%------------------------------------------------------------------------------
% File     : cocATP---0.2.0
% Problem  : SEV022^5 : TPTP v6.1.0. Released v4.0.0.
% Transfm  : none
% Format   : tptp:raw
% Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p

% Computer : n118.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 300.05s
% Output   : None 
% Verified : 
% SZS Type : None (Parsing solution fails)
% Syntax   : Number of formulae    : 0

% Comments : 
%------------------------------------------------------------------------------
%----NO SOLUTION OUTPUT BY SYSTEM
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% % Problem  : SEV022^5 : TPTP v6.1.0. Released v4.0.0.
% % Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n118.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 07:34:56 CDT 2014
% % CPUTime  : 300.05 
% 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 0xa1d3b0>, <kernel.Type object at 0xa1d8c0>) of role type named a_type
% Using role type
% Declaring a:Type
% FOF formula (<kernel.Constant object at 0xa1d830>, <kernel.DependentProduct object at 0xa1c290>) of role type named cP
% Using role type
% Declaring cP:((a->Prop)->Prop)
% FOF formula (((and (forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a) (Xp:(a->Prop)) (Xq:(a->Prop)), (((and ((and ((and (cP Xp)) (cP Xq))) (Xp Xx))) (Xq Xx))->(((eq (a->Prop)) Xp) Xq))))->((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((and (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((R Xx) Xy)))))))) cP))))) of role conjecture named cTHM556_pme
% Conjecture to prove = (((and (forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a) (Xp:(a->Prop)) (Xq:(a->Prop)), (((and ((and ((and (cP Xp)) (cP Xq))) (Xp Xx))) (Xq Xx))->(((eq (a->Prop)) Xp) Xq))))->((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((and (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((R Xx) Xy)))))))) cP))))):Prop
% Parameter a_DUMMY:a.
% We need to prove ['(((and (forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a) (Xp:(a->Prop)) (Xq:(a->Prop)), (((and ((and ((and (cP Xp)) (cP Xq))) (Xp Xx))) (Xq Xx))->(((eq (a->Prop)) Xp) Xq))))->((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((and (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((R Xx) Xy)))))))) cP)))))']
% Parameter a:Type.
% Parameter cP:((a->Prop)->Prop).
% Trying to prove (((and (forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a) (Xp:(a->Prop)) (Xq:(a->Prop)), (((and ((and ((and (cP Xp)) (cP Xq))) (Xp Xx))) (Xq Xx))->(((eq (a->Prop)) Xp) Xq))))->((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((and (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((R Xx) Xy)))))))) cP)))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (R:(a->(a->Prop)))=> ((and ((and (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((R Xx) Xy)))))))) cP)))):(((eq ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> ((and ((and (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((R Xx) Xy)))))))) cP)))) (fun (x:(a->(a->Prop)))=> ((and ((and (forall (Xx:a) (Xy:a), (((x Xx) Xy)->((x Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x Xx) Xy)))))))) cP))))
% Found (eta_expansion_dep00 (fun (R:(a->(a->Prop)))=> ((and ((and (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((R Xx) Xy)))))))) cP)))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> ((and ((and (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((R Xx) Xy)))))))) cP)))) b)
% Found ((eta_expansion_dep0 (fun (x1:(a->(a->Prop)))=> Prop)) (fun (R:(a->(a->Prop)))=> ((and ((and (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((R Xx) Xy)))))))) cP)))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> ((and ((and (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((R Xx) Xy)))))))) cP)))) b)
% Found (((eta_expansion_dep (a->(a->Prop))) (fun (x1:(a->(a->Prop)))=> Prop)) (fun (R:(a->(a->Prop)))=> ((and ((and (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((R Xx) Xy)))))))) cP)))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> ((and ((and (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((R Xx) Xy)))))))) cP)))) b)
% Found (((eta_expansion_dep (a->(a->Prop))) (fun (x1:(a->(a->Prop)))=> Prop)) (fun (R:(a->(a->Prop)))=> ((and ((and (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((R Xx) Xy)))))))) cP)))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> ((and ((and (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((R Xx) Xy)))))))) cP)))) b)
% Found (((eta_expansion_dep (a->(a->Prop))) (fun (x1:(a->(a->Prop)))=> Prop)) (fun (R:(a->(a->Prop)))=> ((and ((and (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((R Xx) Xy)))))))) cP)))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> ((and ((and (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((R Xx) Xy)))))))) cP)))) b)
% Found eq_ref00:=(eq_ref0 (fun (R:(a->(a->Prop)))=> ((and ((and (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((R Xx) Xy)))))))) cP)))):(((eq ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> ((and ((and (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((R Xx) Xy)))))))) cP)))) (fun (R:(a->(a->Prop)))=> ((and ((and (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((R Xx) Xy)))))))) cP))))
% Found (eq_ref0 (fun (R:(a->(a->Prop)))=> ((and ((and (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((R Xx) Xy)))))))) cP)))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> ((and ((and (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((R Xx) Xy)))))))) cP)))) b)
% Found ((eq_ref ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> ((and ((and (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((R Xx) Xy)))))))) cP)))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> ((and ((and (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((R Xx) Xy)))))))) cP)))) b)
% Found ((eq_ref ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> ((and ((and (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((R Xx) Xy)))))))) cP)))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> ((and ((and (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((R Xx) Xy)))))))) cP)))) b)
% Found ((eq_ref ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> ((and ((and (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((R Xx) Xy)))))))) cP)))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> ((and ((and (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((R Xx) Xy)))))))) cP)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq ((a->Prop)->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq ((a->Prop)->Prop)) b) 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x2 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)) ((x2 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)) ((x2 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)) ((x2 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)) ((x2 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)) ((x2 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)) ((x2 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)) ((x2 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)) ((x2 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)) ((x2 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)) ((x2 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)) ((x2 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)) ((x2 