TSTP Solution File: SEV020^5 by cocATP---0.2.0

View Problem - Process Solution 
%------------------------------------------------------------------------------
% File     : cocATP---0.2.0
% Problem  : SEV020^5 : TPTP v6.1.0. Released v4.0.0.
% Transfm  : none
% Format   : tptp:raw
% Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p

% Computer : n105.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 300.11s
% Output   : None 
% Verified : 
% SZS Type : None (Parsing solution fails)
% Syntax   : Number of formulae    : 0

% Comments : 
%------------------------------------------------------------------------------
%----NO SOLUTION OUTPUT BY SYSTEM
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% % Problem  : SEV020^5 : TPTP v6.1.0. Released v4.0.0.
% % Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n105.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 07:31:21 CDT 2014
% % CPUTime  : 300.11 
% 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 0x20cc950>, <kernel.Type object at 0x1e7ca70>) of role type named a_type
% Using role type
% Declaring a:Type
% FOF formula (forall (P:((a->Prop)->Prop)), ((forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp)))))))->((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((Q Xx) Xx))) (forall (Xx:a) (Xy:a), (((Q Xx) Xy)->((Q Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Q Xx) Xy)) ((Q Xy) Xz))->((Q Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy))))))) P)))))) of role conjecture named cTHM262_B_pme
% Conjecture to prove = (forall (P:((a->Prop)->Prop)), ((forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp)))))))->((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((Q Xx) Xx))) (forall (Xx:a) (Xy:a), (((Q Xx) Xy)->((Q Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Q Xx) Xy)) ((Q Xy) Xz))->((Q Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> (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)), ((forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp)))))))->((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((Q Xx) Xx))) (forall (Xx:a) (Xy:a), (((Q Xx) Xy)->((Q Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Q Xx) Xy)) ((Q Xy) Xz))->((Q Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> (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)), ((forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp)))))))->((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((Q Xx) Xx))) (forall (Xx:a) (Xy:a), (((Q Xx) Xy)->((Q Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Q Xx) Xy)) ((Q Xy) Xz))->((Q Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy))))))) P))))))
% Found eta_expansion000:=(eta_expansion00 (fun (Q:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((Q Xx) Xx))) (forall (Xx:a) (Xy:a), (((Q Xx) Xy)->((Q Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Q Xx) Xy)) ((Q Xy) Xz))->((Q Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy))))))) P)))):(((eq ((a->(a->Prop))->Prop)) (fun (Q:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((Q Xx) Xx))) (forall (Xx:a) (Xy:a), (((Q Xx) Xy)->((Q Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Q Xx) Xy)) ((Q Xy) Xz))->((Q Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy))))))) P)))) (fun (x:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((x Xx) Xx))) (forall (Xx:a) (Xy:a), (((x Xx) Xy)->((x Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x Xx) Xy))))))) P))))
% Found (eta_expansion00 (fun (Q:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((Q Xx) Xx))) (forall (Xx:a) (Xy:a), (((Q Xx) Xy)->((Q Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Q Xx) Xy)) ((Q Xy) Xz))->((Q Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy))))))) P)))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (Q:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((Q Xx) Xx))) (forall (Xx:a) (Xy:a), (((Q Xx) Xy)->((Q Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Q Xx) Xy)) ((Q Xy) Xz))->((Q Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy))))))) P)))) b)
% Found ((eta_expansion0 Prop) (fun (Q:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((Q Xx) Xx))) (forall (Xx:a) (Xy:a), (((Q Xx) Xy)->((Q Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Q Xx) Xy)) ((Q Xy) Xz))->((Q Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy))))))) P)))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (Q:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((Q Xx) Xx))) (forall (Xx:a) (Xy:a), (((Q Xx) Xy)->((Q Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Q Xx) Xy)) ((Q Xy) Xz))->((Q Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy))))))) P)))) b)
% Found (((eta_expansion (a->(a->Prop))) Prop) (fun (Q:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((Q Xx) Xx))) (forall (Xx:a) (Xy:a), (((Q Xx) Xy)->((Q Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Q Xx) Xy)) ((Q Xy) Xz))->((Q Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy))))))) P)))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (Q:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((Q Xx) Xx))) (forall (Xx:a) (Xy:a), (((Q Xx) Xy)->((Q Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Q Xx) Xy)) ((Q Xy) Xz))->((Q Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy))))))) P)))) b)
% Found (((eta_expansion (a->(a->Prop))) Prop) (fun (Q:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((Q Xx) Xx))) (forall (Xx:a) (Xy:a), (((Q Xx) Xy)->((Q Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Q Xx) Xy)) ((Q Xy) Xz))->((Q Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy))))))) P)))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (Q:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((Q Xx) Xx))) (forall (Xx:a) (Xy:a), (((Q Xx) Xy)->((Q Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Q Xx) Xy)) ((Q Xy) Xz))->((Q Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy))))))) P)))) b)
% Found (((eta_expansion (a->(a->Prop))) Prop) (fun (Q:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((Q Xx) Xx))) (forall (Xx:a) (Xy:a), (((Q Xx) Xy)->((Q Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Q Xx) Xy)) ((Q Xy) Xz))->((Q Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy))))))) P)))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (Q:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((Q Xx) Xx))) (forall (Xx:a) (Xy:a), (((Q Xx) Xy)->((Q Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Q Xx) Xy)) ((Q Xy) Xz))->((Q Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy))))))) P)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq ((a->Prop)->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq ((a->Prop)->Prop)) b) P)
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) P)
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) P)
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) P)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy))))))):(((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy))))))) (fun (x:(a->Prop))=> (forall (Xx:a), ((x Xx)->(forall (Xy:a), ((iff (x Xy)) ((x0 Xx) Xy)))))))
% Found (eta_expansion_dep00 (fun (Xs:(a->Prop))=> (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))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy))))))) b)
% Found ((eta_expansion_dep0 (fun (x2:(a->Prop))=> Prop)) (fun (Xs:(a->Prop))=> (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))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy))))))) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x2:(a->Prop))=> Prop)) (fun (Xs:(a->Prop))=> (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))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy))))))) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x2:(a->Prop))=> Prop)) (fun (Xs:(a->Prop))=> (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))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy))))))) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x2:(a->Prop))=> Prop)) (fun (Xs:(a->Prop))=> (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))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy))))))) b)
% Found eta_expansion0000:=(eta_expansion000 P0):((P0 (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy)))))))->(P0 (fun (x:(a->Prop))=> (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))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy)))))))
