TSTP Solution File: SEU877^5 by cocATP---0.2.0

%------------------------------------------------------------------------------
% File     : cocATP---0.2.0
% Problem  : SEU877^5 : TPTP v6.1.0. Released v4.0.0.
% Transfm  : none
% Format   : tptp:raw
% Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p

% Computer : n185.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 13:33:19 EDT 2014

% Result   : Timeout 300.02s
% Output   : None 
% Verified : 
% SZS Type : None (Parsing solution fails)
% Syntax   : Number of formulae    : 0

% Comments : 
%------------------------------------------------------------------------------
%----NO SOLUTION OUTPUT BY SYSTEM
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% % Problem  : SEU877^5 : TPTP v6.1.0. Released v4.0.0.
% % Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n185.star.cs.uiowa.edu
% % Model    : x86_64 x86_64
% % CPU      : Intel(R) Xeon(R) CPU E5-2609 0 @ 2.40GHz
% % Memory   : 32286.75MB
% % OS       : Linux 2.6.32-431.20.3.el6.x86_64
% % CPULimit : 300
% % DateTime : Thu Jul 17 11:39:31 CDT 2014
% % CPUTime  : 300.02 
% Python 2.7.5
% Using paths ['/home/cristobal/cocATP/CASC/TPTP/', '/export/starexec/sandbox/benchmark/', '/export/starexec/sandbox/benchmark/']
% FOF formula (<kernel.Constant object at 0x27aecb0>, <kernel.Type object at 0x27aeab8>) of role type named a_type
% Using role type
% Declaring a:Type
% FOF formula (<kernel.Constant object at 0x278f8c0>, <kernel.DependentProduct object at 0x27aedd0>) of role type named cA
% Using role type
% Declaring cA:((a->Prop)->Prop)
% FOF formula (<kernel.Constant object at 0x27ae098>, <kernel.DependentProduct object at 0x27aebd8>) of role type named cB
% Using role type
% Declaring cB:((a->Prop)->Prop)
% FOF formula ((forall (Xx:(a->Prop)), ((cA Xx)->(cB Xx)))->(((eq (((a->Prop)->Prop)->Prop)) (fun (U:((a->Prop)->Prop))=> ((and (forall (Xx:(a->Prop)), ((U Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((U Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(U Xy))))))))))) (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx)))))))))) of role conjecture named cDOMTHM3_pme
% Conjecture to prove = ((forall (Xx:(a->Prop)), ((cA Xx)->(cB Xx)))->(((eq (((a->Prop)->Prop)->Prop)) (fun (U:((a->Prop)->Prop))=> ((and (forall (Xx:(a->Prop)), ((U Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((U Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(U Xy))))))))))) (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx)))))))))):Prop
% Parameter a_DUMMY:a.
% We need to prove ['((forall (Xx:(a->Prop)), ((cA Xx)->(cB Xx)))->(((eq (((a->Prop)->Prop)->Prop)) (fun (U:((a->Prop)->Prop))=> ((and (forall (Xx:(a->Prop)), ((U Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((U Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(U Xy))))))))))) (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx))))))))))']
% Parameter a:Type.
% Parameter cA:((a->Prop)->Prop).
% Parameter cB:((a->Prop)->Prop).
% Trying to prove ((forall (Xx:(a->Prop)), ((cA Xx)->(cB Xx)))->(((eq (((a->Prop)->Prop)->Prop)) (fun (U:((a->Prop)->Prop))=> ((and (forall (Xx:(a->Prop)), ((U Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((U Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(U Xy))))))))))) (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx))))))))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P (fun (U:((a->Prop)->Prop))=> ((and (forall (Xx:(a->Prop)), ((U Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((U Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(U Xy)))))))))))->(P (fun (x:((a->Prop)->Prop))=> ((and (forall (Xx:(a->Prop)), ((x Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((x Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(x Xy))))))))))))
% Found (eta_expansion_dep000 P) as proof of (P0 (fun (U:((a->Prop)->Prop))=> ((and (forall (Xx:(a->Prop)), ((U Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((U Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(U Xy)))))))))))
% Found ((eta_expansion_dep00 (fun (U:((a->Prop)->Prop))=> ((and (forall (Xx:(a->Prop)), ((U Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((U Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(U Xy))))))))))) P) as proof of (P0 (fun (U:((a->Prop)->Prop))=> ((and (forall (Xx:(a->Prop)), ((U Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((U Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(U Xy)))))))))))
% Found (((eta_expansion_dep0 (fun (x1:((a->Prop)->Prop))=> Prop)) (fun (U:((a->Prop)->Prop))=> ((and (forall (Xx:(a->Prop)), ((U Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((U Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(U Xy))))))))))) P) as proof of (P0 (fun (U:((a->Prop)->Prop))=> ((and (forall (Xx:(a->Prop)), ((U Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((U Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(U Xy)))))))))))
% Found ((((eta_expansion_dep ((a->Prop)->Prop)) (fun (x1:((a->Prop)->Prop))=> Prop)) (fun (U:((a->Prop)->Prop))=> ((and (forall (Xx:(a->Prop)), ((U Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((U Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(U Xy))))))))))) P) as proof of (P0 (fun (U:((a->Prop)->Prop))=> ((and (forall (Xx:(a->Prop)), ((U Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((U Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(U Xy)))))))))))
% Found ((((eta_expansion_dep ((a->Prop)->Prop)) (fun (x1:((a->Prop)->Prop))=> Prop)) (fun (U:((a->Prop)->Prop))=> ((and (forall (Xx:(a->Prop)), ((U Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((U Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(U Xy))))))))))) P) as proof of (P0 (fun (U:((a->Prop)->Prop))=> ((and (forall (Xx:(a->Prop)), ((U Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((U Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(U Xy)))))))))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P (fun (U:((a->Prop)->Prop))=> ((and (forall (Xx:(a->Prop)), ((U Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((U Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(U Xy)))))))))))->(P (fun (x:((a->Prop)->Prop))=> ((and (forall (Xx:(a->Prop)), ((x Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((x Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(x Xy))))))))))))
% Found (eta_expansion_dep000 P) as proof of (P0 (fun (U:((a->Prop)->Prop))=> ((and (forall (Xx:(a->Prop)), ((U Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((U Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(U Xy)))))))))))
% Found ((eta_expansion_dep00 (fun (U:((a->Prop)->Prop))=> ((and (forall (Xx:(a->Prop)), ((U Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((U Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(U Xy))))))))))) P) as proof of (P0 (fun (U:((a->Prop)->Prop))=> ((and (forall (Xx:(a->Prop)), ((U Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((U Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(U Xy)))))))))))
% Found (((eta_expansion_dep0 (fun (x1:((a->Prop)->Prop))=> Prop)) (fun (U:((a->Prop)->Prop))=> ((and (forall (Xx:(a->Prop)), ((U Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((U Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(U Xy))))))))))) P) as proof of (P0 (fun (U:((a->Prop)->Prop))=> ((and (forall (Xx:(a->Prop)), ((U Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((U Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(U Xy)))))))))))
% Found ((((eta_expansion_dep ((a->Prop)->Prop)) (fun (x1:((a->Prop)->Prop))=> Prop)) (fun (U:((a->Prop)->Prop))=> ((and (forall (Xx:(a->Prop)), ((U Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((U Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(U Xy))))))))))) P) as proof of (P0 (fun (U:((a->Prop)->Prop))=> ((and (forall (Xx:(a->Prop)), ((U Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((U Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(U Xy)))))))))))
% Found ((((eta_expansion_dep ((a->Prop)->Prop)) (fun (x1:((a->Prop)->Prop))=> Prop)) (fun (U:((a->Prop)->Prop))=> ((and (forall (Xx:(a->Prop)), ((U Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((U Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(U Xy))))))))))) P) as proof of (P0 (fun (U:((a->Prop)->Prop))=> ((and (forall (Xx:(a->Prop)), ((U Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((U Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(U Xy)))))))))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P (fun (U:((a->Prop)->Prop))=> ((and (forall (Xx:(a->Prop)), ((U Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((U Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(U Xy)))))))))))->(P (fun (x:((a->Prop)->Prop))=> ((and (forall (Xx:(a->Prop)), ((x Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((x Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(x Xy))))))))))))
% Found (eta_expansion_dep000 P) as proof of (P0 (fun (U:((a->Prop)->Prop))=> ((and (forall (Xx:(a->Prop)), ((U Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((U Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(U Xy)))))))))))
% Found ((eta_expansion_dep00 (fun (U:((a->Prop)->Prop))=> ((and (forall (Xx:(a->Prop)), ((U Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((U Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(U Xy))))))))))) P) as proof of (P0 (fun (U:((a->Prop)->Prop))=> ((and (forall (Xx:(a->Prop)), ((U Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((U Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(U Xy)))))))))))
% Found (((eta_expansion_dep0 (fun (x1:((a->Prop)->Prop))=> Prop)) (fun (U:((a->Prop)->Prop))=> ((and (forall (Xx:(a->Prop)), ((U Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((U Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(U Xy))))))))))) P) as proof of (P0 (fun (U:((a->Prop)->Prop))=> ((and (forall (Xx:(a->Prop)), ((U Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((U Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(U Xy)))))))))))
% Found ((((eta_expansion_dep ((a->Prop)->Prop)) (fun (x1:((a->Prop)->Prop))=> Prop)) (fun (U:((a->Prop)->Prop))=> ((and (forall (Xx:(a->Prop)), ((U Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((U Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(U Xy))))))))))) P) as proof of (P0 (fun (U:((a->Prop)->Prop))=> ((and (forall (Xx:(a->Prop)), ((U Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((U Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(U Xy)))))))))))
% Found ((((eta_expansion_dep ((a->Prop)->Prop)) (fun (x1:((a->Prop)->Prop))=> Prop)) (fun (U:((a->Prop)->Prop))=> ((and (forall (Xx:(a->Prop)), ((U Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((U Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(U Xy))))))))))) P) as proof of (P0 (fun (U:((a->Prop)->Prop))=> ((and (forall (Xx:(a->Prop)), ((U Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((U Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(U Xy)))))))))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (U:((a->Prop)->Prop))=> ((and (forall (Xx:(a->Prop)), ((U Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((U Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(U Xy))))))))))):(((eq (((a->Prop)->Prop)->Prop)) (fun (U:((a->Prop)->Prop))=> ((and (forall (Xx:(a->Prop)), ((U Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((U Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(U Xy))))))))))) (fun (x:((a->Prop)->Prop))=> ((and (forall (Xx:(a->Prop)), ((x Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((x Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(x Xy)))))))))))
% Found (eta_expansion_dep00 (fun (U:((a->Prop)->Prop))=> ((and (forall (Xx:(a->Prop)), ((U Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((U Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(U Xy))))))))))) as proof of (((eq (((a->Prop)->Prop)->Prop)) (fun (U:((a->Prop)->Prop))=> ((and (forall (Xx:(a->Prop)), ((U Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((U Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(U Xy))))))))))) b)
% Found ((eta_expansion_dep0 (fun (x1:((a->Prop)->Prop))=> Prop)) (fun (U:((a->Prop)->Prop))=> ((and (forall (Xx:(a->Prop)), ((U Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((U Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(U Xy))))))))))) as proof of (((eq (((a->Prop)->Prop)->Prop)) (fun (U:((a->Prop)->Prop))=> ((and (forall (Xx:(a->Prop)), ((U Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((U Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(U Xy))))))))))) b)
