:a), ((iff (x2 Xy)) ((x1 Xx) Xy))))))) ((and ((ex a) (fun (Xz:a)=> (x2 Xz)))) (forall (Xx:a), ((x2 Xx)->(forall (Xy:a), ((iff (x2 Xy)) ((x1 Xx) Xy)))))))
% Found (eq_ref0 ((and ((ex a) (fun (Xz:a)=> (x2 Xz)))) (forall (Xx:a), ((x2 Xx)->(forall (Xy:a), ((iff (x2 Xy)) ((x1 Xx) Xy))))))) as proof of (((eq Prop) ((and ((ex a) (fun (Xz:a)=> (x2 Xz)))) (forall (Xx:a), ((x2 Xx)->(forall (Xy:a), ((iff (x2 Xy)) ((x1 Xx) Xy))))))) b)
% Found ((eq_ref Prop) ((and ((ex a) (fun (Xz:a)=> (x2 Xz)))) (forall (Xx:a), ((x2 Xx)->(forall (Xy:a), ((iff (x2 Xy)) ((x1 Xx) Xy))))))) as proof of (((eq Prop) ((and ((ex a) (fun (Xz:a)=> (x2 Xz)))) (forall (Xx:a), ((x2 Xx)->(forall (Xy:a), ((iff (x2 Xy)) ((x1 Xx) Xy))))))) b)
% Found ((eq_ref Prop) ((and ((ex a) (fun (Xz:a)=> (x2 Xz)))) (forall (Xx:a), ((x2 Xx)->(forall (Xy:a), ((iff (x2 Xy)) ((x1 Xx) Xy))))))) as proof of (((eq Prop) ((and ((ex a) (fun (Xz:a)=> (x2 Xz)))) (forall (Xx:a), ((x2 Xx)->(forall (Xy:a), ((iff (x2 Xy)) ((x1 Xx) Xy))))))) b)
% Found ((eq_ref Prop) ((and ((ex a) (fun (Xz:a)=> (x2 Xz)))) (forall (Xx:a), ((x2 Xx)->(forall (Xy:a), ((iff (x2 Xy)) ((x1 Xx) Xy))))))) as proof of (((eq Prop) ((and ((ex a) (fun (Xz:a)=> (x2 Xz)))) (forall (Xx:a), ((x2 Xx)->(forall (Xy:a), ((iff (x2 Xy)) ((x1 Xx) Xy))))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (P x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (P x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (P x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (P x2))
% Found eq_ref000:=(eq_ref00 P0):((P0 ((and ((ex a) (fun (Xz:a)=> (x2 Xz)))) (forall (Xx:a), ((x2 Xx)->(forall (Xy:a), ((iff (x2 Xy)) ((x1 Xx) Xy)))))))->(P0 ((and ((ex a) (fun (Xz:a)=> (x2 Xz)))) (forall (Xx:a), ((x2 Xx)->(forall (Xy:a), ((iff (x2 Xy)) ((x1 Xx) Xy))))))))
% Found (eq_ref00 P0) as proof of (P00 ((and ((ex a) (fun (Xz:a)=> (x2 Xz)))) (forall (Xx:a), ((x2 Xx)->(forall (Xy:a), ((iff (x2 Xy)) ((x1 Xx) Xy)))))))
% Found ((eq_ref0 ((and ((ex a) (fun (Xz:a)=> (x2 Xz)))) (forall (Xx:a), ((x2 Xx)->(forall (Xy:a), ((iff (x2 Xy)) ((x1 Xx) Xy))))))) P0) as proof of (P00 ((and ((ex a) (fun (Xz:a)=> (x2 Xz)))) (forall (Xx:a), ((x2 Xx)->(forall (Xy:a), ((iff (x2 Xy)) ((x1 Xx) Xy)))))))
% Found (((eq_ref Prop) ((and ((ex a) (fun (Xz:a)=> (x2 Xz)))) (forall (Xx:a), ((x2 Xx)->(forall (Xy:a), ((iff (x2 Xy)) ((x1 Xx) Xy))))))) P0) as proof of (P00 ((and ((ex a) (fun (Xz:a)=> (x2 Xz)))) (forall (Xx:a), ((x2 Xx)->(forall (Xy:a), ((iff (x2 Xy)) ((x1 Xx) Xy)))))))
% Found (((eq_ref Prop) ((and ((ex a) (fun (Xz:a)=> (x2 Xz)))) (forall (Xx:a), ((x2 Xx)->(forall (Xy:a), ((iff (x2 Xy)) ((x1 Xx) Xy))))))) P0) as proof of (P00 ((and ((ex a) (fun (Xz:a)=> (x2 Xz)))) (forall (Xx:a), ((x2 Xx)->(forall (Xy:a), ((iff (x2 Xy)) ((x1 Xx) Xy)))))))
% Found eq_ref000:=(eq_ref00 P0):((P0 ((and ((ex a) (fun (Xz:a)=> (x2 Xz)))) (forall (Xx:a), ((x2 Xx)->(forall (Xy:a), ((iff (x2 Xy)) ((x1 Xx) Xy)))))))->(P0 ((and ((ex a) (fun (Xz:a)=> (x2 Xz)))) (forall (Xx:a), ((x2 Xx)->(forall (Xy:a), ((iff (x2 Xy)) ((x1 Xx) Xy))))))))
% Found (eq_ref00 P0) as proof of (P00 ((and ((ex a) (fun (Xz:a)=> (x2 Xz)))) (forall (Xx:a), ((x2 Xx)->(forall (Xy:a), ((iff (x2 Xy)) ((x1 Xx) Xy)))))))
% Found ((eq_ref0 ((and ((ex a) (fun (Xz:a)=> (x2 Xz)))) (forall (Xx:a), ((x2 Xx)->(forall (Xy:a), ((iff (x2 Xy)) ((x1 Xx) Xy))))))) P0) as proof of (P00 ((and ((ex a) (fun (Xz:a)=> (x2 Xz)))) (forall (Xx:a), ((x2 Xx)->(forall (Xy:a), ((iff (x2 Xy)) ((x1 Xx) Xy)))))))
% Found (((eq_ref Prop) ((and ((ex a) (fun (Xz:a)=> (x2 Xz)))) (forall (Xx:a), ((x2 Xx)->(forall (Xy:a), ((iff (x2 Xy)) ((x1 Xx) Xy))))))) P0) as proof of (P00 ((and ((ex a) (fun (Xz:a)=> (x2 Xz)))) (forall (Xx:a), ((x2 Xx)->(forall (Xy:a), ((iff (x2 Xy)) ((x1 Xx) Xy)))))))
% Found (((eq_ref Prop) ((and ((ex a) (fun (Xz:a)=> (x2 Xz)))) (forall (Xx:a), ((x2 Xx)->(forall (Xy:a), ((iff (x2 Xy)) ((x1 Xx) Xy))))))) P0) as proof of (P00 ((and ((ex a) (fun (Xz:a)=> (x2 Xz)))) (forall (Xx:a), ((x2 Xx)->(forall (Xy:a), ((iff (x2 Xy)) ((x1 Xx) Xy)))))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P0):((P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))->(P0 (fun (x:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (x Xz)))) (forall (Xx:a), ((x Xx)->(forall (Xy:a), ((iff (x Xy)) ((x1 Xx) Xy)))))))))
% Found (eta_expansion_dep000 P0) as proof of (P00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found ((eta_expansion_dep00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) P0) as proof of (P00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found (((eta_expansion_dep0 (fun (x5:(a->Prop))=> Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) P0) as proof of (P00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found ((((eta_expansion_dep (a->Prop)) (fun (x5:(a->Prop))=> Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) P0) as proof of (P00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found ((((eta_expansion_dep (a->Prop)) (fun (x5:(a->Prop))=> Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) P0) as proof of (P00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P0):((P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))->(P0 (fun (x:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (x Xz)))) (forall (Xx:a), ((x Xx)->(forall (Xy:a), ((iff (x Xy)) ((x1 Xx) Xy)))))))))
