%------------------------------------------------------------------------------
% File     : cocATP---0.2.0
% Problem  : SEV018^5 : TPTP v6.1.0. Released v4.0.0.
% Transfm  : none
% Format   : tptp:raw
% Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p

% Computer : n106.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 300.00s
% Output   : None 
% Verified : 
% SZS Type : None (Parsing solution fails)
% Syntax   : Number of formulae    : 0

% Comments : 
%------------------------------------------------------------------------------
%----NO SOLUTION OUTPUT BY SYSTEM
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% % Problem  : SEV018^5 : TPTP v6.1.0. Released v4.0.0.
% % Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n106.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 07:30:36 CDT 2014
% % CPUTime  : 300.00 
% 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 0x1b48680>, <kernel.Type object at 0x1b48cf8>) of role type named a_type
% Using role type
% Declaring a:Type
% FOF formula (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy)))))))) P))))) of role conjecture named cTHM262A_pme
% Conjecture to prove = (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy)))))))) P))))):Prop
% Parameter a_DUMMY:a.
% We need to prove ['(forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy)))))))) P)))))']
% Parameter a:Type.
% Trying to prove (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy)))))))) P)))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Q:(a->(a->Prop)))=> (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy)))))))) P))):(((eq ((a->(a->Prop))->Prop)) (fun (Q:(a->(a->Prop)))=> (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy)))))))) P))) (fun (x:(a->(a->Prop)))=> (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x Xx) Xy)))))))) P)))
% Found (eta_expansion_dep00 (fun (Q:(a->(a->Prop)))=> (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy)))))))) P))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (Q:(a->(a->Prop)))=> (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy)))))))) P))) b)
% Found ((eta_expansion_dep0 (fun (x1:(a->(a->Prop)))=> Prop)) (fun (Q:(a->(a->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)) ((Q Xx) Xy)))))))) P))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (Q:(a->(a->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)) ((Q Xx) Xy)))))))) P))) b)
% Found (((eta_expansion_dep (a->(a->Prop))) (fun (x1:(a->(a->Prop)))=> Prop)) (fun (Q:(a->(a->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)) ((Q Xx) Xy)))))))) P))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (Q:(a->(a->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)) ((Q Xx) Xy)))))))) P))) b)
% Found (((eta_expansion_dep (a->(a->Prop))) (fun (x1:(a->(a->Prop)))=> Prop)) (fun (Q:(a->(a->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)) ((Q Xx) Xy)))))))) P))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (Q:(a->(a->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)) ((Q Xx) Xy)))))))) P))) b)
% Found (((eta_expansion_dep (a->(a->Prop))) (fun (x1:(a->(a->Prop)))=> Prop)) (fun (Q:(a->(a->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)) ((Q Xx) Xy)))))))) P))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (Q:(a->(a->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)) ((Q Xx) Xy)))))))) P))) b)
% 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)) (((eq ((a->Prop)->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))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (((eq ((a->Prop)->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))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (((eq ((a->Prop)->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))
% Found (fun (x0:(a->(a->Prop)))=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (((eq ((a->Prop)->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))
% Found (fun (x0:(a->(a->Prop)))=> ((eq_ref Prop) (f x0))) as proof of (forall (x:(a->(a->Prop))), (((eq Prop) (f x)) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x Xx) Xy)))))))) P)))
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) (((eq ((a->Prop)->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))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (((eq ((a->Prop)->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))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (((eq ((a->Prop)->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))
% Found (fun (x0:(a->(a->Prop)))=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (((eq ((a->Prop)->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))