% Found ((eta_expansion00 (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy))))))) P0) as proof of (P00 (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy)))))))
% Found (((eta_expansion0 Prop) (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy))))))) P0) as proof of (P00 (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy)))))))
% Found ((((eta_expansion (a->Prop)) Prop) (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy))))))) P0) as proof of (P00 (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy)))))))
% Found ((((eta_expansion (a->Prop)) Prop) (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy))))))) P0) as proof of (P00 (fun (Xs:(a->Prop))=> (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))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy)))))))->(P0 (fun (Xs:(a->Prop))=> (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))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy)))))))
% Found ((eq_ref0 (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy))))))) P0) as proof of (P00 (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy)))))))
% Found (((eq_ref ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy))))))) P0) as proof of (P00 (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy)))))))
% Found (((eq_ref ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy))))))) P0) as proof of (P00 (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy)))))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P0):((P0 (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy)))))))->(P0 (fun (x:(a->Prop))=> (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))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy)))))))
% Found ((eta_expansion_dep00 (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy))))))) P0) as proof of (P00 (fun (Xs:(a->Prop))=> (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))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy))))))) P0) as proof of (P00 (fun (Xs:(a->Prop))=> (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))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy))))))) P0) as proof of (P00 (fun (Xs:(a->Prop))=> (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))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy))))))) P0) as proof of (P00 (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy)))))))
% Found eq_ref00:=(eq_ref0 b):(((eq ((a->Prop)->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq ((a->Prop)->Prop)) b) (fun (Xs:(a->Prop))=> (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))=> (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))=> (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))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy)))))))
% Found eq_ref00:=(eq_ref0 P):(((eq ((a->Prop)->Prop)) P) P)
% Found (eq_ref0 P) as proof of (((eq ((a->Prop)->Prop)) P) b)
% Found ((eq_ref ((a->Prop)->Prop)) P) as proof of (((eq ((a->Prop)->Prop)) P) b)
% Found ((eq_ref ((a->Prop)->Prop)) P) as proof of (((eq ((a->Prop)->Prop)) P) b)
% Found ((eq_ref ((a->Prop)->Prop)) P) as proof of (((eq ((a->Prop)->Prop)) P) b)
% Found eq_ref000:=(eq_ref00 P0):((P0 P)->(P0 P))
% Found (eq_ref00 P0) as proof of (P00 P)
% Found ((eq_ref0 P) P0) as proof of (P00 P)
% Found (((eq_ref ((a->Prop)->Prop)) P) P0) as proof of (P00 P)
% Found (((eq_ref ((a->Prop)->Prop)) P) P0) as proof of (P00 P)
% Found eq_ref000:=(eq_ref00 P0):((P0 P)->(P0 P))
% Found (eq_ref00 P0) as proof of (P00 P)
% Found ((eq_ref0 P) P0) as proof of (P00 P)
% Found (((eq_ref ((a->Prop)->Prop)) P) P0) as proof of (P00 P)
% Found (((eq_ref ((a->Prop)->Prop)) P) P0) as proof of (P00 P)
% Found eq_ref00:=(eq_ref0 (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((x0 Xx) Xy)))))):(((eq Prop) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((x0 Xx) Xy)))))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((x0 Xx) Xy))))))
% Found (eq_ref0 (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((x0 Xx) Xy)))))) as proof of (((eq Prop) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((x0 Xx) Xy)))))) b)
% Found ((eq_ref Prop) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((x0 Xx) Xy)))))) as proof of (((eq Prop) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((x0 Xx) Xy)))))) b)
% Found ((eq_ref Prop) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((x0 Xx) Xy)))))) as proof of (((eq Prop) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((x0 Xx) Xy)))))) b)
% Found ((eq_ref Prop) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((x0 Xx) Xy)))))) as proof of (((eq Prop) (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 (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((x0 Xx) Xy)))))):(((eq Prop) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((x0 Xx) Xy)))))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((x0 Xx) Xy))))))
% Found (eq_ref0 (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((x0 Xx) Xy)))))) as proof of (((eq Prop) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((x0 Xx) Xy)))))) b)
% Found ((eq_ref Prop) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((x0 Xx) Xy)))))) as proof of (((eq Prop) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((x0 Xx) Xy)))))) b)
% Found ((eq_ref Prop) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((x0 Xx) Xy)))))) as proof of (((eq Prop) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((x0 Xx) Xy)))))) b)
% Found ((eq_ref Prop) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((x0 Xx) Xy)))))) as proof of (((eq Prop) (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 (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((x0 Xx) Xy))))))->(P0 (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((x0 Xx) Xy)))))))
% Found (eq_ref00 P0) as proof of (P00 (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((x0 Xx) Xy))))))
% Found ((eq_ref0 (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((x0 Xx) Xy)))))) P0) as proof of (P00 (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((x0 Xx) Xy))))))
% Found (((eq_ref Prop) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((x0 Xx) Xy)))))) P0) as proof of (P00 (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((x0 Xx) Xy))))))
% Found (((eq_ref Prop) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((x0 Xx) Xy)))))) P0) as proof of (P00 (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((x0 Xx) Xy))))))
% Found eq_ref000:=(eq_ref00 P0):((P0 (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((x0 Xx) Xy))))))->(P0 (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((x0 Xx) Xy)))))))
% Found (eq_ref00 P0) as proof of (P00 (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((x0 Xx) Xy))))))
% Found ((eq_ref0 (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((x0 Xx) Xy)))))) P0) as proof of (P00 (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((x0 Xx) Xy))))))
% Found (((eq_ref Prop) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((x0 Xx) Xy)))))) P0) as proof of (P00 (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((x0 Xx) Xy))))))
% Found (((eq_ref Prop) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((x0 Xx) Xy)))))) P0) as proof of (P00 (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((x0 Xx) Xy))))))
% Found eq_ref00:=(eq_ref0 b):(((eq ((a->(a->Prop))->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq ((a->(a->Prop))->Prop)) b) (fun (Q:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((Q Xx) Xx))) (forall (Xx:a) (Xy:a), (((Q Xx) Xy)->((Q Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Q Xx) Xy)) ((Q Xy) Xz))->((Q Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy))))))) P))))
% Found ((eq_ref ((a->(a->Prop))->Prop)) b) as proof of (((eq ((a->(a->Prop))->Prop)) b) (fun (Q:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((Q Xx) Xx))) (forall (Xx:a) (Xy:a), (((Q Xx) Xy)->((Q Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Q Xx) Xy)) ((Q Xy) Xz))->((Q Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy))))))) P))))
% Found ((eq_ref ((a->(a->Prop))->Prop)) b) as proof of (((eq ((a->(a->Prop))->Prop)) b) (fun (Q:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((Q Xx) Xx))) (forall (Xx:a) (Xy:a), (((Q Xx) Xy)->((Q Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Q Xx) Xy)) ((Q Xy) Xz))->((Q Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy))))))) P))))