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)) ((x2 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)) ((x2 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)) ((x2 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)) ((x2 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)) ((x2 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)) ((x2 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)) ((x2 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)) ((x2 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)) ((x2 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)) ((x2 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)) ((x2 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x2 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)) ((x2 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)) ((x2 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)) ((x2 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)) ((x2 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)) ((x2 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)) ((x2 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)) ((x2 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)) ((x2 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)) ((x2 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)) ((x2 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)) ((x2 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)) ((x2 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)) ((x2 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)) ((x2 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)) ((x2 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)) ((x2 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)) ((x2 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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_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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 Xx) Xy)))))))) 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 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 Xx) Xy))))))):(((eq Prop) ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((x0 Xx) Xy))))))) ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((x0 Xx) Xy)))))))
% Found (eq_ref0 ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 Xx) Xy))))))):(((eq Prop) ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((x0 Xx) Xy))))))) ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((x0 Xx) Xy)))))))
% Found (eq_ref0 ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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_ref000:=(eq_ref00 P):((P ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((x0 Xx) Xy)))))))->(P ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((x0 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)) ((x0 Xx) Xy)))))))
% Found ((eq_ref0 ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 Xx) Xy)))))))->(P ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((x0 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)) ((x0 Xx) Xy)))))))
% Found ((eq_ref0 ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x2 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)) ((x2 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)) ((x2 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)) ((x2 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 (x4:(a->Prop))=> Prop)) cP) as proof of (((eq ((a->Prop)->Prop)) cP) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x4:(a->Prop))=> Prop)) cP) as proof of (((eq ((a->Prop)->Prop)) cP) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x4:(a->Prop))=> Prop)) cP) as proof of (((eq ((a->Prop)->Prop)) cP) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x4:(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 (((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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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 (x4:(a->Prop))=> Prop)) cP) as proof of (((eq ((a->Prop)->Prop)) cP) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x4:(a->Prop))=> Prop)) cP) as proof of (((eq ((a->Prop)->Prop)) cP) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x4:(a->Prop))=> Prop)) cP) as proof of (((eq ((a->Prop)->Prop)) cP) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x4:(a->Prop))=> Prop)) cP) as proof of (((eq ((a->Prop)->Prop)) cP) b)
% Found eq_ref00:=(eq_ref0 b):(((eq ((a->(a->Prop))->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq ((a->(a->Prop))->Prop)) b) (fun (R:(a->(a->Prop)))=> ((and ((and (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((R Xx) Xy)))))))) cP))))
% Found ((eq_ref ((a->(a->Prop))->Prop)) b) as proof of (((eq ((a->(a->Prop))->Prop)) b) (fun (R:(a->(a->Prop)))=> ((and ((and (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((R Xx) Xy)))))))) cP))))
% Found ((eq_ref ((a->(a->Prop))->Prop)) b) as proof of (((eq ((a->(a->Prop))->Prop)) b) (fun (R:(a->(a->Prop)))=> ((and ((and (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((R Xx) Xy)))))))) cP))))
% Found ((eq_ref ((a->(a->Prop))->Prop)) b) as proof of (((eq ((a->(a->Prop))->Prop)) b) (fun (R:(a->(a->Prop)))=> ((and ((and (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((R Xx) Xy)))))))) cP))))
% Found eta_expansion000:=(eta_expansion00 a0):(((eq ((a->(a->Prop))->Prop)) a0) (fun (x:(a->(a->Prop)))=> (a0 x)))
% Found (eta_expansion00 a0) as proof of (((eq ((a->(a->Prop))->Prop)) a0) b)
% Found ((eta_expansion0 Prop) a0) as proof of (((eq ((a->(a->Prop))->Prop)) a0) b)
% Found (((eta_expansion (a->(a->Prop))) Prop) a0) as proof of (((eq ((a->(a->Prop))->Prop)) a0) b)
% Found (((eta_expansion (a->(a->Prop))) Prop) a0) as proof of (((eq ((a->(a->Prop))->Prop)) a0) b)
% Found (((eta_expansion (a->(a->Prop))) Prop) a0) as proof of (((eq ((a->(a->Prop))->Prop)) a0) b)
% Found eq_ref00:=(eq_ref0 a0):(((eq ((a->(a->Prop))->Prop)) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq ((a->(a->Prop))->Prop)) a0) b)
% Found ((eq_ref ((a->(a->Prop))->Prop)) a0) as proof of (((eq ((a->(a->Prop))->Prop)) a0) b)
% Found ((eq_ref ((a->(a->Prop))->Prop)) a0) as proof of (((eq ((a->(a->Prop))->Prop)) a0) b)
% Found ((eq_ref ((a->(a->Prop))->Prop)) a0) as proof of (((eq ((a->(a->Prop))->Prop)) a0) b)
% Found eq_ref000:=(eq_ref00 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 (((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)) ((x0 Xx) Xy)))))))) cP)):(((eq Prop) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy)))))))) cP)) (((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)) ((x0 Xx) Xy)))))))) cP))
% Found (eq_ref0 (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy)))))))) cP)) as proof of (((eq Prop) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy)))))))) cP)) b)
% Found ((eq_ref Prop) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy)))))))) cP)) as proof of (((eq Prop) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy)))))))) cP)) b)
% Found ((eq_ref Prop) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy)))))))) cP)) as proof of (((eq Prop) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy)))))))) cP)) b)
% Found ((eq_ref Prop) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy)))))))) cP)) as proof of (((eq Prop) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy)))))))) cP)) b)