% Found (((eta_expansion_dep ((a->Prop)->Prop)) (fun (x1:((a->Prop)->Prop))=> Prop)) (fun (U:((a->Prop)->Prop))=> ((and (forall (Xx:(a->Prop)), ((U Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((U Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(U Xy))))))))))) as proof of (((eq (((a->Prop)->Prop)->Prop)) (fun (U:((a->Prop)->Prop))=> ((and (forall (Xx:(a->Prop)), ((U Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((U Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(U Xy))))))))))) b)
% Found (((eta_expansion_dep ((a->Prop)->Prop)) (fun (x1:((a->Prop)->Prop))=> Prop)) (fun (U:((a->Prop)->Prop))=> ((and (forall (Xx:(a->Prop)), ((U Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((U Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(U Xy))))))))))) as proof of (((eq (((a->Prop)->Prop)->Prop)) (fun (U:((a->Prop)->Prop))=> ((and (forall (Xx:(a->Prop)), ((U Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((U Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(U Xy))))))))))) b)
% Found (((eta_expansion_dep ((a->Prop)->Prop)) (fun (x1:((a->Prop)->Prop))=> Prop)) (fun (U:((a->Prop)->Prop))=> ((and (forall (Xx:(a->Prop)), ((U Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((U Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(U Xy))))))))))) as proof of (((eq (((a->Prop)->Prop)->Prop)) (fun (U:((a->Prop)->Prop))=> ((and (forall (Xx:(a->Prop)), ((U Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((U Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(U Xy))))))))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (((a->Prop)->Prop)->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (((a->Prop)->Prop)->Prop)) b) (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx)))))))))
% Found ((eq_ref (((a->Prop)->Prop)->Prop)) b) as proof of (((eq (((a->Prop)->Prop)->Prop)) b) (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx)))))))))
% Found ((eq_ref (((a->Prop)->Prop)->Prop)) b) as proof of (((eq (((a->Prop)->Prop)->Prop)) b) (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx)))))))))
% Found ((eq_ref (((a->Prop)->Prop)->Prop)) b) as proof of (((eq (((a->Prop)->Prop)->Prop)) b) (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx)))))))))
% Found eq_ref00:=(eq_ref0 b):(((eq (((a->Prop)->Prop)->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (((a->Prop)->Prop)->Prop)) b) (fun (U:((a->Prop)->Prop))=> ((and (forall (Xx:(a->Prop)), ((U Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((U Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(U Xy)))))))))))
% Found ((eq_ref (((a->Prop)->Prop)->Prop)) b) as proof of (((eq (((a->Prop)->Prop)->Prop)) b) (fun (U:((a->Prop)->Prop))=> ((and (forall (Xx:(a->Prop)), ((U Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((U Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(U Xy)))))))))))
% Found ((eq_ref (((a->Prop)->Prop)->Prop)) b) as proof of (((eq (((a->Prop)->Prop)->Prop)) b) (fun (U:((a->Prop)->Prop))=> ((and (forall (Xx:(a->Prop)), ((U Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((U Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(U Xy)))))))))))
% Found ((eq_ref (((a->Prop)->Prop)->Prop)) b) as proof of (((eq (((a->Prop)->Prop)->Prop)) b) (fun (U:((a->Prop)->Prop))=> ((and (forall (Xx:(a->Prop)), ((U Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((U Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(U Xy)))))))))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx))))))))):(((eq (((a->Prop)->Prop)->Prop)) (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx))))))))) (fun (x:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) x) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx)))))))))
% Found (eta_expansion_dep00 (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx))))))))) as proof of (((eq (((a->Prop)->Prop)->Prop)) (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx))))))))) b)
% Found ((eta_expansion_dep0 (fun (x1:((a->Prop)->Prop))=> Prop)) (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx))))))))) as proof of (((eq (((a->Prop)->Prop)->Prop)) (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx))))))))) b)
% Found (((eta_expansion_dep ((a->Prop)->Prop)) (fun (x1:((a->Prop)->Prop))=> Prop)) (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx))))))))) as proof of (((eq (((a->Prop)->Prop)->Prop)) (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx))))))))) b)
% Found (((eta_expansion_dep ((a->Prop)->Prop)) (fun (x1:((a->Prop)->Prop))=> Prop)) (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx))))))))) as proof of (((eq (((a->Prop)->Prop)->Prop)) (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx))))))))) b)
% Found (((eta_expansion_dep ((a->Prop)->Prop)) (fun (x1:((a->Prop)->Prop))=> Prop)) (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx))))))))) as proof of (((eq (((a->Prop)->Prop)->Prop)) (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx))))))))) b)
% Found eq_ref000:=(eq_ref00 P):((P (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx)))))))))->(P (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx))))))))))
% Found (eq_ref00 P) as proof of (P0 (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx)))))))))
% Found ((eq_ref0 (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx))))))))) P) as proof of (P0 (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx)))))))))
% Found (((eq_ref (((a->Prop)->Prop)->Prop)) (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx))))))))) P) as proof of (P0 (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx)))))))))
% Found (((eq_ref (((a->Prop)->Prop)->Prop)) (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx))))))))) P) as proof of (P0 (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx)))))))))
% Found eq_ref000:=(eq_ref00 P):((P (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx)))))))))->(P (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx))))))))))
% Found (eq_ref00 P) as proof of (P0 (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx)))))))))
% Found ((eq_ref0 (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx))))))))) P) as proof of (P0 (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx)))))))))
% Found (((eq_ref (((a->Prop)->Prop)->Prop)) (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx))))))))) P) as proof of (P0 (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx)))))))))
% Found (((eq_ref (((a->Prop)->Prop)->Prop)) (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx))))))))) P) as proof of (P0 (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx)))))))))
% Found eq_ref000:=(eq_ref00 P):((P ((and (forall (Xx:(a->Prop)), ((x0 Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((x0 Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(x0 Xy))))))))))->(P ((and (forall (Xx:(a->Prop)), ((x0 Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((x0 Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(x0 Xy)))))))))))
% Found (eq_ref00 P) as proof of (P0 ((and (forall (Xx:(a->Prop)), ((x0 Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((x0 Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(x0 Xy))))))))))
% Found ((eq_ref0 ((and (forall (Xx:(a->Prop)), ((x0 Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((x0 Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(x0 Xy)))))))))) P) as proof of (P0 ((and (forall (Xx:(a->Prop)), ((x0 Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((x0 Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(x0 Xy))))))))))
% Found (((eq_ref Prop) ((and (forall (Xx:(a->Prop)), ((x0 Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((x0 Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(x0 Xy)))))))))) P) as proof of (P0 ((and (forall (Xx:(a->Prop)), ((x0 Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((x0 Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(x0 Xy))))))))))
% Found (((eq_ref Prop) ((and (forall (Xx:(a->Prop)), ((x0 Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((x0 Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(x0 Xy)))))))))) P) as proof of (P0 ((and (forall (Xx:(a->Prop)), ((x0 Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((x0 Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(x0 Xy))))))))))
% Found eq_ref000:=(eq_ref00 P):((P ((and (forall (Xx:(a->Prop)), ((x0 Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((x0 Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(x0 Xy))))))))))->(P ((and (forall (Xx:(a->Prop)), ((x0 Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((x0 Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(x0 Xy)))))))))))
% Found (eq_ref00 P) as proof of (P0 ((and (forall (Xx:(a->Prop)), ((x0 Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((x0 Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(x0 Xy))))))))))
% Found ((eq_ref0 ((and (forall (Xx:(a->Prop)), ((x0 Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((x0 Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(x0 Xy)))))))))) P) as proof of (P0 ((and (forall (Xx:(a->Prop)), ((x0 Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((x0 Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(x0 Xy))))))))))
% Found (((eq_ref Prop) ((and (forall (Xx:(a->Prop)), ((x0 Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((x0 Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(x0 Xy)))))))))) P) as proof of (P0 ((and (forall (Xx:(a->Prop)), ((x0 Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((x0 Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(x0 Xy))))))))))
% Found (((eq_ref Prop) ((and (forall (Xx:(a->Prop)), ((x0 Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((x0 Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(x0 Xy)))))))))) P) as proof of (P0 ((and (forall (Xx:(a->Prop)), ((x0 Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((x0 Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(x0 Xy))))))))))
% Found eq_ref00:=(eq_ref0 ((and (forall (Xx:(a->Prop)), ((x0 Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((x0 Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(x0 Xy)))))))))):(((eq Prop) ((and (forall (Xx:(a->Prop)), ((x0 Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((x0 Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(x0 Xy)))))))))) ((and (forall (Xx:(a->Prop)), ((x0 Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((x0 Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(x0 Xy))))))))))
% Found (eq_ref0 ((and (forall (Xx:(a->Prop)), ((x0 Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((x0 Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(x0 Xy)))))))))) as proof of (((eq Prop) ((and (forall (Xx:(a->Prop)), ((x0 Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((x0 Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(x0 Xy)))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx:(a->Prop)), ((x0 Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((x0 Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(x0 Xy)))))))))) as proof of (((eq Prop) ((and (forall (Xx:(a->Prop)), ((x0 Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((x0 Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(x0 Xy)))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx:(a->Prop)), ((x0 Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((x0 Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(x0 Xy)))))))))) as proof of (((eq Prop) ((and (forall (Xx:(a->Prop)), ((x0 Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((x0 Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(x0 Xy)))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx:(a->Prop)), ((x0 Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((x0 Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(x0 Xy)))))))))) as proof of (((eq Prop) ((and (forall (Xx:(a->Prop)), ((x0 Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((x0 Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(x0 Xy)))))))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) x0) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) x0) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) x0) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) x0) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx))))))))