% Found (fun (x0:(a->(a->Prop)))=> ((eq_ref Prop) (f x0))) as proof of (forall (x:(a->(a->Prop))), (((eq Prop) (f x)) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x Xx) Xy)))))))) P)))
% Found eta_expansion000:=(eta_expansion00 (fun (Q:(a->(a->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)) ((Q Xx) Xy)))))))) P))):(((eq ((a->(a->Prop))->Prop)) (fun (Q:(a->(a->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)) ((Q Xx) Xy)))))))) P))) (fun (x:(a->(a->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)) ((x Xx) Xy)))))))) P)))
% Found (eta_expansion00 (fun (Q:(a->(a->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)) ((Q Xx) Xy)))))))) P))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (Q:(a->(a->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)) ((Q Xx) Xy)))))))) P))) b)
% Found ((eta_expansion0 Prop) (fun (Q:(a->(a->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)) ((Q Xx) Xy)))))))) P))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (Q:(a->(a->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)) ((Q Xx) Xy)))))))) P))) b)
% Found (((eta_expansion (a->(a->Prop))) Prop) (fun (Q:(a->(a->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)) ((Q Xx) Xy)))))))) P))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (Q:(a->(a->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)) ((Q Xx) Xy)))))))) P))) b)
% Found (((eta_expansion (a->(a->Prop))) Prop) (fun (Q:(a->(a->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)) ((Q Xx) Xy)))))))) P))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (Q:(a->(a->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)) ((Q Xx) Xy)))))))) P))) b)
% Found (((eta_expansion (a->(a->Prop))) Prop) (fun (Q:(a->(a->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)) ((Q Xx) Xy)))))))) P))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (Q:(a->(a->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)) ((Q Xx) Xy)))))))) P))) b)
% Found eq_ref000:=(eq_ref00 P0):((P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy))))))))->(P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy)))))))))
% Found (eq_ref00 P0) as proof of (P00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((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)))))))) P0) as proof of (P00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((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)))))))) P0) as proof of (P00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((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)))))))) P0) as proof of (P00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((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) P)
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) P)
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) P)
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) P)
% Found eq_ref00:=(eq_ref0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((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 eta_expansion_dep0000:=(eta_expansion_dep000 P0):((P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy))))))))->(P0 (fun (x:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (x Xz)))) (forall (Xx:a), ((x Xx)->(forall (Xy:a), ((iff (x Xy)) ((x0 Xx) Xy)))))))))
% Found (eta_expansion_dep000 P0) as proof of (P00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((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)))))))) P0) as proof of (P00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy))))))))
% Found (((eta_expansion_dep0 (fun (x2:(a->Prop))=> Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy)))))))) P0) as proof of (P00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy))))))))
% Found ((((eta_expansion_dep (a->Prop)) (fun (x2:(a->Prop))=> Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy)))))))) P0) as proof of (P00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy))))))))
% Found ((((eta_expansion_dep (a->Prop)) (fun (x2:(a->Prop))=> Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy)))))))) P0) as proof of (P00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy))))))))
% Found eq_ref000:=(eq_ref00 P0):((P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy))))))))->(P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy)))))))))
% Found (eq_ref00 P0) as proof of (P00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((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)))))))) P0) as proof of (P00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((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)))))))) P0) as proof of (P00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((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)))))))) P0) as proof of (P00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy))))))))
