%------------------------------------------------------------------------------
% File     : cocATP---0.2.0
% Problem  : SEV021^6 : TPTP v6.1.0. Released v5.5.0.
% Transfm  : none
% Format   : tptp:raw
% Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p

% Computer : n100.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 13:33:35 EDT 2014

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

% Comments : 
%------------------------------------------------------------------------------
%----NO SOLUTION OUTPUT BY SYSTEM
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% % Problem  : SEV021^6 : TPTP v6.1.0. Released v5.5.0.
% % Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n100.star.cs.uiowa.edu
% % Model    : x86_64 x86_64
% % CPU      : Intel(R) Xeon(R) CPU E5-2609 0 @ 2.40GHz
% % Memory   : 32286.75MB
% % OS       : Linux 2.6.32-431.20.3.el6.x86_64
% % CPULimit : 300
% % DateTime : Thu Jul 17 07:32:21 CDT 2014
% % CPUTime  : 300.10 
% Python 2.7.5
% Using paths ['/home/cristobal/cocATP/CASC/TPTP/', '/export/starexec/sandbox/benchmark/', '/export/starexec/sandbox/benchmark/']
% FOF formula (<kernel.Constant object at 0x2b88200>, <kernel.Type object at 0x2b2bc68>) of role type named a_type
% Using role type
% Declaring a:Type
% FOF formula (<kernel.Constant object at 0x2b88b00>, <kernel.DependentProduct object at 0x2b2be60>) of role type named cP
% Using role type
% Declaring cP:((a->Prop)->Prop)
% FOF formula (<kernel.Constant object at 0x2b88200>, <kernel.DependentProduct object at 0x2b2be18>) of role type named cQ
% Using role type
% Declaring cQ:(a->(a->Prop))
% FOF formula (((eq (a->(a->Prop))) cQ) (fun (X:a) (Y:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (cP S)) (S X))) (S Y)))))) of role definition named cQ_def
% A new definition: (((eq (a->(a->Prop))) cQ) (fun (X:a) (Y:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (cP S)) (S X))) (S Y))))))
% Defined: cQ:=(fun (X:a) (Y:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (cP S)) (S X))) (S Y)))))
% FOF formula ((forall (Xq1:(a->Prop)) (Xq2:(a->Prop)), (((and (((eq (a->Prop)) Xq1) Xq2)) (cP Xq1))->(cP Xq2)))->(((and ((and (forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx))))))) (forall (Xx:a) (Xy:a) (Xp:(a->Prop)) (Xq:(a->Prop)), (((and ((and ((and ((and (cP Xp)) (cP Xq))) (Xp Xx))) (Xq Xx))) (Xp Xy))->(Xq Xy))))->(((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy)))))))) cP))) of role conjecture named cTHM262_D_EXT2_pme
% Conjecture to prove = ((forall (Xq1:(a->Prop)) (Xq2:(a->Prop)), (((and (((eq (a->Prop)) Xq1) Xq2)) (cP Xq1))->(cP Xq2)))->(((and ((and (forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx))))))) (forall (Xx:a) (Xy:a) (Xp:(a->Prop)) (Xq:(a->Prop)), (((and ((and ((and ((and (cP Xp)) (cP Xq))) (Xp Xx))) (Xq Xx))) (Xp Xy))->(Xq Xy))))->(((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy)))))))) cP))):Prop
% Parameter a_DUMMY:a.
% We need to prove ['((forall (Xq1:(a->Prop)) (Xq2:(a->Prop)), (((and (((eq (a->Prop)) Xq1) Xq2)) (cP Xq1))->(cP Xq2)))->(((and ((and (forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx))))))) (forall (Xx:a) (Xy:a) (Xp:(a->Prop)) (Xq:(a->Prop)), (((and ((and ((and ((and (cP Xp)) (cP Xq))) (Xp Xx))) (Xq Xx))) (Xp Xy))->(Xq Xy))))->(((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy)))))))) cP)))']
% Parameter a:Type.
% Parameter cP:((a->Prop)->Prop).
% Definition cQ:=(fun (X:a) (Y:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (cP S)) (S X))) (S Y))))):(a->(a->Prop)).
% Trying to prove ((forall (Xq1:(a->Prop)) (Xq2:(a->Prop)), (((and (((eq (a->Prop)) Xq1) Xq2)) (cP Xq1))->(cP Xq2)))->(((and ((and (forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx))))))) (forall (Xx:a) (Xy:a) (Xp:(a->Prop)) (Xq:(a->Prop)), (((and ((and ((and ((and (cP Xp)) (cP Xq))) (Xp Xx))) (Xq Xx))) (Xp Xy))->(Xq Xy))))->(((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy)))))))) cP)))
% Found eq_ref000:=(eq_ref00 P):((P (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy))))))))->(P (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy)))))))))
% Found (eq_ref00 P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy))))))))
% Found ((eq_ref0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy)))))))) P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy))))))))
% Found (((eq_ref ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy)))))))) P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy))))))))
% Found (((eq_ref ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy)))))))) P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy))))))))
% Found eq_ref000:=(eq_ref00 P):((P (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy))))))))->(P (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy)))))))))
% Found (eq_ref00 P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy))))))))
% Found ((eq_ref0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy)))))))) P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy))))))))
% Found (((eq_ref ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy)))))))) P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy))))))))
% Found (((eq_ref ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy)))))))) P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy))))))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy))))))))->(P (fun (x:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (x Xz)))) (forall (Xx:a), ((x Xx)->(forall (Xy:a), ((iff (x Xy)) ((cQ Xx) Xy)))))))))
% Found (eta_expansion_dep000 P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy))))))))
% Found ((eta_expansion_dep00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy)))))))) P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy))))))))
% Found (((eta_expansion_dep0 (fun (x2:(a->Prop))=> Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy)))))))) P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy))))))))
% Found ((((eta_expansion_dep (a->Prop)) (fun (x2:(a->Prop))=> Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy)))))))) P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy))))))))
% Found ((((eta_expansion_dep (a->Prop)) (fun (x2:(a->Prop))=> Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy)))))))) P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy))))))))
% Found eq_ref00:=(eq_ref0 b):(((eq ((a->Prop)->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq ((a->Prop)->Prop)) b) cP)
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) cP)
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) cP)
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) cP)
% Found eta_expansion000:=(eta_expansion00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy)))))))):(((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy)))))))) (fun (x:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (x Xz)))) (forall (Xx:a), ((x Xx)->(forall (Xy:a), ((iff (x Xy)) ((cQ Xx) Xy))))))))
% Found (eta_expansion00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy)))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy)))))))) b)
% Found ((eta_expansion0 Prop) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy)))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy)))))))) b)
% Found (((eta_expansion (a->Prop)) Prop) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy)))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy)))))))) b)
% Found (((eta_expansion (a->Prop)) Prop) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy)))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy)))))))) b)
% Found (((eta_expansion (a->Prop)) Prop) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy)))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy)))))))) b)
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy))))))))->(P (fun (x:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (x Xz)))) (forall (Xx:a), ((x Xx)->(forall (Xy:a), ((iff (x Xy)) ((cQ Xx) Xy)))))))))
% Found (eta_expansion_dep000 P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy))))))))
% Found ((eta_expansion_dep00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy)))))))) P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy))))))))
% Found (((eta_expansion_dep0 (fun (x4:(a->Prop))=> Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy)))))))) P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy))))))))
% Found ((((eta_expansion_dep (a->Prop)) (fun (x4:(a->Prop))=> Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy)))))))) P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy))))))))
% Found ((((eta_expansion_dep (a->Prop)) (fun (x4:(a->Prop))=> Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy)))))))) P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy))))))))
% Found eq_ref00:=(eq_ref0 b):(((eq ((a->Prop)->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq ((a->Prop)->Prop)) b) cP)
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) cP)
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) cP)
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) cP)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy)))))))):(((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy)))))))) (fun (x:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (x Xz)))) (forall (Xx:a), ((x Xx)->(forall (Xy:a), ((iff (x Xy)) ((cQ Xx) Xy))))))))
% Found (eta_expansion_dep00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy)))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy)))))))) b)