% Found eq_ref00:=(eq_ref0 ((and ((ex a) (fun (Xz:a)=> (x3 Xz)))) (forall (Xx:a), ((x3 Xx)->(forall (Xy:a), ((iff (x3 Xy)) ((x2 Xx) Xy))))))):(((eq Prop) ((and ((ex a) (fun (Xz:a)=> (x3 Xz)))) (forall (Xx:a), ((x3 Xx)->(forall (Xy:a), ((iff (x3 Xy)) ((x2 Xx) Xy))))))) ((and ((ex a) (fun (Xz:a)=> (x3 Xz)))) (forall (Xx:a), ((x3 Xx)->(forall (Xy:a), ((iff (x3 Xy)) ((x2 Xx) Xy)))))))
% Found (eq_ref0 ((and ((ex a) (fun (Xz:a)=> (x3 Xz)))) (forall (Xx:a), ((x3 Xx)->(forall (Xy:a), ((iff (x3 Xy)) ((x2 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)) ((x2 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)) ((x2 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)) ((x2 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)) ((x2 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)) ((x2 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)) ((x2 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)) ((x2 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)) ((x2 Xx) Xy))))))):(((eq Prop) ((and ((ex a) (fun (Xz:a)=> (x3 Xz)))) (forall (Xx:a), ((x3 Xx)->(forall (Xy:a), ((iff (x3 Xy)) ((x2 Xx) Xy))))))) ((and ((ex a) (fun (Xz:a)=> (x3 Xz)))) (forall (Xx:a), ((x3 Xx)->(forall (Xy:a), ((iff (x3 Xy)) ((x2 Xx) Xy)))))))
% Found (eq_ref0 ((and ((ex a) (fun (Xz:a)=> (x3 Xz)))) (forall (Xx:a), ((x3 Xx)->(forall (Xy:a), ((iff (x3 Xy)) ((x2 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)) ((x2 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)) ((x2 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)) ((x2 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)) ((x2 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)) ((x2 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)) ((x2 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)) ((x2 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_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)) ((x2 Xx) Xy)))))))->(P ((and ((ex a) (fun (Xz:a)=> (x3 Xz)))) (forall (Xx:a), ((x3 Xx)->(forall (Xy:a), ((iff (x3 Xy)) ((x2 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)) ((x2 Xx) Xy)))))))
% Found ((eq_ref0 ((and ((ex a) (fun (Xz:a)=> (x3 Xz)))) (forall (Xx:a), ((x3 Xx)->(forall (Xy:a), ((iff (x3 Xy)) ((x2 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)) ((x2 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)) ((x2 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)) ((x2 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)) ((x2 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)) ((x2 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)) ((x2 Xx) Xy)))))))->(P ((and ((ex a) (fun (Xz:a)=> (x3 Xz)))) (forall (Xx:a), ((x3 Xx)->(forall (Xy:a), ((iff (x3 Xy)) ((x2 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)) ((x2 Xx) Xy)))))))
% Found ((eq_ref0 ((and ((ex a) (fun (Xz:a)=> (x3 Xz)))) (forall (Xx:a), ((x3 Xx)->(forall (Xy:a), ((iff (x3 Xy)) ((x2 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)) ((x2 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)) ((x2 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)) ((x2 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)) ((x2 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)) ((x2 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)) ((x0 Xx) Xy))))))):(((eq Prop) ((and ((ex a) (fun (Xz:a)=> (x3 Xz)))) (forall (Xx:a), ((x3 Xx)->(forall (Xy:a), ((iff (x3 Xy)) ((x0 Xx) Xy))))))) ((and ((ex a) (fun (Xz:a)=> (x3 Xz)))) (forall (Xx:a), ((x3 Xx)->(forall (Xy:a), ((iff (x3 Xy)) ((x0 Xx) Xy)))))))
% Found (eq_ref0 ((and ((ex a) (fun (Xz:a)=> (x3 Xz)))) (forall (Xx:a), ((x3 Xx)->(forall (Xy:a), ((iff (x3 Xy)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 Xx) Xy))))))):(((eq Prop) ((and ((ex a) (fun (Xz:a)=> (x3 Xz)))) (forall (Xx:a), ((x3 Xx)->(forall (Xy:a), ((iff (x3 Xy)) ((x0 Xx) Xy))))))) ((and ((ex a) (fun (Xz:a)=> (x3 Xz)))) (forall (Xx:a), ((x3 Xx)->(forall (Xy:a), ((iff (x3 Xy)) ((x0 Xx) Xy)))))))
% Found (eq_ref0 ((and ((ex a) (fun (Xz:a)=> (x3 Xz)))) (forall (Xx:a), ((x3 Xx)->(forall (Xy:a), ((iff (x3 Xy)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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->(a->Prop))->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq ((a->(a->Prop))->Prop)) b) (fun (R:(a->(a->Prop)))=> ((and ((and (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((R Xx) Xy)))))))) cP))))
% Found ((eq_ref ((a->(a->Prop))->Prop)) b) as proof of (((eq ((a->(a->Prop))->Prop)) b) (fun (R:(a->(a->Prop)))=> ((and ((and (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((R Xx) Xy)))))))) cP))))
% Found ((eq_ref ((a->(a->Prop))->Prop)) b) as proof of (((eq ((a->(a->Prop))->Prop)) b) (fun (R:(a->(a->Prop)))=> ((and ((and (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((R Xx) Xy)))))))) cP))))
% Found ((eq_ref ((a->(a->Prop))->Prop)) b) as proof of (((eq ((a->(a->Prop))->Prop)) b) (fun (R:(a->(a->Prop)))=> ((and ((and (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((R Xx) Xy)))))))) cP))))
% Found eta_expansion000:=(eta_expansion00 a0):(((eq ((a->(a->Prop))->Prop)) a0) (fun (x:(a->(a->Prop)))=> (a0 x)))
% Found (eta_expansion00 a0) as proof of (((eq ((a->(a->Prop))->Prop)) a0) b)
% Found ((eta_expansion0 Prop) a0) as proof of (((eq ((a->(a->Prop))->Prop)) a0) b)
% Found (((eta_expansion (a->(a->Prop))) Prop) a0) as proof of (((eq ((a->(a->Prop))->Prop)) a0) b)
% Found (((eta_expansion (a->(a->Prop))) Prop) a0) as proof of (((eq ((a->(a->Prop))->Prop)) a0) b)
% Found (((eta_expansion (a->(a->Prop))) Prop) a0) as proof of (((eq ((a->(a->Prop))->Prop)) a0) b)
% Found eq_ref00:=(eq_ref0 a0):(((eq ((a->(a->Prop))->Prop)) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq ((a->(a->Prop))->Prop)) a0) b)
% Found ((eq_ref ((a->(a->Prop))->Prop)) a0) as proof of (((eq ((a->(a->Prop))->Prop)) a0) b)
% Found ((eq_ref ((a->(a->Prop))->Prop)) a0) as proof of (((eq ((a->(a->Prop))->Prop)) a0) b)
% Found ((eq_ref ((a->(a->Prop))->Prop)) a0) as proof of (((eq ((a->(a->Prop))->Prop)) a0) b)
% Found eq_ref000:=(eq_ref00 P):((P ((and ((ex a) (fun (Xz:a)=> (x3 Xz)))) (forall (Xx:a), ((x3 Xx)->(forall (Xy:a), ((iff (x3 Xy)) ((x0 Xx) Xy)))))))->(P ((and ((ex a) (fun (Xz:a)=> (x3 Xz)))) (forall (Xx:a), ((x3 Xx)->(forall (Xy:a), ((iff (x3 Xy)) ((x0 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)) ((x0 Xx) Xy)))))))
% Found ((eq_ref0 ((and ((ex a) (fun (Xz:a)=> (x3 Xz)))) (forall (Xx:a), ((x3 Xx)->(forall (Xy:a), ((iff (x3 Xy)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 Xx) Xy)))))))->(P ((and ((ex a) (fun (Xz:a)=> (x3 Xz)))) (forall (Xx:a), ((x3 Xx)->(forall (Xy:a), ((iff (x3 Xy)) ((x0 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)) ((x0 Xx) Xy)))))))
% Found ((eq_ref0 ((and ((ex a) (fun (Xz:a)=> (x3 Xz)))) (forall (Xx:a), ((x3 Xx)->(forall (Xy:a), ((iff (x3 Xy)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 Xx) Xy)))))))
% Found iff_sym:=(fun (A:Prop) (B:Prop) (H:((iff A) B))=> ((((conj (B->A)) (A->B)) (((proj2 (A->B)) (B->A)) H)) (((proj1 (A->B)) (B->A)) H))):(forall (A:Prop) (B:Prop), (((iff A) B)->((iff B) A)))
% Instantiate: b:=(forall (A:Prop) (B:Prop), (((iff A) B)->((iff B) A))):Prop
% Found iff_sym as proof of b
% Found eq_ref00:=(eq_ref0 (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x2 Xx) Xy)))))))) cP)):(((eq Prop) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x2 Xx) Xy)))))))) cP)) (((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)) ((x2 Xx) Xy)))))))) cP))