% Found eq_ref00:=(eq_ref0 ((and (forall (Xx:(a->Prop)), ((x0 Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((x0 Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(x0 Xy)))))))))):(((eq Prop) ((and (forall (Xx:(a->Prop)), ((x0 Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((x0 Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(x0 Xy)))))))))) ((and (forall (Xx:(a->Prop)), ((x0 Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((x0 Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(x0 Xy))))))))))
% Found (eq_ref0 ((and (forall (Xx:(a->Prop)), ((x0 Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((x0 Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(x0 Xy)))))))))) as proof of (((eq Prop) ((and (forall (Xx:(a->Prop)), ((x0 Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((x0 Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(x0 Xy)))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx:(a->Prop)), ((x0 Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((x0 Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(x0 Xy)))))))))) as proof of (((eq Prop) ((and (forall (Xx:(a->Prop)), ((x0 Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((x0 Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(x0 Xy)))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx:(a->Prop)), ((x0 Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((x0 Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(x0 Xy)))))))))) as proof of (((eq Prop) ((and (forall (Xx:(a->Prop)), ((x0 Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((x0 Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(x0 Xy)))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx:(a->Prop)), ((x0 Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((x0 Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(x0 Xy)))))))))) as proof of (((eq Prop) ((and (forall (Xx:(a->Prop)), ((x0 Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((x0 Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(x0 Xy)))))))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) x0) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) x0) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) x0) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) x0) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx))))))))
% Found eq_ref000:=(eq_ref00 P):((P ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) x0) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx))))))))->(P ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) x0) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx)))))))))
% Found (eq_ref00 P) as proof of (P0 ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) x0) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx))))))))
% Found ((eq_ref0 ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) x0) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx)))))))) P) as proof of (P0 ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) x0) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx))))))))
% Found (((eq_ref Prop) ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) x0) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx)))))))) P) as proof of (P0 ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) x0) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx))))))))
% Found (((eq_ref Prop) ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) x0) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx)))))))) P) as proof of (P0 ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) x0) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx))))))))
% Found eq_ref000:=(eq_ref00 P):((P ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) x0) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx))))))))->(P ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) x0) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx)))))))))
% Found (eq_ref00 P) as proof of (P0 ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) x0) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx))))))))
% Found ((eq_ref0 ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) x0) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx)))))))) P) as proof of (P0 ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) x0) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx))))))))
% Found (((eq_ref Prop) ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) x0) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx)))))))) P) as proof of (P0 ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) x0) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx))))))))
% Found (((eq_ref Prop) ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) x0) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx)))))))) P) as proof of (P0 ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) x0) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx))))))))
% Found eq_ref00:=(eq_ref0 ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) x0) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx)))))))):(((eq Prop) ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) x0) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx)))))))) ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) x0) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx))))))))
% Found (eq_ref0 ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) x0) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx)))))))) as proof of (((eq Prop) ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) x0) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx)))))))) b)
% Found ((eq_ref Prop) ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) x0) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx)))))))) as proof of (((eq Prop) ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) x0) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx)))))))) b)
% Found ((eq_ref Prop) ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) x0) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx)))))))) as proof of (((eq Prop) ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) x0) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx)))))))) b)
% Found ((eq_ref Prop) ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) x0) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx)))))))) as proof of (((eq Prop) ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) x0) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx)))))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (forall (Xx:(a->Prop)), ((x0 Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((x0 Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(x0 Xy))))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx:(a->Prop)), ((x0 Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((x0 Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(x0 Xy))))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx:(a->Prop)), ((x0 Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((x0 Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(x0 Xy))))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx:(a->Prop)), ((x0 Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((x0 Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(x0 Xy))))))))))
% Found eq_ref00:=(eq_ref0 ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) x0) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx)))))))):(((eq Prop) ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) x0) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx)))))))) ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) x0) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx))))))))
% Found (eq_ref0 ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) x0) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx)))))))) as proof of (((eq Prop) ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) x0) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx)))))))) b)
% Found ((eq_ref Prop) ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) x0) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx)))))))) as proof of (((eq Prop) ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) x0) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx)))))))) b)
% Found ((eq_ref Prop) ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) x0) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx)))))))) as proof of (((eq Prop) ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) x0) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx)))))))) b)
% Found ((eq_ref Prop) ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) x0) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx)))))))) as proof of (((eq Prop) ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) x0) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx)))))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (forall (Xx:(a->Prop)), ((x0 Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((x0 Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(x0 Xy))))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx:(a->Prop)), ((x0 Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((x0 Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(x0 Xy))))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx:(a->Prop)), ((x0 Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((x0 Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(x0 Xy))))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx:(a->Prop)), ((x0 Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((x0 Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(x0 Xy))))))))))
% Found eq_ref00:=(eq_ref0 ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) x0) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx)))))))):(((eq Prop) ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) x0) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx)))))))) ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) x0) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx))))))))
% Found (eq_ref0 ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) x0) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx)))))))) as proof of (((eq Prop) ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) x0) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx)))))))) b)
% Found ((eq_ref Prop) ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) x0) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx)))))))) as proof of (((eq Prop) ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) x0) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx)))))))) b)
% Found ((eq_ref Prop) ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) x0) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx)))))))) as proof of (((eq Prop) ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) x0) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx)))))))) b)
% Found ((eq_ref Prop) ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) x0) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx)))))))) as proof of (((eq Prop) ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) x0) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx)))))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (forall (Xx:(a->Prop)), ((x0 Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((x0 Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(x0 Xy))))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx:(a->Prop)), ((x0 Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((x0 Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(x0 Xy))))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx:(a->Prop)), ((x0 Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((x0 Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(x0 Xy))))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx:(a->Prop)), ((x0 Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((x0 Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(x0 Xy))))))))))
% Found eq_ref00:=(eq_ref0 ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) x0) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx)))))))):(((eq Prop) ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) x0) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx)))))))) ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) x0) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx))))))))