% 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)) (((eq ((a->Prop)->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))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (((eq ((a->Prop)->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))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (((eq ((a->Prop)->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))
% Found (fun (x2:(a->(a->Prop)))=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (((eq ((a->Prop)->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))
% Found (fun (x2:(a->(a->Prop)))=> ((eq_ref Prop) (f x2))) as proof of (forall (x:(a->(a->Prop))), (((eq Prop) (f x)) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x Xx) Xy)))))))) P)))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (((eq ((a->Prop)->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))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (((eq ((a->Prop)->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))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (((eq ((a->Prop)->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))
% Found (fun (x2:(a->(a->Prop)))=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (((eq ((a->Prop)->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))
% Found (fun (x2:(a->(a->Prop)))=> ((eq_ref Prop) (f x2))) as proof of (forall (x:(a->(a->Prop))), (((eq Prop) (f x)) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x Xx) Xy)))))))) P)))
% Found eq_ref000:=(eq_ref00 P0):((P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x2 Xx) Xy))))))))->(P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x2 Xx) Xy)))))))))
% Found (eq_ref00 P0) as proof of (P00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((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)))))))) P0) as proof of (P00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((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)))))))) P0) as proof of (P00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((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)))))))) P0) as proof of (P00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((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) P)
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) P)
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) P)
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) P)
% Found eta_expansion000:=(eta_expansion00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((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_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)))))))) 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_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)))))))) 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 (a->Prop)) 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 (a->Prop)) 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 (a->Prop)) 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 P0):((P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x2 Xx) Xy))))))))->(P0 (fun (x:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (x Xz)))) (forall (Xx:a), ((x Xx)->(forall (Xy:a), ((iff (x Xy)) ((x2 Xx) Xy)))))))))
% Found (eta_expansion000 P0) as proof of (P00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((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)))))))) P0) as proof of (P00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((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)))))))) P0) as proof of (P00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((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)))))))) P0) as proof of (P00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((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)))))))) P0) as proof of (P00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x2 Xx) Xy))))))))
% Found eq_ref000:=(eq_ref00 P0):((P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x2 Xx) Xy))))))))->(P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x2 Xx) Xy)))))))))
% Found (eq_ref00 P0) as proof of (P00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((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)))))))) P0) as proof of (P00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((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)))))))) P0) as proof of (P00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((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)))))))) P0) as proof of (P00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x2 Xx) Xy))))))))