% Found (eq_ref0 (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x2 Xx) Xy)))))))) cP)) as proof of (((eq Prop) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x2 Xx) Xy)))))))) cP)) b)
% Found ((eq_ref Prop) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x2 Xx) Xy)))))))) cP)) as proof of (((eq Prop) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x2 Xx) Xy)))))))) cP)) b)
% Found ((eq_ref Prop) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x2 Xx) Xy)))))))) cP)) as proof of (((eq Prop) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x2 Xx) Xy)))))))) cP)) b)
% Found ((eq_ref Prop) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x2 Xx) Xy)))))))) cP)) as proof of (((eq Prop) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x2 Xx) Xy)))))))) cP)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq ((a->Prop)->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq ((a->Prop)->Prop)) b) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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 (x4:(a->Prop))=> Prop)) cP) as proof of (((eq ((a->Prop)->Prop)) cP) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x4:(a->Prop))=> Prop)) cP) as proof of (((eq ((a->Prop)->Prop)) cP) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x4:(a->Prop))=> Prop)) cP) as proof of (((eq ((a->Prop)->Prop)) cP) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x4:(a->Prop))=> Prop)) cP) as proof of (((eq ((a->Prop)->Prop)) cP) b)
% Found eq_ref00:=(eq_ref0 (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy)))))))) cP)):(((eq Prop) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy)))))))) cP)) (((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)) ((x0 Xx) Xy)))))))) cP))
% Found (eq_ref0 (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy)))))))) cP)) as proof of (((eq Prop) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy)))))))) cP)) b)
% Found ((eq_ref Prop) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy)))))))) cP)) as proof of (((eq Prop) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy)))))))) cP)) b)
% Found ((eq_ref Prop) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy)))))))) cP)) as proof of (((eq Prop) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy)))))))) cP)) b)
% Found ((eq_ref Prop) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy)))))))) cP)) as proof of (((eq Prop) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy)))))))) 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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 ((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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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 (x4:(a->Prop))=> Prop)) cP) as proof of (((eq ((a->Prop)->Prop)) cP) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x4:(a->Prop))=> Prop)) cP) as proof of (((eq ((a->Prop)->Prop)) cP) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x4:(a->Prop))=> Prop)) cP) as proof of (((eq ((a->Prop)->Prop)) cP) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x4:(a->Prop))=> Prop)) cP) as proof of (((eq ((a->Prop)->Prop)) cP) 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 Xx) Xy)))))))
% 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_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_sym:=(fun (T:Type) (a:T) (b:T) (H:(((eq T) a) b))=> ((H (fun (x:T)=> (((eq T) x) a))) ((eq_ref T) a))):(forall (T:Type) (a:T) (b:T), ((((eq T) a) b)->(((eq T) b) a)))
% Instantiate: b:=(forall (T:Type) (a:T) (b:T), ((((eq T) a) b)->(((eq T) b) a))):Prop
% Found eq_sym as proof of b
% Found eq_ref00:=(eq_ref0 ((and ((ex a) (fun (Xz:a)=> (x3 Xz)))) (forall (Xx:a), ((x3 Xx)->(forall (Xy:a), ((iff (x3 Xy)) ((x0 Xx) Xy))))))):(((eq Prop) ((and ((ex a) (fun (Xz:a)=> (x3 Xz)))) (forall (Xx:a), ((x3 Xx)->(forall (Xy:a), ((iff (x3 Xy)) ((x0 Xx) Xy))))))) ((and ((ex a) (fun (Xz:a)=> (x3 Xz)))) (forall (Xx:a), ((x3 Xx)->(forall (Xy:a), ((iff (x3 Xy)) ((x0 Xx) Xy)))))))
% Found (eq_ref0 ((and ((ex a) (fun (Xz:a)=> (x3 Xz)))) (forall (Xx:a), ((x3 Xx)->(forall (Xy:a), ((iff (x3 Xy)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 Xx) Xy))))))):(((eq Prop) ((and ((ex a) (fun (Xz:a)=> (x3 Xz)))) (forall (Xx:a), ((x3 Xx)->(forall (Xy:a), ((iff (x3 Xy)) ((x0 Xx) Xy))))))) ((and ((ex a) (fun (Xz:a)=> (x3 Xz)))) (forall (Xx:a), ((x3 Xx)->(forall (Xy:a), ((iff (x3 Xy)) ((x0 Xx) Xy)))))))
% Found (eq_ref0 ((and ((ex a) (fun (Xz:a)=> (x3 Xz)))) (forall (Xx:a), ((x3 Xx)->(forall (Xy:a), ((iff (x3 Xy)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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_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)) ((x0 Xx) Xy)))))))->(P ((and ((ex a) (fun (Xz:a)=> (x3 Xz)))) (forall (Xx:a), ((x3 Xx)->(forall (Xy:a), ((iff (x3 Xy)) ((x0 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)) ((x0 Xx) Xy)))))))
% Found ((eq_ref0 ((and ((ex a) (fun (Xz:a)=> (x3 Xz)))) (forall (Xx:a), ((x3 Xx)->(forall (Xy:a), ((iff (x3 Xy)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 Xx) Xy)))))))->(P ((and ((ex a) (fun (Xz:a)=> (x3 Xz)))) (forall (Xx:a), ((x3 Xx)->(forall (Xy:a), ((iff (x3 Xy)) ((x0 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)) ((x0 Xx) Xy)))))))
% Found ((eq_ref0 ((and ((ex a) (fun (Xz:a)=> (x3 Xz)))) (forall (Xx:a), ((x3 Xx)->(forall (Xy:a), ((iff (x3 Xy)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 Xx) Xy)))))))
% Found eq_sym:=(fun (T:Type) (a:T) (b:T) (H:(((eq T) a) b))=> ((H (fun (x:T)=> (((eq T) x) a))) ((eq_ref T) a))):(forall (T:Type) (a:T) (b:T), ((((eq T) a) b)->(((eq T) b) a)))
% Instantiate: b:=(forall (T:Type) (a:T) (b:T), ((((eq T) a) b)->(((eq T) b) a))):Prop
% Found eq_sym as proof of 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)) ((x0 Xx) Xy))))))):(((eq Prop) ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((x0 Xx) Xy))))))) ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((x0 Xx) Xy)))))))
% Found (eq_ref0 ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 Xx) Xy))))))):(((eq Prop) ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((x0 Xx) Xy))))))) ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((x0 Xx) Xy)))))))
% Found (eq_ref0 ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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_ref000:=(eq_ref00 P):((P ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((x0 Xx) Xy)))))))->(P ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((x0 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)) ((x0 Xx) Xy)))))))
% Found ((eq_ref0 ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 Xx) Xy)))))))->(P ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((x0 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)) ((x0 Xx) Xy)))))))
% Found ((eq_ref0 ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 Xx) Xy)))))))