% Found ((eta_expansion_dep0 (fun (x4:(a->Prop))=> Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy)))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy)))))))) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x4:(a->Prop))=> Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy)))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy)))))))) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x4:(a->Prop))=> Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy)))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy)))))))) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x4:(a->Prop))=> Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy)))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy)))))))) b)
% Found eta_expansion0000:=(eta_expansion000 P):((P (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy))))))))->(P (fun (x:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (x Xz)))) (forall (Xx:a), ((x Xx)->(forall (Xy:a), ((iff (x Xy)) ((cQ Xx) Xy)))))))))
% Found (eta_expansion000 P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy))))))))
% Found ((eta_expansion00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy)))))))) P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy))))))))
% Found (((eta_expansion0 Prop) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy)))))))) P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy))))))))
% Found ((((eta_expansion (a->Prop)) Prop) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy)))))))) P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy))))))))
% Found ((((eta_expansion (a->Prop)) Prop) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy)))))))) P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy))))))))
% Found eta_expansion0000:=(eta_expansion000 P):((P (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy))))))))->(P (fun (x:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (x Xz)))) (forall (Xx:a), ((x Xx)->(forall (Xy:a), ((iff (x Xy)) ((cQ Xx) Xy)))))))))
% Found (eta_expansion000 P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy))))))))
% Found ((eta_expansion00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy)))))))) P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy))))))))
% Found (((eta_expansion0 Prop) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy)))))))) P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy))))))))
% Found ((((eta_expansion (a->Prop)) Prop) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy)))))))) P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy))))))))
% Found ((((eta_expansion (a->Prop)) Prop) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy)))))))) P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy))))))))
% Found eq_ref00:=(eq_ref0 b):(((eq ((a->Prop)->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq ((a->Prop)->Prop)) b) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy))))))))
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy))))))))
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy))))))))
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy))))))))
% Found eta_expansion000:=(eta_expansion00 cP):(((eq ((a->Prop)->Prop)) cP) (fun (x:(a->Prop))=> (cP x)))
% Found (eta_expansion00 cP) as proof of (((eq ((a->Prop)->Prop)) cP) b)
% Found ((eta_expansion0 Prop) cP) as proof of (((eq ((a->Prop)->Prop)) cP) b)
% Found (((eta_expansion (a->Prop)) Prop) cP) as proof of (((eq ((a->Prop)->Prop)) cP) b)
% Found (((eta_expansion (a->Prop)) Prop) cP) as proof of (((eq ((a->Prop)->Prop)) cP) b)
% Found (((eta_expansion (a->Prop)) Prop) cP) as proof of (((eq ((a->Prop)->Prop)) cP) b)
% Found eq_ref000:=(eq_ref00 P):((P cP)->(P cP))
% Found (eq_ref00 P) as proof of (P0 cP)
% Found ((eq_ref0 cP) P) as proof of (P0 cP)
% Found (((eq_ref ((a->Prop)->Prop)) cP) P) as proof of (P0 cP)
% Found (((eq_ref ((a->Prop)->Prop)) cP) P) as proof of (P0 cP)
% Found eq_ref000:=(eq_ref00 P):((P cP)->(P cP))
% Found (eq_ref00 P) as proof of (P0 cP)
% Found ((eq_ref0 cP) P) as proof of (P0 cP)
% Found (((eq_ref ((a->Prop)->Prop)) cP) P) as proof of (P0 cP)
% Found (((eq_ref ((a->Prop)->Prop)) cP) P) as proof of (P0 cP)
% Found eq_ref000:=(eq_ref00 P):((P ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy)))))))->(P ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy))))))))
% Found (eq_ref00 P) as proof of (P0 ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy)))))))
% Found ((eq_ref0 ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy))))))) P) as proof of (P0 ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy)))))))
% Found (((eq_ref Prop) ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy))))))) P) as proof of (P0 ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy)))))))
% Found (((eq_ref Prop) ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy))))))) P) as proof of (P0 ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy)))))))
% Found eq_ref000:=(eq_ref00 P):((P ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy)))))))->(P ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy))))))))
% Found (eq_ref00 P) as proof of (P0 ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy)))))))
% Found ((eq_ref0 ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy))))))) P) as proof of (P0 ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy)))))))
% Found (((eq_ref Prop) ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy))))))) P) as proof of (P0 ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy)))))))
% Found (((eq_ref Prop) ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy))))))) P) as proof of (P0 ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy)))))))
% Found eq_ref00:=(eq_ref0 ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy))))))):(((eq Prop) ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy))))))) ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy)))))))
% Found (eq_ref0 ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy))))))) as proof of (((eq Prop) ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy))))))) b)
% Found ((eq_ref Prop) ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy))))))) as proof of (((eq Prop) ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy))))))) b)
% Found ((eq_ref Prop) ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy))))))) as proof of (((eq Prop) ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy))))))) b)
% Found ((eq_ref Prop) ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy))))))) as proof of (((eq Prop) ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy))))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (cP x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cP x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cP x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cP x1))
% Found eq_ref00:=(eq_ref0 ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy))))))):(((eq Prop) ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy))))))) ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy)))))))
% Found (eq_ref0 ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy))))))) as proof of (((eq Prop) ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy))))))) b)
% Found ((eq_ref Prop) ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy))))))) as proof of (((eq Prop) ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy))))))) b)
% Found ((eq_ref Prop) ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy))))))) as proof of (((eq Prop) ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy))))))) b)
% Found ((eq_ref Prop) ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy))))))) as proof of (((eq Prop) ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy))))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (cP x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cP x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cP x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cP x1))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy))))))))->(P (fun (x:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (x Xz)))) (forall (Xx:a), ((x Xx)->(forall (Xy:a), ((iff (x Xy)) ((cQ Xx) Xy)))))))))
% Found (eta_expansion_dep000 P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy))))))))
% Found ((eta_expansion_dep00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy)))))))) P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy))))))))
% Found (((eta_expansion_dep0 (fun (x6:(a->Prop))=> Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy)))))))) P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy))))))))
% Found ((((eta_expansion_dep (a->Prop)) (fun (x6:(a->Prop))=> Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy)))))))) P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy))))))))
% Found ((((eta_expansion_dep (a->Prop)) (fun (x6:(a->Prop))=> Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy)))))))) P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy))))))))
% Found eq_ref00:=(eq_ref0 b):(((eq ((a->Prop)->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq ((a->Prop)->Prop)) b) cP)
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) cP)
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) cP)
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) cP)
% Found eq_ref00:=(eq_ref0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy)))))))):(((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy)))))))) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy))))))))
% Found (eq_ref0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy)))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy)))))))) b)
% Found ((eq_ref ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy)))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy)))))))) b)