% Found eq_ref000:=(eq_ref00 P0):((P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy))))))))->(P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy)))))))))
% Found (eq_ref00 P0) as proof of (P00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((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)))))))) P0) as proof of (P00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((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)))))))) P0) as proof of (P00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((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)))))))) P0) as proof of (P00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((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) P)
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) P)
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) P)
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) P)
% Found eta_expansion000:=(eta_expansion00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((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_expansion00 (fun (Xs:(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_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)) ((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 (a->Prop)) 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 (a->Prop)) 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 (a->Prop)) 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) (fun (Xs:(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_expansion000:=(eta_expansion00 P):(((eq ((a->Prop)->Prop)) P) (fun (x:(a->Prop))=> (P x)))
% Found (eta_expansion00 P) as proof of (((eq ((a->Prop)->Prop)) P) b)
% Found ((eta_expansion0 Prop) P) as proof of (((eq ((a->Prop)->Prop)) P) b)
% Found (((eta_expansion (a->Prop)) Prop) P) as proof of (((eq ((a->Prop)->Prop)) P) b)
% Found (((eta_expansion (a->Prop)) Prop) P) as proof of (((eq ((a->Prop)->Prop)) P) b)
% Found (((eta_expansion (a->Prop)) Prop) P) as proof of (((eq ((a->Prop)->Prop)) P) b)
% Found eq_ref000:=(eq_ref00 P0):((P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy))))))))->(P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy)))))))))
% Found (eq_ref00 P0) as proof of (P00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((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)))))))) P0) as proof of (P00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((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)))))))) P0) as proof of (P00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((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)))))))) P0) as proof of (P00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy))))))))
% Found eta_expansion0000:=(eta_expansion000 P0):((P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy))))))))->(P0 (fun (x:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (x Xz)))) (forall (Xx:a), ((x Xx)->(forall (Xy:a), ((iff (x Xy)) ((x0 Xx) Xy)))))))))
% Found (eta_expansion000 P0) as proof of (P00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 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)) ((x0 Xx) Xy)))))))) P0) as proof of (P00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 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)) ((x0 Xx) Xy)))))))) P0) as proof of (P00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 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)) ((x0 Xx) Xy)))))))) P0) as proof of (P00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 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)) ((x0 Xx) Xy)))))))) P0) as proof of (P00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy))))))))
% Found eq_ref000:=(eq_ref00 P0):((P0 P)->(P0 P))
% Found (eq_ref00 P0) as proof of (P00 P)
% Found ((eq_ref0 P) P0) as proof of (P00 P)
% Found (((eq_ref ((a->Prop)->Prop)) P) P0) as proof of (P00 P)
% Found (((eq_ref ((a->Prop)->Prop)) P) P0) as proof of (P00 P)
% Found eq_ref000:=(eq_ref00 P0):((P0 P)->(P0 P))
% Found (eq_ref00 P0) as proof of (P00 P)
% Found ((eq_ref0 P) P0) as proof of (P00 P)
% Found (((eq_ref ((a->Prop)->Prop)) P) P0) as proof of (P00 P)
% Found (((eq_ref ((a->Prop)->Prop)) P) P0) as proof of (P00 P)