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((and (forall (Xx:a) (Xy:a), (((x0 Xx) Xy)->((x0 Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xz))->((x0 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy)))))))) cP)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (forall (Xx:a) (Xy:a), (((x0 Xx) Xy)->((x0 Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xz))->((x0 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy)))))))) cP)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (forall (Xx:a) (Xy:a), (((x0 Xx) Xy)->((x0 Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xz))->((x0 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy)))))))) cP)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (forall (Xx:a) (Xy:a), (((x0 Xx) Xy)->((x0 Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xz))->((x0 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy)))))))) cP)))
% Found ((eq_trans0000 ((eq_ref Prop) (f x0))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f x0)) ((and ((and (forall (Xx:a) (Xy:a), (((x0 Xx) Xy)->((x0 Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xz))->((x0 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy)))))))) cP)))
% Found (((eq_trans000 ((and ((and (forall (Xx:a) (Xy:a), (((x0 Xx) Xy)->((x0 Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xz))->((x0 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy)))))))) cP))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f x0)) ((and ((and (forall (Xx:a) (Xy:a), (((x0 Xx) Xy)->((x0 Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xz))->((x0 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy)))))))) cP)))
% Found ((((eq_trans00 ((and ((and (forall (Xx:a) (Xy:a), (((x0 Xx) Xy)->((x0 Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xz))->((x0 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy)))))))) cP))) ((and ((and (forall (Xx:a) (Xy:a), (((x0 Xx) Xy)->((x0 Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xz))->((x0 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy)))))))) cP))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((and (forall (Xx:a) (Xy:a), (((x0 Xx) Xy)->((x0 Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xz))->((x0 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy)))))))) cP)))) as proof of (((eq Prop) (f x0)) ((and ((and (forall (Xx:a) (Xy:a), (((x0 Xx) Xy)->((x0 Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xz))->((x0 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy)))))))) cP)))
% Found (((((eq_trans0 (f x0)) ((and ((and (forall (Xx:a) (Xy:a), (((x0 Xx) Xy)->((x0 Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xz))->((x0 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy)))))))) cP))) ((and ((and (forall (Xx:a) (Xy:a), (((x0 Xx) Xy)->((x0 Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xz))->((x0 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy)))))))) cP))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((and (forall (Xx:a) (Xy:a), (((x0 Xx) Xy)->((x0 Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xz))->((x0 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy)))))))) cP)))) as proof of (((eq Prop) (f x0)) ((and ((and (forall (Xx:a) (Xy:a), (((x0 Xx) Xy)->((x0 Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xz))->((x0 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy)))))))) cP)))
% Found ((((((eq_trans Prop) (f x0)) ((and ((and (forall (Xx:a) (Xy:a), (((x0 Xx) Xy)->((x0 Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xz))->((x0 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy)))))))) cP))) ((and ((and (forall (Xx:a) (Xy:a), (((x0 Xx) Xy)->((x0 Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xz))->((x0 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy)))))))) cP))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((and (forall (Xx:a) (Xy:a), (((x0 Xx) Xy)->((x0 Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xz))->((x0 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy)))))))) cP)))) as proof of (((eq Prop) (f x0)) ((and ((and (forall (Xx:a) (Xy:a), (((x0 Xx) Xy)->((x0 Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xz))->((x0 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy)))))))) cP)))
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((and (forall (Xx:a) (Xy:a), (((x0 Xx) Xy)->((x0 Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xz))->((x0 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy)))))))) cP)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (forall (Xx:a) (Xy:a), (((x0 Xx) Xy)->((x0 Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xz))->((x0 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy)))))))) cP)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (forall (Xx:a) (Xy:a), (((x0 Xx) Xy)->((x0 Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xz))->((x0 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy)))))))) cP)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (forall (Xx:a) (Xy:a), (((x0 Xx) Xy)->((x0 Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xz))->((x0 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy)))))))) cP)))
% Found ((eq_trans0000 ((eq_ref Prop) (f x0))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f x0)) ((and ((and (forall (Xx:a) (Xy:a), (((x0 Xx) Xy)->((x0 Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xz))->((x0 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy)))))))) cP)))
% Found (((eq_trans000 ((and ((and (forall (Xx:a) (Xy:a), (((x0 Xx) Xy)->((x0 Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xz))->((x0 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy)))))))) cP))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f x0)) ((and ((and (forall (Xx:a) (Xy:a), (((x0 Xx) Xy)->((x0 Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xz))->((x0 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy)))))))) cP)))
% Found ((((eq_trans00 ((and ((and (forall (Xx:a) (Xy:a), (((x0 Xx) Xy)->((x0 Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xz))->((x0 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy)))))))) cP))) ((and ((and (forall (Xx:a) (Xy:a), (((x0 Xx) Xy)->((x0 Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xz))->((x0 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy)))))))) cP))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((and (forall (Xx:a) (Xy:a), (((x0 Xx) Xy)->((x0 Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xz))->((x0 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy)))))))) cP)))) as proof of (((eq Prop) (f x0)) ((and ((and (forall (Xx:a) (Xy:a), (((x0 Xx) Xy)->((x0 Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xz))->((x0 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy)))))))) cP)))
% Found (((((eq_trans0 (f x0)) ((and ((and (forall (Xx:a) (Xy:a), (((x0 Xx) Xy)->((x0 Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xz))->((x0 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy)))))))) cP))) ((and ((and (forall (Xx:a) (Xy:a), (((x0 Xx) Xy)->((x0 Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xz))->((x0 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy)))))))) cP))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((and (forall (Xx:a) (Xy:a), (((x0 Xx) Xy)->((x0 Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xz))->((x0 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy)))))))) cP)))) as proof of (((eq Prop) (f x0)) ((and ((and (forall (Xx:a) (Xy:a), (((x0 Xx) Xy)->((x0 Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xz))->((x0 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy)))))))) cP)))