% Found (eq_ref0 ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) x0) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx)))))))) as proof of (((eq Prop) ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) x0) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx)))))))) b)
% Found ((eq_ref Prop) ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) x0) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx)))))))) as proof of (((eq Prop) ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) x0) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx)))))))) b)
% Found ((eq_ref Prop) ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) x0) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx)))))))) as proof of (((eq Prop) ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) x0) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx)))))))) b)
% Found ((eq_ref Prop) ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) x0) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx)))))))) as proof of (((eq Prop) ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) x0) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx)))))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (forall (Xx:(a->Prop)), ((x0 Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((x0 Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(x0 Xy))))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx:(a->Prop)), ((x0 Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((x0 Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(x0 Xy))))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx:(a->Prop)), ((x0 Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((x0 Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(x0 Xy))))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx:(a->Prop)), ((x0 Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((x0 Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(x0 Xy))))))))))
% Found x0:(P (fun (U:((a->Prop)->Prop))=> ((and (forall (Xx:(a->Prop)), ((U Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((U Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(U Xy)))))))))))
% Instantiate: b:=(fun (U:((a->Prop)->Prop))=> ((and (forall (Xx:(a->Prop)), ((U Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((U Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(U Xy)))))))))):(((a->Prop)->Prop)->Prop)
% Found x0 as proof of (P0 b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx))))))))):(((eq (((a->Prop)->Prop)->Prop)) (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx))))))))) (fun (x:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) x) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx)))))))))
% Found (eta_expansion_dep00 (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx))))))))) as proof of (((eq (((a->Prop)->Prop)->Prop)) (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx))))))))) b)
% Found ((eta_expansion_dep0 (fun (x2:((a->Prop)->Prop))=> Prop)) (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx))))))))) as proof of (((eq (((a->Prop)->Prop)->Prop)) (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx))))))))) b)
% Found (((eta_expansion_dep ((a->Prop)->Prop)) (fun (x2:((a->Prop)->Prop))=> Prop)) (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx))))))))) as proof of (((eq (((a->Prop)->Prop)->Prop)) (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx))))))))) b)
% Found (((eta_expansion_dep ((a->Prop)->Prop)) (fun (x2:((a->Prop)->Prop))=> Prop)) (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx))))))))) as proof of (((eq (((a->Prop)->Prop)->Prop)) (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx))))))))) b)
% Found (((eta_expansion_dep ((a->Prop)->Prop)) (fun (x2:((a->Prop)->Prop))=> Prop)) (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx))))))))) as proof of (((eq (((a->Prop)->Prop)->Prop)) (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx))))))))) b)
% Found x0:(P (fun (U:((a->Prop)->Prop))=> ((and (forall (Xx:(a->Prop)), ((U Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((U Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(U Xy)))))))))))
% Instantiate: f:=(fun (U:((a->Prop)->Prop))=> ((and (forall (Xx:(a->Prop)), ((U Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((U Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(U Xy)))))))))):(((a->Prop)->Prop)->Prop)
% Found x0 as proof of (P0 f)
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) x1) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx))))))))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) x1) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx))))))))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) x1) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx))))))))
% Found (fun (x1:((a->Prop)->Prop))=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) x1) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx))))))))
% Found (fun (x1:((a->Prop)->Prop))=> ((eq_ref Prop) (f x1))) as proof of (forall (x:((a->Prop)->Prop)), (((eq Prop) (f x)) ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) x) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx)))))))))
% Found x0:(P (fun (U:((a->Prop)->Prop))=> ((and (forall (Xx:(a->Prop)), ((U Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((U Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(U Xy)))))))))))
% Instantiate: f:=(fun (U:((a->Prop)->Prop))=> ((and (forall (Xx:(a->Prop)), ((U Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((U Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(U Xy)))))))))):(((a->Prop)->Prop)->Prop)
% Found x0 as proof of (P0 f)
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) x1) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx))))))))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) x1) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx))))))))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) x1) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx))))))))
% Found (fun (x1:((a->Prop)->Prop))=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) x1) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx))))))))
% Found (fun (x1:((a->Prop)->Prop))=> ((eq_ref Prop) (f x1))) as proof of (forall (x:((a->Prop)->Prop)), (((eq Prop) (f x)) ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) x) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx)))))))))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) x0) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) x0) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) x0) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) x0) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx))))))))
% Found eq_ref00:=(eq_ref0 ((and (forall (Xx:(a->Prop)), ((x0 Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((x0 Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(x0 Xy)))))))))):(((eq Prop) ((and (forall (Xx:(a->Prop)), ((x0 Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((x0 Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(x0 Xy)))))))))) ((and (forall (Xx:(a->Prop)), ((x0 Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((x0 Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(x0 Xy))))))))))
% Found (eq_ref0 ((and (forall (Xx:(a->Prop)), ((x0 Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((x0 Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(x0 Xy)))))))))) as proof of (((eq Prop) ((and (forall (Xx:(a->Prop)), ((x0 Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((x0 Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(x0 Xy)))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx:(a->Prop)), ((x0 Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((x0 Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(x0 Xy)))))))))) as proof of (((eq Prop) ((and (forall (Xx:(a->Prop)), ((x0 Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((x0 Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(x0 Xy)))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx:(a->Prop)), ((x0 Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((x0 Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(x0 Xy)))))))))) as proof of (((eq Prop) ((and (forall (Xx:(a->Prop)), ((x0 Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((x0 Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(x0 Xy)))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx:(a->Prop)), ((x0 Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((x0 Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(x0 Xy)))))))))) as proof of (((eq Prop) ((and (forall (Xx:(a->Prop)), ((x0 Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((x0 Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(x0 Xy)))))))))) b)
% Found eq_ref00:=(eq_ref0 ((and (forall (Xx:(a->Prop)), ((x0 Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((x0 Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(x0 Xy)))))))))):(((eq Prop) ((and (forall (Xx:(a->Prop)), ((x0 Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((x0 Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(x0 Xy)))))))))) ((and (forall (Xx:(a->Prop)), ((x0 Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((x0 Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(x0 Xy))))))))))
% Found (eq_ref0 ((and (forall (Xx:(a->Prop)), ((x0 Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((x0 Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(x0 Xy)))))))))) as proof of (((eq Prop) ((and (forall (Xx:(a->Prop)), ((x0 Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((x0 Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(x0 Xy)))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx:(a->Prop)), ((x0 Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((x0 Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(x0 Xy)))))))))) as proof of (((eq Prop) ((and (forall (Xx:(a->Prop)), ((x0 Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((x0 Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(x0 Xy)))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx:(a->Prop)), ((x0 Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((x0 Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(x0 Xy)))))))))) as proof of (((eq Prop) ((and (forall (Xx:(a->Prop)), ((x0 Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((x0 Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(x0 Xy)))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx:(a->Prop)), ((x0 Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((x0 Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(x0 Xy)))))))))) as proof of (((eq Prop) ((and (forall (Xx:(a->Prop)), ((x0 Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((x0 Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(x0 Xy)))))))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) x0) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) x0) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) x0) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) x0) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx))))))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (U:((a->Prop)->Prop))=> ((and (forall (Xx:(a->Prop)), ((U Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((U Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(U Xy))))))))))):(((eq (((a->Prop)->Prop)->Prop)) (fun (U:((a->Prop)->Prop))=> ((and (forall (Xx:(a->Prop)), ((U Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((U Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(U Xy))))))))))) (fun (x:((a->Prop)->Prop))=> ((and (forall (Xx:(a->Prop)), ((x Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((x Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(x Xy)))))))))))
% Found (eta_expansion_dep00 (fun (U:((a->Prop)->Prop))=> ((and (forall (Xx:(a->Prop)), ((U Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((U Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(U Xy))))))))))) as proof of (((eq (((a->Prop)->Prop)->Prop)) (fun (U:((a->Prop)->Prop))=> ((and (forall (Xx:(a->Prop)), ((U Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((U Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(U Xy))))))))))) b)