% Found (eta_expansion_dep000 P0) as proof of (P00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found ((eta_expansion_dep00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) P0) as proof of (P00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found (((eta_expansion_dep0 (fun (x5:(a->Prop))=> Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) P0) as proof of (P00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found ((((eta_expansion_dep (a->Prop)) (fun (x5:(a->Prop))=> Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) P0) as proof of (P00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found ((((eta_expansion_dep (a->Prop)) (fun (x5:(a->Prop))=> Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) P0) as proof of (P00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found eq_ref00:=(eq_ref0 b):(((eq ((a->Prop)->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq ((a->Prop)->Prop)) b) P)
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) P)
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) P)
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) P)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))):(((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) (fun (x:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (x Xz)))) (forall (Xx:a), ((x Xx)->(forall (Xy:a), ((iff (x Xy)) ((x1 Xx) Xy))))))))
% Found (eta_expansion_dep00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) b)
% Found ((eta_expansion_dep0 (fun (x5:(a->Prop))=> Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x5:(a->Prop))=> Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x5:(a->Prop))=> Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x5:(a->Prop))=> Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) b)
% Found eq_ref000:=(eq_ref00 P0):((P0 P)->(P0 P))
% Found (eq_ref00 P0) as proof of (P00 P)
% Found ((eq_ref0 P) P0) as proof of (P00 P)
% Found (((eq_ref ((a->Prop)->Prop)) P) P0) as proof of (P00 P)
% Found (((eq_ref ((a->Prop)->Prop)) P) P0) as proof of (P00 P)
% Found eq_ref000:=(eq_ref00 P0):((P0 P)->(P0 P))
% Found (eq_ref00 P0) as proof of (P00 P)
% Found ((eq_ref0 P) P0) as proof of (P00 P)
% Found (((eq_ref ((a->Prop)->Prop)) P) P0) as proof of (P00 P)
% Found (((eq_ref ((a->Prop)->Prop)) P) P0) as proof of (P00 P)
% Found eq_ref000:=(eq_ref00 P0):((P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))->(P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))))
% Found (eq_ref00 P0) as proof of (P00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found ((eq_ref0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) P0) as proof of (P00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found (((eq_ref ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) P0) as proof of (P00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found (((eq_ref ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) P0) as proof of (P00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found eq_ref00:=(eq_ref0 b):(((eq ((a->Prop)->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq ((a->Prop)->Prop)) b) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy))))))))
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy))))))))
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy))))))))
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy))))))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 P):(((eq ((a->Prop)->Prop)) P) (fun (x:(a->Prop))=> (P x)))
% Found (eta_expansion_dep00 P) as proof of (((eq ((a->Prop)->Prop)) P) b)
% Found ((eta_expansion_dep0 (fun (x5:(a->Prop))=> Prop)) P) as proof of (((eq ((a->Prop)->Prop)) P) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x5:(a->Prop))=> Prop)) P) as proof of (((eq ((a->Prop)->Prop)) P) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x5:(a->Prop))=> Prop)) P) as proof of (((eq ((a->Prop)->Prop)) P) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x5:(a->Prop))=> Prop)) P) as proof of (((eq ((a->Prop)->Prop)) P) b)
% Found eq_ref00:=(eq_ref0 ((and ((ex a) (fun (Xz:a)=> (x2 Xz)))) (forall (Xx:a), ((x2 Xx)->(forall (Xy:a), ((iff (x2 Xy)) ((x1 Xx) Xy))))))):(((eq Prop) ((and ((ex a) (fun (Xz:a)=> (x2 Xz)))) (forall (Xx:a), ((x2 Xx)->(forall (Xy:a), ((iff (x2 Xy)) ((x1 Xx) Xy))))))) ((and ((ex a) (fun (Xz:a)=> (x2 Xz)))) (forall (Xx:a), ((x2 Xx)->(forall (Xy:a), ((iff (x2 Xy)) ((x1 Xx) Xy)))))))
% Found (eq_ref0 ((and ((ex a) (fun (Xz:a)=> (x2 Xz)))) (forall (Xx:a), ((x2 Xx)->(forall (Xy:a), ((iff (x2 Xy)) ((x1 Xx) Xy))))))) as proof of (((eq Prop) ((and ((ex a) (fun (Xz:a)=> (x2 Xz)))) (forall (Xx:a), ((x2 Xx)->(forall (Xy:a), ((iff (x2 Xy)) ((x1 Xx) Xy))))))) b)
% Found ((eq_ref Prop) ((and ((ex a) (fun (Xz:a)=> (x2 Xz)))) (forall (Xx:a), ((x2 Xx)->(forall (Xy:a), ((iff (x2 Xy)) ((x1 Xx) Xy))))))) as proof of (((eq Prop) ((and ((ex a) (fun (Xz:a)=> (x2 Xz)))) (forall (Xx:a), ((x2 Xx)->(forall (Xy:a), ((iff (x2 Xy)) ((x1 Xx) Xy))))))) b)
% Found ((eq_ref Prop) ((and ((ex a) (fun (Xz:a)=> (x2 Xz)))) (forall (Xx:a), ((x2 Xx)->(forall (Xy:a), ((iff (x2 Xy)) ((x1 Xx) Xy))))))) as proof of (((eq Prop) ((and ((ex a) (fun (Xz:a)=> (x2 Xz)))) (forall (Xx:a), ((x2 Xx)->(forall (Xy:a), ((iff (x2 Xy)) ((x1 Xx) Xy))))))) b)