% Found ((eq_ref ((a->(a->Prop))->Prop)) b) as proof of (((eq ((a->(a->Prop))->Prop)) b) (fun (Q:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((Q Xx) Xx))) (forall (Xx:a) (Xy:a), (((Q Xx) Xy)->((Q Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Q Xx) Xy)) ((Q Xy) Xz))->((Q Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy))))))) P))))
% Found eta_expansion000:=(eta_expansion00 a0):(((eq ((a->(a->Prop))->Prop)) a0) (fun (x:(a->(a->Prop)))=> (a0 x)))
% Found (eta_expansion00 a0) as proof of (((eq ((a->(a->Prop))->Prop)) a0) b)
% Found ((eta_expansion0 Prop) a0) as proof of (((eq ((a->(a->Prop))->Prop)) a0) b)
% Found (((eta_expansion (a->(a->Prop))) Prop) a0) as proof of (((eq ((a->(a->Prop))->Prop)) a0) b)
% Found (((eta_expansion (a->(a->Prop))) Prop) a0) as proof of (((eq ((a->(a->Prop))->Prop)) a0) b)
% Found (((eta_expansion (a->(a->Prop))) Prop) a0) as proof of (((eq ((a->(a->Prop))->Prop)) a0) b)
% Found eq_ref00:=(eq_ref0 a0):(((eq ((a->(a->Prop))->Prop)) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq ((a->(a->Prop))->Prop)) a0) b)
% Found ((eq_ref ((a->(a->Prop))->Prop)) a0) as proof of (((eq ((a->(a->Prop))->Prop)) a0) b)
% Found ((eq_ref ((a->(a->Prop))->Prop)) a0) as proof of (((eq ((a->(a->Prop))->Prop)) a0) b)
% Found ((eq_ref ((a->(a->Prop))->Prop)) a0) as proof of (((eq ((a->(a->Prop))->Prop)) a0) b)
% Found eq_ref00:=(eq_ref0 (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy))))))) P)):(((eq Prop) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy))))))) P)) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy))))))) P))
% Found (eq_ref0 (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy))))))) P)) as proof of (((eq Prop) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy))))))) P)) b)
% Found ((eq_ref Prop) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy))))))) P)) as proof of (((eq Prop) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy))))))) P)) b)
% Found ((eq_ref Prop) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy))))))) P)) as proof of (((eq Prop) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy))))))) P)) b)
% Found ((eq_ref Prop) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy))))))) P)) as proof of (((eq Prop) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy))))))) P)) b)
% Found eta_expansion000:=(eta_expansion00 (fun (Q:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((Q Xx) Xx))) (forall (Xx:a) (Xy:a), (((Q Xx) Xy)->((Q Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Q Xx) Xy)) ((Q Xy) Xz))->((Q Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy))))))) P)))):(((eq ((a->(a->Prop))->Prop)) (fun (Q:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((Q Xx) Xx))) (forall (Xx:a) (Xy:a), (((Q Xx) Xy)->((Q Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Q Xx) Xy)) ((Q Xy) Xz))->((Q Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy))))))) P)))) (fun (x:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((x Xx) Xx))) (forall (Xx:a) (Xy:a), (((x Xx) Xy)->((x Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x Xx) Xy))))))) P))))
% Found (eta_expansion00 (fun (Q:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((Q Xx) Xx))) (forall (Xx:a) (Xy:a), (((Q Xx) Xy)->((Q Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Q Xx) Xy)) ((Q Xy) Xz))->((Q Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy))))))) P)))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (Q:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((Q Xx) Xx))) (forall (Xx:a) (Xy:a), (((Q Xx) Xy)->((Q Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Q Xx) Xy)) ((Q Xy) Xz))->((Q Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy))))))) P)))) b)
% Found ((eta_expansion0 Prop) (fun (Q:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((Q Xx) Xx))) (forall (Xx:a) (Xy:a), (((Q Xx) Xy)->((Q Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Q Xx) Xy)) ((Q Xy) Xz))->((Q Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy))))))) P)))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (Q:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((Q Xx) Xx))) (forall (Xx:a) (Xy:a), (((Q Xx) Xy)->((Q Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Q Xx) Xy)) ((Q Xy) Xz))->((Q Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy))))))) P)))) b)
% Found (((eta_expansion (a->(a->Prop))) Prop) (fun (Q:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((Q Xx) Xx))) (forall (Xx:a) (Xy:a), (((Q Xx) Xy)->((Q Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Q Xx) Xy)) ((Q Xy) Xz))->((Q Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy))))))) P)))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (Q:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((Q Xx) Xx))) (forall (Xx:a) (Xy:a), (((Q Xx) Xy)->((Q Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Q Xx) Xy)) ((Q Xy) Xz))->((Q Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy))))))) P)))) b)
% Found (((eta_expansion (a->(a->Prop))) Prop) (fun (Q:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((Q Xx) Xx))) (forall (Xx:a) (Xy:a), (((Q Xx) Xy)->((Q Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Q Xx) Xy)) ((Q Xy) Xz))->((Q Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy))))))) P)))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (Q:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((Q Xx) Xx))) (forall (Xx:a) (Xy:a), (((Q Xx) Xy)->((Q Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Q Xx) Xy)) ((Q Xy) Xz))->((Q Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy))))))) P)))) b)
% Found (((eta_expansion (a->(a->Prop))) Prop) (fun (Q:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((Q Xx) Xx))) (forall (Xx:a) (Xy:a), (((Q Xx) Xy)->((Q Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Q Xx) Xy)) ((Q Xy) Xz))->((Q Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy))))))) P)))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (Q:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((Q Xx) Xx))) (forall (Xx:a) (Xy:a), (((Q Xx) Xy)->((Q Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Q Xx) Xy)) ((Q Xy) Xz))->((Q Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy))))))) P)))) b)
% Found iff_sym:=(fun (A:Prop) (B:Prop) (H:((iff A) B))=> ((((conj (B->A)) (A->B)) (((proj2 (A->B)) (B->A)) H)) (((proj1 (A->B)) (B->A)) H))):(forall (A:Prop) (B:Prop), (((iff A) B)->((iff B) A)))
% Instantiate: b:=(forall (A:Prop) (B:Prop), (((iff A) B)->((iff B) A))):Prop
% Found iff_sym as proof of b
% Found eta_expansion000:=(eta_expansion00 (fun (Q:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((Q Xx) Xx))) (forall (Xx:a) (Xy:a), (((Q Xx) Xy)->((Q Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Q Xx) Xy)) ((Q Xy) Xz))->((Q Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy))))))) P)))):(((eq ((a->(a->Prop))->Prop)) (fun (Q:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((Q Xx) Xx))) (forall (Xx:a) (Xy:a), (((Q Xx) Xy)->((Q Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Q Xx) Xy)) ((Q Xy) Xz))->((Q Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy))))))) P)))) (fun (x:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((x Xx) Xx))) (forall (Xx:a) (Xy:a), (((x Xx) Xy)->((x Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x Xx) Xy))))))) P))))
% Found (eta_expansion00 (fun (Q:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((Q Xx) Xx))) (forall (Xx:a) (Xy:a), (((Q Xx) Xy)->((Q Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Q Xx) Xy)) ((Q Xy) Xz))->((Q Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy))))))) P)))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (Q:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((Q Xx) Xx))) (forall (Xx:a) (Xy:a), (((Q Xx) Xy)->((Q Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Q Xx) Xy)) ((Q Xy) Xz))->((Q Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy))))))) P)))) b)
% Found ((eta_expansion0 Prop) (fun (Q:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((Q Xx) Xx))) (forall (Xx:a) (Xy:a), (((Q Xx) Xy)->((Q Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Q Xx) Xy)) ((Q Xy) Xz))->((Q Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy))))))) P)))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (Q:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((Q Xx) Xx))) (forall (Xx:a) (Xy:a), (((Q Xx) Xy)->((Q Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Q Xx) Xy)) ((Q Xy) Xz))->((Q Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy))))))) P)))) b)