% Found eq_ref000:=(eq_ref00 P0):((P0 ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((x0 Xx) Xy)))))))->(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 P0) as proof of (P00 ((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))))))) P0) as proof of (P00 ((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))))))) P0) as proof of (P00 ((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))))))) P0) as proof of (P00 ((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 P0):((P0 ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((x0 Xx) Xy)))))))->(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 P0) as proof of (P00 ((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))))))) P0) as proof of (P00 ((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))))))) P0) as proof of (P00 ((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))))))) P0) as proof of (P00 ((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 ((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) (P x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (P x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (P x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (P 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) (P x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (P x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (P x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (P x1))
% Found eq_ref00:=(eq_ref0 b):(((eq ((a->Prop)->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq ((a->Prop)->Prop)) b) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((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 P):(((eq ((a->Prop)->Prop)) P) (fun (x:(a->Prop))=> (P x)))
% Found (eta_expansion_dep00 P) as proof of (((eq ((a->Prop)->Prop)) P) b)
% Found ((eta_expansion_dep0 (fun (x4:(a->Prop))=> Prop)) P) as proof of (((eq ((a->Prop)->Prop)) P) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x4:(a->Prop))=> Prop)) P) as proof of (((eq ((a->Prop)->Prop)) P) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x4:(a->Prop))=> Prop)) P) as proof of (((eq ((a->Prop)->Prop)) P) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x4:(a->Prop))=> Prop)) P) as proof of (((eq ((a->Prop)->Prop)) P) b)
% Found eq_ref000:=(eq_ref00 P0):((P0 P)->(P0 P))
% Found (eq_ref00 P0) as proof of (P00 P)
% Found ((eq_ref0 P) P0) as proof of (P00 P)
% Found (((eq_ref ((a->Prop)->Prop)) P) P0) as proof of (P00 P)
% Found (((eq_ref ((a->Prop)->Prop)) P) P0) as proof of (P00 P)
% Found eq_ref000:=(eq_ref00 P0):((P0 P)->(P0 P))
% Found (eq_ref00 P0) as proof of (P00 P)
% Found ((eq_ref0 P) P0) as proof of (P00 P)
% Found (((eq_ref ((a->Prop)->Prop)) P) P0) as proof of (P00 P)
% Found (((eq_ref ((a->Prop)->Prop)) P) P0) as proof of (P00 P)
% Found eq_ref000:=(eq_ref00 P0):((P0 ((and ((ex a) (fun (Xz:a)=> (x3 Xz)))) (forall (Xx:a), ((x3 Xx)->(forall (Xy:a), ((iff (x3 Xy)) ((x2 Xx) Xy)))))))->(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 P0) as proof of (P00 ((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))))))) P0) as proof of (P00 ((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))))))) P0) as proof of (P00 ((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))))))) P0) as proof of (P00 ((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 P0):((P0 ((and ((ex a) (fun (Xz:a)=> (x3 Xz)))) (forall (Xx:a), ((x3 Xx)->(forall (Xy:a), ((iff (x3 Xy)) ((x2 Xx) Xy)))))))->(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 P0) as proof of (P00 ((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))))))) P0) as proof of (P00 ((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))))))) P0) as proof of (P00 ((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))))))) P0) as proof of (P00 ((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 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 P):(((eq ((a->Prop)->Prop)) P) (fun (x:(a->Prop))=> (P x)))
% Found (eta_expansion_dep00 P) as proof of (((eq ((a->Prop)->Prop)) P) b)
% Found ((eta_expansion_dep0 (fun (x4:(a->Prop))=> Prop)) P) as proof of (((eq ((a->Prop)->Prop)) P) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x4:(a->Prop))=> Prop)) P) as proof of (((eq ((a->Prop)->Prop)) P) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x4:(a->Prop))=> Prop)) P) as proof of (((eq ((a->Prop)->Prop)) P) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x4:(a->Prop))=> Prop)) P) as proof of (((eq ((a->Prop)->Prop)) P) 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) (P x3))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (P x3))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (P x3))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (P 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) (P x3))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (P x3))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (P x3))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (P x3))
% Found eq_ref000:=(eq_ref00 P0):((P0 P)->(P0 P))