% Found ((eq_ref ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy)))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy)))))))) b)
% Found ((eq_ref ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy)))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy)))))))) b)
% Found eq_ref000:=(eq_ref00 P):((P (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy))))))))->(P (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy)))))))))
% Found (eq_ref00 P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy))))))))
% Found ((eq_ref0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy)))))))) P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy))))))))
% Found (((eq_ref ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy)))))))) P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy))))))))
% Found (((eq_ref ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy)))))))) P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy))))))))
% Found eq_ref000:=(eq_ref00 P):((P (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy))))))))->(P (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy)))))))))
% Found (eq_ref00 P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy))))))))
% Found ((eq_ref0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy)))))))) P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy))))))))
% Found (((eq_ref ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy)))))))) P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy))))))))
% Found (((eq_ref ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy)))))))) P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy))))))))
% Found eq_ref00:=(eq_ref0 b):(((eq ((a->Prop)->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq ((a->Prop)->Prop)) b) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy))))))))
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy))))))))
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy))))))))
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy))))))))
% Found eq_ref00:=(eq_ref0 cP):(((eq ((a->Prop)->Prop)) cP) cP)
% Found (eq_ref0 cP) as proof of (((eq ((a->Prop)->Prop)) cP) b)
% Found ((eq_ref ((a->Prop)->Prop)) cP) as proof of (((eq ((a->Prop)->Prop)) cP) b)
% Found ((eq_ref ((a->Prop)->Prop)) cP) as proof of (((eq ((a->Prop)->Prop)) cP) b)
% Found ((eq_ref ((a->Prop)->Prop)) cP) as proof of (((eq ((a->Prop)->Prop)) cP) b)
% Found eq_ref000:=(eq_ref00 P):((P cP)->(P cP))
% Found (eq_ref00 P) as proof of (P0 cP)
% Found ((eq_ref0 cP) P) as proof of (P0 cP)
% Found (((eq_ref ((a->Prop)->Prop)) cP) P) as proof of (P0 cP)
% Found (((eq_ref ((a->Prop)->Prop)) cP) P) as proof of (P0 cP)
% Found eq_ref000:=(eq_ref00 P):((P cP)->(P cP))
% Found (eq_ref00 P) as proof of (P0 cP)
% Found ((eq_ref0 cP) P) as proof of (P0 cP)
% Found (((eq_ref ((a->Prop)->Prop)) cP) P) as proof of (P0 cP)
% Found (((eq_ref ((a->Prop)->Prop)) cP) P) as proof of (P0 cP)
% Found eq_ref000:=(eq_ref00 P):((P ((and ((ex a) (fun (Xz:a)=> (x3 Xz)))) (forall (Xx:a), ((x3 Xx)->(forall (Xy:a), ((iff (x3 Xy)) ((cQ Xx) Xy)))))))->(P ((and ((ex a) (fun (Xz:a)=> (x3 Xz)))) (forall (Xx:a), ((x3 Xx)->(forall (Xy:a), ((iff (x3 Xy)) ((cQ Xx) Xy))))))))
% Found (eq_ref00 P) as proof of (P0 ((and ((ex a) (fun (Xz:a)=> (x3 Xz)))) (forall (Xx:a), ((x3 Xx)->(forall (Xy:a), ((iff (x3 Xy)) ((cQ Xx) Xy)))))))
% Found ((eq_ref0 ((and ((ex a) (fun (Xz:a)=> (x3 Xz)))) (forall (Xx:a), ((x3 Xx)->(forall (Xy:a), ((iff (x3 Xy)) ((cQ Xx) Xy))))))) P) as proof of (P0 ((and ((ex a) (fun (Xz:a)=> (x3 Xz)))) (forall (Xx:a), ((x3 Xx)->(forall (Xy:a), ((iff (x3 Xy)) ((cQ Xx) Xy)))))))
% Found (((eq_ref Prop) ((and ((ex a) (fun (Xz:a)=> (x3 Xz)))) (forall (Xx:a), ((x3 Xx)->(forall (Xy:a), ((iff (x3 Xy)) ((cQ Xx) Xy))))))) P) as proof of (P0 ((and ((ex a) (fun (Xz:a)=> (x3 Xz)))) (forall (Xx:a), ((x3 Xx)->(forall (Xy:a), ((iff (x3 Xy)) ((cQ Xx) Xy)))))))
% Found (((eq_ref Prop) ((and ((ex a) (fun (Xz:a)=> (x3 Xz)))) (forall (Xx:a), ((x3 Xx)->(forall (Xy:a), ((iff (x3 Xy)) ((cQ Xx) Xy))))))) P) as proof of (P0 ((and ((ex a) (fun (Xz:a)=> (x3 Xz)))) (forall (Xx:a), ((x3 Xx)->(forall (Xy:a), ((iff (x3 Xy)) ((cQ Xx) Xy)))))))
% Found eq_ref000:=(eq_ref00 P):((P ((and ((ex a) (fun (Xz:a)=> (x3 Xz)))) (forall (Xx:a), ((x3 Xx)->(forall (Xy:a), ((iff (x3 Xy)) ((cQ Xx) Xy)))))))->(P ((and ((ex a) (fun (Xz:a)=> (x3 Xz)))) (forall (Xx:a), ((x3 Xx)->(forall (Xy:a), ((iff (x3 Xy)) ((cQ Xx) Xy))))))))
% Found (eq_ref00 P) as proof of (P0 ((and ((ex a) (fun (Xz:a)=> (x3 Xz)))) (forall (Xx:a), ((x3 Xx)->(forall (Xy:a), ((iff (x3 Xy)) ((cQ Xx) Xy)))))))
% Found ((eq_ref0 ((and ((ex a) (fun (Xz:a)=> (x3 Xz)))) (forall (Xx:a), ((x3 Xx)->(forall (Xy:a), ((iff (x3 Xy)) ((cQ Xx) Xy))))))) P) as proof of (P0 ((and ((ex a) (fun (Xz:a)=> (x3 Xz)))) (forall (Xx:a), ((x3 Xx)->(forall (Xy:a), ((iff (x3 Xy)) ((cQ Xx) Xy)))))))
% Found (((eq_ref Prop) ((and ((ex a) (fun (Xz:a)=> (x3 Xz)))) (forall (Xx:a), ((x3 Xx)->(forall (Xy:a), ((iff (x3 Xy)) ((cQ Xx) Xy))))))) P) as proof of (P0 ((and ((ex a) (fun (Xz:a)=> (x3 Xz)))) (forall (Xx:a), ((x3 Xx)->(forall (Xy:a), ((iff (x3 Xy)) ((cQ Xx) Xy)))))))
% Found (((eq_ref Prop) ((and ((ex a) (fun (Xz:a)=> (x3 Xz)))) (forall (Xx:a), ((x3 Xx)->(forall (Xy:a), ((iff (x3 Xy)) ((cQ Xx) Xy))))))) P) as proof of (P0 ((and ((ex a) (fun (Xz:a)=> (x3 Xz)))) (forall (Xx:a), ((x3 Xx)->(forall (Xy:a), ((iff (x3 Xy)) ((cQ Xx) Xy)))))))
% Found eq_ref00:=(eq_ref0 ((and ((ex a) (fun (Xz:a)=> (x3 Xz)))) (forall (Xx:a), ((x3 Xx)->(forall (Xy:a), ((iff (x3 Xy)) ((cQ Xx) Xy))))))):(((eq Prop) ((and ((ex a) (fun (Xz:a)=> (x3 Xz)))) (forall (Xx:a), ((x3 Xx)->(forall (Xy:a), ((iff (x3 Xy)) ((cQ Xx) Xy))))))) ((and ((ex a) (fun (Xz:a)=> (x3 Xz)))) (forall (Xx:a), ((x3 Xx)->(forall (Xy:a), ((iff (x3 Xy)) ((cQ Xx) Xy)))))))
% Found (eq_ref0 ((and ((ex a) (fun (Xz:a)=> (x3 Xz)))) (forall (Xx:a), ((x3 Xx)->(forall (Xy:a), ((iff (x3 Xy)) ((cQ Xx) Xy))))))) as proof of (((eq Prop) ((and ((ex a) (fun (Xz:a)=> (x3 Xz)))) (forall (Xx:a), ((x3 Xx)->(forall (Xy:a), ((iff (x3 Xy)) ((cQ Xx) Xy))))))) b)
% Found ((eq_ref Prop) ((and ((ex a) (fun (Xz:a)=> (x3 Xz)))) (forall (Xx:a), ((x3 Xx)->(forall (Xy:a), ((iff (x3 Xy)) ((cQ Xx) Xy))))))) as proof of (((eq Prop) ((and ((ex a) (fun (Xz:a)=> (x3 Xz)))) (forall (Xx:a), ((x3 Xx)->(forall (Xy:a), ((iff (x3 Xy)) ((cQ Xx) Xy))))))) b)
% Found ((eq_ref Prop) ((and ((ex a) (fun (Xz:a)=> (x3 Xz)))) (forall (Xx:a), ((x3 Xx)->(forall (Xy:a), ((iff (x3 Xy)) ((cQ Xx) Xy))))))) as proof of (((eq Prop) ((and ((ex a) (fun (Xz:a)=> (x3 Xz)))) (forall (Xx:a), ((x3 Xx)->(forall (Xy:a), ((iff (x3 Xy)) ((cQ Xx) Xy))))))) b)
% Found ((eq_ref Prop) ((and ((ex a) (fun (Xz:a)=> (x3 Xz)))) (forall (Xx:a), ((x3 Xx)->(forall (Xy:a), ((iff (x3 Xy)) ((cQ Xx) Xy))))))) as proof of (((eq Prop) ((and ((ex a) (fun (Xz:a)=> (x3 Xz)))) (forall (Xx:a), ((x3 Xx)->(forall (Xy:a), ((iff (x3 Xy)) ((cQ Xx) Xy))))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (cP x3))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cP x3))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cP x3))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cP x3))
% Found eq_ref00:=(eq_ref0 ((and ((ex a) (fun (Xz:a)=> (x3 Xz)))) (forall (Xx:a), ((x3 Xx)->(forall (Xy:a), ((iff (x3 Xy)) ((cQ Xx) Xy))))))):(((eq Prop) ((and ((ex a) (fun (Xz:a)=> (x3 Xz)))) (forall (Xx:a), ((x3 Xx)->(forall (Xy:a), ((iff (x3 Xy)) ((cQ Xx) Xy))))))) ((and ((ex a) (fun (Xz:a)=> (x3 Xz)))) (forall (Xx:a), ((x3 Xx)->(forall (Xy:a), ((iff (x3 Xy)) ((cQ Xx) Xy)))))))
% Found (eq_ref0 ((and ((ex a) (fun (Xz:a)=> (x3 Xz)))) (forall (Xx:a), ((x3 Xx)->(forall (Xy:a), ((iff (x3 Xy)) ((cQ Xx) Xy))))))) as proof of (((eq Prop) ((and ((ex a) (fun (Xz:a)=> (x3 Xz)))) (forall (Xx:a), ((x3 Xx)->(forall (Xy:a), ((iff (x3 Xy)) ((cQ Xx) Xy))))))) b)
% Found ((eq_ref Prop) ((and ((ex a) (fun (Xz:a)=> (x3 Xz)))) (forall (Xx:a), ((x3 Xx)->(forall (Xy:a), ((iff (x3 Xy)) ((cQ Xx) Xy))))))) as proof of (((eq Prop) ((and ((ex a) (fun (Xz:a)=> (x3 Xz)))) (forall (Xx:a), ((x3 Xx)->(forall (Xy:a), ((iff (x3 Xy)) ((cQ Xx) Xy))))))) b)
% Found ((eq_ref Prop) ((and ((ex a) (fun (Xz:a)=> (x3 Xz)))) (forall (Xx:a), ((x3 Xx)->(forall (Xy:a), ((iff (x3 Xy)) ((cQ Xx) Xy))))))) as proof of (((eq Prop) ((and ((ex a) (fun (Xz:a)=> (x3 Xz)))) (forall (Xx:a), ((x3 Xx)->(forall (Xy:a), ((iff (x3 Xy)) ((cQ Xx) Xy))))))) b)
% Found ((eq_ref Prop) ((and ((ex a) (fun (Xz:a)=> (x3 Xz)))) (forall (Xx:a), ((x3 Xx)->(forall (Xy:a), ((iff (x3 Xy)) ((cQ Xx) Xy))))))) as proof of (((eq Prop) ((and ((ex a) (fun (Xz:a)=> (x3 Xz)))) (forall (Xx:a), ((x3 Xx)->(forall (Xy:a), ((iff (x3 Xy)) ((cQ Xx) Xy))))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (cP x3))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cP x3))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cP x3))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cP x3))
% Found eq_ref00:=(eq_ref0 b):(((eq ((a->Prop)->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq ((a->Prop)->Prop)) b) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy))))))))
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy))))))))
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy))))))))
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy))))))))
% Found eq_ref00:=(eq_ref0 cP):(((eq ((a->Prop)->Prop)) cP) cP)
% Found (eq_ref0 cP) as proof of (((eq ((a->Prop)->Prop)) cP) b)
% Found ((eq_ref ((a->Prop)->Prop)) cP) as proof of (((eq ((a->Prop)->Prop)) cP) b)
% Found ((eq_ref ((a->Prop)->Prop)) cP) as proof of (((eq ((a->Prop)->Prop)) cP) b)
% Found ((eq_ref ((a->Prop)->Prop)) cP) as proof of (((eq ((a->Prop)->Prop)) cP) b)
% Found eq_ref000:=(eq_ref00 P):((P cP)->(P cP))