% Found ((eq_ref Prop) ((and ((ex a) (fun (Xz:a)=> (x2 Xz)))) (forall (Xx:a), ((x2 Xx)->(forall (Xy:a), ((iff (x2 Xy)) ((x1 Xx) Xy))))))) as proof of (((eq Prop) ((and ((ex a) (fun (Xz:a)=> (x2 Xz)))) (forall (Xx:a), ((x2 Xx)->(forall (Xy:a), ((iff (x2 Xy)) ((x1 Xx) Xy))))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (P x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (P x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (P x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (P x2))
% Found eq_ref00:=(eq_ref0 ((and ((ex a) (fun (Xz:a)=> (x2 Xz)))) (forall (Xx:a), ((x2 Xx)->(forall (Xy:a), ((iff (x2 Xy)) ((x1 Xx) Xy))))))):(((eq Prop) ((and ((ex a) (fun (Xz:a)=> (x2 Xz)))) (forall (Xx:a), ((x2 Xx)->(forall (Xy:a), ((iff (x2 Xy)) ((x1 Xx) Xy))))))) ((and ((ex a) (fun (Xz:a)=> (x2 Xz)))) (forall (Xx:a), ((x2 Xx)->(forall (Xy:a), ((iff (x2 Xy)) ((x1 Xx) Xy)))))))
% Found (eq_ref0 ((and ((ex a) (fun (Xz:a)=> (x2 Xz)))) (forall (Xx:a), ((x2 Xx)->(forall (Xy:a), ((iff (x2 Xy)) ((x1 Xx) Xy))))))) as proof of (((eq Prop) ((and ((ex a) (fun (Xz:a)=> (x2 Xz)))) (forall (Xx:a), ((x2 Xx)->(forall (Xy:a), ((iff (x2 Xy)) ((x1 Xx) Xy))))))) b)
% Found ((eq_ref Prop) ((and ((ex a) (fun (Xz:a)=> (x2 Xz)))) (forall (Xx:a), ((x2 Xx)->(forall (Xy:a), ((iff (x2 Xy)) ((x1 Xx) Xy))))))) as proof of (((eq Prop) ((and ((ex a) (fun (Xz:a)=> (x2 Xz)))) (forall (Xx:a), ((x2 Xx)->(forall (Xy:a), ((iff (x2 Xy)) ((x1 Xx) Xy))))))) b)
% Found ((eq_ref Prop) ((and ((ex a) (fun (Xz:a)=> (x2 Xz)))) (forall (Xx:a), ((x2 Xx)->(forall (Xy:a), ((iff (x2 Xy)) ((x1 Xx) Xy))))))) as proof of (((eq Prop) ((and ((ex a) (fun (Xz:a)=> (x2 Xz)))) (forall (Xx:a), ((x2 Xx)->(forall (Xy:a), ((iff (x2 Xy)) ((x1 Xx) Xy))))))) b)
% Found ((eq_ref Prop) ((and ((ex a) (fun (Xz:a)=> (x2 Xz)))) (forall (Xx:a), ((x2 Xx)->(forall (Xy:a), ((iff (x2 Xy)) ((x1 Xx) Xy))))))) as proof of (((eq Prop) ((and ((ex a) (fun (Xz:a)=> (x2 Xz)))) (forall (Xx:a), ((x2 Xx)->(forall (Xy:a), ((iff (x2 Xy)) ((x1 Xx) Xy))))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (P x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (P x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (P x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (P x2))
% Found eq_ref00:=(eq_ref0 b):(((eq ((a->(a->Prop))->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq ((a->(a->Prop))->Prop)) b) (fun (Q:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((Q Xx) Xx))) (forall (Xx:a) (Xy:a), (((Q Xx) Xy)->((Q Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Q Xx) Xy)) ((Q Xy) Xz))->((Q Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy)))))))) P))))
% Found ((eq_ref ((a->(a->Prop))->Prop)) b) as proof of (((eq ((a->(a->Prop))->Prop)) b) (fun (Q:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((Q Xx) Xx))) (forall (Xx:a) (Xy:a), (((Q Xx) Xy)->((Q Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Q Xx) Xy)) ((Q Xy) Xz))->((Q Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy)))))))) P))))
% Found ((eq_ref ((a->(a->Prop))->Prop)) b) as proof of (((eq ((a->(a->Prop))->Prop)) b) (fun (Q:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((Q Xx) Xx))) (forall (Xx:a) (Xy:a), (((Q Xx) Xy)->((Q Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Q Xx) Xy)) ((Q Xy) Xz))->((Q Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy)))))))) P))))
% Found ((eq_ref ((a->(a->Prop))->Prop)) b) as proof of (((eq ((a->(a->Prop))->Prop)) b) (fun (Q:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((Q Xx) Xx))) (forall (Xx:a) (Xy:a), (((Q Xx) Xy)->((Q Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Q Xx) Xy)) ((Q Xy) Xz))->((Q Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy)))))))) P))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 a0):(((eq ((a->(a->Prop))->Prop)) a0) (fun (x:(a->(a->Prop)))=> (a0 x)))
% Found (eta_expansion_dep00 a0) as proof of (((eq ((a->(a->Prop))->Prop)) a0) b)
% Found ((eta_expansion_dep0 (fun (x2:(a->(a->Prop)))=> Prop)) a0) as proof of (((eq ((a->(a->Prop))->Prop)) a0) b)
% Found (((eta_expansion_dep (a->(a->Prop))) (fun (x2:(a->(a->Prop)))=> Prop)) a0) as proof of (((eq ((a->(a->Prop))->Prop)) a0) b)
% Found (((eta_expansion_dep (a->(a->Prop))) (fun (x2:(a->(a->Prop)))=> Prop)) a0) as proof of (((eq ((a->(a->Prop))->Prop)) a0) b)
% Found (((eta_expansion_dep (a->(a->Prop))) (fun (x2:(a->(a->Prop)))=> Prop)) a0) as proof of (((eq ((a->(a->Prop))->Prop)) a0) b)
% Found eq_ref00:=(eq_ref0 a0):(((eq ((a->(a->Prop))->Prop)) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq ((a->(a->Prop))->Prop)) a0) b)
% Found ((eq_ref ((a->(a->Prop))->Prop)) a0) as proof of (((eq ((a->(a->Prop))->Prop)) a0) b)
% Found ((eq_ref ((a->(a->Prop))->Prop)) a0) as proof of (((eq ((a->(a->Prop))->Prop)) a0) b)
% Found ((eq_ref ((a->(a->Prop))->Prop)) a0) as proof of (((eq ((a->(a->Prop))->Prop)) a0) b)
% Found eq_ref000:=(eq_ref00 P0):((P0 ((and ((ex a) (fun (Xz:a)=> (x2 Xz)))) (forall (Xx:a), ((x2 Xx)->(forall (Xy:a), ((iff (x2 Xy)) ((x1 Xx) Xy)))))))->(P0 ((and ((ex a) (fun (Xz:a)=> (x2 Xz)))) (forall (Xx:a), ((x2 Xx)->(forall (Xy:a), ((iff (x2 Xy)) ((x1 Xx) Xy))))))))
% Found (eq_ref00 P0) as proof of (P00 ((and ((ex a) (fun (Xz:a)=> (x2 Xz)))) (forall (Xx:a), ((x2 Xx)->(forall (Xy:a), ((iff (x2 Xy)) ((x1 Xx) Xy)))))))