% Found ((eta_expansion_dep0 (fun (x1:((a->Prop)->Prop))=> Prop)) (fun (U:((a->Prop)->Prop))=> ((and (forall (Xx:(a->Prop)), ((U Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((U Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(U Xy))))))))))) as proof of (((eq (((a->Prop)->Prop)->Prop)) (fun (U:((a->Prop)->Prop))=> ((and (forall (Xx:(a->Prop)), ((U Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((U Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(U Xy))))))))))) b)
% Found (((eta_expansion_dep ((a->Prop)->Prop)) (fun (x1:((a->Prop)->Prop))=> Prop)) (fun (U:((a->Prop)->Prop))=> ((and (forall (Xx:(a->Prop)), ((U Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((U Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(U Xy))))))))))) as proof of (((eq (((a->Prop)->Prop)->Prop)) (fun (U:((a->Prop)->Prop))=> ((and (forall (Xx:(a->Prop)), ((U Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((U Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(U Xy))))))))))) b)
% Found (((eta_expansion_dep ((a->Prop)->Prop)) (fun (x1:((a->Prop)->Prop))=> Prop)) (fun (U:((a->Prop)->Prop))=> ((and (forall (Xx:(a->Prop)), ((U Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((U Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(U Xy))))))))))) as proof of (((eq (((a->Prop)->Prop)->Prop)) (fun (U:((a->Prop)->Prop))=> ((and (forall (Xx:(a->Prop)), ((U Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((U Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(U Xy))))))))))) b)
% Found (((eta_expansion_dep ((a->Prop)->Prop)) (fun (x1:((a->Prop)->Prop))=> Prop)) (fun (U:((a->Prop)->Prop))=> ((and (forall (Xx:(a->Prop)), ((U Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((U Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(U Xy))))))))))) as proof of (((eq (((a->Prop)->Prop)->Prop)) (fun (U:((a->Prop)->Prop))=> ((and (forall (Xx:(a->Prop)), ((U Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((U Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(U Xy))))))))))) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq (((a->Prop)->Prop)->Prop)) b) (fun (x:((a->Prop)->Prop))=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq (((a->Prop)->Prop)->Prop)) b) (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx)))))))))
% Found ((eta_expansion_dep0 (fun (x1:((a->Prop)->Prop))=> Prop)) b) as proof of (((eq (((a->Prop)->Prop)->Prop)) b) (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx)))))))))
% Found (((eta_expansion_dep ((a->Prop)->Prop)) (fun (x1:((a->Prop)->Prop))=> Prop)) b) as proof of (((eq (((a->Prop)->Prop)->Prop)) b) (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx)))))))))
% Found (((eta_expansion_dep ((a->Prop)->Prop)) (fun (x1:((a->Prop)->Prop))=> Prop)) b) as proof of (((eq (((a->Prop)->Prop)->Prop)) b) (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx)))))))))
% Found (((eta_expansion_dep ((a->Prop)->Prop)) (fun (x1:((a->Prop)->Prop))=> Prop)) b) as proof of (((eq (((a->Prop)->Prop)->Prop)) b) (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx)))))))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P (fun (U:((a->Prop)->Prop))=> ((and (forall (Xx:(a->Prop)), ((U Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((U Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(U Xy)))))))))))->(P (fun (x:((a->Prop)->Prop))=> ((and (forall (Xx:(a->Prop)), ((x Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((x Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(x Xy))))))))))))
% Found (eta_expansion_dep000 P) as proof of (P0 (fun (U:((a->Prop)->Prop))=> ((and (forall (Xx:(a->Prop)), ((U Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((U Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(U Xy)))))))))))
% Found ((eta_expansion_dep00 (fun (U:((a->Prop)->Prop))=> ((and (forall (Xx:(a->Prop)), ((U Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((U Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(U Xy))))))))))) P) as proof of (P0 (fun (U:((a->Prop)->Prop))=> ((and (forall (Xx:(a->Prop)), ((U Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((U Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(U Xy)))))))))))
% Found (((eta_expansion_dep0 (fun (x1:((a->Prop)->Prop))=> Prop)) (fun (U:((a->Prop)->Prop))=> ((and (forall (Xx:(a->Prop)), ((U Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((U Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(U Xy))))))))))) P) as proof of (P0 (fun (U:((a->Prop)->Prop))=> ((and (forall (Xx:(a->Prop)), ((U Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((U Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(U Xy)))))))))))
% Found ((((eta_expansion_dep ((a->Prop)->Prop)) (fun (x1:((a->Prop)->Prop))=> Prop)) (fun (U:((a->Prop)->Prop))=> ((and (forall (Xx:(a->Prop)), ((U Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((U Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(U Xy))))))))))) P) as proof of (P0 (fun (U:((a->Prop)->Prop))=> ((and (forall (Xx:(a->Prop)), ((U Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((U Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(U Xy)))))))))))
% Found ((((eta_expansion_dep ((a->Prop)->Prop)) (fun (x1:((a->Prop)->Prop))=> Prop)) (fun (U:((a->Prop)->Prop))=> ((and (forall (Xx:(a->Prop)), ((U Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((U Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(U Xy))))))))))) P) as proof of (P0 (fun (U:((a->Prop)->Prop))=> ((and (forall (Xx:(a->Prop)), ((U Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((U Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(U Xy)))))))))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (U:((a->Prop)->Prop))=> ((and (forall (Xx:(a->Prop)), ((U Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((U Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(U Xy))))))))))):(((eq (((a->Prop)->Prop)->Prop)) (fun (U:((a->Prop)->Prop))=> ((and (forall (Xx:(a->Prop)), ((U Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((U Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(U Xy))))))))))) (fun (x:((a->Prop)->Prop))=> ((and (forall (Xx:(a->Prop)), ((x Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((x Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(x Xy)))))))))))
% Found (eta_expansion_dep00 (fun (U:((a->Prop)->Prop))=> ((and (forall (Xx:(a->Prop)), ((U Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((U Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(U Xy))))))))))) as proof of (((eq (((a->Prop)->Prop)->Prop)) (fun (U:((a->Prop)->Prop))=> ((and (forall (Xx:(a->Prop)), ((U Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((U Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(U Xy))))))))))) b)
% Found ((eta_expansion_dep0 (fun (x1:((a->Prop)->Prop))=> Prop)) (fun (U:((a->Prop)->Prop))=> ((and (forall (Xx:(a->Prop)), ((U Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((U Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(U Xy))))))))))) as proof of (((eq (((a->Prop)->Prop)->Prop)) (fun (U:((a->Prop)->Prop))=> ((and (forall (Xx:(a->Prop)), ((U Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((U Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(U Xy))))))))))) b)
% Found (((eta_expansion_dep ((a->Prop)->Prop)) (fun (x1:((a->Prop)->Prop))=> Prop)) (fun (U:((a->Prop)->Prop))=> ((and (forall (Xx:(a->Prop)), ((U Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((U Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(U Xy))))))))))) as proof of (((eq (((a->Prop)->Prop)->Prop)) (fun (U:((a->Prop)->Prop))=> ((and (forall (Xx:(a->Prop)), ((U Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((U Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(U Xy))))))))))) b)
% Found (((eta_expansion_dep ((a->Prop)->Prop)) (fun (x1:((a->Prop)->Prop))=> Prop)) (fun (U:((a->Prop)->Prop))=> ((and (forall (Xx:(a->Prop)), ((U Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((U Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(U Xy))))))))))) as proof of (((eq (((a->Prop)->Prop)->Prop)) (fun (U:((a->Prop)->Prop))=> ((and (forall (Xx:(a->Prop)), ((U Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((U Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(U Xy))))))))))) b)
% Found (((eta_expansion_dep ((a->Prop)->Prop)) (fun (x1:((a->Prop)->Prop))=> Prop)) (fun (U:((a->Prop)->Prop))=> ((and (forall (Xx:(a->Prop)), ((U Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((U Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(U Xy))))))))))) as proof of (((eq (((a->Prop)->Prop)->Prop)) (fun (U:((a->Prop)->Prop))=> ((and (forall (Xx:(a->Prop)), ((U Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((U Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(U Xy))))))))))) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq (((a->Prop)->Prop)->Prop)) b) (fun (x:((a->Prop)->Prop))=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq (((a->Prop)->Prop)->Prop)) b) (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx)))))))))
% Found ((eta_expansion_dep0 (fun (x1:((a->Prop)->Prop))=> Prop)) b) as proof of (((eq (((a->Prop)->Prop)->Prop)) b) (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx)))))))))
% Found (((eta_expansion_dep ((a->Prop)->Prop)) (fun (x1:((a->Prop)->Prop))=> Prop)) b) as proof of (((eq (((a->Prop)->Prop)->Prop)) b) (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx)))))))))
% Found (((eta_expansion_dep ((a->Prop)->Prop)) (fun (x1:((a->Prop)->Prop))=> Prop)) b) as proof of (((eq (((a->Prop)->Prop)->Prop)) b) (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx)))))))))
% Found (((eta_expansion_dep ((a->Prop)->Prop)) (fun (x1:((a->Prop)->Prop))=> Prop)) b) as proof of (((eq (((a->Prop)->Prop)->Prop)) b) (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx)))))))))
% Found eq_ref00:=(eq_ref0 (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx))))))))):(((eq (((a->Prop)->Prop)->Prop)) (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx))))))))) (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx)))))))))
% Found (eq_ref0 (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx))))))))) as proof of (((eq (((a->Prop)->Prop)->Prop)) (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx))))))))) b)
% Found ((eq_ref (((a->Prop)->Prop)->Prop)) (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx))))))))) as proof of (((eq (((a->Prop)->Prop)->Prop)) (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx))))))))) b)
% Found ((eq_ref (((a->Prop)->Prop)->Prop)) (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx))))))))) as proof of (((eq (((a->Prop)->Prop)->Prop)) (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx))))))))) b)
% Found ((eq_ref (((a->Prop)->Prop)->Prop)) (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx))))))))) as proof of (((eq (((a->Prop)->Prop)->Prop)) (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx))))))))) b)
% Found x0:(P0 b)
% Instantiate: b:=(fun (U:((a->Prop)->Prop))=> ((and (forall (Xx:(a->Prop)), ((U Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((U Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(U Xy)))))))))):(((a->Prop)->Prop)->Prop)
% Found (fun (x0:(P0 b))=> x0) as proof of (P0 (fun (U:((a->Prop)->Prop))=> ((and (forall (Xx:(a->Prop)), ((U Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((U Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(U Xy)))))))))))
% Found (fun (P0:((((a->Prop)->Prop)->Prop)->Prop)) (x0:(P0 b))=> x0) as proof of ((P0 b)->(P0 (fun (U:((a->Prop)->Prop))=> ((and (forall (Xx:(a->Prop)), ((U Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((U Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(U Xy))))))))))))
% Found (fun (P0:((((a->Prop)->Prop)->Prop)->Prop)) (x0:(P0 b))=> x0) as proof of (P b)
% Found x0:(P (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx)))))))))
% Instantiate: b:=(fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx)))))))):(((a->Prop)->Prop)->Prop)
% Found x0 as proof of (P0 b)
% Found eta_expansion000:=(eta_expansion00 (fun (U:((a->Prop)->Prop))=> ((and (forall (Xx:(a->Prop)), ((U Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((U Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(U Xy))))))))))):(((eq (((a->Prop)->Prop)->Prop)) (fun (U:((a->Prop)->Prop))=> ((and (forall (Xx:(a->Prop)), ((U Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((U Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(U Xy))))))))))) (fun (x:((a->Prop)->Prop))=> ((and (forall (Xx:(a->Prop)), ((x Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((x Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(x Xy)))))))))))