% Found (eq_ref00 P) as proof of (P0 cP)
% Found ((eq_ref0 cP) P) as proof of (P0 cP)
% Found (((eq_ref ((a->Prop)->Prop)) cP) P) as proof of (P0 cP)
% Found (((eq_ref ((a->Prop)->Prop)) cP) P) as proof of (P0 cP)
% Found eq_ref000:=(eq_ref00 P):((P cP)->(P cP))
% Found (eq_ref00 P) as proof of (P0 cP)
% Found ((eq_ref0 cP) P) as proof of (P0 cP)
% Found (((eq_ref ((a->Prop)->Prop)) cP) P) as proof of (P0 cP)
% Found (((eq_ref ((a->Prop)->Prop)) cP) P) as proof of (P0 cP)
% Found eq_ref000:=(eq_ref00 P):((P (cP x1))->(P (cP x1)))
% Found (eq_ref00 P) as proof of (P0 (cP x1))
% Found ((eq_ref0 (cP x1)) P) as proof of (P0 (cP x1))
% Found (((eq_ref Prop) (cP x1)) P) as proof of (P0 (cP x1))
% Found (((eq_ref Prop) (cP x1)) P) as proof of (P0 (cP x1))
% Found eq_ref000:=(eq_ref00 P):((P (cP x1))->(P (cP x1)))
% Found (eq_ref00 P) as proof of (P0 (cP x1))
% Found ((eq_ref0 (cP x1)) P) as proof of (P0 (cP x1))
% Found (((eq_ref Prop) (cP x1)) P) as proof of (P0 (cP x1))
% Found (((eq_ref Prop) (cP x1)) P) as proof of (P0 (cP x1))
% Found eq_ref00:=(eq_ref0 (cP x1)):(((eq Prop) (cP x1)) (cP x1))
% Found (eq_ref0 (cP x1)) as proof of (((eq Prop) (cP x1)) b)
% Found ((eq_ref Prop) (cP x1)) as proof of (((eq Prop) (cP x1)) b)
% Found ((eq_ref Prop) (cP x1)) as proof of (((eq Prop) (cP x1)) b)
% Found ((eq_ref Prop) (cP x1)) as proof of (((eq Prop) (cP x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy)))))))
% Found eq_ref00:=(eq_ref0 (cP x1)):(((eq Prop) (cP x1)) (cP x1))
% Found (eq_ref0 (cP x1)) as proof of (((eq Prop) (cP x1)) b)
% Found ((eq_ref Prop) (cP x1)) as proof of (((eq Prop) (cP x1)) b)
% Found ((eq_ref Prop) (cP x1)) as proof of (((eq Prop) (cP x1)) b)
% Found ((eq_ref Prop) (cP x1)) as proof of (((eq Prop) (cP x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy)))))))
% Found eq_ref00:=(eq_ref0 (cP x1)):(((eq Prop) (cP x1)) (cP x1))
% Found (eq_ref0 (cP x1)) as proof of (((eq Prop) (cP x1)) b)
% Found ((eq_ref Prop) (cP x1)) as proof of (((eq Prop) (cP x1)) b)
% Found ((eq_ref Prop) (cP x1)) as proof of (((eq Prop) (cP x1)) b)
% Found ((eq_ref Prop) (cP x1)) as proof of (((eq Prop) (cP x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy)))))))
% Found eq_ref00:=(eq_ref0 (cP x1)):(((eq Prop) (cP x1)) (cP x1))
% Found (eq_ref0 (cP x1)) as proof of (((eq Prop) (cP x1)) b)
% Found ((eq_ref Prop) (cP x1)) as proof of (((eq Prop) (cP x1)) b)
% Found ((eq_ref Prop) (cP x1)) as proof of (((eq Prop) (cP x1)) b)
% Found ((eq_ref Prop) (cP x1)) as proof of (((eq Prop) (cP x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy)))))))
% Found eq_ref000:=(eq_ref00 P):((P ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy)))))))->(P ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy))))))))
% Found (eq_ref00 P) as proof of (P0 ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy)))))))
% Found ((eq_ref0 ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy))))))) P) as proof of (P0 ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy)))))))
% Found (((eq_ref Prop) ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy))))))) P) as proof of (P0 ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy)))))))
% Found (((eq_ref Prop) ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy))))))) P) as proof of (P0 ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy)))))))
% Found eq_ref000:=(eq_ref00 P):((P ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy)))))))->(P ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy))))))))
% Found (eq_ref00 P) as proof of (P0 ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy)))))))
% Found ((eq_ref0 ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy))))))) P) as proof of (P0 ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy)))))))
% Found (((eq_ref Prop) ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy))))))) P) as proof of (P0 ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy)))))))
% Found (((eq_ref Prop) ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy))))))) P) as proof of (P0 ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy)))))))
% Found eq_ref00:=(eq_ref0 ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy))))))):(((eq Prop) ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy))))))) ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy)))))))
% Found (eq_ref0 ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy))))))) as proof of (((eq Prop) ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy))))))) b)
% Found ((eq_ref Prop) ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy))))))) as proof of (((eq Prop) ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy))))))) b)
% Found ((eq_ref Prop) ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy))))))) as proof of (((eq Prop) ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy))))))) b)
% Found ((eq_ref Prop) ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy))))))) as proof of (((eq Prop) ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy))))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (cP x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cP x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cP x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cP x1))
% Found eq_ref00:=(eq_ref0 ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy))))))):(((eq Prop) ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy))))))) ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy)))))))
% Found (eq_ref0 ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy))))))) as proof of (((eq Prop) ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy))))))) b)
% Found ((eq_ref Prop) ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy))))))) as proof of (((eq Prop) ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy))))))) b)
% Found ((eq_ref Prop) ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy))))))) as proof of (((eq Prop) ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy))))))) b)
% Found ((eq_ref Prop) ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy))))))) as proof of (((eq Prop) ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy))))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (cP x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cP x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cP x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cP x1))
% Found eq_ref00:=(eq_ref0 b):(((eq ((a->Prop)->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq ((a->Prop)->Prop)) b) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy))))))))
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy))))))))
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy))))))))
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy))))))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 cP):(((eq ((a->Prop)->Prop)) cP) (fun (x:(a->Prop))=> (cP x)))
% Found (eta_expansion_dep00 cP) as proof of (((eq ((a->Prop)->Prop)) cP) b)
% Found ((eta_expansion_dep0 (fun (x6:(a->Prop))=> Prop)) cP) as proof of (((eq ((a->Prop)->Prop)) cP) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x6:(a->Prop))=> Prop)) cP) as proof of (((eq ((a->Prop)->Prop)) cP) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x6:(a->Prop))=> Prop)) cP) as proof of (((eq ((a->Prop)->Prop)) cP) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x6:(a->Prop))=> Prop)) cP) as proof of (((eq ((a->Prop)->Prop)) cP) b)
% Found eq_ref000:=(eq_ref00 P):((P cP)->(P cP))
% Found (eq_ref00 P) as proof of (P0 cP)
% Found ((eq_ref0 cP) P) as proof of (P0 cP)
% Found (((eq_ref ((a->Prop)->Prop)) cP) P) as proof of (P0 cP)
% Found (((eq_ref ((a->Prop)->Prop)) cP) P) as proof of (P0 cP)
% Found eq_ref000:=(eq_ref00 P):((P cP)->(P cP))
% Found (eq_ref00 P) as proof of (P0 cP)
% Found ((eq_ref0 cP) P) as proof of (P0 cP)
% Found (((eq_ref ((a->Prop)->Prop)) cP) P) as proof of (P0 cP)
% Found (((eq_ref ((a->Prop)->Prop)) cP) P) as proof of (P0 cP)
% Found eq_ref00:=(eq_ref0 b):(((eq ((a->Prop)->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq ((a->Prop)->Prop)) b) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy))))))))
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy))))))))
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy))))))))
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy))))))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 cP):(((eq ((a->Prop)->Prop)) cP) (fun (x:(a->Prop))=> (cP x)))
% Found (eta_expansion_dep00 cP) as proof of (((eq ((a->Prop)->Prop)) cP) b)
% Found ((eta_expansion_dep0 (fun (x6:(a->Prop))=> Prop)) cP) as proof of (((eq ((a->Prop)->Prop)) cP) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x6:(a->Prop))=> Prop)) cP) as proof of (((eq ((a->Prop)->Prop)) cP) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x6:(a->Prop))=> Prop)) cP) as proof of (((eq ((a->Prop)->Prop)) cP) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x6:(a->Prop))=> Prop)) cP) as proof of (((eq ((a->Prop)->Prop)) cP) b)
% Found eq_ref000:=(eq_ref00 P):((P ((and ((ex a) (fun (Xz:a)=> (x5 Xz)))) (forall (Xx:a), ((x5 Xx)->(forall (Xy:a), ((iff (x5 Xy)) ((cQ Xx) Xy)))))))->(P ((and ((ex a) (fun (Xz:a)=> (x5 Xz)))) (forall (Xx:a), ((x5 Xx)->(forall (Xy:a), ((iff (x5 Xy)) ((cQ Xx) Xy))))))))
% Found (eq_ref00 P) as proof of (P0 ((and ((ex a) (fun (Xz:a)=> (x5 Xz)))) (forall (Xx:a), ((x5 Xx)->(forall (Xy:a), ((iff (x5 Xy)) ((cQ Xx) Xy)))))))
% Found ((eq_ref0 ((and ((ex a) (fun (Xz:a)=> (x5 Xz)))) (forall (Xx:a), ((x5 Xx)->(forall (Xy:a), ((iff (x5 Xy)) ((cQ Xx) Xy))))))) P) as proof of (P0 ((and ((ex a) (fun (Xz:a)=> (x5 Xz)))) (forall (Xx:a), ((x5 Xx)->(forall (Xy:a), ((iff (x5 Xy)) ((cQ Xx) Xy)))))))