% Found (((eta_expansion (a->(a->Prop))) Prop) (fun (Q:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((Q Xx) Xx))) (forall (Xx:a) (Xy:a), (((Q Xx) Xy)->((Q Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Q Xx) Xy)) ((Q Xy) Xz))->((Q Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy))))))) P)))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (Q:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((Q Xx) Xx))) (forall (Xx:a) (Xy:a), (((Q Xx) Xy)->((Q Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Q Xx) Xy)) ((Q Xy) Xz))->((Q Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy))))))) P)))) b)
% Found (((eta_expansion (a->(a->Prop))) Prop) (fun (Q:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((Q Xx) Xx))) (forall (Xx:a) (Xy:a), (((Q Xx) Xy)->((Q Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Q Xx) Xy)) ((Q Xy) Xz))->((Q Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy))))))) P)))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (Q:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((Q Xx) Xx))) (forall (Xx:a) (Xy:a), (((Q Xx) Xy)->((Q Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Q Xx) Xy)) ((Q Xy) Xz))->((Q Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy))))))) P)))) b)
% Found (((eta_expansion (a->(a->Prop))) Prop) (fun (Q:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((Q Xx) Xx))) (forall (Xx:a) (Xy:a), (((Q Xx) Xy)->((Q Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Q Xx) Xy)) ((Q Xy) Xz))->((Q Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy))))))) P)))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (Q:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((Q Xx) Xx))) (forall (Xx:a) (Xy:a), (((Q Xx) Xy)->((Q Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Q Xx) Xy)) ((Q Xy) Xz))->((Q Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy))))))) P)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (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) (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) (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) (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) (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) (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) (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) (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) (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) (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) (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) (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) (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) (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) (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) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((x0 Xx) Xy))))))
% 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 eta_expansion000:=(eta_expansion00 (fun (Q:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((Q Xx) Xx))) (forall (Xx:a) (Xy:a), (((Q Xx) Xy)->((Q Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Q Xx) Xy)) ((Q Xy) Xz))->((Q Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy))))))) P)))):(((eq ((a->(a->Prop))->Prop)) (fun (Q:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((Q Xx) Xx))) (forall (Xx:a) (Xy:a), (((Q Xx) Xy)->((Q Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Q Xx) Xy)) ((Q Xy) Xz))->((Q Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy))))))) P)))) (fun (x:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((x Xx) Xx))) (forall (Xx:a) (Xy:a), (((x Xx) Xy)->((x Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x Xx) Xy))))))) P))))
% Found (eta_expansion00 (fun (Q:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((Q Xx) Xx))) (forall (Xx:a) (Xy:a), (((Q Xx) Xy)->((Q Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Q Xx) Xy)) ((Q Xy) Xz))->((Q Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy))))))) P)))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (Q:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((Q Xx) Xx))) (forall (Xx:a) (Xy:a), (((Q Xx) Xy)->((Q Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Q Xx) Xy)) ((Q Xy) Xz))->((Q Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy))))))) P)))) b)
% Found ((eta_expansion0 Prop) (fun (Q:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((Q Xx) Xx))) (forall (Xx:a) (Xy:a), (((Q Xx) Xy)->((Q Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Q Xx) Xy)) ((Q Xy) Xz))->((Q Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy))))))) P)))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (Q:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((Q Xx) Xx))) (forall (Xx:a) (Xy:a), (((Q Xx) Xy)->((Q Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Q Xx) Xy)) ((Q Xy) Xz))->((Q Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy))))))) P)))) b)
% Found (((eta_expansion (a->(a->Prop))) Prop) (fun (Q:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((Q Xx) Xx))) (forall (Xx:a) (Xy:a), (((Q Xx) Xy)->((Q Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Q Xx) Xy)) ((Q Xy) Xz))->((Q Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy))))))) P)))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (Q:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((Q Xx) Xx))) (forall (Xx:a) (Xy:a), (((Q Xx) Xy)->((Q Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Q Xx) Xy)) ((Q Xy) Xz))->((Q Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy))))))) P)))) b)
% Found (((eta_expansion (a->(a->Prop))) Prop) (fun (Q:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((Q Xx) Xx))) (forall (Xx:a) (Xy:a), (((Q Xx) Xy)->((Q Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Q Xx) Xy)) ((Q Xy) Xz))->((Q Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy))))))) P)))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (Q:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((Q Xx) Xx))) (forall (Xx:a) (Xy:a), (((Q Xx) Xy)->((Q Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Q Xx) Xy)) ((Q Xy) Xz))->((Q Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy))))))) P)))) b)
% Found (((eta_expansion (a->(a->Prop))) Prop) (fun (Q:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((Q Xx) Xx))) (forall (Xx:a) (Xy:a), (((Q Xx) Xy)->((Q Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Q Xx) Xy)) ((Q Xy) Xz))->((Q Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy))))))) P)))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (Q:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((Q Xx) Xx))) (forall (Xx:a) (Xy:a), (((Q Xx) Xy)->((Q Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Q Xx) Xy)) ((Q Xy) Xz))->((Q Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> (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)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((and ((and (forall (Xx:a), ((x0 Xx) Xx))) (forall (Xx:a) (Xy:a), (((x0 Xx) Xy)->((x0 Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xz))->((x0 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy))))))) P)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and ((and (forall (Xx:a), ((x0 Xx) Xx))) (forall (Xx:a) (Xy:a), (((x0 Xx) Xy)->((x0 Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xz))->((x0 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy))))))) P)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and ((and (forall (Xx:a), ((x0 Xx) Xx))) (forall (Xx:a) (Xy:a), (((x0 Xx) Xy)->((x0 Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xz))->((x0 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy))))))) P)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and ((and (forall (Xx:a), ((x0 Xx) Xx))) (forall (Xx:a) (Xy:a), (((x0 Xx) Xy)->((x0 Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xz))->((x0 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy))))))) P)))
% Found ((eq_trans0000 ((eq_ref Prop) (f x0))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f x0)) ((and ((and ((and (forall (Xx:a), ((x0 Xx) Xx))) (forall (Xx:a) (Xy:a), (((x0 Xx) Xy)->((x0 Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xz))->((x0 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy))))))) P)))