% Found ((eq_ref0 ((and ((ex a) (fun (Xz:a)=> (x2 Xz)))) (forall (Xx:a), ((x2 Xx)->(forall (Xy:a), ((iff (x2 Xy)) ((x1 Xx) Xy))))))) P0) as proof of (P00 ((and ((ex a) (fun (Xz:a)=> (x2 Xz)))) (forall (Xx:a), ((x2 Xx)->(forall (Xy:a), ((iff (x2 Xy)) ((x1 Xx) Xy)))))))
% Found (((eq_ref Prop) ((and ((ex a) (fun (Xz:a)=> (x2 Xz)))) (forall (Xx:a), ((x2 Xx)->(forall (Xy:a), ((iff (x2 Xy)) ((x1 Xx) Xy))))))) P0) as proof of (P00 ((and ((ex a) (fun (Xz:a)=> (x2 Xz)))) (forall (Xx:a), ((x2 Xx)->(forall (Xy:a), ((iff (x2 Xy)) ((x1 Xx) Xy)))))))
% Found (((eq_ref Prop) ((and ((ex a) (fun (Xz:a)=> (x2 Xz)))) (forall (Xx:a), ((x2 Xx)->(forall (Xy:a), ((iff (x2 Xy)) ((x1 Xx) Xy))))))) P0) as proof of (P00 ((and ((ex a) (fun (Xz:a)=> (x2 Xz)))) (forall (Xx:a), ((x2 Xx)->(forall (Xy:a), ((iff (x2 Xy)) ((x1 Xx) Xy)))))))
% Found eq_ref000:=(eq_ref00 P0):((P0 ((and ((ex a) (fun (Xz:a)=> (x2 Xz)))) (forall (Xx:a), ((x2 Xx)->(forall (Xy:a), ((iff (x2 Xy)) ((x1 Xx) Xy)))))))->(P0 ((and ((ex a) (fun (Xz:a)=> (x2 Xz)))) (forall (Xx:a), ((x2 Xx)->(forall (Xy:a), ((iff (x2 Xy)) ((x1 Xx) Xy))))))))
% Found (eq_ref00 P0) as proof of (P00 ((and ((ex a) (fun (Xz:a)=> (x2 Xz)))) (forall (Xx:a), ((x2 Xx)->(forall (Xy:a), ((iff (x2 Xy)) ((x1 Xx) Xy)))))))
% Found ((eq_ref0 ((and ((ex a) (fun (Xz:a)=> (x2 Xz)))) (forall (Xx:a), ((x2 Xx)->(forall (Xy:a), ((iff (x2 Xy)) ((x1 Xx) Xy))))))) P0) as proof of (P00 ((and ((ex a) (fun (Xz:a)=> (x2 Xz)))) (forall (Xx:a), ((x2 Xx)->(forall (Xy:a), ((iff (x2 Xy)) ((x1 Xx) Xy)))))))
% Found (((eq_ref Prop) ((and ((ex a) (fun (Xz:a)=> (x2 Xz)))) (forall (Xx:a), ((x2 Xx)->(forall (Xy:a), ((iff (x2 Xy)) ((x1 Xx) Xy))))))) P0) as proof of (P00 ((and ((ex a) (fun (Xz:a)=> (x2 Xz)))) (forall (Xx:a), ((x2 Xx)->(forall (Xy:a), ((iff (x2 Xy)) ((x1 Xx) Xy)))))))
% Found (((eq_ref Prop) ((and ((ex a) (fun (Xz:a)=> (x2 Xz)))) (forall (Xx:a), ((x2 Xx)->(forall (Xy:a), ((iff (x2 Xy)) ((x1 Xx) Xy))))))) P0) as proof of (P00 ((and ((ex a) (fun (Xz:a)=> (x2 Xz)))) (forall (Xx:a), ((x2 Xx)->(forall (Xy:a), ((iff (x2 Xy)) ((x1 Xx) Xy)))))))
% Found eq_ref00:=(eq_ref0 b):(((eq ((a->Prop)->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq ((a->Prop)->Prop)) b) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 P):(((eq ((a->Prop)->Prop)) P) (fun (x:(a->Prop))=> (P x)))
% Found (eta_expansion_dep00 P) as proof of (((eq ((a->Prop)->Prop)) P) b)
% Found ((eta_expansion_dep0 (fun (x5:(a->Prop))=> Prop)) P) as proof of (((eq ((a->Prop)->Prop)) P) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x5:(a->Prop))=> Prop)) P) as proof of (((eq ((a->Prop)->Prop)) P) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x5:(a->Prop))=> Prop)) P) as proof of (((eq ((a->Prop)->Prop)) P) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x5:(a->Prop))=> Prop)) P) as proof of (((eq ((a->Prop)->Prop)) P) b)
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P0):((P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))->(P0 (fun (x:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (x Xz)))) (forall (Xx:a), ((x Xx)->(forall (Xy:a), ((iff (x Xy)) ((x1 Xx) Xy)))))))))
% Found (eta_expansion_dep000 P0) as proof of (P00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found ((eta_expansion_dep00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) P0) as proof of (P00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found (((eta_expansion_dep0 (fun (x5:(a->Prop))=> Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) P0) as proof of (P00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found ((((eta_expansion_dep (a->Prop)) (fun (x5:(a->Prop))=> Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) P0) as proof of (P00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found ((((eta_expansion_dep (a->Prop)) (fun (x5:(a->Prop))=> Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) P0) as proof of (P00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found eta_expansion0000:=(eta_expansion000 P0):((P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))->(P0 (fun (x:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (x Xz)))) (forall (Xx:a), ((x Xx)->(forall (Xy:a), ((iff (x Xy)) ((x1 Xx) Xy)))))))))
% Found (eta_expansion000 P0) as proof of (P00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found ((eta_expansion00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) P0) as proof of (P00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found (((eta_expansion0 Prop) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) P0) as proof of (P00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found ((((eta_expansion (a->Prop)) Prop) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) P0) as proof of (P00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found ((((eta_expansion (a->Prop)) Prop) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) P0) as proof of (P00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found eq_ref00:=(eq_ref0 (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) P)):(((eq Prop) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) P)) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) P))