% Found (((eq_ref Prop) ((and ((ex a) (fun (Xz:a)=> (x5 Xz)))) (forall (Xx:a), ((x5 Xx)->(forall (Xy:a), ((iff (x5 Xy)) ((cQ Xx) Xy))))))) P) as proof of (P0 ((and ((ex a) (fun (Xz:a)=> (x5 Xz)))) (forall (Xx:a), ((x5 Xx)->(forall (Xy:a), ((iff (x5 Xy)) ((cQ Xx) Xy)))))))
% Found (((eq_ref Prop) ((and ((ex a) (fun (Xz:a)=> (x5 Xz)))) (forall (Xx:a), ((x5 Xx)->(forall (Xy:a), ((iff (x5 Xy)) ((cQ Xx) Xy))))))) P) as proof of (P0 ((and ((ex a) (fun (Xz:a)=> (x5 Xz)))) (forall (Xx:a), ((x5 Xx)->(forall (Xy:a), ((iff (x5 Xy)) ((cQ Xx) Xy)))))))
% Found eq_ref000:=(eq_ref00 P):((P ((and ((ex a) (fun (Xz:a)=> (x5 Xz)))) (forall (Xx:a), ((x5 Xx)->(forall (Xy:a), ((iff (x5 Xy)) ((cQ Xx) Xy)))))))->(P ((and ((ex a) (fun (Xz:a)=> (x5 Xz)))) (forall (Xx:a), ((x5 Xx)->(forall (Xy:a), ((iff (x5 Xy)) ((cQ Xx) Xy))))))))
% Found (eq_ref00 P) as proof of (P0 ((and ((ex a) (fun (Xz:a)=> (x5 Xz)))) (forall (Xx:a), ((x5 Xx)->(forall (Xy:a), ((iff (x5 Xy)) ((cQ Xx) Xy)))))))
% Found ((eq_ref0 ((and ((ex a) (fun (Xz:a)=> (x5 Xz)))) (forall (Xx:a), ((x5 Xx)->(forall (Xy:a), ((iff (x5 Xy)) ((cQ Xx) Xy))))))) P) as proof of (P0 ((and ((ex a) (fun (Xz:a)=> (x5 Xz)))) (forall (Xx:a), ((x5 Xx)->(forall (Xy:a), ((iff (x5 Xy)) ((cQ Xx) Xy)))))))
% Found (((eq_ref Prop) ((and ((ex a) (fun (Xz:a)=> (x5 Xz)))) (forall (Xx:a), ((x5 Xx)->(forall (Xy:a), ((iff (x5 Xy)) ((cQ Xx) Xy))))))) P) as proof of (P0 ((and ((ex a) (fun (Xz:a)=> (x5 Xz)))) (forall (Xx:a), ((x5 Xx)->(forall (Xy:a), ((iff (x5 Xy)) ((cQ Xx) Xy)))))))
% Found (((eq_ref Prop) ((and ((ex a) (fun (Xz:a)=> (x5 Xz)))) (forall (Xx:a), ((x5 Xx)->(forall (Xy:a), ((iff (x5 Xy)) ((cQ Xx) Xy))))))) P) as proof of (P0 ((and ((ex a) (fun (Xz:a)=> (x5 Xz)))) (forall (Xx:a), ((x5 Xx)->(forall (Xy:a), ((iff (x5 Xy)) ((cQ Xx) Xy)))))))
% Found eq_ref00:=(eq_ref0 b):(((eq ((a->Prop)->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq ((a->Prop)->Prop)) b) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy))))))))
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy))))))))
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy))))))))
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy))))))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 cP):(((eq ((a->Prop)->Prop)) cP) (fun (x:(a->Prop))=> (cP x)))
% Found (eta_expansion_dep00 cP) as proof of (((eq ((a->Prop)->Prop)) cP) b)
% Found ((eta_expansion_dep0 (fun (x6:(a->Prop))=> Prop)) cP) as proof of (((eq ((a->Prop)->Prop)) cP) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x6:(a->Prop))=> Prop)) cP) as proof of (((eq ((a->Prop)->Prop)) cP) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x6:(a->Prop))=> Prop)) cP) as proof of (((eq ((a->Prop)->Prop)) cP) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x6:(a->Prop))=> Prop)) cP) as proof of (((eq ((a->Prop)->Prop)) cP) b)
% Found eq_ref00:=(eq_ref0 ((and ((ex a) (fun (Xz:a)=> (x5 Xz)))) (forall (Xx:a), ((x5 Xx)->(forall (Xy:a), ((iff (x5 Xy)) ((cQ Xx) Xy))))))):(((eq Prop) ((and ((ex a) (fun (Xz:a)=> (x5 Xz)))) (forall (Xx:a), ((x5 Xx)->(forall (Xy:a), ((iff (x5 Xy)) ((cQ Xx) Xy))))))) ((and ((ex a) (fun (Xz:a)=> (x5 Xz)))) (forall (Xx:a), ((x5 Xx)->(forall (Xy:a), ((iff (x5 Xy)) ((cQ Xx) Xy)))))))
% Found (eq_ref0 ((and ((ex a) (fun (Xz:a)=> (x5 Xz)))) (forall (Xx:a), ((x5 Xx)->(forall (Xy:a), ((iff (x5 Xy)) ((cQ Xx) Xy))))))) as proof of (((eq Prop) ((and ((ex a) (fun (Xz:a)=> (x5 Xz)))) (forall (Xx:a), ((x5 Xx)->(forall (Xy:a), ((iff (x5 Xy)) ((cQ Xx) Xy))))))) b)
% Found ((eq_ref Prop) ((and ((ex a) (fun (Xz:a)=> (x5 Xz)))) (forall (Xx:a), ((x5 Xx)->(forall (Xy:a), ((iff (x5 Xy)) ((cQ Xx) Xy))))))) as proof of (((eq Prop) ((and ((ex a) (fun (Xz:a)=> (x5 Xz)))) (forall (Xx:a), ((x5 Xx)->(forall (Xy:a), ((iff (x5 Xy)) ((cQ Xx) Xy))))))) b)
% Found ((eq_ref Prop) ((and ((ex a) (fun (Xz:a)=> (x5 Xz)))) (forall (Xx:a), ((x5 Xx)->(forall (Xy:a), ((iff (x5 Xy)) ((cQ Xx) Xy))))))) as proof of (((eq Prop) ((and ((ex a) (fun (Xz:a)=> (x5 Xz)))) (forall (Xx:a), ((x5 Xx)->(forall (Xy:a), ((iff (x5 Xy)) ((cQ Xx) Xy))))))) b)
% Found ((eq_ref Prop) ((and ((ex a) (fun (Xz:a)=> (x5 Xz)))) (forall (Xx:a), ((x5 Xx)->(forall (Xy:a), ((iff (x5 Xy)) ((cQ Xx) Xy))))))) as proof of (((eq Prop) ((and ((ex a) (fun (Xz:a)=> (x5 Xz)))) (forall (Xx:a), ((x5 Xx)->(forall (Xy:a), ((iff (x5 Xy)) ((cQ Xx) Xy))))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (cP x5))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cP x5))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cP x5))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cP x5))
% Found eq_ref00:=(eq_ref0 ((and ((ex a) (fun (Xz:a)=> (x5 Xz)))) (forall (Xx:a), ((x5 Xx)->(forall (Xy:a), ((iff (x5 Xy)) ((cQ Xx) Xy))))))):(((eq Prop) ((and ((ex a) (fun (Xz:a)=> (x5 Xz)))) (forall (Xx:a), ((x5 Xx)->(forall (Xy:a), ((iff (x5 Xy)) ((cQ Xx) Xy))))))) ((and ((ex a) (fun (Xz:a)=> (x5 Xz)))) (forall (Xx:a), ((x5 Xx)->(forall (Xy:a), ((iff (x5 Xy)) ((cQ Xx) Xy)))))))
% Found (eq_ref0 ((and ((ex a) (fun (Xz:a)=> (x5 Xz)))) (forall (Xx:a), ((x5 Xx)->(forall (Xy:a), ((iff (x5 Xy)) ((cQ Xx) Xy))))))) as proof of (((eq Prop) ((and ((ex a) (fun (Xz:a)=> (x5 Xz)))) (forall (Xx:a), ((x5 Xx)->(forall (Xy:a), ((iff (x5 Xy)) ((cQ Xx) Xy))))))) b)
% Found ((eq_ref Prop) ((and ((ex a) (fun (Xz:a)=> (x5 Xz)))) (forall (Xx:a), ((x5 Xx)->(forall (Xy:a), ((iff (x5 Xy)) ((cQ Xx) Xy))))))) as proof of (((eq Prop) ((and ((ex a) (fun (Xz:a)=> (x5 Xz)))) (forall (Xx:a), ((x5 Xx)->(forall (Xy:a), ((iff (x5 Xy)) ((cQ Xx) Xy))))))) b)
% Found ((eq_ref Prop) ((and ((ex a) (fun (Xz:a)=> (x5 Xz)))) (forall (Xx:a), ((x5 Xx)->(forall (Xy:a), ((iff (x5 Xy)) ((cQ Xx) Xy))))))) as proof of (((eq Prop) ((and ((ex a) (fun (Xz:a)=> (x5 Xz)))) (forall (Xx:a), ((x5 Xx)->(forall (Xy:a), ((iff (x5 Xy)) ((cQ Xx) Xy))))))) b)
% Found ((eq_ref Prop) ((and ((ex a) (fun (Xz:a)=> (x5 Xz)))) (forall (Xx:a), ((x5 Xx)->(forall (Xy:a), ((iff (x5 Xy)) ((cQ Xx) Xy))))))) as proof of (((eq Prop) ((and ((ex a) (fun (Xz:a)=> (x5 Xz)))) (forall (Xx:a), ((x5 Xx)->(forall (Xy:a), ((iff (x5 Xy)) ((cQ Xx) Xy))))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (cP x5))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cP x5))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cP x5))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cP x5))
% Found x1:(P (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy))))))))
% Instantiate: b:=(fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy))))))):((a->Prop)->Prop)
% Found x1 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 cP):(((eq ((a->Prop)->Prop)) cP) cP)
% Found (eq_ref0 cP) as proof of (((eq ((a->Prop)->Prop)) cP) b)
% Found ((eq_ref ((a->Prop)->Prop)) cP) as proof of (((eq ((a->Prop)->Prop)) cP) b)
% Found ((eq_ref ((a->Prop)->Prop)) cP) as proof of (((eq ((a->Prop)->Prop)) cP) b)
% Found ((eq_ref ((a->Prop)->Prop)) cP) as proof of (((eq ((a->Prop)->Prop)) cP) b)
% Found eq_ref000:=(eq_ref00 P):((P cP)->(P cP))
% Found (eq_ref00 P) as proof of (P0 cP)
% Found ((eq_ref0 cP) P) as proof of (P0 cP)
% Found (((eq_ref ((a->Prop)->Prop)) cP) P) as proof of (P0 cP)
% Found (((eq_ref ((a->Prop)->Prop)) cP) P) as proof of (P0 cP)
% Found eq_ref000:=(eq_ref00 P):((P cP)->(P cP))
% Found (eq_ref00 P) as proof of (P0 cP)
% Found ((eq_ref0 cP) P) as proof of (P0 cP)
% Found (((eq_ref ((a->Prop)->Prop)) cP) P) as proof of (P0 cP)
% Found (((eq_ref ((a->Prop)->Prop)) cP) P) as proof of (P0 cP)
% Found eq_ref000:=(eq_ref00 P):((P cP)->(P cP))
% Found (eq_ref00 P) as proof of (P0 cP)
% Found ((eq_ref0 cP) P) as proof of (P0 cP)
% Found (((eq_ref ((a->Prop)->Prop)) cP) P) as proof of (P0 cP)
% Found (((eq_ref ((a->Prop)->Prop)) cP) P) as proof of (P0 cP)
% Found eq_ref000:=(eq_ref00 P):((P cP)->(P cP))
% Found (eq_ref00 P) as proof of (P0 cP)
% Found ((eq_ref0 cP) P) as proof of (P0 cP)
% Found (((eq_ref ((a->Prop)->Prop)) cP) P) as proof of (P0 cP)
% Found (((eq_ref ((a->Prop)->Prop)) cP) P) as proof of (P0 cP)
% Found eq_ref000:=(eq_ref00 P):((P (cP x3))->(P (cP x3)))
% Found (eq_ref00 P) as proof of (P0 (cP x3))
% Found ((eq_ref0 (cP x3)) P) as proof of (P0 (cP x3))
% Found (((eq_ref Prop) (cP x3)) P) as proof of (P0 (cP x3))
% Found (((eq_ref Prop) (cP x3)) P) as proof of (P0 (cP x3))
% Found eq_ref000:=(eq_ref00 P):((P (cP x3))->(P (cP x3)))
% Found (eq_ref00 P) as proof of (P0 (cP x3))
% Found ((eq_ref0 (cP x3)) P) as proof of (P0 (cP x3))
% Found (((eq_ref Prop) (cP x3)) P) as proof of (P0 (cP x3))
% Found (((eq_ref Prop) (cP x3)) P) as proof of (P0 (cP x3))
% Found x1:(P (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy))))))))
% Instantiate: f:=(fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy))))))):((a->Prop)->Prop)
% Found x1 as proof of (P0 f)
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (cP x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (cP x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (cP x2))
% Found (fun (x2:(a->Prop))=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (cP x2))
% Found (fun (x2:(a->Prop))=> ((eq_ref Prop) (f x2))) as proof of (forall (x:(a->Prop)), (((eq Prop) (f x)) (cP x)))
% Found x1:(P (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy))))))))
% Instantiate: f:=(fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy))))))):((a->Prop)->Prop)
% Found x1 as proof of (P0 f)
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (cP x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (cP x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (cP x2))
% Found (fun (x2:(a->Prop))=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (cP x2))