% Found (((eq_trans000 ((and ((and ((and (forall (Xx:a), ((x0 Xx) Xx))) (forall (Xx:a) (Xy:a), (((x0 Xx) Xy)->((x0 Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xz))->((x0 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy))))))) P))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f x0)) ((and ((and ((and (forall (Xx:a), ((x0 Xx) Xx))) (forall (Xx:a) (Xy:a), (((x0 Xx) Xy)->((x0 Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xz))->((x0 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy))))))) P)))
% Found ((((eq_trans00 ((and ((and ((and (forall (Xx:a), ((x0 Xx) Xx))) (forall (Xx:a) (Xy:a), (((x0 Xx) Xy)->((x0 Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xz))->((x0 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy))))))) P))) ((and ((and ((and (forall (Xx:a), ((x0 Xx) Xx))) (forall (Xx:a) (Xy:a), (((x0 Xx) Xy)->((x0 Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xz))->((x0 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy))))))) P))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((and ((and (forall (Xx:a), ((x0 Xx) Xx))) (forall (Xx:a) (Xy:a), (((x0 Xx) Xy)->((x0 Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xz))->((x0 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy))))))) P)))) as proof of (((eq Prop) (f x0)) ((and ((and ((and (forall (Xx:a), ((x0 Xx) Xx))) (forall (Xx:a) (Xy:a), (((x0 Xx) Xy)->((x0 Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xz))->((x0 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy))))))) P)))
% Found (((((eq_trans0 (f x0)) ((and ((and ((and (forall (Xx:a), ((x0 Xx) Xx))) (forall (Xx:a) (Xy:a), (((x0 Xx) Xy)->((x0 Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xz))->((x0 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy))))))) P))) ((and ((and ((and (forall (Xx:a), ((x0 Xx) Xx))) (forall (Xx:a) (Xy:a), (((x0 Xx) Xy)->((x0 Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xz))->((x0 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy))))))) P))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((and ((and (forall (Xx:a), ((x0 Xx) Xx))) (forall (Xx:a) (Xy:a), (((x0 Xx) Xy)->((x0 Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xz))->((x0 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy))))))) P)))) as proof of (((eq Prop) (f x0)) ((and ((and ((and (forall (Xx:a), ((x0 Xx) Xx))) (forall (Xx:a) (Xy:a), (((x0 Xx) Xy)->((x0 Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xz))->((x0 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy))))))) P)))
% Found ((((((eq_trans Prop) (f x0)) ((and ((and ((and (forall (Xx:a), ((x0 Xx) Xx))) (forall (Xx:a) (Xy:a), (((x0 Xx) Xy)->((x0 Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xz))->((x0 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy))))))) P))) ((and ((and ((and (forall (Xx:a), ((x0 Xx) Xx))) (forall (Xx:a) (Xy:a), (((x0 Xx) Xy)->((x0 Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xz))->((x0 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy))))))) P))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((and ((and (forall (Xx:a), ((x0 Xx) Xx))) (forall (Xx:a) (Xy:a), (((x0 Xx) Xy)->((x0 Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xz))->((x0 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy))))))) P)))) as proof of (((eq Prop) (f x0)) ((and ((and ((and (forall (Xx:a), ((x0 Xx) Xx))) (forall (Xx:a) (Xy:a), (((x0 Xx) Xy)->((x0 Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xz))->((x0 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 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)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((and ((and (forall (Xx:a), ((x0 Xx) Xx))) (forall (Xx:a) (Xy:a), (((x0 Xx) Xy)->((x0 Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xz))->((x0 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy))))))) P)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and ((and (forall (Xx:a), ((x0 Xx) Xx))) (forall (Xx:a) (Xy:a), (((x0 Xx) Xy)->((x0 Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xz))->((x0 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy))))))) P)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and ((and (forall (Xx:a), ((x0 Xx) Xx))) (forall (Xx:a) (Xy:a), (((x0 Xx) Xy)->((x0 Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xz))->((x0 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy))))))) P)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and ((and (forall (Xx:a), ((x0 Xx) Xx))) (forall (Xx:a) (Xy:a), (((x0 Xx) Xy)->((x0 Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xz))->((x0 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy))))))) P)))
% Found ((eq_trans0000 ((eq_ref Prop) (f x0))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f x0)) ((and ((and ((and (forall (Xx:a), ((x0 Xx) Xx))) (forall (Xx:a) (Xy:a), (((x0 Xx) Xy)->((x0 Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xz))->((x0 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy))))))) P)))
% Found (((eq_trans000 ((and ((and ((and (forall (Xx:a), ((x0 Xx) Xx))) (forall (Xx:a) (Xy:a), (((x0 Xx) Xy)->((x0 Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xz))->((x0 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy))))))) P))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f x0)) ((and ((and ((and (forall (Xx:a), ((x0 Xx) Xx))) (forall (Xx:a) (Xy:a), (((x0 Xx) Xy)->((x0 Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xz))->((x0 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy))))))) P)))
% Found ((((eq_trans00 ((and ((and ((and (forall (Xx:a), ((x0 Xx) Xx))) (forall (Xx:a) (Xy:a), (((x0 Xx) Xy)->((x0 Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xz))->((x0 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy))))))) P))) ((and ((and ((and (forall (Xx:a), ((x0 Xx) Xx))) (forall (Xx:a) (Xy:a), (((x0 Xx) Xy)->((x0 Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xz))->((x0 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy))))))) P))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((and ((and (forall (Xx:a), ((x0 Xx) Xx))) (forall (Xx:a) (Xy:a), (((x0 Xx) Xy)->((x0 Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xz))->((x0 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy))))))) P)))) as proof of (((eq Prop) (f x0)) ((and ((and ((and (forall (Xx:a), ((x0 Xx) Xx))) (forall (Xx:a) (Xy:a), (((x0 Xx) Xy)->((x0 Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xz))->((x0 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy))))))) P)))
% Found (((((eq_trans0 (f x0)) ((and ((and ((and (forall (Xx:a), ((x0 Xx) Xx))) (forall (Xx:a) (Xy:a), (((x0 Xx) Xy)->((x0 Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xz))->((x0 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy))))))) P))) ((and ((and ((and (forall (Xx:a), ((x0 Xx) Xx))) (forall (Xx:a) (Xy:a), (((x0 Xx) Xy)->((x0 Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xz))->((x0 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy))))))) P))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((and ((and (forall (Xx:a), ((x0 Xx) Xx))) (forall (Xx:a) (Xy:a), (((x0 Xx) Xy)->((x0 Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xz))->((x0 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy))))))) P)))) as proof of (((eq Prop) (f x0)) ((and ((and ((and (forall (Xx:a), ((x0 Xx) Xx))) (forall (Xx:a) (Xy:a), (((x0 Xx) Xy)->((x0 Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xz))->((x0 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy))))))) P)))
% Found ((((((eq_trans Prop) (f x0)) ((and ((and ((and (forall (Xx:a), ((x0 Xx) Xx))) (forall (Xx:a) (Xy:a), (((x0 Xx) Xy)->((x0 Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xz))->((x0 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy))))))) P))) ((and ((and ((and (forall (Xx:a), ((x0 Xx) Xx))) (forall (Xx:a) (Xy:a), (((x0 Xx) Xy)->((x0 Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xz))->((x0 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy))))))) P))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((and ((and (forall (Xx:a), ((x0 Xx) Xx))) (forall (Xx:a) (Xy:a), (((x0 Xx) Xy)->((x0 Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xz))->((x0 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy))))))) P)))) as proof of (((eq Prop) (f x0)) ((and ((and ((and (forall (Xx:a), ((x0 Xx) Xx))) (forall (Xx:a) (Xy:a), (((x0 Xx) Xy)->((x0 Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xz))->((x0 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy))))))) P)))
% Found eq_ref00:=(eq_ref0 b):(((eq ((a->Prop)->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq ((a->Prop)->Prop)) b) P)