% Found (eq_ref0 (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) P)) as proof of (((eq Prop) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) P)) b)
% Found ((eq_ref Prop) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) P)) as proof of (((eq Prop) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) P)) b)
% Found ((eq_ref Prop) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) P)) as proof of (((eq Prop) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) P)) b)
% Found ((eq_ref Prop) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) P)) as proof of (((eq Prop) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) P)) b)
% Found eq_ref000:=(eq_ref00 P0):((P0 P)->(P0 P))
% Found (eq_ref00 P0) as proof of (P00 P)
% Found ((eq_ref0 P) P0) as proof of (P00 P)
% Found (((eq_ref ((a->Prop)->Prop)) P) P0) as proof of (P00 P)
% Found (((eq_ref ((a->Prop)->Prop)) P) P0) as proof of (P00 P)
% Found eq_ref000:=(eq_ref00 P0):((P0 P)->(P0 P))
% Found (eq_ref00 P0) as proof of (P00 P)
% Found ((eq_ref0 P) P0) as proof of (P00 P)
% Found (((eq_ref ((a->Prop)->Prop)) P) P0) as proof of (P00 P)
% Found (((eq_ref ((a->Prop)->Prop)) P) P0) as proof of (P00 P)
% Found eq_ref00:=(eq_ref0 b):(((eq ((a->Prop)->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq ((a->Prop)->Prop)) b) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy))))))))
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy))))))))
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy))))))))
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy))))))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 P):(((eq ((a->Prop)->Prop)) P) (fun (x:(a->Prop))=> (P x)))
% Found (eta_expansion_dep00 P) as proof of (((eq ((a->Prop)->Prop)) P) b)
% Found ((eta_expansion_dep0 (fun (x5:(a->Prop))=> Prop)) P) as proof of (((eq ((a->Prop)->Prop)) P) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x5:(a->Prop))=> Prop)) P) as proof of (((eq ((a->Prop)->Prop)) P) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x5:(a->Prop))=> Prop)) P) as proof of (((eq ((a->Prop)->Prop)) P) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x5:(a->Prop))=> Prop)) P) as proof of (((eq ((a->Prop)->Prop)) P) b)
% Found eq_ref00:=(eq_ref0 b):(((eq ((a->(a->Prop))->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq ((a->(a->Prop))->Prop)) b) (fun (Q:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((Q Xx) Xx))) (forall (Xx:a) (Xy:a), (((Q Xx) Xy)->((Q Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Q Xx) Xy)) ((Q Xy) Xz))->((Q Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy)))))))) P))))
% Found ((eq_ref ((a->(a->Prop))->Prop)) b) as proof of (((eq ((a->(a->Prop))->Prop)) b) (fun (Q:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((Q Xx) Xx))) (forall (Xx:a) (Xy:a), (((Q Xx) Xy)->((Q Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Q Xx) Xy)) ((Q Xy) Xz))->((Q Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy)))))))) P))))
% Found ((eq_ref ((a->(a->Prop))->Prop)) b) as proof of (((eq ((a->(a->Prop))->Prop)) b) (fun (Q:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((Q Xx) Xx))) (forall (Xx:a) (Xy:a), (((Q Xx) Xy)->((Q Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Q Xx) Xy)) ((Q Xy) Xz))->((Q Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy)))))))) P))))
% Found ((eq_ref ((a->(a->Prop))->Prop)) b) as proof of (((eq ((a->(a->Prop))->Prop)) b) (fun (Q:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((Q Xx) Xx))) (forall (Xx:a) (Xy:a), (((Q Xx) Xy)->((Q Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Q Xx) Xy)) ((Q Xy) Xz))->((Q Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy)))))))) P))))
% Found eq_ref00:=(eq_ref0 a0):(((eq ((a->(a->Prop))->Prop)) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq ((a->(a->Prop))->Prop)) a0) b)
% Found ((eq_ref ((a->(a->Prop))->Prop)) a0) as proof of (((eq ((a->(a->Prop))->Prop)) a0) b)
% Found ((eq_ref ((a->(a->Prop))->Prop)) a0) as proof of (((eq ((a->(a->Prop))->Prop)) a0) b)
% Found ((eq_ref ((a->(a->Prop))->Prop)) a0) as proof of (((eq ((a->(a->Prop))->Prop)) a0) b)
% Found eq_ref00:=(eq_ref0 ((and ((ex a) (fun (Xz:a)=> (x4 Xz)))) (forall (Xx:a), ((x4 Xx)->(forall (Xy:a), ((iff (x4 Xy)) ((x3 Xx) Xy))))))):(((eq Prop) ((and ((ex a) (fun (Xz:a)=> (x4 Xz)))) (forall (Xx:a), ((x4 Xx)->(forall (Xy:a), ((iff (x4 Xy)) ((x3 Xx) Xy))))))) ((and ((ex a) (fun (Xz:a)=> (x4 Xz)))) (forall (Xx:a), ((x4 Xx)->(forall (Xy:a), ((iff (x4 Xy)) ((x3 Xx) Xy)))))))
% Found (eq_ref0 ((and ((ex a) (fun (Xz:a)=> (x4 Xz)))) (forall (Xx:a), ((x4 Xx)->(forall (Xy:a), ((iff (x4 Xy)) ((x3 Xx) Xy))))))) as proof of (((eq Prop) ((and ((ex a) (fun (Xz:a)=> (x4 Xz)))) (forall (Xx:a), ((x4 Xx)->(forall (Xy:a), ((iff (x4 Xy)) ((x3 Xx) Xy))))))) b)
% Found ((eq_ref Prop) ((and ((ex a) (fun (Xz:a)=> (x4 Xz)))) (forall (Xx:a), ((x4 Xx)->(forall (Xy:a), ((iff (x4 Xy)) ((x3 Xx) Xy))))))) as proof of (((eq Prop) ((and ((ex a) (fun (Xz:a)=> (x4 Xz)))) (forall (Xx:a), ((x4 Xx)->(forall (Xy:a), ((iff (x4 Xy)) ((x3 Xx) Xy))))))) b)
% Found ((eq_ref Prop) ((and ((ex a) (fun (Xz:a)=> (x4 Xz)))) (forall (Xx:a), ((x4 Xx)->(forall (Xy:a), ((iff (x4 Xy)) ((x3 Xx) Xy))))))) as proof of (((eq Prop) ((and ((ex a) (fun (Xz:a)=> (x4 Xz)))) (forall (Xx:a), ((x4 Xx)->(forall (Xy:a), ((iff (x4 Xy)) ((x3 Xx) Xy))))))) b)