% Found (eq_ref00 P0) as proof of (P00 P)
% Found ((eq_ref0 P) P0) as proof of (P00 P)
% Found (((eq_ref ((a->Prop)->Prop)) P) P0) as proof of (P00 P)
% Found (((eq_ref ((a->Prop)->Prop)) P) P0) as proof of (P00 P)
% Found eq_ref000:=(eq_ref00 P0):((P0 P)->(P0 P))
% Found (eq_ref00 P0) as proof of (P00 P)
% Found ((eq_ref0 P) P0) as proof of (P00 P)
% Found (((eq_ref ((a->Prop)->Prop)) P) P0) as proof of (P00 P)
% Found (((eq_ref ((a->Prop)->Prop)) P) P0) as proof of (P00 P)
% Found eq_ref000:=(eq_ref00 P0):((P0 (P x1))->(P0 (P x1)))
% Found (eq_ref00 P0) as proof of (P00 (P x1))
% Found ((eq_ref0 (P x1)) P0) as proof of (P00 (P x1))
% Found (((eq_ref Prop) (P x1)) P0) as proof of (P00 (P x1))
% Found (((eq_ref Prop) (P x1)) P0) as proof of (P00 (P x1))
% Found eq_ref000:=(eq_ref00 P0):((P0 (P x1))->(P0 (P x1)))
% Found (eq_ref00 P0) as proof of (P00 (P x1))
% Found ((eq_ref0 (P x1)) P0) as proof of (P00 (P x1))
% Found (((eq_ref Prop) (P x1)) P0) as proof of (P00 (P x1))
% Found (((eq_ref Prop) (P x1)) P0) as proof of (P00 (P x1))
% Found eq_ref000:=(eq_ref00 P0):((P0 ((and ((ex a) (fun (Xz:a)=> (x3 Xz)))) (forall (Xx:a), ((x3 Xx)->(forall (Xy:a), ((iff (x3 Xy)) ((x0 Xx) Xy)))))))->(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_ref00 P0) as proof of (P00 ((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))))))) P0) as proof of (P00 ((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))))))) P0) as proof of (P00 ((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))))))) P0) as proof of (P00 ((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 P0):((P0 ((and ((ex a) (fun (Xz:a)=> (x3 Xz)))) (forall (Xx:a), ((x3 Xx)->(forall (Xy:a), ((iff (x3 Xy)) ((x0 Xx) Xy)))))))->(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_ref00 P0) as proof of (P00 ((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))))))) P0) as proof of (P00 ((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))))))) P0) as proof of (P00 ((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))))))) P0) as proof of (P00 ((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 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 P):(((eq ((a->Prop)->Prop)) P) (fun (x:(a->Prop))=> (P x)))
% Found (eta_expansion_dep00 P) as proof of (((eq ((a->Prop)->Prop)) P) b)
% Found ((eta_expansion_dep0 (fun (x4:(a->Prop))=> Prop)) P) as proof of (((eq ((a->Prop)->Prop)) P) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x4:(a->Prop))=> Prop)) P) as proof of (((eq ((a->Prop)->Prop)) P) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x4:(a->Prop))=> Prop)) P) as proof of (((eq ((a->Prop)->Prop)) P) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x4:(a->Prop))=> Prop)) P) as proof of (((eq ((a->Prop)->Prop)) P) 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 (P x1)):(((eq Prop) (P x1)) (P x1))
% Found (eq_ref0 (P x1)) as proof of (((eq Prop) (P x1)) b)
% Found ((eq_ref Prop) (P x1)) as proof of (((eq Prop) (P x1)) b)
% Found ((eq_ref Prop) (P x1)) as proof of (((eq Prop) (P x1)) b)
% Found ((eq_ref Prop) (P x1)) as proof of (((eq Prop) (P x1)) b)
% Found eq_ref00:=(eq_ref0 (P x1)):(((eq Prop) (P x1)) (P x1))
% Found (eq_ref0 (P x1)) as proof of (((eq Prop) (P x1)) b)
% Found ((eq_ref Prop) (P x1)) as proof of (((eq Prop) (P x1)) b)
% Found ((eq_ref Prop) (P x1)) as proof of (((eq Prop) (P x1)) b)
% Found ((eq_ref Prop) (P x1)) as proof of (((eq Prop) (P 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 (P x1)):(((eq Prop) (P x1)) (P x1))
% Found (eq_ref0 (P x1)) as proof of (((eq Prop) (P x1)) b)
% Found ((eq_ref Prop) (P x1)) as proof of (((eq Prop) (P x1)) b)
% Found ((eq_ref Prop) (P x1)) as proof of (((eq Prop) (P x1)) b)
% Found ((eq_ref Prop) (P x1)) as proof of (((eq Prop) (P 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 (P x1)):(((eq Prop) (P x1)) (P x1))
% Found (eq_ref0 (P x1)) as proof of (((eq Prop) (P x1)) b)
% Found ((eq_ref Prop) (P x1)) as proof of (((eq Prop) (P x1)) b)
% Found ((eq_ref Prop) (P x1)) as proof of (((eq Prop) (P x1)) b)
% Found ((eq_ref Prop) (P x1)) as proof of (((eq Prop) (P 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 ((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) (P x3))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (P x3))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (P x3))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (P 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) (P x3))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (P x3))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (P x3))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (P x3))