% Found (eta_expansion00 (fun (U:((a->Prop)->Prop))=> ((and (forall (Xx:(a->Prop)), ((U Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((U Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(U Xy))))))))))) as proof of (((eq (((a->Prop)->Prop)->Prop)) (fun (U:((a->Prop)->Prop))=> ((and (forall (Xx:(a->Prop)), ((U Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((U Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(U Xy))))))))))) b)
% Found ((eta_expansion0 Prop) (fun (U:((a->Prop)->Prop))=> ((and (forall (Xx:(a->Prop)), ((U Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((U Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(U Xy))))))))))) as proof of (((eq (((a->Prop)->Prop)->Prop)) (fun (U:((a->Prop)->Prop))=> ((and (forall (Xx:(a->Prop)), ((U Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((U Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(U Xy))))))))))) b)
% Found (((eta_expansion ((a->Prop)->Prop)) Prop) (fun (U:((a->Prop)->Prop))=> ((and (forall (Xx:(a->Prop)), ((U Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((U Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(U Xy))))))))))) as proof of (((eq (((a->Prop)->Prop)->Prop)) (fun (U:((a->Prop)->Prop))=> ((and (forall (Xx:(a->Prop)), ((U Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((U Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(U Xy))))))))))) b)
% Found (((eta_expansion ((a->Prop)->Prop)) Prop) (fun (U:((a->Prop)->Prop))=> ((and (forall (Xx:(a->Prop)), ((U Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((U Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(U Xy))))))))))) as proof of (((eq (((a->Prop)->Prop)->Prop)) (fun (U:((a->Prop)->Prop))=> ((and (forall (Xx:(a->Prop)), ((U Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((U Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(U Xy))))))))))) b)
% Found (((eta_expansion ((a->Prop)->Prop)) Prop) (fun (U:((a->Prop)->Prop))=> ((and (forall (Xx:(a->Prop)), ((U Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((U Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(U Xy))))))))))) as proof of (((eq (((a->Prop)->Prop)->Prop)) (fun (U:((a->Prop)->Prop))=> ((and (forall (Xx:(a->Prop)), ((U Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((U Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(U Xy))))))))))) b)
% Found eta_expansion000:=(eta_expansion00 (fun (U:((a->Prop)->Prop))=> ((and (forall (Xx:(a->Prop)), ((U Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((U Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(U Xy))))))))))):(((eq (((a->Prop)->Prop)->Prop)) (fun (U:((a->Prop)->Prop))=> ((and (forall (Xx:(a->Prop)), ((U Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((U Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(U Xy))))))))))) (fun (x:((a->Prop)->Prop))=> ((and (forall (Xx:(a->Prop)), ((x Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((x Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(x Xy)))))))))))
% Found (eta_expansion00 (fun (U:((a->Prop)->Prop))=> ((and (forall (Xx:(a->Prop)), ((U Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((U Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(U Xy))))))))))) as proof of (((eq (((a->Prop)->Prop)->Prop)) (fun (U:((a->Prop)->Prop))=> ((and (forall (Xx:(a->Prop)), ((U Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((U Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(U Xy))))))))))) b0)
% Found ((eta_expansion0 Prop) (fun (U:((a->Prop)->Prop))=> ((and (forall (Xx:(a->Prop)), ((U Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((U Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(U Xy))))))))))) as proof of (((eq (((a->Prop)->Prop)->Prop)) (fun (U:((a->Prop)->Prop))=> ((and (forall (Xx:(a->Prop)), ((U Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((U Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(U Xy))))))))))) b0)
% Found (((eta_expansion ((a->Prop)->Prop)) Prop) (fun (U:((a->Prop)->Prop))=> ((and (forall (Xx:(a->Prop)), ((U Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((U Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(U Xy))))))))))) as proof of (((eq (((a->Prop)->Prop)->Prop)) (fun (U:((a->Prop)->Prop))=> ((and (forall (Xx:(a->Prop)), ((U Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((U Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(U Xy))))))))))) b0)
% Found (((eta_expansion ((a->Prop)->Prop)) Prop) (fun (U:((a->Prop)->Prop))=> ((and (forall (Xx:(a->Prop)), ((U Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((U Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(U Xy))))))))))) as proof of (((eq (((a->Prop)->Prop)->Prop)) (fun (U:((a->Prop)->Prop))=> ((and (forall (Xx:(a->Prop)), ((U Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((U Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(U Xy))))))))))) b0)
% Found (((eta_expansion ((a->Prop)->Prop)) Prop) (fun (U:((a->Prop)->Prop))=> ((and (forall (Xx:(a->Prop)), ((U Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((U Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(U Xy))))))))))) as proof of (((eq (((a->Prop)->Prop)->Prop)) (fun (U:((a->Prop)->Prop))=> ((and (forall (Xx:(a->Prop)), ((U Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((U Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(U Xy))))))))))) b0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b0):(((eq (((a->Prop)->Prop)->Prop)) b0) (fun (x:((a->Prop)->Prop))=> (b0 x)))
% Found (eta_expansion_dep00 b0) as proof of (((eq (((a->Prop)->Prop)->Prop)) b0) (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx)))))))))
% Found ((eta_expansion_dep0 (fun (x1:((a->Prop)->Prop))=> Prop)) b0) as proof of (((eq (((a->Prop)->Prop)->Prop)) b0) (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx)))))))))
% Found (((eta_expansion_dep ((a->Prop)->Prop)) (fun (x1:((a->Prop)->Prop))=> Prop)) b0) as proof of (((eq (((a->Prop)->Prop)->Prop)) b0) (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx)))))))))
% Found (((eta_expansion_dep ((a->Prop)->Prop)) (fun (x1:((a->Prop)->Prop))=> Prop)) b0) as proof of (((eq (((a->Prop)->Prop)->Prop)) b0) (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx)))))))))
% Found (((eta_expansion_dep ((a->Prop)->Prop)) (fun (x1:((a->Prop)->Prop))=> Prop)) b0) as proof of (((eq (((a->Prop)->Prop)->Prop)) b0) (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx)))))))))
% Found x0:(P (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx)))))))))
% Instantiate: f:=(fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx)))))))):(((a->Prop)->Prop)->Prop)
% Found x0 as proof of (P0 f)
% Found x0:(P (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx)))))))))
% Instantiate: f:=(fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx)))))))):(((a->Prop)->Prop)->Prop)
% Found x0 as proof of (P0 f)
% Found x1:(P ((and (forall (Xx:(a->Prop)), ((x0 Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((x0 Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(x0 Xy))))))))))
% Instantiate: b:=((and (forall (Xx:(a->Prop)), ((x0 Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((x0 Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(x0 Xy))))))))):Prop
% Found x1 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) x0) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx)))))))):(((eq Prop) ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) x0) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx)))))))) ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) x0) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx))))))))
% Found (eq_ref0 ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) x0) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx)))))))) as proof of (((eq Prop) ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) x0) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx)))))))) b)
% Found ((eq_ref Prop) ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) x0) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx)))))))) as proof of (((eq Prop) ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) x0) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx)))))))) b)
% Found ((eq_ref Prop) ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) x0) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx)))))))) as proof of (((eq Prop) ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) x0) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx)))))))) b)
% Found ((eq_ref Prop) ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) x0) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx)))))))) as proof of (((eq Prop) ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) x0) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx)))))))) b)
% Found x1:(P ((and (forall (Xx:(a->Prop)), ((x0 Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((x0 Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(x0 Xy))))))))))
% Instantiate: b:=((and (forall (Xx:(a->Prop)), ((x0 Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((x0 Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(x0 Xy))))))))):Prop
% Found x1 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) x0) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx)))))))):(((eq Prop) ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) x0) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx)))))))) ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) x0) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx))))))))
% Found (eq_ref0 ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) x0) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx)))))))) as proof of (((eq Prop) ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) x0) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx)))))))) b)
% Found ((eq_ref Prop) ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) x0) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx)))))))) as proof of (((eq Prop) ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) x0) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx)))))))) b)
% Found ((eq_ref Prop) ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) x0) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx)))))))) as proof of (((eq Prop) ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) x0) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx)))))))) b)
% Found ((eq_ref Prop) ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) x0) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx)))))))) as proof of (((eq Prop) ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) x0) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx)))))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (((a->Prop)->Prop)->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (((a->Prop)->Prop)->Prop)) b) (fun (U:((a->Prop)->Prop))=> ((and (forall (Xx:(a->Prop)), ((U Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((U Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(U Xy)))))))))))
% Found ((eq_ref (((a->Prop)->Prop)->Prop)) b) as proof of (((eq (((a->Prop)->Prop)->Prop)) b) (fun (U:((a->Prop)->Prop))=> ((and (forall (Xx:(a->Prop)), ((U Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((U Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(U Xy)))))))))))
% Found ((eq_ref (((a->Prop)->Prop)->Prop)) b) as proof of (((eq (((a->Prop)->Prop)->Prop)) b) (fun (U:((a->Prop)->Prop))=> ((and (forall (Xx:(a->Prop)), ((U Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((U Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(U Xy)))))))))))
% Found ((eq_ref (((a->Prop)->Prop)->Prop)) b) as proof of (((eq (((a->Prop)->Prop)->Prop)) b) (fun (U:((a->Prop)->Prop))=> ((and (forall (Xx:(a->Prop)), ((U Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((U Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(U Xy)))))))))))
% Found eq_ref00:=(eq_ref0 (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx))))))))):(((eq (((a->Prop)->Prop)->Prop)) (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx))))))))) (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx)))))))))
% Found (eq_ref0 (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx))))))))) as proof of (((eq (((a->Prop)->Prop)->Prop)) (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx))))))))) b)