% Found ((eq_ref Prop) ((and ((ex a) (fun (Xz:a)=> (x4 Xz)))) (forall (Xx:a), ((x4 Xx)->(forall (Xy:a), ((iff (x4 Xy)) ((x3 Xx) Xy))))))) as proof of (((eq Prop) ((and ((ex a) (fun (Xz:a)=> (x4 Xz)))) (forall (Xx:a), ((x4 Xx)->(forall (Xy:a), ((iff (x4 Xy)) ((x3 Xx) Xy))))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (P x4))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (P x4))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (P x4))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (P x4))
% Found eq_ref00:=(eq_ref0 ((and ((ex a) (fun (Xz:a)=> (x4 Xz)))) (forall (Xx:a), ((x4 Xx)->(forall (Xy:a), ((iff (x4 Xy)) ((x3 Xx) Xy))))))):(((eq Prop) ((and ((ex a) (fun (Xz:a)=> (x4 Xz)))) (forall (Xx:a), ((x4 Xx)->(forall (Xy:a), ((iff (x4 Xy)) ((x3 Xx) Xy))))))) ((and ((ex a) (fun (Xz:a)=> (x4 Xz)))) (forall (Xx:a), ((x4 Xx)->(forall (Xy:a), ((iff (x4 Xy)) ((x3 Xx) Xy)))))))
% Found (eq_ref0 ((and ((ex a) (fun (Xz:a)=> (x4 Xz)))) (forall (Xx:a), ((x4 Xx)->(forall (Xy:a), ((iff (x4 Xy)) ((x3 Xx) Xy))))))) as proof of (((eq Prop) ((and ((ex a) (fun (Xz:a)=> (x4 Xz)))) (forall (Xx:a), ((x4 Xx)->(forall (Xy:a), ((iff (x4 Xy)) ((x3 Xx) Xy))))))) b)
% Found ((eq_ref Prop) ((and ((ex a) (fun (Xz:a)=> (x4 Xz)))) (forall (Xx:a), ((x4 Xx)->(forall (Xy:a), ((iff (x4 Xy)) ((x3 Xx) Xy))))))) as proof of (((eq Prop) ((and ((ex a) (fun (Xz:a)=> (x4 Xz)))) (forall (Xx:a), ((x4 Xx)->(forall (Xy:a), ((iff (x4 Xy)) ((x3 Xx) Xy))))))) b)
% Found ((eq_ref Prop) ((and ((ex a) (fun (Xz:a)=> (x4 Xz)))) (forall (Xx:a), ((x4 Xx)->(forall (Xy:a), ((iff (x4 Xy)) ((x3 Xx) Xy))))))) as proof of (((eq Prop) ((and ((ex a) (fun (Xz:a)=> (x4 Xz)))) (forall (Xx:a), ((x4 Xx)->(forall (Xy:a), ((iff (x4 Xy)) ((x3 Xx) Xy))))))) b)
% Found ((eq_ref Prop) ((and ((ex a) (fun (Xz:a)=> (x4 Xz)))) (forall (Xx:a), ((x4 Xx)->(forall (Xy:a), ((iff (x4 Xy)) ((x3 Xx) Xy))))))) as proof of (((eq Prop) ((and ((ex a) (fun (Xz:a)=> (x4 Xz)))) (forall (Xx:a), ((x4 Xx)->(forall (Xy:a), ((iff (x4 Xy)) ((x3 Xx) Xy))))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (P x4))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (P x4))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (P x4))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (P x4))
% Found eq_ref000:=(eq_ref00 P0):((P0 P)->(P0 P))
% Found (eq_ref00 P0) as proof of (P00 P)
% Found ((eq_ref0 P) P0) as proof of (P00 P)
% Found (((eq_ref ((a->Prop)->Prop)) P) P0) as proof of (P00 P)
% Found (((eq_ref ((a->Prop)->Prop)) P) P0) as proof of (P00 P)
% Found eq_ref000:=(eq_ref00 P0):((P0 P)->(P0 P))
% Found (eq_ref00 P0) as proof of (P00 P)
% Found ((eq_ref0 P) P0) as proof of (P00 P)
% Found (((eq_ref ((a->Prop)->Prop)) P) P0) as proof of (P00 P)
% Found (((eq_ref ((a->Prop)->Prop)) P) P0) as proof of (P00 P)
% Found eq_ref00:=(eq_ref0 b):(((eq ((a->Prop)->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq ((a->Prop)->Prop)) b) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 P):(((eq ((a->Prop)->Prop)) P) (fun (x:(a->Prop))=> (P x)))
% Found (eta_expansion_dep00 P) as proof of (((eq ((a->Prop)->Prop)) P) b)
% Found ((eta_expansion_dep0 (fun (x5:(a->Prop))=> Prop)) P) as proof of (((eq ((a->Prop)->Prop)) P) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x5:(a->Prop))=> Prop)) P) as proof of (((eq ((a->Prop)->Prop)) P) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x5:(a->Prop))=> Prop)) P) as proof of (((eq ((a->Prop)->Prop)) P) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x5:(a->Prop))=> Prop)) P) as proof of (((eq ((a->Prop)->Prop)) P) b)
% Found eq_ref00:=(eq_ref0 b):(((eq ((a->(a->Prop))->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq ((a->(a->Prop))->Prop)) b) (fun (Q:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((Q Xx) Xx))) (forall (Xx:a) (Xy:a), (((Q Xx) Xy)->((Q Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Q Xx) Xy)) ((Q Xy) Xz))->((Q Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy)))))))) P))))
% Found ((eq_ref ((a->(a->Prop))->Prop)) b) as proof of (((eq ((a->(a->Prop))->Prop)) b) (fun (Q:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((Q Xx) Xx))) (forall (Xx:a) (Xy:a), (((Q Xx) Xy)->((Q Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Q Xx) Xy)) ((Q Xy) Xz))->((Q Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy)))))))) P))))
% Found ((eq_ref ((a->(a->Prop))->Prop)) b) as proof of (((eq ((a->(a->Prop))->Prop)) b) (fun (Q:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((Q Xx) Xx))) (forall (Xx:a) (Xy:a), (((Q Xx) Xy)->((Q Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Q Xx) Xy)) ((Q Xy) Xz))->((Q Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy)))))))) P))))
% Found ((eq_ref ((a->(a->Prop))->Prop)) b) as proof of (((eq ((a->(a->Prop))->Prop)) b) (fun (Q:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((Q Xx) Xx))) (forall (Xx:a) (Xy:a), (((Q Xx) Xy)->((Q Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Q Xx) Xy)) ((Q Xy) Xz))->((Q Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy)))))))) P))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 a0):(((eq ((a->(a->Prop))->Prop)) a0) (fun (x:(a->(a->Prop)))=> (a0 x)))
% Found (eta_expansion_dep00 a0) as proof of (((eq ((a->(a->Prop))->Prop)) a0) b)