% Found eq_ref000:=(eq_ref00 P0):((P0 P)->(P0 P))
% Found (eq_ref00 P0) as proof of (P00 P)
% Found ((eq_ref0 P) P0) as proof of (P00 P)
% Found (((eq_ref ((a->Prop)->Prop)) P) P0) as proof of (P00 P)
% Found (((eq_ref ((a->Prop)->Prop)) P) P0) as proof of (P00 P)
% Found eq_ref000:=(eq_ref00 P0):((P0 P)->(P0 P))
% Found (eq_ref00 P0) as proof of (P00 P)
% Found ((eq_ref0 P) P0) as proof of (P00 P)
% Found (((eq_ref ((a->Prop)->Prop)) P) P0) as proof of (P00 P)
% Found (((eq_ref ((a->Prop)->Prop)) P) P0) as proof of (P00 P)
% Found eq_ref000:=(eq_ref00 P0):((P0 ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((x0 Xx) Xy)))))))->(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 P0) as proof of (P00 ((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))))))) P0) as proof of (P00 ((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))))))) P0) as proof of (P00 ((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))))))) P0) as proof of (P00 ((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 P0):((P0 ((and ((ex a) (fun (Xz:a)=> (x1 Xz)))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((x0 Xx) Xy)))))))->(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 P0) as proof of (P00 ((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))))))) P0) as proof of (P00 ((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))))))) P0) as proof of (P00 ((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))))))) P0) as proof of (P00 ((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 ((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) (P x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (P x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (P x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (P 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) (P x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (P x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (P x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (P x1))
% Found eq_ref000:=(eq_ref00 P0):((P0 (P x3))->(P0 (P x3)))
% Found (eq_ref00 P0) as proof of (P00 (P x3))
% Found ((eq_ref0 (P x3)) P0) as proof of (P00 (P x3))
% Found (((eq_ref Prop) (P x3)) P0) as proof of (P00 (P x3))
% Found (((eq_ref Prop) (P x3)) P0) as proof of (P00 (P x3))
% Found eq_ref000:=(eq_ref00 P0):((P0 (P x3))->(P0 (P x3)))
% Found (eq_ref00 P0) as proof of (P00 (P x3))
% Found ((eq_ref0 (P x3)) P0) as proof of (P00 (P x3))
% Found (((eq_ref Prop) (P x3)) P0) as proof of (P00 (P x3))
% Found (((eq_ref Prop) (P x3)) P0) as proof of (P00 (P 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)) ((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 (P x3)):(((eq Prop) (P x3)) (P x3))
% Found (eq_ref0 (P x3)) as proof of (((eq Prop) (P x3)) b)
% Found ((eq_ref Prop) (P x3)) as proof of (((eq Prop) (P x3)) b)
% Found ((eq_ref Prop) (P x3)) as proof of (((eq Prop) (P x3)) b)
% Found ((eq_ref Prop) (P x3)) as proof of (((eq Prop) (P x3)) b)
% Found eq_ref00:=(eq_ref0 (P x3)):(((eq Prop) (P x3)) (P x3))
% Found (eq_ref0 (P x3)) as proof of (((eq Prop) (P x3)) b)
% Found ((eq_ref Prop) (P x3)) as proof of (((eq Prop) (P x3)) b)
% Found ((eq_ref Prop) (P x3)) as proof of (((eq Prop) (P x3)) b)
% Found ((eq_ref Prop) (P x3)) as proof of (((eq Prop) (P 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 (P x3)):(((eq Prop) (P x3)) (P x3))
% Found (eq_ref0 (P x3)) as proof of (((eq Prop) (P x3)) b)
% Found ((eq_ref Prop) (P x3)) as proof of (((eq Prop) (P x3)) b)
% Found ((eq_ref Prop) (P x3)) as proof of (((eq Prop) (P x3)) b)
% Found ((eq_ref Prop) (P x3)) as proof of (((eq Prop) (P 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 (P x3)):(((eq Prop) (P x3)) (P x3))
% Found (eq_ref0 (P x3)) as proof of (((eq Prop) (P x3)) b)
% Found ((eq_ref Prop) (P x3)) as proof of (((eq Prop) (P x3)) b)
% Found ((eq_ref Prop) (P x3)) as proof of (((eq Prop) (P x3)) b)
% Found ((eq_ref Prop) (P x3)) as proof of (((eq Prop) (P 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 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (P x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (P x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (P x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (P 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 ((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) (P x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (P x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (P x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (P x1))
% Found eq_ref000:=(eq_ref00 P0):((P0 (P x3))->(P0 (P x3)))
% Found (eq_ref00 P0) as proof of (P00 (P x3))