% Found (fun (x2:(a->Prop))=> ((eq_ref Prop) (f x2))) as proof of (forall (x:(a->Prop)), (((eq Prop) (f x)) (cP x)))
% Found eq_ref00:=(eq_ref0 (cP x3)):(((eq Prop) (cP x3)) (cP x3))
% Found (eq_ref0 (cP x3)) as proof of (((eq Prop) (cP x3)) b)
% Found ((eq_ref Prop) (cP x3)) as proof of (((eq Prop) (cP x3)) b)
% Found ((eq_ref Prop) (cP x3)) as proof of (((eq Prop) (cP x3)) b)
% Found ((eq_ref Prop) (cP x3)) as proof of (((eq Prop) (cP x3)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((ex a) (fun (Xz:a)=> (x3 Xz)))) (forall (Xx:a), ((x3 Xx)->(forall (Xy:a), ((iff (x3 Xy)) ((cQ Xx) Xy)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((ex a) (fun (Xz:a)=> (x3 Xz)))) (forall (Xx:a), ((x3 Xx)->(forall (Xy:a), ((iff (x3 Xy)) ((cQ Xx) Xy)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((ex a) (fun (Xz:a)=> (x3 Xz)))) (forall (Xx:a), ((x3 Xx)->(forall (Xy:a), ((iff (x3 Xy)) ((cQ Xx) Xy)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((ex a) (fun (Xz:a)=> (x3 Xz)))) (forall (Xx:a), ((x3 Xx)->(forall (Xy:a), ((iff (x3 Xy)) ((cQ Xx) Xy)))))))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((ex a) (fun (Xz:a)=> (x3 Xz)))) (forall (Xx:a), ((x3 Xx)->(forall (Xy:a), ((iff (x3 Xy)) ((cQ Xx) Xy)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((ex a) (fun (Xz:a)=> (x3 Xz)))) (forall (Xx:a), ((x3 Xx)->(forall (Xy:a), ((iff (x3 Xy)) ((cQ Xx) Xy)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((ex a) (fun (Xz:a)=> (x3 Xz)))) (forall (Xx:a), ((x3 Xx)->(forall (Xy:a), ((iff (x3 Xy)) ((cQ Xx) Xy)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((ex a) (fun (Xz:a)=> (x3 Xz)))) (forall (Xx:a), ((x3 Xx)->(forall (Xy:a), ((iff (x3 Xy)) ((cQ Xx) Xy)))))))
% Found eq_ref00:=(eq_ref0 (cP x3)):(((eq Prop) (cP x3)) (cP x3))
% Found (eq_ref0 (cP x3)) as proof of (((eq Prop) (cP x3)) b)
% Found ((eq_ref Prop) (cP x3)) as proof of (((eq Prop) (cP x3)) b)
% Found ((eq_ref Prop) (cP x3)) as proof of (((eq Prop) (cP x3)) b)
% Found ((eq_ref Prop) (cP x3)) as proof of (((eq Prop) (cP x3)) b)
% Found eq_ref00:=(eq_ref0 (cP x3)):(((eq Prop) (cP x3)) (cP x3))
% Found (eq_ref0 (cP x3)) as proof of (((eq Prop) (cP x3)) b)
% Found ((eq_ref Prop) (cP x3)) as proof of (((eq Prop) (cP x3)) b)
% Found ((eq_ref Prop) (cP x3)) as proof of (((eq Prop) (cP x3)) b)
% Found ((eq_ref Prop) (cP x3)) as proof of (((eq Prop) (cP x3)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((ex a) (fun (Xz:a)=> (x3 Xz)))) (forall (Xx:a), ((x3 Xx)->(forall (Xy:a), ((iff (x3 Xy)) ((cQ Xx) Xy)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((ex a) (fun (Xz:a)=> (x3 Xz)))) (forall (Xx:a), ((x3 Xx)->(forall (Xy:a), ((iff (x3 Xy)) ((cQ Xx) Xy)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((ex a) (fun (Xz:a)=> (x3 Xz)))) (forall (Xx:a), ((x3 Xx)->(forall (Xy:a), ((iff (x3 Xy)) ((cQ Xx) Xy)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((ex a) (fun (Xz:a)=> (x3 Xz)))) (forall (Xx:a), ((x3 Xx)->(forall (Xy:a), ((iff (x3 Xy)) ((cQ Xx) Xy)))))))
% Found eq_ref00:=(eq_ref0 (cP x3)):(((eq Prop) (cP x3)) (cP x3))
% Found (eq_ref0 (cP x3)) as proof of (((eq Prop) (cP x3)) b)
% Found ((eq_ref Prop) (cP x3)) as proof of (((eq Prop) (cP x3)) b)
% Found ((eq_ref Prop) (cP x3)) as proof of (((eq Prop) (cP x3)) b)
% Found ((eq_ref Prop) (cP x3)) as proof of (((eq Prop) (cP x3)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((ex a) (fun (Xz:a)=> (x3 Xz)))) (forall (Xx:a), ((x3 Xx)->(forall (Xy:a), ((iff (x3 Xy)) ((cQ Xx) Xy)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((ex a) (fun (Xz:a)=> (x3 Xz)))) (forall (Xx:a), ((x3 Xx)->(forall (Xy:a), ((iff (x3 Xy)) ((cQ Xx) Xy)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((ex a) (fun (Xz:a)=> (x3 Xz)))) (forall (Xx:a), ((x3 Xx)->(forall (Xy:a), ((iff (x3 Xy)) ((cQ Xx) Xy)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((ex a) (fun (Xz:a)=> (x3 Xz)))) (forall (Xx:a), ((x3 Xx)->(forall (Xy:a), ((iff (x3 Xy)) ((cQ Xx) Xy)))))))
% Found eq_ref000:=(eq_ref00 P):((P ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy)))))))->(P ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy))))))))
% Found (eq_ref00 P) as proof of (P0 ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy)))))))
% Found ((eq_ref0 ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy))))))) P) as proof of (P0 ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy)))))))
% Found (((eq_ref Prop) ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy))))))) P) as proof of (P0 ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy)))))))
% Found (((eq_ref Prop) ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy))))))) P) as proof of (P0 ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy)))))))
% Found eq_ref000:=(eq_ref00 P):((P ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy)))))))->(P ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy))))))))
% Found (eq_ref00 P) as proof of (P0 ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy)))))))
% Found ((eq_ref0 ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy))))))) P) as proof of (P0 ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy)))))))
% Found (((eq_ref Prop) ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy))))))) P) as proof of (P0 ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy)))))))
% Found (((eq_ref Prop) ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy))))))) P) as proof of (P0 ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy)))))))
% Found eq_ref000:=(eq_ref00 P):((P ((and ((ex a) (fun (Xz:a)=> (x3 Xz)))) (forall (Xx:a), ((x3 Xx)->(forall (Xy:a), ((iff (x3 Xy)) ((cQ Xx) Xy)))))))->(P ((and ((ex a) (fun (Xz:a)=> (x3 Xz)))) (forall (Xx:a), ((x3 Xx)->(forall (Xy:a), ((iff (x3 Xy)) ((cQ Xx) Xy))))))))
% Found (eq_ref00 P) as proof of (P0 ((and ((ex a) (fun (Xz:a)=> (x3 Xz)))) (forall (Xx:a), ((x3 Xx)->(forall (Xy:a), ((iff (x3 Xy)) ((cQ Xx) Xy)))))))
% Found ((eq_ref0 ((and ((ex a) (fun (Xz:a)=> (x3 Xz)))) (forall (Xx:a), ((x3 Xx)->(forall (Xy:a), ((iff (x3 Xy)) ((cQ Xx) Xy))))))) P) as proof of (P0 ((and ((ex a) (fun (Xz:a)=> (x3 Xz)))) (forall (Xx:a), ((x3 Xx)->(forall (Xy:a), ((iff (x3 Xy)) ((cQ Xx) Xy)))))))
% Found (((eq_ref Prop) ((and ((ex a) (fun (Xz:a)=> (x3 Xz)))) (forall (Xx:a), ((x3 Xx)->(forall (Xy:a), ((iff (x3 Xy)) ((cQ Xx) Xy))))))) P) as proof of (P0 ((and ((ex a) (fun (Xz:a)=> (x3 Xz)))) (forall (Xx:a), ((x3 Xx)->(forall (Xy:a), ((iff (x3 Xy)) ((cQ Xx) Xy)))))))
% Found (((eq_ref Prop) ((and ((ex a) (fun (Xz:a)=> (x3 Xz)))) (forall (Xx:a), ((x3 Xx)->(forall (Xy:a), ((iff (x3 Xy)) ((cQ Xx) Xy))))))) P) as proof of (P0 ((and ((ex a) (fun (Xz:a)=> (x3 Xz)))) (forall (Xx:a), ((x3 Xx)->(forall (Xy:a), ((iff (x3 Xy)) ((cQ Xx) Xy)))))))
% Found eq_ref000:=(eq_ref00 P):((P ((and ((ex a) (fun (Xz:a)=> (x3 Xz)))) (forall (Xx:a), ((x3 Xx)->(forall (Xy:a), ((iff (x3 Xy)) ((cQ Xx) Xy)))))))->(P ((and ((ex a) (fun (Xz:a)=> (x3 Xz)))) (forall (Xx:a), ((x3 Xx)->(forall (Xy:a), ((iff (x3 Xy)) ((cQ Xx) Xy))))))))
% Found (eq_ref00 P) as proof of (P0 ((and ((ex a) (fun (Xz:a)=> (x3 Xz)))) (forall (Xx:a), ((x3 Xx)->(forall (Xy:a), ((iff (x3 Xy)) ((cQ Xx) Xy)))))))
% Found ((eq_ref0 ((and ((ex a) (fun (Xz:a)=> (x3 Xz)))) (forall (Xx:a), ((x3 Xx)->(forall (Xy:a), ((iff (x3 Xy)) ((cQ Xx) Xy))))))) P) as proof of (P0 ((and ((ex a) (fun (Xz:a)=> (x3 Xz)))) (forall (Xx:a), ((x3 Xx)->(forall (Xy:a), ((iff (x3 Xy)) ((cQ Xx) Xy)))))))
% Found (((eq_ref Prop) ((and ((ex a) (fun (Xz:a)=> (x3 Xz)))) (forall (Xx:a), ((x3 Xx)->(forall (Xy:a), ((iff (x3 Xy)) ((cQ Xx) Xy))))))) P) as proof of (P0 ((and ((ex a) (fun (Xz:a)=> (x3 Xz)))) (forall (Xx:a), ((x3 Xx)->(forall (Xy:a), ((iff (x3 Xy)) ((cQ Xx) Xy)))))))
% Found (((eq_ref Prop) ((and ((ex a) (fun (Xz:a)=> (x3 Xz)))) (forall (Xx:a), ((x3 Xx)->(forall (Xy:a), ((iff (x3 Xy)) ((cQ Xx) Xy))))))) P) as proof of (P0 ((and ((ex a) (fun (Xz:a)=> (x3 Xz)))) (forall (Xx:a), ((x3 Xx)->(forall (Xy:a), ((iff (x3 Xy)) ((cQ Xx) Xy)))))))
% Found eq_ref00:=(eq_ref0 ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy))))))):(((eq Prop) ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy))))))) ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy)))))))
% Found (eq_ref0 ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy))))))) as proof of (((eq Prop) ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy))))))) b)
% Found ((eq_ref Prop) ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy))))))) as proof of (((eq Prop) ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy))))))) b)
% Found ((eq_ref Prop) ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy))))))) as proof of (((eq Prop) ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy))))))) b)
% Found ((eq_ref Prop) ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy))))))) as proof of (((eq Prop) ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy))))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (cP x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cP x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cP x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cP x1))
% Found eq_ref00:=(eq_ref0 ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy))))))):(((eq Prop) ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy))))))) ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy)))))))
% Found (eq_ref0 ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy))))))) as proof of (((eq Prop) ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy))))))) b)
% Found ((eq_ref Prop) ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy))))))) as proof of (((eq Prop) ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy))))))) b)