% Found ((eta_expansion_dep0 (fun (x4:(a->(a->Prop)))=> Prop)) a0) as proof of (((eq ((a->(a->Prop))->Prop)) a0) b)
% Found (((eta_expansion_dep (a->(a->Prop))) (fun (x4:(a->(a->Prop)))=> Prop)) a0) as proof of (((eq ((a->(a->Prop))->Prop)) a0) b)
% Found (((eta_expansion_dep (a->(a->Prop))) (fun (x4:(a->(a->Prop)))=> Prop)) a0) as proof of (((eq ((a->(a->Prop))->Prop)) a0) b)
% Found (((eta_expansion_dep (a->(a->Prop))) (fun (x4:(a->(a->Prop)))=> Prop)) a0) as proof of (((eq ((a->(a->Prop))->Prop)) a0) b)
% Found eq_ref00:=(eq_ref0 a0):(((eq ((a->(a->Prop))->Prop)) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq ((a->(a->Prop))->Prop)) a0) b)
% Found ((eq_ref ((a->(a->Prop))->Prop)) a0) as proof of (((eq ((a->(a->Prop))->Prop)) a0) b)
% Found ((eq_ref ((a->(a->Prop))->Prop)) a0) as proof of (((eq ((a->(a->Prop))->Prop)) a0) b)
% Found ((eq_ref ((a->(a->Prop))->Prop)) a0) as proof of (((eq ((a->(a->Prop))->Prop)) a0) b)
% Found eq_ref000:=(eq_ref00 P0):((P0 ((and ((ex a) (fun (Xz:a)=> (x4 Xz)))) (forall (Xx:a), ((x4 Xx)->(forall (Xy:a), ((iff (x4 Xy)) ((x3 Xx) Xy)))))))->(P0 ((and ((ex a) (fun (Xz:a)=> (x4 Xz)))) (forall (Xx:a), ((x4 Xx)->(forall (Xy:a), ((iff (x4 Xy)) ((x3 Xx) Xy))))))))
% Found (eq_ref00 P0) as proof of (P00 ((and ((ex a) (fun (Xz:a)=> (x4 Xz)))) (forall (Xx:a), ((x4 Xx)->(forall (Xy:a), ((iff (x4 Xy)) ((x3 Xx) Xy)))))))
% Found ((eq_ref0 ((and ((ex a) (fun (Xz:a)=> (x4 Xz)))) (forall (Xx:a), ((x4 Xx)->(forall (Xy:a), ((iff (x4 Xy)) ((x3 Xx) Xy))))))) P0) as proof of (P00 ((and ((ex a) (fun (Xz:a)=> (x4 Xz)))) (forall (Xx:a), ((x4 Xx)->(forall (Xy:a), ((iff (x4 Xy)) ((x3 Xx) Xy)))))))
% Found (((eq_ref Prop) ((and ((ex a) (fun (Xz:a)=> (x4 Xz)))) (forall (Xx:a), ((x4 Xx)->(forall (Xy:a), ((iff (x4 Xy)) ((x3 Xx) Xy))))))) P0) as proof of (P00 ((and ((ex a) (fun (Xz:a)=> (x4 Xz)))) (forall (Xx:a), ((x4 Xx)->(forall (Xy:a), ((iff (x4 Xy)) ((x3 Xx) Xy)))))))
% Found (((eq_ref Prop) ((and ((ex a) (fun (Xz:a)=> (x4 Xz)))) (forall (Xx:a), ((x4 Xx)->(forall (Xy:a), ((iff (x4 Xy)) ((x3 Xx) Xy))))))) P0) as proof of (P00 ((and ((ex a) (fun (Xz:a)=> (x4 Xz)))) (forall (Xx:a), ((x4 Xx)->(forall (Xy:a), ((iff (x4 Xy)) ((x3 Xx) Xy)))))))
% Found eq_ref000:=(eq_ref00 P0):((P0 ((and ((ex a) (fun (Xz:a)=> (x4 Xz)))) (forall (Xx:a), ((x4 Xx)->(forall (Xy:a), ((iff (x4 Xy)) ((x3 Xx) Xy)))))))->(P0 ((and ((ex a) (fun (Xz:a)=> (x4 Xz)))) (forall (Xx:a), ((x4 Xx)->(forall (Xy:a), ((iff (x4 Xy)) ((x3 Xx) Xy))))))))
% Found (eq_ref00 P0) as proof of (P00 ((and ((ex a) (fun (Xz:a)=> (x4 Xz)))) (forall (Xx:a), ((x4 Xx)->(forall (Xy:a), ((iff (x4 Xy)) ((x3 Xx) Xy)))))))
% Found ((eq_ref0 ((and ((ex a) (fun (Xz:a)=> (x4 Xz)))) (forall (Xx:a), ((x4 Xx)->(forall (Xy:a), ((iff (x4 Xy)) ((x3 Xx) Xy))))))) P0) as proof of (P00 ((and ((ex a) (fun (Xz:a)=> (x4 Xz)))) (forall (Xx:a), ((x4 Xx)->(forall (Xy:a), ((iff (x4 Xy)) ((x3 Xx) Xy)))))))
% Found (((eq_ref Prop) ((and ((ex a) (fun (Xz:a)=> (x4 Xz)))) (forall (Xx:a), ((x4 Xx)->(forall (Xy:a), ((iff (x4 Xy)) ((x3 Xx) Xy))))))) P0) as proof of (P00 ((and ((ex a) (fun (Xz:a)=> (x4 Xz)))) (forall (Xx:a), ((x4 Xx)->(forall (Xy:a), ((iff (x4 Xy)) ((x3 Xx) Xy)))))))
% Found (((eq_ref Prop) ((and ((ex a) (fun (Xz:a)=> (x4 Xz)))) (forall (Xx:a), ((x4 Xx)->(forall (Xy:a), ((iff (x4 Xy)) ((x3 Xx) Xy))))))) P0) as proof of (P00 ((and ((ex a) (fun (Xz:a)=> (x4 Xz)))) (forall (Xx:a), ((x4 Xx)->(forall (Xy:a), ((iff (x4 Xy)) ((x3 Xx) Xy)))))))
% Found eq_ref00:=(eq_ref0 ((and ((ex a) (fun (Xz:a)=> (x4 Xz)))) (forall (Xx:a), ((x4 Xx)->(forall (Xy:a), ((iff (x4 Xy)) ((x1 Xx) Xy))))))):(((eq Prop) ((and ((ex a) (fun (Xz:a)=> (x4 Xz)))) (forall (Xx:a), ((x4 Xx)->(forall (Xy:a), ((iff (x4 Xy)) ((x1 Xx) Xy))))))) ((and ((ex a) (fun (Xz:a)=> (x4 Xz)))) (forall (Xx:a), ((x4 Xx)->(forall (Xy:a), ((iff (x4 Xy)) ((x1 Xx) Xy)))))))
% Found (eq_ref0 ((and ((ex a) (fun (Xz:a)=> (x4 Xz)))) (forall (Xx:a), ((x4 Xx)->(forall (Xy:a), ((iff (x4 Xy)) ((x1 Xx) Xy))))))) as proof of (((eq Prop) ((and ((ex a) (fun (Xz:a)=> (x4 Xz)))) (forall (Xx:a), ((x4 Xx)->(forall (Xy:a), ((iff (x4 Xy)) ((x1 Xx) Xy))))))) b)
% Found ((eq_ref Prop) ((and ((ex a) (fun (Xz:a)=> (x4 Xz)))) (forall (Xx:a), ((x4 Xx)->(forall (Xy:a), ((iff (x4 Xy)) ((x1 Xx) Xy))))))) as proof of (((eq Prop) ((and ((ex a) (fun (Xz:a)=> (x4 Xz)))) (forall (Xx:a), ((x4 Xx)->(forall (Xy:a), ((iff (x4 Xy)) ((x1 Xx) Xy))))))) b)
% Found ((eq_ref Prop) ((and ((ex a) (fun (Xz:a)=> (x4 Xz)))) (forall (Xx:a), ((x4 Xx)->(forall (Xy:a), ((iff (x4 Xy)) ((x1 Xx) Xy))))))) as proof of (((eq Prop) ((and ((ex a) (fun (Xz:a)=> (x4 Xz)))) (forall (Xx:a), ((x4 Xx)->(forall (Xy:a), ((iff (x4 Xy)) ((x1 Xx) Xy))))))) b)
% Found ((eq_ref Prop) ((and ((ex a) (fun (Xz:a)=> (x4 Xz)))) (forall (Xx:a), ((x4 Xx)->(forall (Xy:a), ((iff (x4 Xy)) ((x1 Xx) Xy))))))) as proof of (((eq Prop) ((and ((ex a) (fun (Xz:a)=> (x4 Xz)))) (forall (Xx:a), ((x4 Xx)->(forall (Xy:a), ((iff (x4 Xy)) ((x1 Xx) Xy))))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (P x4))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (P x4))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (P x4))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (P x4))