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) P)
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) P)
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) P)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy))))))):(((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy))))))) (fun (x:(a->Prop))=> (forall (Xx:a), ((x Xx)->(forall (Xy:a), ((iff (x Xy)) ((x0 Xx) Xy)))))))
% Found (eta_expansion_dep00 (fun (Xs:(a->Prop))=> (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))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy))))))) b)
% Found ((eta_expansion_dep0 (fun (x2:(a->Prop))=> Prop)) (fun (Xs:(a->Prop))=> (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))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy))))))) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x2:(a->Prop))=> Prop)) (fun (Xs:(a->Prop))=> (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))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy))))))) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x2:(a->Prop))=> Prop)) (fun (Xs:(a->Prop))=> (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))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy))))))) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x2:(a->Prop))=> Prop)) (fun (Xs:(a->Prop))=> (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))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy))))))) b)
% Found eq_ref000:=(eq_ref00 P00):((P00 (f x0))->(P00 (f x0)))
% Found (eq_ref00 P00) as proof of (P01 (f x0))
% Found ((eq_ref0 (f x0)) P00) as proof of (P01 (f x0))
% Found (((eq_ref Prop) (f x0)) P00) as proof of (P01 (f x0))
% Found (((eq_ref Prop) (f x0)) P00) as proof of (P01 (f x0))
% Found eq_ref000:=(eq_ref00 P00):((P00 (f x0))->(P00 (f x0)))
% Found (eq_ref00 P00) as proof of (P01 (f x0))
% Found ((eq_ref0 (f x0)) P00) as proof of (P01 (f x0))
% Found (((eq_ref Prop) (f x0)) P00) as proof of (P01 (f x0))
% Found (((eq_ref Prop) (f x0)) P00) as proof of (P01 (f x0))
% Found eta_expansion0000:=(eta_expansion000 P0):((P0 (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy)))))))->(P0 (fun (x:(a->Prop))=> (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))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy)))))))
% Found ((eta_expansion00 (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy))))))) P0) as proof of (P00 (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy)))))))
% Found (((eta_expansion0 Prop) (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy))))))) P0) as proof of (P00 (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy)))))))
% Found ((((eta_expansion (a->Prop)) Prop) (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy))))))) P0) as proof of (P00 (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy)))))))
% Found ((((eta_expansion (a->Prop)) Prop) (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy))))))) P0) as proof of (P00 (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy)))))))
% Found eq_ref00:=(eq_ref0 (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xz))->((x0 Xx) Xz)))):(((eq Prop) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xz))->((x0 Xx) Xz)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xz))->((x0 Xx) Xz))))
% Found (eq_ref0 (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xz))->((x0 Xx) Xz)))) as proof of (((eq Prop) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xz))->((x0 Xx) Xz)))) b)
% Found ((eq_ref Prop) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xz))->((x0 Xx) Xz)))) as proof of (((eq Prop) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xz))->((x0 Xx) Xz)))) b)
% Found ((eq_ref Prop) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xz))->((x0 Xx) Xz)))) as proof of (((eq Prop) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xz))->((x0 Xx) Xz)))) b)
% Found ((eq_ref Prop) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xz))->((x0 Xx) Xz)))) as proof of (((eq Prop) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xz))->((x0 Xx) Xz)))) b)
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P0):((P0 (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy)))))))->(P0 (fun (x:(a->Prop))=> (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))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy)))))))
% Found ((eta_expansion_dep00 (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy))))))) P0) as proof of (P00 (fun (Xs:(a->Prop))=> (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))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy))))))) P0) as proof of (P00 (fun (Xs:(a->Prop))=> (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))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy))))))) P0) as proof of (P00 (fun (Xs:(a->Prop))=> (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))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy))))))) P0) as proof of (P00 (fun (Xs:(a->Prop))=> (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))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy)))))))->(P0 (fun (Xs:(a->Prop))=> (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))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy)))))))
% Found ((eq_ref0 (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy))))))) P0) as proof of (P00 (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy)))))))
% Found (((eq_ref ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy))))))) P0) as proof of (P00 (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy)))))))
% Found (((eq_ref ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy))))))) P0) as proof of (P00 (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy)))))))
% Found x1:(P0 (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy)))))))
% Instantiate: b:=(fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy)))))):((a->Prop)->Prop)
% Found x1 as proof of (P00 b)
% Found eq_ref00:=(eq_ref0 P):(((eq ((a->Prop)->Prop)) P) P)
% Found (eq_ref0 P) as proof of (((eq ((a->Prop)->Prop)) P) b)
% Found ((eq_ref ((a->Prop)->Prop)) P) as proof of (((eq ((a->Prop)->Prop)) P) b)
% Found ((eq_ref ((a->Prop)->Prop)) P) as proof of (((eq ((a->Prop)->Prop)) P) b)
% Found ((eq_ref ((a->Prop)->Prop)) P) as proof of (((eq ((a->Prop)->Prop)) P) b)
% Found x1:(P0 (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy)))))))
% Instantiate: f:=(fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy)))))):((a->Prop)->Prop)
% Found x1 as proof of (P00 f)
% Found x1:(P0 (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy)))))))
% Instantiate: f:=(fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy)))))):((a->Prop)->Prop)
% Found x1 as proof of (P00 f)
% 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))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy))))))):(((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy))))))) (fun (x:(a->Prop))=> (forall (Xx:a), ((x Xx)->(forall (Xy:a), ((iff (x Xy)) ((x0 Xx) Xy)))))))
% Found (eta_expansion00 (fun (Xs:(a->Prop))=> (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))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy))))))) b)
% Found ((eta_expansion0 Prop) (fun (Xs:(a->Prop))=> (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))=> (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))=> (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))=> (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))=> (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))=> (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))=> (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))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy))))))) b)
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (P x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (P x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (P x2))
% Found (fun (x2:(a->Prop))=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (P x2))
% Found (fun (x2:(a->Prop))=> ((eq_ref Prop) (f x2))) as proof of (forall (x:(a->Prop)), (((eq Prop) (f x)) (P x)))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (P x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (P x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (P x2))
% Found (fun (x2:(a->Prop))=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (P x2))
% Found (fun (x2:(a->Prop))=> ((eq_ref Prop) (f x2))) as proof of (forall (x:(a->Prop)), (((eq Prop) (f x)) (P x)))