% Found ((eq_ref (((a->Prop)->Prop)->Prop)) (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx))))))))) as proof of (((eq (((a->Prop)->Prop)->Prop)) (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx))))))))) b)
% Found ((eq_ref (((a->Prop)->Prop)->Prop)) (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx))))))))) as proof of (((eq (((a->Prop)->Prop)->Prop)) (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx))))))))) b)
% Found ((eq_ref (((a->Prop)->Prop)->Prop)) (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx))))))))) as proof of (((eq (((a->Prop)->Prop)->Prop)) (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx))))))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (((a->Prop)->Prop)->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (((a->Prop)->Prop)->Prop)) b) (fun (U:((a->Prop)->Prop))=> ((and (forall (Xx:(a->Prop)), ((U Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((U Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(U Xy)))))))))))
% Found ((eq_ref (((a->Prop)->Prop)->Prop)) b) as proof of (((eq (((a->Prop)->Prop)->Prop)) b) (fun (U:((a->Prop)->Prop))=> ((and (forall (Xx:(a->Prop)), ((U Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((U Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(U Xy)))))))))))
% Found ((eq_ref (((a->Prop)->Prop)->Prop)) b) as proof of (((eq (((a->Prop)->Prop)->Prop)) b) (fun (U:((a->Prop)->Prop))=> ((and (forall (Xx:(a->Prop)), ((U Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((U Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(U Xy)))))))))))
% Found ((eq_ref (((a->Prop)->Prop)->Prop)) b) as proof of (((eq (((a->Prop)->Prop)->Prop)) b) (fun (U:((a->Prop)->Prop))=> ((and (forall (Xx:(a->Prop)), ((U Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((U Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(U Xy)))))))))))
% Found eq_ref00:=(eq_ref0 (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx))))))))):(((eq (((a->Prop)->Prop)->Prop)) (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx))))))))) (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx)))))))))
% Found (eq_ref0 (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx))))))))) as proof of (((eq (((a->Prop)->Prop)->Prop)) (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx))))))))) b)
% Found ((eq_ref (((a->Prop)->Prop)->Prop)) (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx))))))))) as proof of (((eq (((a->Prop)->Prop)->Prop)) (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx))))))))) b)
% Found ((eq_ref (((a->Prop)->Prop)->Prop)) (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx))))))))) as proof of (((eq (((a->Prop)->Prop)->Prop)) (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx))))))))) b)
% Found ((eq_ref (((a->Prop)->Prop)->Prop)) (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx))))))))) as proof of (((eq (((a->Prop)->Prop)->Prop)) (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx))))))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (((a->Prop)->Prop)->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (((a->Prop)->Prop)->Prop)) b) (fun (U:((a->Prop)->Prop))=> ((and (forall (Xx:(a->Prop)), ((U Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((U Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(U Xy)))))))))))
% Found ((eq_ref (((a->Prop)->Prop)->Prop)) b) as proof of (((eq (((a->Prop)->Prop)->Prop)) b) (fun (U:((a->Prop)->Prop))=> ((and (forall (Xx:(a->Prop)), ((U Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((U Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(U Xy)))))))))))
% Found ((eq_ref (((a->Prop)->Prop)->Prop)) b) as proof of (((eq (((a->Prop)->Prop)->Prop)) b) (fun (U:((a->Prop)->Prop))=> ((and (forall (Xx:(a->Prop)), ((U Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((U Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(U Xy)))))))))))
% Found ((eq_ref (((a->Prop)->Prop)->Prop)) b) as proof of (((eq (((a->Prop)->Prop)->Prop)) b) (fun (U:((a->Prop)->Prop))=> ((and (forall (Xx:(a->Prop)), ((U Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((U Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(U Xy)))))))))))
% Found eta_expansion000:=(eta_expansion00 (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx))))))))):(((eq (((a->Prop)->Prop)->Prop)) (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx))))))))) (fun (x:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) x) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx)))))))))
% Found (eta_expansion00 (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx))))))))) as proof of (((eq (((a->Prop)->Prop)->Prop)) (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx))))))))) b)
% Found ((eta_expansion0 Prop) (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx))))))))) as proof of (((eq (((a->Prop)->Prop)->Prop)) (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx))))))))) b)
% Found (((eta_expansion ((a->Prop)->Prop)) Prop) (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx))))))))) as proof of (((eq (((a->Prop)->Prop)->Prop)) (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx))))))))) b)
% Found (((eta_expansion ((a->Prop)->Prop)) Prop) (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx))))))))) as proof of (((eq (((a->Prop)->Prop)->Prop)) (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx))))))))) b)
% Found (((eta_expansion ((a->Prop)->Prop)) Prop) (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx))))))))) as proof of (((eq (((a->Prop)->Prop)->Prop)) (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx))))))))) b)
% Found eq_ref000:=(eq_ref00 P):((P (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx)))))))))->(P (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx))))))))))
% Found (eq_ref00 P) as proof of (P0 (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx)))))))))
% Found ((eq_ref0 (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx))))))))) P) as proof of (P0 (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx)))))))))
% Found (((eq_ref (((a->Prop)->Prop)->Prop)) (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx))))))))) P) as proof of (P0 (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx)))))))))
% Found (((eq_ref (((a->Prop)->Prop)->Prop)) (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx))))))))) P) as proof of (P0 (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx)))))))))
% Found eq_ref00:=(eq_ref0 b):(((eq (((a->Prop)->Prop)->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (((a->Prop)->Prop)->Prop)) b) (fun (U:((a->Prop)->Prop))=> ((and (forall (Xx:(a->Prop)), ((U Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((U Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(U Xy)))))))))))
% Found ((eq_ref (((a->Prop)->Prop)->Prop)) b) as proof of (((eq (((a->Prop)->Prop)->Prop)) b) (fun (U:((a->Prop)->Prop))=> ((and (forall (Xx:(a->Prop)), ((U Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((U Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(U Xy)))))))))))
% Found ((eq_ref (((a->Prop)->Prop)->Prop)) b) as proof of (((eq (((a->Prop)->Prop)->Prop)) b) (fun (U:((a->Prop)->Prop))=> ((and (forall (Xx:(a->Prop)), ((U Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((U Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(U Xy)))))))))))
% Found ((eq_ref (((a->Prop)->Prop)->Prop)) b) as proof of (((eq (((a->Prop)->Prop)->Prop)) b) (fun (U:((a->Prop)->Prop))=> ((and (forall (Xx:(a->Prop)), ((U Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((U Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(U Xy)))))))))))
% Found eq_ref00:=(eq_ref0 (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx))))))))):(((eq (((a->Prop)->Prop)->Prop)) (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx))))))))) (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx)))))))))
% Found (eq_ref0 (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx))))))))) as proof of (((eq (((a->Prop)->Prop)->Prop)) (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx))))))))) b)
% Found ((eq_ref (((a->Prop)->Prop)->Prop)) (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx))))))))) as proof of (((eq (((a->Prop)->Prop)->Prop)) (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx))))))))) b)
% Found ((eq_ref (((a->Prop)->Prop)->Prop)) (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx))))))))) as proof of (((eq (((a->Prop)->Prop)->Prop)) (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx))))))))) b)
% Found ((eq_ref (((a->Prop)->Prop)->Prop)) (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx))))))))) as proof of (((eq (((a->Prop)->Prop)->Prop)) (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx))))))))) b)
% Found eta_expansion0000:=(eta_expansion000 P1):((P1 (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx)))))))))->(P1 (fun (x:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) x) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx))))))))))
% Found (eta_expansion000 P1) as proof of (P2 (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx)))))))))
% Found ((eta_expansion00 (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx))))))))) P1) as proof of (P2 (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx)))))))))
% Found (((eta_expansion0 Prop) (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx))))))))) P1) as proof of (P2 (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx)))))))))
% Found ((((eta_expansion ((a->Prop)->Prop)) Prop) (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx))))))))) P1) as proof of (P2 (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx)))))))))
% Found ((((eta_expansion ((a->Prop)->Prop)) Prop) (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx))))))))) P1) as proof of (P2 (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx)))))))))
% Found eta_expansion0000:=(eta_expansion000 P1):((P1 (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx)))))))))->(P1 (fun (x:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) x) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx))))))))))
% Found (eta_expansion000 P1) as proof of (P2 (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx)))))))))
% Found ((eta_expansion00 (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx))))))))) P1) as proof of (P2 (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx)))))))))
% Found (((eta_expansion0 Prop) (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx))))))))) P1) as proof of (P2 (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx)))))))))
% Found ((((eta_expansion ((a->Prop)->Prop)) Prop) (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx))))))))) P1) as proof of (P2 (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx)))))))))
% Found ((((eta_expansion ((a->Prop)->Prop)) Prop) (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx))))))))) P1) as proof of (P2 (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx)))))))))
% Found eta_expansion0000:=(eta_expansion000 P1):((P1 (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx)))))))))->(P1 (fun (x:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) x) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx))))))))))
% Found (eta_expansion000 P1) as proof of (P2 (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx)))))))))
% Found ((eta_expansion00 (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx))))))))) P1) as proof of (P2 (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx)))))))))
% Found (((eta_expansion0 Prop) (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx))))))))) P1) as proof of (P2 (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx)))))))))
% Found ((((eta_expansion ((a->Prop)->Prop)) Prop) (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx))))))))) P1) as proof of (P2 (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx)))))))))
% Found ((((eta_expansion ((a->Prop)->Prop)) Prop) (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx))))))))) P1) as proof of (P2 (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx)))))))))
% Found eta_expansion0000:=(eta_expansion000 P1):((P1 (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx)))))))))->(P1 (fun (x:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) x) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx))))))))))
% Found (eta_expansion000 P1) as proof of (P2 (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx)))))))))