% Found ((((((eq_trans Prop) (f x0)) ((and ((and (forall (Xx:a) (Xy:a), (((x0 Xx) Xy)->((x0 Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xz))->((x0 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy)))))))) cP))) ((and ((and (forall (Xx:a) (Xy:a), (((x0 Xx) Xy)->((x0 Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xz))->((x0 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy)))))))) cP))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((and (forall (Xx:a) (Xy:a), (((x0 Xx) Xy)->((x0 Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xz))->((x0 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy)))))))) cP)))) as proof of (((eq Prop) (f x0)) ((and ((and (forall (Xx:a) (Xy:a), (((x0 Xx) Xy)->((x0 Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xz))->((x0 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy)))))))) 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) 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 Xx) Xy)))))))) b)
% Found eq_sym:=(fun (T:Type) (a:T) (b:T) (H:(((eq T) a) b))=> ((H (fun (x:T)=> (((eq T) x) a))) ((eq_ref T) a))):(forall (T:Type) (a:T) (b:T), ((((eq T) a) b)->(((eq T) b) a)))
% Instantiate: b:=(forall (T:Type) (a:T) (b:T), ((((eq T) a) b)->(((eq T) b) a))):Prop
% Found eq_sym as proof of b
% Found eq_ref000:=(eq_ref00 P0):((P0 (f x0))->(P0 (f x0)))
% Found (eq_ref00 P0) as proof of (P1 (f x0))
% Found ((eq_ref0 (f x0)) P0) as proof of (P1 (f x0))
% Found (((eq_ref Prop) (f x0)) P0) as proof of (P1 (f x0))
% Found (((eq_ref Prop) (f x0)) P0) as proof of (P1 (f x0))
% Found eq_ref000:=(eq_ref00 P0):((P0 (f x0))->(P0 (f x0)))
% Found (eq_ref00 P0) as proof of (P1 (f x0))
% Found ((eq_ref0 (f x0)) P0) as proof of (P1 (f x0))
% Found (((eq_ref Prop) (f x0)) P0) as proof of (P1 (f x0))
% Found (((eq_ref Prop) (f x0)) P0) as proof of (P1 (f x0))
% 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x2 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)) ((x2 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)) ((x2 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)) ((x2 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)) ((x2 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)) ((x2 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)) ((x2 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)) ((x2 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)) ((x2 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)) ((x2 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)) ((x2 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)) ((x2 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)) ((x2 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)) ((x2 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)) ((x2 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)) ((x2 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 Xx) Xy))))))))
% Found eq_ref00:=(eq_ref0 (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xz))->((x0 Xx) Xz)))):(((eq Prop) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xz))->((x0 Xx) Xz)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xz))->((x0 Xx) Xz))))
% Found (eq_ref0 (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xz))->((x0 Xx) Xz)))) as proof of (((eq Prop) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xz))->((x0 Xx) Xz)))) b)
% Found ((eq_ref Prop) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xz))->((x0 Xx) Xz)))) as proof of (((eq Prop) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xz))->((x0 Xx) Xz)))) b)
% Found ((eq_ref Prop) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xz))->((x0 Xx) Xz)))) as proof of (((eq Prop) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xz))->((x0 Xx) Xz)))) b)
% Found ((eq_ref Prop) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xz))->((x0 Xx) Xz)))) as proof of (((eq Prop) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xz))->((x0 Xx) Xz)))) 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) ((and ((ex a) (fun (Xz:a)=> (x3 Xz)))) (forall (Xx:a), ((x3 Xx)->(forall (Xy:a), ((iff (x3 Xy)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 Xx) Xy)))))))
% 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)) ((x0 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)) ((x0 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 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 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) b)
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) b)
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) b)
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((and (forall (Xx:a) (Xy:a), (((x2 Xx) Xy)->((x2 Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xz))->((x2 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x2 Xx) Xy)))))))) cP)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (forall (Xx:a) (Xy:a), (((x2 Xx) Xy)->((x2 Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xz))->((x2 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x2 Xx) Xy)))))))) cP)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (forall (Xx:a) (Xy:a), (((x2 Xx) Xy)->((x2 Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xz))->((x2 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x2 Xx) Xy)))))))) cP)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (forall (Xx:a) (Xy:a), (((x2 Xx) Xy)->((x2 Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xz))->((x2 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x2 Xx) Xy)))))))) cP)))
% Found ((eq_trans0000 ((eq_ref Prop) (f x2))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f x2)) ((and ((and (forall (Xx:a) (Xy:a), (((x2 Xx) Xy)->((x2 Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xz))->((x2 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x2 Xx) Xy)))))))) cP)))
% Found (((eq_trans000 ((and ((and (forall (Xx:a) (Xy:a), (((x2 Xx) Xy)->((x2 Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xz))->((x2 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x2 Xx) Xy)))))))) cP))) ((eq_ref Prop) (f x2))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f x2)) ((and ((and (forall (Xx:a) (Xy:a), (((x2 Xx) Xy)->((x2 Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xz))->((x2 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x2 Xx) Xy)))))))) cP)))
% Found ((((eq_trans00 ((and ((and (forall (Xx:a) (Xy:a), (((x2 Xx) Xy)->((x2 Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xz))->((x2 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x2 Xx) Xy)))))))) cP))) ((and ((and (forall (Xx:a) (Xy:a), (((x2 Xx) Xy)->((x2 Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xz))->((x2 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x2 Xx) Xy)))))))) cP))) ((eq_ref Prop) (f x2))) ((eq_ref Prop) ((and ((and (forall (Xx:a) (Xy:a), (((x2 Xx) Xy)->((x2 Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xz))->((x2 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x2 Xx) Xy)))))))) cP)))) as proof of (((eq Prop) (f x2)) ((and ((and (forall (Xx:a) (Xy:a), (((x2 Xx) Xy)->((x2 Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xz))->((x2 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x2 Xx) Xy)))))))) cP)))