% Found eq_ref000:=(eq_ref00 P0):((P0 (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy)))))))->(P0 (fun (Xs:(a->Prop))=> (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))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy)))))))
% Found ((eq_ref0 (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy))))))) P0) as proof of (P00 (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy)))))))
% Found (((eq_ref ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy))))))) P0) as proof of (P00 (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy)))))))
% Found (((eq_ref ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy))))))) P0) as proof of (P00 (fun (Xs:(a->Prop))=> (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))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy))))))):(((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy))))))) (fun (x:(a->Prop))=> (forall (Xx:a), ((x Xx)->(forall (Xy:a), ((iff (x Xy)) ((x0 Xx) Xy)))))))
% Found (eta_expansion00 (fun (Xs:(a->Prop))=> (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))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy))))))) b)
% Found ((eta_expansion0 Prop) (fun (Xs:(a->Prop))=> (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))=> (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))=> (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))=> (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))=> (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))=> (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))=> (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))=> (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) 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))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy))))))):(((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy))))))) (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy)))))))
% Found (eq_ref0 (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy))))))) b)
% Found ((eq_ref ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy))))))) b)
% Found ((eq_ref ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy))))))) b)
% Found ((eq_ref ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy))))))) b)
% Found eq_ref00:=(eq_ref0 (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((x0 Xx) Xy)))))):(((eq Prop) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((x0 Xx) Xy)))))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((x0 Xx) Xy))))))
% Found (eq_ref0 (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((x0 Xx) Xy)))))) as proof of (((eq Prop) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((x0 Xx) Xy)))))) b)
% Found ((eq_ref Prop) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((x0 Xx) Xy)))))) as proof of (((eq Prop) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((x0 Xx) Xy)))))) b)
% Found ((eq_ref Prop) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((x0 Xx) Xy)))))) as proof of (((eq Prop) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((x0 Xx) Xy)))))) b)
% Found ((eq_ref Prop) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((x0 Xx) Xy)))))) as proof of (((eq Prop) (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 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 (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((x0 Xx) Xy)))))):(((eq Prop) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((x0 Xx) Xy)))))) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((x0 Xx) Xy))))))
% Found (eq_ref0 (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((x0 Xx) Xy)))))) as proof of (((eq Prop) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((x0 Xx) Xy)))))) b)
% Found ((eq_ref Prop) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((x0 Xx) Xy)))))) as proof of (((eq Prop) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((x0 Xx) Xy)))))) b)
% Found ((eq_ref Prop) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((x0 Xx) Xy)))))) as proof of (((eq Prop) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((x0 Xx) Xy)))))) b)
% Found ((eq_ref Prop) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((x0 Xx) Xy)))))) as proof of (((eq Prop) (forall (Xx:a), ((x1 Xx)->(forall (Xy:a), ((iff (x1 Xy)) ((x0 Xx) Xy)))))) b)
% Found eta_expansion0000:=(eta_expansion000 P0):((P0 (fun (Xs:(a->Prop))=> (forall (Xx0:a), ((Xs Xx0)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx0) Xy)))))))->(P0 (fun (x:(a->Prop))=> (forall (Xx0:a), ((x Xx0)->(forall (Xy:a), ((iff (x Xy)) ((x3 Xx0) Xy))))))))
% Found (eta_expansion000 P0) as proof of (P00 (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy)))))))
% Found ((eta_expansion00 (fun (Xs:(a->Prop))=> (forall (Xx0:a), ((Xs Xx0)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx0) Xy))))))) P0) as proof of (P00 (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy)))))))
% Found (((eta_expansion0 Prop) (fun (Xs:(a->Prop))=> (forall (Xx0:a), ((Xs Xx0)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx0) Xy))))))) P0) as proof of (P00 (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy)))))))
% Found ((((eta_expansion (a->Prop)) Prop) (fun (Xs:(a->Prop))=> (forall (Xx0:a), ((Xs Xx0)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx0) Xy))))))) P0) as proof of (P00 (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy)))))))
% Found ((((eta_expansion (a->Prop)) Prop) (fun (Xs:(a->Prop))=> (forall (Xx0:a), ((Xs Xx0)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx0) Xy))))))) P0) as proof of (P00 (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy)))))))
% Found iff_refl:=(fun (A:Prop)=> ((((conj (A->A)) (A->A)) (fun (H:A)=> H)) (fun (H:A)=> H))):(forall (P:Prop), ((iff P) P))
% Instantiate: b:=(forall (P:Prop), ((iff P) P)):Prop
% Found iff_refl as proof of b
% Found eq_ref00:=(eq_ref0 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))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy))))))):(((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy))))))) (fun (x:(a->Prop))=> (forall (Xx:a), ((x Xx)->(forall (Xy:a), ((iff (x Xy)) ((x0 Xx) Xy)))))))
% Found (eta_expansion00 (fun (Xs:(a->Prop))=> (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))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy))))))) b)
% Found ((eta_expansion0 Prop) (fun (Xs:(a->Prop))=> (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))=> (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))=> (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))=> (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))=> (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))=> (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))=> (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))=> (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 (P0 b)
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (P0 b)
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (P0 b)
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (P0 b)
% 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 eta_expansion0000:=(eta_expansion000 P0):((P0 (fun (Xs:(a->Prop))=> (forall (Xx0:a), ((Xs Xx0)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx0) Xy)))))))->(P0 (fun (x:(a->Prop))=> (forall (Xx0:a), ((x Xx0)->(forall (Xy:a), ((iff (x Xy)) ((x0 Xx0) Xy))))))))
% Found (eta_expansion000 P0) as proof of (P00 (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy)))))))
% Found ((eta_expansion00 (fun (Xs:(a->Prop))=> (forall (Xx0:a), ((Xs Xx0)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx0) Xy))))))) P0) as proof of (P00 (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy)))))))
% Found (((eta_expansion0 Prop) (fun (Xs:(a->Prop))=> (forall (Xx0:a), ((Xs Xx0)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx0) Xy))))))) P0) as proof of (P00 (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy)))))))
% Found ((((eta_expansion (a->Prop)) Prop) (fun (Xs:(a->Prop))=> (forall (Xx0:a), ((Xs Xx0)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx0) Xy))))))) P0) as proof of (P00 (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy)))))))
% Found ((((eta_expansion (a->Prop)) Prop) (fun (Xs:(a->Prop))=> (forall (Xx0:a), ((Xs Xx0)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx0) Xy))))))) P0) as proof of (P00 (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy)))))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P0):((P0 (fun (Xs:(a->Prop))=> (forall (Xx0:a), ((Xs Xx0)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx0) Xy)))))))->(P0 (fun (x:(a->Prop))=> (forall (Xx0:a), ((x Xx0)->(forall (Xy:a), ((iff (x Xy)) ((x3 Xx0) Xy))))))))