% Found ((eq_ref Prop) ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy))))))) as proof of (((eq Prop) ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy))))))) b)
% Found ((eq_ref Prop) ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy))))))) as proof of (((eq Prop) ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy))))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (cP x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cP x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cP x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cP x1))
% Found eq_ref00:=(eq_ref0 ((and ((ex a) (fun (Xz:a)=> (x3 Xz)))) (forall (Xx:a), ((x3 Xx)->(forall (Xy:a), ((iff (x3 Xy)) ((cQ Xx) Xy))))))):(((eq Prop) ((and ((ex a) (fun (Xz:a)=> (x3 Xz)))) (forall (Xx:a), ((x3 Xx)->(forall (Xy:a), ((iff (x3 Xy)) ((cQ Xx) Xy))))))) ((and ((ex a) (fun (Xz:a)=> (x3 Xz)))) (forall (Xx:a), ((x3 Xx)->(forall (Xy:a), ((iff (x3 Xy)) ((cQ Xx) Xy)))))))
% Found (eq_ref0 ((and ((ex a) (fun (Xz:a)=> (x3 Xz)))) (forall (Xx:a), ((x3 Xx)->(forall (Xy:a), ((iff (x3 Xy)) ((cQ Xx) Xy))))))) as proof of (((eq Prop) ((and ((ex a) (fun (Xz:a)=> (x3 Xz)))) (forall (Xx:a), ((x3 Xx)->(forall (Xy:a), ((iff (x3 Xy)) ((cQ Xx) Xy))))))) b)
% Found ((eq_ref Prop) ((and ((ex a) (fun (Xz:a)=> (x3 Xz)))) (forall (Xx:a), ((x3 Xx)->(forall (Xy:a), ((iff (x3 Xy)) ((cQ Xx) Xy))))))) as proof of (((eq Prop) ((and ((ex a) (fun (Xz:a)=> (x3 Xz)))) (forall (Xx:a), ((x3 Xx)->(forall (Xy:a), ((iff (x3 Xy)) ((cQ Xx) Xy))))))) b)
% Found ((eq_ref Prop) ((and ((ex a) (fun (Xz:a)=> (x3 Xz)))) (forall (Xx:a), ((x3 Xx)->(forall (Xy:a), ((iff (x3 Xy)) ((cQ Xx) Xy))))))) as proof of (((eq Prop) ((and ((ex a) (fun (Xz:a)=> (x3 Xz)))) (forall (Xx:a), ((x3 Xx)->(forall (Xy:a), ((iff (x3 Xy)) ((cQ Xx) Xy))))))) b)
% Found ((eq_ref Prop) ((and ((ex a) (fun (Xz:a)=> (x3 Xz)))) (forall (Xx:a), ((x3 Xx)->(forall (Xy:a), ((iff (x3 Xy)) ((cQ Xx) Xy))))))) as proof of (((eq Prop) ((and ((ex a) (fun (Xz:a)=> (x3 Xz)))) (forall (Xx:a), ((x3 Xx)->(forall (Xy:a), ((iff (x3 Xy)) ((cQ Xx) Xy))))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (cP x3))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cP x3))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cP x3))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cP x3))
% Found eq_ref00:=(eq_ref0 ((and ((ex a) (fun (Xz:a)=> (x3 Xz)))) (forall (Xx:a), ((x3 Xx)->(forall (Xy:a), ((iff (x3 Xy)) ((cQ Xx) Xy))))))):(((eq Prop) ((and ((ex a) (fun (Xz:a)=> (x3 Xz)))) (forall (Xx:a), ((x3 Xx)->(forall (Xy:a), ((iff (x3 Xy)) ((cQ Xx) Xy))))))) ((and ((ex a) (fun (Xz:a)=> (x3 Xz)))) (forall (Xx:a), ((x3 Xx)->(forall (Xy:a), ((iff (x3 Xy)) ((cQ Xx) Xy)))))))
% Found (eq_ref0 ((and ((ex a) (fun (Xz:a)=> (x3 Xz)))) (forall (Xx:a), ((x3 Xx)->(forall (Xy:a), ((iff (x3 Xy)) ((cQ Xx) Xy))))))) as proof of (((eq Prop) ((and ((ex a) (fun (Xz:a)=> (x3 Xz)))) (forall (Xx:a), ((x3 Xx)->(forall (Xy:a), ((iff (x3 Xy)) ((cQ Xx) Xy))))))) b)
% Found ((eq_ref Prop) ((and ((ex a) (fun (Xz:a)=> (x3 Xz)))) (forall (Xx:a), ((x3 Xx)->(forall (Xy:a), ((iff (x3 Xy)) ((cQ Xx) Xy))))))) as proof of (((eq Prop) ((and ((ex a) (fun (Xz:a)=> (x3 Xz)))) (forall (Xx:a), ((x3 Xx)->(forall (Xy:a), ((iff (x3 Xy)) ((cQ Xx) Xy))))))) b)
% Found ((eq_ref Prop) ((and ((ex a) (fun (Xz:a)=> (x3 Xz)))) (forall (Xx:a), ((x3 Xx)->(forall (Xy:a), ((iff (x3 Xy)) ((cQ Xx) Xy))))))) as proof of (((eq Prop) ((and ((ex a) (fun (Xz:a)=> (x3 Xz)))) (forall (Xx:a), ((x3 Xx)->(forall (Xy:a), ((iff (x3 Xy)) ((cQ Xx) Xy))))))) b)
% Found ((eq_ref Prop) ((and ((ex a) (fun (Xz:a)=> (x3 Xz)))) (forall (Xx:a), ((x3 Xx)->(forall (Xy:a), ((iff (x3 Xy)) ((cQ Xx) Xy))))))) as proof of (((eq Prop) ((and ((ex a) (fun (Xz:a)=> (x3 Xz)))) (forall (Xx:a), ((x3 Xx)->(forall (Xy:a), ((iff (x3 Xy)) ((cQ Xx) Xy))))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (cP x3))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cP x3))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cP x3))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cP x3))
% Found eq_ref00:=(eq_ref0 b):(((eq ((a->Prop)->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq ((a->Prop)->Prop)) b) cP)
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) cP)
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) cP)
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) cP)
% Found eta_expansion000:=(eta_expansion00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy)))))))):(((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy)))))))) (fun (x:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (x Xz)))) (forall (Xx:a), ((x Xx)->(forall (Xy:a), ((iff (x Xy)) ((cQ Xx) Xy))))))))
% Found (eta_expansion00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy)))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy)))))))) b)
% Found ((eta_expansion0 Prop) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy)))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy)))))))) b)
% Found (((eta_expansion (a->Prop)) Prop) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy)))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy)))))))) b)
% Found (((eta_expansion (a->Prop)) Prop) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy)))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy)))))))) b)
% Found (((eta_expansion (a->Prop)) Prop) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy)))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy)))))))) b)
% Found eq_ref000:=(eq_ref00 P):((P (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy))))))))->(P (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy)))))))))
% Found (eq_ref00 P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy))))))))
% Found ((eq_ref0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy)))))))) P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy))))))))
% Found (((eq_ref ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy)))))))) P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy))))))))
% Found (((eq_ref ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy)))))))) P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy))))))))
% Found eq_ref00:=(eq_ref0 b):(((eq ((a->Prop)->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq ((a->Prop)->Prop)) b) cP)
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) cP)
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) cP)
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) cP)
% Found eq_ref00:=(eq_ref0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy)))))))):(((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy)))))))) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy))))))))
% Found (eq_ref0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy)))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy)))))))) b)
% Found ((eq_ref ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy)))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy)))))))) b)
% Found ((eq_ref ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy)))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy)))))))) b)
% Found ((eq_ref ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy)))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy)))))))) b)
% Found eq_ref000:=(eq_ref00 P):((P (cP x3))->(P (cP x3)))
% Found (eq_ref00 P) as proof of (P0 (cP x3))
% Found ((eq_ref0 (cP x3)) P) as proof of (P0 (cP x3))
% Found (((eq_ref Prop) (cP x3)) P) as proof of (P0 (cP x3))
% Found (((eq_ref Prop) (cP x3)) P) as proof of (P0 (cP x3))
% Found eq_ref000:=(eq_ref00 P):((P (cP x3))->(P (cP x3)))
% Found (eq_ref00 P) as proof of (P0 (cP x3))
% Found ((eq_ref0 (cP x3)) P) as proof of (P0 (cP x3))
% Found (((eq_ref Prop) (cP x3)) P) as proof of (P0 (cP x3))
% Found (((eq_ref Prop) (cP x3)) P) as proof of (P0 (cP x3))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (cP x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cP x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cP x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cP x1))
% Found eq_ref00:=(eq_ref0 ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy))))))):(((eq Prop) ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy))))))) ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy)))))))
% Found (eq_ref0 ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy))))))) as proof of (((eq Prop) ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy))))))) b)
% Found ((eq_ref Prop) ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy))))))) as proof of (((eq Prop) ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy))))))) b)
% Found ((eq_ref Prop) ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy))))))) as proof of (((eq Prop) ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy))))))) b)
% Found ((eq_ref Prop) ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy))))))) as proof of (((eq Prop) ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy))))))) b)
% Found eq_ref00:=(eq_ref0 ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy))))))):(((eq Prop) ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy))))))) ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy)))))))
% Found (eq_ref0 ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy))))))) as proof of (((eq Prop) ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy))))))) b)
% Found ((eq_ref Prop) ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy))))))) as proof of (((eq Prop) ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy))))))) b)
% Found ((eq_ref Prop) ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy))))))) as proof of (((eq Prop) ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy))))))) b)