% Found eq_ref00:=(eq_ref0 ((and ((ex a) (fun (Xz:a)=> (x4 Xz)))) (forall (Xx:a), ((x4 Xx)->(forall (Xy:a), ((iff (x4 Xy)) ((x1 Xx) Xy))))))):(((eq Prop) ((and ((ex a) (fun (Xz:a)=> (x4 Xz)))) (forall (Xx:a), ((x4 Xx)->(forall (Xy:a), ((iff (x4 Xy)) ((x1 Xx) Xy))))))) ((and ((ex a) (fun (Xz:a)=> (x4 Xz)))) (forall (Xx:a), ((x4 Xx)->(forall (Xy:a), ((iff (x4 Xy)) ((x1 Xx) Xy)))))))
% Found (eq_ref0 ((and ((ex a) (fun (Xz:a)=> (x4 Xz)))) (forall (Xx:a), ((x4 Xx)->(forall (Xy:a), ((iff (x4 Xy)) ((x1 Xx) Xy))))))) as proof of (((eq Prop) ((and ((ex a) (fun (Xz:a)=> (x4 Xz)))) (forall (Xx:a), ((x4 Xx)->(forall (Xy:a), ((iff (x4 Xy)) ((x1 Xx) Xy))))))) b)
% Found ((eq_ref Prop) ((and ((ex a) (fun (Xz:a)=> (x4 Xz)))) (forall (Xx:a), ((x4 Xx)->(forall (Xy:a), ((iff (x4 Xy)) ((x1 Xx) Xy))))))) as proof of (((eq Prop) ((and ((ex a) (fun (Xz:a)=> (x4 Xz)))) (forall (Xx:a), ((x4 Xx)->(forall (Xy:a), ((iff (x4 Xy)) ((x1 Xx) Xy))))))) b)
% Found ((eq_ref Prop) ((and ((ex a) (fun (Xz:a)=> (x4 Xz)))) (forall (Xx:a), ((x4 Xx)->(forall (Xy:a), ((iff (x4 Xy)) ((x1 Xx) Xy))))))) as proof of (((eq Prop) ((and ((ex a) (fun (Xz:a)=> (x4 Xz)))) (forall (Xx:a), ((x4 Xx)->(forall (Xy:a), ((iff (x4 Xy)) ((x1 Xx) Xy))))))) b)
% Found ((eq_ref Prop) ((and ((ex a) (fun (Xz:a)=> (x4 Xz)))) (forall (Xx:a), ((x4 Xx)->(forall (Xy:a), ((iff (x4 Xy)) ((x1 Xx) Xy))))))) as proof of (((eq Prop) ((and ((ex a) (fun (Xz:a)=> (x4 Xz)))) (forall (Xx:a), ((x4 Xx)->(forall (Xy:a), ((iff (x4 Xy)) ((x1 Xx) Xy))))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (P x4))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (P x4))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (P x4))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (P x4))
% Found eq_ref00:=(eq_ref0 (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) P)):(((eq Prop) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) P)) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) P))
% Found (eq_ref0 (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) P)) as proof of (((eq Prop) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) P)) b)
% Found ((eq_ref Prop) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) P)) as proof of (((eq Prop) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) P)) b)
% Found ((eq_ref Prop) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) P)) as proof of (((eq Prop) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) P)) b)
% Found ((eq_ref Prop) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) P)) as proof of (((eq Prop) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) P)) b)
% Found iff_sym:=(fun (A:Prop) (B:Prop) (H:((iff A) B))=> ((((conj (B->A)) (A->B)) (((proj2 (A->B)) (B->A)) H)) (((proj1 (A->B)) (B->A)) H))):(forall (A:Prop) (B:Prop), (((iff A) B)->((iff B) A)))
% Instantiate: b:=(forall (A:Prop) (B:Prop), (((iff A) B)->((iff B) A))):Prop
% Found iff_sym as proof of b
% Found eq_ref000:=(eq_ref00 P0):((P0 P)->(P0 P))
% Found (eq_ref00 P0) as proof of (P00 P)
% Found ((eq_ref0 P) P0) as proof of (P00 P)
% Found (((eq_ref ((a->Prop)->Prop)) P) P0) as proof of (P00 P)
% Found (((eq_ref ((a->Prop)->Prop)) P) P0) as proof of (P00 P)
% Found eq_ref000:=(eq_ref00 P0):((P0 P)->(P0 P))
% Found (eq_ref00 P0) as proof of (P00 P)
% Found ((eq_ref0 P) P0) as proof of (P00 P)
% Found (((eq_ref ((a->Prop)->Prop)) P) P0) as proof of (P00 P)
% Found (((eq_ref ((a->Prop)->Prop)) P) P0) as proof of (P00 P)
% Found eq_ref000:=(eq_ref00 P0):((P0 ((and ((ex a) (fun (Xz:a)=> (x4 Xz)))) (forall (Xx:a), ((x4 Xx)->(forall (Xy:a), ((iff (x4 Xy)) ((x1 Xx) Xy)))))))->(P0 ((and ((ex a) (fun (Xz:a)=> (x4 Xz)))) (forall (Xx:a), ((x4 Xx)->(forall (Xy:a), ((iff (x4 Xy)) ((x1 Xx) Xy))))))))
% Found (eq_ref00 P0) as proof of (P00 ((and ((ex a) (fun (Xz:a)=> (x4 Xz)))) (forall (Xx:a), ((x4 Xx)->(forall (Xy:a), ((iff (x4 Xy)) ((x1 Xx) Xy)))))))
% Found ((eq_ref0 ((and ((ex a) (fun (Xz:a)=> (x4 Xz)))) (forall (Xx:a), ((x4 Xx)->(forall (Xy:a), ((iff (x4 Xy)) ((x1 Xx) Xy))))))) P0) as proof of (P00 ((and ((ex a) (fun (Xz:a)=> (x4 Xz)))) (forall (Xx:a), ((x4 Xx)->(forall (Xy:a), ((iff (x4 Xy)) ((x1 Xx) Xy)))))))
% Found (((eq_ref Prop) ((and ((ex a) (fun (Xz:a)=> (x4 Xz)))) (forall (Xx:a), ((x4 Xx)->(forall (Xy:a), ((iff (x4 Xy)) ((x1 Xx) Xy))))))) P0) as proof of (P00 ((and ((ex a) (fun (Xz:a)=> (x4 Xz)))) (forall (Xx:a), ((x4 Xx)->(forall (Xy:a), ((iff (x4 Xy)) ((x1 Xx) Xy)))))))
% Found (((eq_ref Prop) ((and ((ex a) (fun (Xz:a)=> (x4 Xz)))) (forall (Xx:a), ((x4 Xx)->(forall (Xy:a), ((iff (x4 Xy)) ((x1 Xx) Xy))))))) P0) as proof of (P00 ((and ((ex a) (fun (Xz:a)=> (x4 Xz)))) (forall (Xx:a), ((x4 Xx)->(forall (Xy:a), ((iff (x4 Xy)) ((x1 Xx) Xy)))))))
% Found eq_ref000:=(eq_ref00 P0):((P0 ((and ((ex a) (fun (Xz:a)=> (x4 Xz)))) (forall (Xx:a), ((x4 Xx)->(forall (Xy:a), ((iff (x4 Xy)) ((x1 Xx) Xy)))))))->(P0 ((and ((ex a) (fun (Xz:a)=> (x4 Xz)))) (forall (Xx:a), ((x4 Xx)->(forall (Xy:a), ((iff (x4 Xy)) ((x1 Xx) Xy))))))))
% Found (eq_ref00 P0) as proof of (P00 ((and ((ex a) (fun (Xz:a)=> (x4 Xz)))) (forall (Xx:a), ((x4 Xx)->(forall (Xy:a), ((iff (x4 Xy)) ((x1 Xx) Xy)))))))
% Found ((eq_ref0 ((and ((ex a) (fun (Xz:a)=> (x4 Xz)))) (forall (Xx:a), ((x4 Xx)->(forall (Xy:a), ((if
% EOF
%------------------------------------------------------------------------------