% Found ((eq_ref0 (P x3)) P0) as proof of (P00 (P x3))
% Found (((eq_ref Prop) (P x3)) P0) as proof of (P00 (P x3))
% Found (((eq_ref Prop) (P x3)) P0) as proof of (P00 (P x3))
% Found eq_ref000:=(eq_ref00 P0):((P0 (P x3))->(P0 (P x3)))
% Found (eq_ref00 P0) as proof of (P00 (P x3))
% Found ((eq_ref0 (P x3)) P0) as proof of (P00 (P x3))
% Found (((eq_ref Prop) (P x3)) P0) as proof of (P00 (P x3))
% Found (((eq_ref Prop) (P x3)) P0) as proof of (P00 (P 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 (P x3)):(((eq Prop) (P x3)) (P x3))
% Found (eq_ref0 (P x3)) as proof of (((eq Prop) (P x3)) b)
% Found ((eq_ref Prop) (P x3)) as proof of (((eq Prop) (P x3)) b)
% Found ((eq_ref Prop) (P x3)) as proof of (((eq Prop) (P x3)) b)
% Found ((eq_ref Prop) (P x3)) as proof of (((eq Prop) (P x3)) b)
% Found eq_ref00:=(eq_ref0 (P x3)):(((eq Prop) (P x3)) (P x3))
% Found (eq_ref0 (P x3)) as proof of (((eq Prop) (P x3)) b)
% Found ((eq_ref Prop) (P x3)) as proof of (((eq Prop) (P x3)) b)
% Found ((eq_ref Prop) (P x3)) as proof of (((eq Prop) (P x3)) b)
% Found ((eq_ref Prop) (P x3)) as proof of (((eq Prop) (P 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 (P x3)):(((eq Prop) (P x3)) (P x3))
% Found (eq_ref0 (P x3)) as proof of (((eq Prop) (P x3)) b)
% Found ((eq_ref Prop) (P x3)) as proof of (((eq Prop) (P x3)) b)
% Found ((eq_ref Prop) (P x3)) as proof of (((eq Prop) (P x3)) b)
% Found ((eq_ref Prop) (P x3)) as proof of (((eq Prop) (P 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 (P x3)):(((eq Prop) (P x3)) (P x3))
% Found (eq_ref0 (P x3)) as proof of (((eq Prop) (P x3)) b)
% Found ((eq_ref Prop) (P x3)) as proof of (((eq Prop) (P x3)) b)
% Found ((eq_ref Prop) (P x3)) as proof of (((eq Prop) (P x3)) b)
% Found ((eq_ref Prop) (P x3)) as proof of (((eq Prop) (P 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_ref000:=(eq_ref00 P0):((P0 (P x3))->(P0 (P x3)))
% Found (eq_ref00 P0) as proof of (P00 (P x3))
% Found ((eq_ref0 (P x3)) P0) as proof of (P00 (P x3))
% Found (((eq_ref Prop) (P x3)) P0) as proof of (P00 (P x3))
% Found (((eq_ref Prop) (P x3)) P0) as proof of (P00 (P x3))
% Found eq_ref000:=(eq_ref00 P0):((P0 (P x3))->(P0 (P x3)))
% Found (eq_ref00 P0) as proof of (P00 (P x3))
% Found ((eq_ref0 (P x3)) P0) as proof of (P00 (P x3))
% Found (((eq_ref Prop) (P x3)) P0) as proof of (P00 (P x3))
% Found (((eq_ref Prop) (P x3)) P0) as proof of (P00 (P x3))
% Found eq_ref00:=(eq_ref0 (P x3)):(((eq Prop) (P x3)) (P x3))
% Found (eq_ref0 (P x3)) as proof of (((eq Prop) (P x3)) b)
% Found ((eq_ref Prop) (P x3)) as proof of (((eq Prop) (P x3)) b)
% Found ((eq_ref Prop) (P x3)) as proof of (((eq Prop) (P x3)) b)
% Found ((eq_ref Prop) (P x3)) as proof of (((eq Prop) (P 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 (P x3)):(((eq Prop) (P x3)) (P x3))
% Found (eq_ref0 (P x3)) as proof of (((eq Prop) (P x3)) b)
% Found ((eq_ref Prop) (P x3)) as proof of (((eq Prop) (P x3)) b)
% Found ((eq_ref Prop) (P x3)) as proof of (((eq Prop) (P x3)) b)
% Found ((eq_ref Prop) (P x3)) as proof of (((eq Prop) (P 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 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 (P x1)):(((eq Prop) (P x1)) (P x1))
% Found (eq_ref0 (P x1)) as proof of (((eq Prop) (P x1)) b)
% Found ((eq_ref Prop) (P x1)) as proof of (((eq Prop) (P x1)) b)
% Found ((eq_ref Prop) (P x1)) as proof of (((eq Prop) (P x1)) b)
% Found ((eq_ref Prop) (P x1)) as proof of (((eq Prop) (P 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 (P x1)):(((eq Prop) (P x1)) (P x1))
% Found (eq_ref0 (P x1)) as proof of (((eq Prop) (P x1)) b)
% Found ((eq_ref Prop) (P x1)) as proof of (((eq Prop) (P x1)) b)
% Found ((eq_ref Prop) (P x1)) as proof of (((eq Prop) (P x1)) b)
% Found ((eq_ref Prop) (P x1)) as proof of (((eq Prop) (P x1)) b)
% Found eq_ref000:=(eq_ref00 P0):((P0 (P x1))->(P0 (P x1)))
% Found (eq_ref00 P0) as proof of (P00 (P x1))
% Found ((eq_ref0 (P x1)) P0) as proof of (P00 (P x1))
% Found (((eq_ref Prop) (P x1)) P0) as proof of (P00 (P x1))
% Found (((eq_ref Prop) (P x1)) P0) as proof of (P00 (P x1))
% Found eq_ref000:=(eq_ref00 P0):((P0 (P x1))->(P0 (P x1)))
% Found (eq_ref00 P0) as proof of (P00 (P x1))
% Found ((eq_ref0 (P x1)) P0) as proof of (P00 (P x1))
% Found (((eq_ref Prop) (P x1)) P0) as proof of (P00 (P x1))
% Found (((eq_ref Prop) (P x1)) P0) as proof of (P00 (P x1))
% Found eq_ref00:=(eq_ref0 (P x1)):(((eq Prop) (P x1)) (P x1))
% Found (eq_ref0 (P x1)) as proof of (((eq Prop) (P x1)) b)
% Found ((eq_ref Prop) (P x1)) as proof of 
% EOF
%------------------------------------------------------------------------------