% Found ((eq_ref Prop) ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy))))))) as proof of (((eq Prop) ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy))))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (cP x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cP x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cP x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cP x1))
% Found eq_ref00:=(eq_ref0 (cP x3)):(((eq Prop) (cP x3)) (cP x3))
% Found (eq_ref0 (cP x3)) as proof of (((eq Prop) (cP x3)) b)
% Found ((eq_ref Prop) (cP x3)) as proof of (((eq Prop) (cP x3)) b)
% Found ((eq_ref Prop) (cP x3)) as proof of (((eq Prop) (cP x3)) b)
% Found ((eq_ref Prop) (cP x3)) as proof of (((eq Prop) (cP x3)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((ex a) (fun (Xz:a)=> (x3 Xz)))) (forall (Xx:a), ((x3 Xx)->(forall (Xy:a), ((iff (x3 Xy)) ((cQ Xx) Xy)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((ex a) (fun (Xz:a)=> (x3 Xz)))) (forall (Xx:a), ((x3 Xx)->(forall (Xy:a), ((iff (x3 Xy)) ((cQ Xx) Xy)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((ex a) (fun (Xz:a)=> (x3 Xz)))) (forall (Xx:a), ((x3 Xx)->(forall (Xy:a), ((iff (x3 Xy)) ((cQ Xx) Xy)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((ex a) (fun (Xz:a)=> (x3 Xz)))) (forall (Xx:a), ((x3 Xx)->(forall (Xy:a), ((iff (x3 Xy)) ((cQ Xx) Xy)))))))
% Found eq_ref00:=(eq_ref0 (cP x3)):(((eq Prop) (cP x3)) (cP x3))
% Found (eq_ref0 (cP x3)) as proof of (((eq Prop) (cP x3)) b)
% Found ((eq_ref Prop) (cP x3)) as proof of (((eq Prop) (cP x3)) b)
% Found ((eq_ref Prop) (cP x3)) as proof of (((eq Prop) (cP x3)) b)
% Found ((eq_ref Prop) (cP x3)) as proof of (((eq Prop) (cP x3)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((ex a) (fun (Xz:a)=> (x3 Xz)))) (forall (Xx:a), ((x3 Xx)->(forall (Xy:a), ((iff (x3 Xy)) ((cQ Xx) Xy)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((ex a) (fun (Xz:a)=> (x3 Xz)))) (forall (Xx:a), ((x3 Xx)->(forall (Xy:a), ((iff (x3 Xy)) ((cQ Xx) Xy)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((ex a) (fun (Xz:a)=> (x3 Xz)))) (forall (Xx:a), ((x3 Xx)->(forall (Xy:a), ((iff (x3 Xy)) ((cQ Xx) Xy)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((ex a) (fun (Xz:a)=> (x3 Xz)))) (forall (Xx:a), ((x3 Xx)->(forall (Xy:a), ((iff (x3 Xy)) ((cQ Xx) Xy)))))))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy)))))))
% Found eq_ref00:=(eq_ref0 (cP x1)):(((eq Prop) (cP x1)) (cP x1))
% Found (eq_ref0 (cP x1)) as proof of (((eq Prop) (cP x1)) b)
% Found ((eq_ref Prop) (cP x1)) as proof of (((eq Prop) (cP x1)) b)
% Found ((eq_ref Prop) (cP x1)) as proof of (((eq Prop) (cP x1)) b)
% Found ((eq_ref Prop) (cP x1)) as proof of (((eq Prop) (cP x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy)))))))
% Found eq_ref00:=(eq_ref0 (cP x1)):(((eq Prop) (cP x1)) (cP x1))
% Found (eq_ref0 (cP x1)) as proof of (((eq Prop) (cP x1)) b)
% Found ((eq_ref Prop) (cP x1)) as proof of (((eq Prop) (cP x1)) b)
% Found ((eq_ref Prop) (cP x1)) as proof of (((eq Prop) (cP x1)) b)
% Found ((eq_ref Prop) (cP x1)) as proof of (((eq Prop) (cP x1)) b)
% Found x3:(P (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy))))))))
% Instantiate: b:=(fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy))))))):((a->Prop)->Prop)
% Found x3 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 cP):(((eq ((a->Prop)->Prop)) cP) cP)
% Found (eq_ref0 cP) as proof of (((eq ((a->Prop)->Prop)) cP) b)
% Found ((eq_ref ((a->Prop)->Prop)) cP) as proof of (((eq ((a->Prop)->Prop)) cP) b)
% Found ((eq_ref ((a->Prop)->Prop)) cP) as proof of (((eq ((a->Prop)->Prop)) cP) b)
% Found ((eq_ref ((a->Prop)->Prop)) cP) as proof of (((eq ((a->Prop)->Prop)) cP) b)
% Found eq_ref000:=(eq_ref00 P):((P (cP x1))->(P (cP x1)))
% Found (eq_ref00 P) as proof of (P0 (cP x1))
% Found ((eq_ref0 (cP x1)) P) as proof of (P0 (cP x1))
% Found (((eq_ref Prop) (cP x1)) P) as proof of (P0 (cP x1))
% Found (((eq_ref Prop) (cP x1)) P) as proof of (P0 (cP x1))
% Found eq_ref000:=(eq_ref00 P):((P (cP x1))->(P (cP x1)))
% Found (eq_ref00 P) as proof of (P0 (cP x1))
% Found ((eq_ref0 (cP x1)) P) as proof of (P0 (cP x1))
% Found (((eq_ref Prop) (cP x1)) P) as proof of (P0 (cP x1))
% Found (((eq_ref Prop) (cP x1)) P) as proof of (P0 (cP x1))
% Found x1:(P (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy))))))))
% Instantiate: b:=(fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy))))))):((a->Prop)->Prop)
% Found x1 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 cP):(((eq ((a->Prop)->Prop)) cP) cP)
% Found (eq_ref0 cP) as proof of (((eq ((a->Prop)->Prop)) cP) b)
% Found ((eq_ref ((a->Prop)->Prop)) cP) as proof of (((eq ((a->Prop)->Prop)) cP) b)
% Found ((eq_ref ((a->Prop)->Prop)) cP) as proof of (((eq ((a->Prop)->Prop)) cP) b)
% Found ((eq_ref ((a->Prop)->Prop)) cP) as proof of (((eq ((a->Prop)->Prop)) cP) b)
% Found eq_ref00:=(eq_ref0 b):(((eq ((a->Prop)->Prop)) b) b)
% Found (eq_ref0 b) as proof of (P b)
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (P b)
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (P b)
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (P b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 cP):(((eq ((a->Prop)->Prop)) cP) (fun (x:(a->Prop))=> (cP x)))
% Found (eta_expansion_dep00 cP) as proof of (((eq ((a->Prop)->Prop)) cP) b)
% Found ((eta_expansion_dep0 (fun (x2:(a->Prop))=> Prop)) cP) as proof of (((eq ((a->Prop)->Prop)) cP) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x2:(a->Prop))=> Prop)) cP) as proof of (((eq ((a->Prop)->Prop)) cP) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x2:(a->Prop))=> Prop)) cP) as proof of (((eq ((a->Prop)->Prop)) cP) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x2:(a->Prop))=> Prop)) cP) as proof of (((eq ((a->Prop)->Prop)) cP) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy)))))))
% Found eq_ref00:=(eq_ref0 (cP x1)):(((eq Prop) (cP x1)) (cP x1))
% Found (eq_ref0 (cP x1)) as proof of (((eq Prop) (cP x1)) b)
% Found ((eq_ref Prop) (cP x1)) as proof of (((eq Prop) (cP x1)) b)
% Found ((eq_ref Prop) (cP x1)) as proof of (((eq Prop) (cP x1)) b)
% Found ((eq_ref Prop) (cP x1)) as proof of (((eq Prop) (cP x1)) b)
% Found eq_ref00:=(eq_ref0 (cP x1)):(((eq Prop) (cP x1)) (cP x1))
% Found (eq_ref0 (cP x1)) as proof of (((eq Prop) (cP x1)) b)
% Found ((eq_ref Prop) (cP x1)) as proof of (((eq Prop) (cP x1)) b)
% Found ((eq_ref Prop) (cP x1)) as proof of (((eq Prop) (cP x1)) b)
% Found ((eq_ref Prop) (cP x1)) as proof of (((eq Prop) (cP x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy)))))))
% Found eq_ref00:=(eq_ref0 (cP x1)):(((eq Prop) (cP x1)) (cP x1))
% Found (eq_ref0 (cP x1)) as proof of (((eq Prop) (cP x1)) b)
% Found ((eq_ref Prop) (cP x1)) as proof of (((eq Prop) (cP x1)) b)
% Found ((eq_ref Prop) (cP x1)) as proof of (((eq Prop) (cP x1)) b)
% Found ((eq_ref Prop) (cP x1)) as proof of (((eq Prop) (cP x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy)))))))
% Found eq_ref00:=(eq_ref0 (cP x1)):(((eq Prop) (cP x1)) (cP x1))
% Found (eq_ref0 (cP x1)) as proof of (((eq Prop) (cP x1)) b)
% Found ((eq_ref Prop) (cP x1)) as proof of (((eq Prop) (cP x1)) b)
% Found ((eq_ref Prop) (cP x1)) as proof of (((eq Prop) (cP x1)) b)
% Found ((eq_ref Prop) (cP x1)) as proof of (((eq Prop) (cP x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((cQ Xx) Xy)))))))
% Found x1:(P cP)
% Instantiate: b:=cP:((a->Prop)->Prop)
% Found x1 as proof of (P0 b)
% Found eta_expansion000:=(eta_expansion00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy)))))))):(((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy)))))))) (fun (x:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (x Xz)))) (forall (Xx:a), ((x Xx)->(forall (Xy:a), ((iff (x Xy)) ((cQ Xx) Xy))))))))
% Found (eta_expansion00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy)))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy)))))))) b)
% Found ((eta_expansion0 Prop) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy)))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy)))))))) b)
% Found (((eta_expansion (a->Prop)) Prop) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy)))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy)))))))) b)
% Found (((eta_expansion (a->Prop)) Prop) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy)))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy)))))))) b)
% Found (((eta_expansion (a->Prop)) Prop) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy)))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy)))))))) b)
% Found x3:(P (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy))))))))
% Instantiate: f:=(fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy))))))):((a->Prop)->Prop)
% Found x3 as proof of (P0 f)
% Found eq_ref00:=(eq_ref0 (f x4)):(((eq Prop) (f x4)) (f x4))
% Found (eq_ref0 (f x4)) as proof of (((eq Prop) (f x4)) (cP x4))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) (cP x4))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) (cP x4))
% Found (fun (x4:(a->Prop))=> ((eq_ref Prop) (f x4))) as proof of (((eq Prop) (f x4)) (cP x4))
% Found (fun (x4:(a->Prop))=> ((eq_ref Prop) (f x4))) as proof of (forall (x:(a->Prop)), (((eq Prop) (f x)) (cP x)))
% Found x3:(P (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy))))))))
% Instantiate: f:=(fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((cQ Xx) Xy))))))):((a->Prop)->Prop)
% Found x3 as proof of (P0 f)
% Found eq_ref00:=(eq_ref0 (f x4)):(((eq Prop) (f x4)) (f x4))
% Found (eq_ref0 (f x4)) as proof of (((eq Prop) 
% EOF
%------------------------------------------------------------------------------