% Found (((((eq_trans0 (f x2)) ((and ((and (forall (Xx:a) (Xy:a), (((x2 Xx) Xy)->((x2 Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xz))->((x2 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x2 Xx) Xy)))))))) cP))) ((and ((and (forall (Xx:a) (Xy:a), (((x2 Xx) Xy)->((x2 Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xz))->((x2 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x2 Xx) Xy)))))))) cP))) ((eq_ref Prop) (f x2))) ((eq_ref Prop) ((and ((and (forall (Xx:a) (Xy:a), (((x2 Xx) Xy)->((x2 Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xz))->((x2 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x2 Xx) Xy)))))))) cP)))) as proof of (((eq Prop) (f x2)) ((and ((and (forall (Xx:a) (Xy:a), (((x2 Xx) Xy)->((x2 Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xz))->((x2 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x2 Xx) Xy)))))))) cP)))
% Found ((((((eq_trans Prop) (f x2)) ((and ((and (forall (Xx:a) (Xy:a), (((x2 Xx) Xy)->((x2 Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xz))->((x2 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x2 Xx) Xy)))))))) cP))) ((and ((and (forall (Xx:a) (Xy:a), (((x2 Xx) Xy)->((x2 Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xz))->((x2 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x2 Xx) Xy)))))))) cP))) ((eq_ref Prop) (f x2))) ((eq_ref Prop) ((and ((and (forall (Xx:a) (Xy:a), (((x2 Xx) Xy)->((x2 Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xz))->((x2 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x2 Xx) Xy)))))))) cP)))) as proof of (((eq Prop) (f x2)) ((and ((and (forall (Xx:a) (Xy:a), (((x2 Xx) Xy)->((x2 Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xz))->((x2 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x2 Xx) Xy)))))))) cP)))
% 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)) b)
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) b)
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) b)
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((and (forall (Xx:a) (Xy:a), (((x2 Xx) Xy)->((x2 Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xz))->((x2 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x2 Xx) Xy)))))))) cP)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (forall (Xx:a) (Xy:a), (((x2 Xx) Xy)->((x2 Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xz))->((x2 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x2 Xx) Xy)))))))) cP)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (forall (Xx:a) (Xy:a), (((x2 Xx) Xy)->((x2 Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xz))->((x2 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x2 Xx) Xy)))))))) cP)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (forall (Xx:a) (Xy:a), (((x2 Xx) Xy)->((x2 Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xz))->((x2 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x2 Xx) Xy)))))))) cP)))
% Found ((eq_trans0000 ((eq_ref Prop) (f x2))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f x2)) ((and ((and (forall (Xx:a) (Xy:a), (((x2 Xx) Xy)->((x2 Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xz))->((x2 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x2 Xx) Xy)))))))) cP)))
% Found (((eq_trans000 ((and ((and (forall (Xx:a) (Xy:a), (((x2 Xx) Xy)->((x2 Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xz))->((x2 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x2 Xx) Xy)))))))) cP))) ((eq_ref Prop) (f x2))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f x2)) ((and ((and (forall (Xx:a) (Xy:a), (((x2 Xx) Xy)->((x2 Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xz))->((x2 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x2 Xx) Xy)))))))) cP)))
% Found ((((eq_trans00 ((and ((and (forall (Xx:a) (Xy:a), (((x2 Xx) Xy)->((x2 Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xz))->((x2 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x2 Xx) Xy)))))))) cP))) ((and ((and (forall (Xx:a) (Xy:a), (((x2 Xx) Xy)->((x2 Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xz))->((x2 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x2 Xx) Xy)))))))) cP))) ((eq_ref Prop) (f x2))) ((eq_ref Prop) ((and ((and (forall (Xx:a) (Xy:a), (((x2 Xx) Xy)->((x2 Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xz))->((x2 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x2 Xx) Xy)))))))) cP)))) as proof of (((eq Prop) (f x2)) ((and ((and (forall (Xx:a) (Xy:a), (((x2 Xx) Xy)->((x2 Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xz))->((x2 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x2 Xx) Xy)))))))) cP)))
% Found (((((eq_trans0 (f x2)) ((and ((and (forall (Xx:a) (Xy:a), (((x2 Xx) Xy)->((x2 Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xz))->((x2 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x2 Xx) Xy)))))))) cP))) ((and ((and (forall (Xx:a) (Xy:a), (((x2 Xx) Xy)->((x2 Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xz))->((x2 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x2 Xx) Xy)))))))) cP))) ((eq_ref Prop) (f x2))) ((eq_ref Prop) ((and ((and (forall (Xx:a) (Xy:a), (((x2 Xx) Xy)->((x2 Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xz))->((x2 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x2 Xx) Xy)))))))) cP)))) as proof of (((eq Prop) (f x2)) ((and ((and (forall (Xx:a) (Xy:a), (((x2 Xx) Xy)->((x2 Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xz))->((x2 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x2 Xx) Xy)))))))) cP)))
% Found ((((((eq_trans Prop) (f x2)) ((and ((and (forall (Xx:a) (Xy:a), (((x2 Xx) Xy)->((x2 Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xz))->((x2 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x2 Xx) Xy)))))))) cP))) ((and ((and (forall (Xx:a) (Xy:a), (((x2 Xx) Xy)->((x2 Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xz))->((x2 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x2 Xx) Xy)))))))) cP))) ((eq_ref Prop) (f x2))) ((eq_ref Prop) ((and ((and (forall (Xx:a) (Xy:a), (((x2 Xx) Xy)->((x2 Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xz))->((x2 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x2 Xx) Xy)))))))) cP)))) as proof of (((eq Prop) (f x2)) ((and ((and (forall (Xx:a) (Xy:a), (((x2 Xx) Xy)->((x2 Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xz))->((x2 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x2 Xx) Xy)))))))) 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) 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)) ((x2 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)) ((x2 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)) ((x2 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)) ((x2 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)) ((x2 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)) ((x2 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)) ((x2 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)) ((x2 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)) ((x2 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)) ((x2 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)) ((x2 Xx) Xy)))))))) b)
% 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)) ((x0 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)) ((x0 Xx) Xy))))))):((a->Prop)->Prop)
% Found x1 as proof of (P0 f)
% 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)) ((x0 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)) ((x0 Xx) Xy))))))):((a->Prop)->Prop)
% Found x1 as proof of (P0 f)
% Found eq_ref000:=(eq_ref00 P0):((P0 (f x2))->(P0 (f x2)))
% Found (eq_ref00 P0) as proof of (P1 (f x2))
% Found ((eq_ref0 (f x2)) P0) as proof of (P1 (f x2))