% Found ((eta_expansion00 (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx))))))))) P1) as proof of (P2 (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx)))))))))
% Found (((eta_expansion0 Prop) (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx))))))))) P1) as proof of (P2 (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx)))))))))
% Found ((((eta_expansion ((a->Prop)->Prop)) Prop) (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx))))))))) P1) as proof of (P2 (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx)))))))))
% Found ((((eta_expansion ((a->Prop)->Prop)) Prop) (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx))))))))) P1) as proof of (P2 (fun (U:((a->Prop)->Prop))=> ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) U) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx)))))))))
% Found x00:(P b)
% Found (fun (x00:(P b))=> x00) as proof of (P b)
% Found (fun (x00:(P b))=> x00) as proof of (P0 b)
% Found eq_ref000:=(eq_ref00 P1):((P1 ((and (forall (Xx:(a->Prop)), ((x0 Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((x0 Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(x0 Xy))))))))))->(P1 ((and (forall (Xx:(a->Prop)), ((x0 Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((x0 Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(x0 Xy)))))))))))
% Found (eq_ref00 P1) as proof of (P2 ((and (forall (Xx:(a->Prop)), ((x0 Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((x0 Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(x0 Xy))))))))))
% Found ((eq_ref0 ((and (forall (Xx:(a->Prop)), ((x0 Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((x0 Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(x0 Xy)))))))))) P1) as proof of (P2 ((and (forall (Xx:(a->Prop)), ((x0 Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((x0 Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(x0 Xy))))))))))
% Found (((eq_ref Prop) ((and (forall (Xx:(a->Prop)), ((x0 Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((x0 Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(x0 Xy)))))))))) P1) as proof of (P2 ((and (forall (Xx:(a->Prop)), ((x0 Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((x0 Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(x0 Xy))))))))))
% Found (((eq_ref Prop) ((and (forall (Xx:(a->Prop)), ((x0 Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((x0 Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(x0 Xy)))))))))) P1) as proof of (P2 ((and (forall (Xx:(a->Prop)), ((x0 Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((x0 Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(x0 Xy))))))))))
% Found eq_ref000:=(eq_ref00 P1):((P1 ((and (forall (Xx:(a->Prop)), ((x0 Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((x0 Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(x0 Xy))))))))))->(P1 ((and (forall (Xx:(a->Prop)), ((x0 Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((x0 Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(x0 Xy)))))))))))
% Found (eq_ref00 P1) as proof of (P2 ((and (forall (Xx:(a->Prop)), ((x0 Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((x0 Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(x0 Xy))))))))))
% Found ((eq_ref0 ((and (forall (Xx:(a->Prop)), ((x0 Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((x0 Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(x0 Xy)))))))))) P1) as proof of (P2 ((and (forall (Xx:(a->Prop)), ((x0 Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((x0 Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(x0 Xy))))))))))
% Found (((eq_ref Prop) ((and (forall (Xx:(a->Prop)), ((x0 Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((x0 Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(x0 Xy)))))))))) P1) as proof of (P2 ((and (forall (Xx:(a->Prop)), ((x0 Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((x0 Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(x0 Xy))))))))))
% Found (((eq_ref Prop) ((and (forall (Xx:(a->Prop)), ((x0 Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((x0 Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(x0 Xy)))))))))) P1) as proof of (P2 ((and (forall (Xx:(a->Prop)), ((x0 Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((x0 Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(x0 Xy))))))))))
% Found eq_ref000:=(eq_ref00 P1):((P1 ((and (forall (Xx:(a->Prop)), ((x0 Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((x0 Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(x0 Xy))))))))))->(P1 ((and (forall (Xx:(a->Prop)), ((x0 Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((x0 Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(x0 Xy)))))))))))
% Found (eq_ref00 P1) as proof of (P2 ((and (forall (Xx:(a->Prop)), ((x0 Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((x0 Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(x0 Xy))))))))))
% Found ((eq_ref0 ((and (forall (Xx:(a->Prop)), ((x0 Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((x0 Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(x0 Xy)))))))))) P1) as proof of (P2 ((and (forall (Xx:(a->Prop)), ((x0 Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((x0 Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(x0 Xy))))))))))
% Found (((eq_ref Prop) ((and (forall (Xx:(a->Prop)), ((x0 Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((x0 Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(x0 Xy)))))))))) P1) as proof of (P2 ((and (forall (Xx:(a->Prop)), ((x0 Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((x0 Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(x0 Xy))))))))))
% Found (((eq_ref Prop) ((and (forall (Xx:(a->Prop)), ((x0 Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((x0 Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(x0 Xy)))))))))) P1) as proof of (P2 ((and (forall (Xx:(a->Prop)), ((x0 Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((x0 Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(x0 Xy))))))))))
% Found eq_ref000:=(eq_ref00 P1):((P1 ((and (forall (Xx:(a->Prop)), ((x0 Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((x0 Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(x0 Xy))))))))))->(P1 ((and (forall (Xx:(a->Prop)), ((x0 Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((x0 Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(x0 Xy)))))))))))
% Found (eq_ref00 P1) as proof of (P2 ((and (forall (Xx:(a->Prop)), ((x0 Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((x0 Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(x0 Xy))))))))))
% Found ((eq_ref0 ((and (forall (Xx:(a->Prop)), ((x0 Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((x0 Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(x0 Xy)))))))))) P1) as proof of (P2 ((and (forall (Xx:(a->Prop)), ((x0 Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((x0 Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(x0 Xy))))))))))
% Found (((eq_ref Prop) ((and (forall (Xx:(a->Prop)), ((x0 Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((x0 Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(x0 Xy)))))))))) P1) as proof of (P2 ((and (forall (Xx:(a->Prop)), ((x0 Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((x0 Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(x0 Xy))))))))))
% Found (((eq_ref Prop) ((and (forall (Xx:(a->Prop)), ((x0 Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((x0 Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(x0 Xy)))))))))) P1) as proof of (P2 ((and (forall (Xx:(a->Prop)), ((x0 Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((x0 Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(x0 Xy))))))))))
% Found eq_ref00:=(eq_ref0 ((and (forall (Xx:(a->Prop)), ((x0 Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((x0 Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(x0 Xy)))))))))):(((eq Prop) ((and (forall (Xx:(a->Prop)), ((x0 Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((x0 Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(x0 Xy)))))))))) ((and (forall (Xx:(a->Prop)), ((x0 Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((x0 Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(x0 Xy))))))))))
% Found (eq_ref0 ((and (forall (Xx:(a->Prop)), ((x0 Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((x0 Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(x0 Xy)))))))))) as proof of (((eq Prop) ((and (forall (Xx:(a->Prop)), ((x0 Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((x0 Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(x0 Xy)))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx:(a->Prop)), ((x0 Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((x0 Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(x0 Xy)))))))))) as proof of (((eq Prop) ((and (forall (Xx:(a->Prop)), ((x0 Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((x0 Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(x0 Xy)))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx:(a->Prop)), ((x0 Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((x0 Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(x0 Xy)))))))))) as proof of (((eq Prop) ((and (forall (Xx:(a->Prop)), ((x0 Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((x0 Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(x0 Xy)))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx:(a->Prop)), ((x0 Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((x0 Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(x0 Xy)))))))))) as proof of (((eq Prop) ((and (forall (Xx:(a->Prop)), ((x0 Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((x0 Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(x0 Xy)))))))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) x0) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) x0) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) x0) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex ((a->Prop)->Prop)) (fun (V:((a->Prop)->Prop))=> ((and ((and (forall (Xx:(a->Prop)), ((V Xx)->(cB Xx)))) (forall (Xx:(a->Prop)), ((V Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cB Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(V Xy)))))))))) (((eq ((a->Prop)->Prop)) x0) (fun (Xx:(a->Prop))=> ((and (V Xx)) (cA Xx))))))))
% Found eq_ref00:=(eq_ref0 ((and (forall (Xx:(a->Prop)), ((x0 Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((x0 Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(x0 Xy)))))))))):(((eq Prop) ((and (forall (Xx:(a->Prop)), ((x0 Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((x0 Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(x0 Xy)))))))))) ((and (forall (Xx:(a->Prop)), ((x0 Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((x0 Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(x0 Xy))))))))))
% Found (eq_ref0 ((and (forall (Xx:(a->Prop)), ((x0 Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((x0 Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(x0 Xy)))))))))) as proof of (((eq Prop) ((and (forall (Xx:(a->Prop)), ((x0 Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((x0 Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(x0 Xy)))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx:(a->Prop)), ((x0 Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((x0 Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(x0 Xy)))))))))) as proof of (((eq Prop) ((and (forall (Xx:(a->Prop)), ((x0 Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((x0 Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(x0 Xy)))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx:(a->Prop)), ((x0 Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((x0 Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(x0 Xy)))))))))) as proof of (((eq Prop) ((and (forall (Xx:(a->Prop)), ((x0 Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((x0 Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(x0 Xy)))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx:(a->Prop)), ((x0 Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((x0 Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->(x0 Xy)))))))))) as proof of (((eq Prop) ((and (forall (Xx:(a->Prop)), ((x0 Xx)->(cA Xx)))) (forall (Xx:(a->Prop)), ((x0 Xx)->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), (
% EOF
%------------------------------------------------------------------------------