% Found (((eq_ref Prop) (f x2)) P0) as proof of (P1 (f x2))
% Found (((eq_ref Prop) (f x2)) P0) as proof of (P1 (f x2))
% Found eq_ref000:=(eq_ref00 P0):((P0 (f x2))->(P0 (f x2)))
% Found (eq_ref00 P0) as proof of (P1 (f x2))
% Found ((eq_ref0 (f x2)) P0) as proof of (P1 (f x2))
% Found (((eq_ref Prop) (f x2)) P0) as proof of (P1 (f x2))
% Found (((eq_ref Prop) (f x2)) P0) as proof of (P1 (f x2))
% 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)) ((x2 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)) ((x2 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)) ((x2 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)) ((x2 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)) ((x2 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)) ((x2 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)) ((x2 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)) ((x2 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)) ((x2 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)) ((x2 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)) ((x2 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 Xx) Xy)))))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq ((a->Prop)->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq ((a->Prop)->Prop)) b) 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 Xx) Xy)))))))) b)
% 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 (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_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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 Xx) Xy))))))))
% Found eq_ref00:=(eq_ref0 (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xz))->((x2 Xx) Xz)))):(((eq Prop) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xz))->((x2 Xx) Xz)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xz))->((x2 Xx) Xz))))
% Found (eq_ref0 (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xz))->((x2 Xx) Xz)))) as proof of (((eq Prop) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xz))->((x2 Xx) Xz)))) b)
% Found ((eq_ref Prop) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xz))->((x2 Xx) Xz)))) as proof of (((eq Prop) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xz))->((x2 Xx) Xz)))) b)
% Found ((eq_ref Prop) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xz))->((x2 Xx) Xz)))) as proof of (((eq Prop) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xz))->((x2 Xx) Xz)))) b)
% Found ((eq_ref Prop) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xz))->((x2 Xx) Xz)))) as proof of (((eq Prop) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xz))->((x2 Xx) Xz)))) 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)) ((x0 Xx) Xy))))))):(((eq Prop) ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((x0 Xx) Xy))))))) ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((x0 Xx) Xy)))))))
% Found (eq_ref0 ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 Xx) Xy))))))):(((eq Prop) ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((x0 Xx) Xy))))))) ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((x0 Xx) Xy)))))))
% Found (eq_ref0 ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x2 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)) ((x2 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)) ((x2 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)) ((x2 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)) ((x2 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)) ((x2 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)) ((x2 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)) ((x2 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)) ((x2 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)) ((x2 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)) ((x2 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)) ((x2 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)) ((x2 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)) ((x2 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)) ((x2 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)) ((x2 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)) ((x2 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)) ((x2 Xx) Xy))))))))
% 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)) ((x2 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)) ((x2 Xx) Xy))))))):((a->Prop)->Prop)
% Found x3 as proof of (P0 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 (x5:(a->Prop))=> Prop)) cP) as proof of (((eq ((a->Prop)->Prop)) cP) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x5:(a->Prop))=> Prop)) cP) as proof of (((eq ((a->Prop)->Prop)) cP) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x5:(a->Prop))=> Prop)) cP) as proof of (((eq ((a->Prop)->Prop)) cP) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x5:(a->Prop))=> Prop)) cP) as proof of (((eq ((a->Prop)->Prop)) cP) b)
% Found iff_refl:=(fun (A:Prop)=> ((((conj (A->A)) (A->A)) (fun (H:A)=> H)) (fun (H:A)=> H))):(forall (P:Prop), ((iff P) P))
% Instantiate: b:=(forall (P:Prop), ((iff P) P)):Prop
% Found iff_refl as proof of b
% Found eq_ref00:=(eq_ref0 (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xz))->((x0 Xx) Xz)))):(((eq Prop) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xz))->((x0 Xx) Xz)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xz))->((x0 Xx) Xz))))
% Found (eq_ref0 (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xz))->((x0 Xx) Xz)))) as proof of (((eq Prop) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xz))->((x0 Xx) Xz)))) b)
% Found ((eq_ref Prop) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xz))->((x0 Xx) Xz)))) as proof of (((eq Prop) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xz))->((x0 Xx) Xz)))) b)
% Found ((eq_ref Prop) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xz))->((x0 Xx) Xz)))) as proof of (((eq Prop) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xz))->((x0 Xx) Xz)))) b)
% Found ((eq_ref Prop) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xz))->((x0 Xx) Xz)))) as proof of (((eq Prop) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xz))->((x0 Xx) Xz)))) 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 Xx) Xy))))))))
% 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)) ((x0 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)) ((x0 Xx) Xy))))))):((a->Prop)->Prop)
% Found x3 as proof of (P0 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_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_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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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)) ((x0 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 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 (x5:(a->Prop))=> Prop)) cP) as proof of (((eq ((a->Prop)->Prop)) cP) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x5:(a->Prop))=> Prop)) cP) as proof of (((eq ((a->Prop)->Prop)) cP) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x5:(a->Prop))=> Prop)) cP) as proof of (((eq ((a->Prop)->Prop)) cP) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x5:(a->Prop))=> Prop)) cP) as proof of (((eq ((a->Prop)->Prop)) cP) b)
% Found eq_ref00:=(eq_r
% EOF
%------------------------------------------------------------------------------