% Found (eta_expansion_dep000 P0) as proof of (P00 (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy)))))))
% Found ((eta_expansion_dep00 (fun (Xs:(a->Prop))=> (forall (Xx0:a), ((Xs Xx0)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx0) Xy))))))) P0) as proof of (P00 (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy)))))))
% Found (((eta_expansion_dep0 (fun (x5:(a->Prop))=> Prop)) (fun (Xs:(a->Prop))=> (forall (Xx0:a), ((Xs Xx0)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx0) Xy))))))) P0) as proof of (P00 (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy)))))))
% Found ((((eta_expansion_dep (a->Prop)) (fun (x5:(a->Prop))=> Prop)) (fun (Xs:(a->Prop))=> (forall (Xx0:a), ((Xs Xx0)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx0) Xy))))))) P0) as proof of (P00 (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy)))))))
% Found ((((eta_expansion_dep (a->Prop)) (fun (x5:(a->Prop))=> Prop)) (fun (Xs:(a->Prop))=> (forall (Xx0:a), ((Xs Xx0)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx0) Xy))))))) P0) as proof of (P00 (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy)))))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P0):((P0 (fun (Xs:(a->Prop))=> (forall (Xx0:a), ((Xs Xx0)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx0) Xy)))))))->(P0 (fun (x:(a->Prop))=> (forall (Xx0:a), ((x Xx0)->(forall (Xy:a), ((iff (x Xy)) ((x3 Xx0) Xy))))))))
% Found (eta_expansion_dep000 P0) as proof of (P00 (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy)))))))
% Found ((eta_expansion_dep00 (fun (Xs:(a->Prop))=> (forall (Xx0:a), ((Xs Xx0)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx0) Xy))))))) P0) as proof of (P00 (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy)))))))
% Found (((eta_expansion_dep0 (fun (x5:(a->Prop))=> Prop)) (fun (Xs:(a->Prop))=> (forall (Xx0:a), ((Xs Xx0)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx0) Xy))))))) P0) as proof of (P00 (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy)))))))
% Found ((((eta_expansion_dep (a->Prop)) (fun (x5:(a->Prop))=> Prop)) (fun (Xs:(a->Prop))=> (forall (Xx0:a), ((Xs Xx0)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx0) Xy))))))) P0) as proof of (P00 (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy)))))))
% Found ((((eta_expansion_dep (a->Prop)) (fun (x5:(a->Prop))=> Prop)) (fun (Xs:(a->Prop))=> (forall (Xx0:a), ((Xs Xx0)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx0) Xy))))))) P0) as proof of (P00 (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy)))))))
% Found eq_ref00:=(eq_ref0 b0):(((eq ((a->Prop)->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq ((a->Prop)->Prop)) b0) b)
% Found ((eq_ref ((a->Prop)->Prop)) b0) as proof of (((eq ((a->Prop)->Prop)) b0) b)
% Found ((eq_ref ((a->Prop)->Prop)) b0) as proof of (((eq ((a->Prop)->Prop)) b0) b)
% Found ((eq_ref ((a->Prop)->Prop)) b0) as proof of (((eq ((a->Prop)->Prop)) b0) b)
% Found eq_ref00:=(eq_ref0 (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy))))))):(((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy))))))) (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy)))))))
% Found (eq_ref0 (fun (Xs:(a->Prop))=> (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))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy))))))) b0)
% Found ((eq_ref ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> (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))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy))))))) b0)
% Found ((eq_ref ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> (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))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy))))))) b0)
% Found ((eq_ref ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> (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))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy))))))) b0)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (f x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% Found eq_ref00:=(eq_ref0 ((and ((and ((and (forall (Xx:a), ((x0 Xx) Xx))) (forall (Xx:a) (Xy:a), (((x0 Xx) Xy)->((x0 Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xz))->((x0 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy))))))) P))):(((eq Prop) ((and ((and ((and (forall (Xx:a), ((x0 Xx) Xx))) (forall (Xx:a) (Xy:a), (((x0 Xx) Xy)->((x0 Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xz))->((x0 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy))))))) P))) ((and ((and ((and (forall (Xx:a), ((x0 Xx) Xx))) (forall (Xx:a) (Xy:a), (((x0 Xx) Xy)->((x0 Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xz))->((x0 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy))))))) P)))
% Found (eq_ref0 ((and ((and ((and (forall (Xx:a), ((x0 Xx) Xx))) (forall (Xx:a) (Xy:a), (((x0 Xx) Xy)->((x0 Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xz))->((x0 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy))))))) P))) as proof of (((eq Prop) ((and ((and ((and (forall (Xx:a), ((x0 Xx) Xx))) (forall (Xx:a) (Xy:a), (((x0 Xx) Xy)->((x0 Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xz))->((x0 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy))))))) P))) b)
% Found ((eq_ref Prop) ((and ((and ((and (forall (Xx:a), ((x0 Xx) Xx))) (forall (Xx:a) (Xy:a), (((x0 Xx) Xy)->((x0 Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xz))->((x0 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy))))))) P))) as proof of (((eq Prop) ((and ((and ((and (forall (Xx:a), ((x0 Xx) Xx))) (forall (Xx:a) (Xy:a), (((x0 Xx) Xy)->((x0 Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xz))->((x0 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy))))))) P))) b)
% Found ((eq_ref Prop) ((and ((and ((and (forall (Xx:a), ((x0 Xx) Xx))) (forall (Xx:a) (Xy:a), (((x0 Xx) Xy)->((x0 Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xz))->((x0 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy))))))) P))) as proof of (((eq Prop) ((and ((and ((and (forall (Xx:a), ((x0 Xx) Xx))) (forall (Xx:a) (Xy:a), (((x0 Xx) Xy)->((x0 Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xz))->((x0 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy))))))) P))) b)
% Found ((eq_ref Prop) ((and ((and ((and (forall (Xx:a), ((x0 Xx) Xx))) (forall (Xx:a) (Xy:a), (((x0 Xx) Xy)->((x0 Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xz))->((x0 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy))))))) P))) as proof of (((eq Prop) ((and ((and ((and (forall (Xx:a), ((x0 Xx) Xx))) (forall (Xx:a) (Xy:a), (((x0 Xx) Xy)->((x0 Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xz))->((x0 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy))))))) P))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (f x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% Found eq_ref00:=(eq_ref0 ((and ((and ((and (forall (Xx:a), ((x0 Xx) Xx))) (forall (Xx:a) (Xy:a), (((x0 Xx) Xy)->((x0 Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xz))->((x0 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy))))))) P))):(((eq Prop) ((and ((and ((and (forall (Xx:a), ((x0 Xx) Xx))) (forall (Xx:a) (Xy:a), (((x0 Xx) Xy)->((x0 Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xz))->((x0 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy))))))) P))) ((and ((and ((and (forall (Xx:a), ((x0 Xx) Xx))) (forall (Xx:a) (Xy:a), (((x0 Xx) Xy)->((x0 Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xz))->((x0 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy))))))) P)))
% Found (eq_ref0 ((and ((and ((and (forall (Xx:a), ((x0 Xx) Xx))) (forall (Xx:a) (Xy:a), (((x0 Xx) Xy)->((x0 Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xz))->((x0 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy))))))) P))) as proof of (((eq Prop) ((and ((and ((and (forall (Xx:a), ((x0 Xx) Xx))) (forall (Xx:a) (Xy:a), (((x0 Xx) Xy)->((x0 Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xz))->((x0 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy))))))) P))) b)
% Found ((eq_ref Prop) ((and ((and ((and (forall (Xx:a), ((x0 Xx) Xx))) (forall (Xx:a) (Xy:a), (((x0 Xx) Xy)->((x0 Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xz))->((x0 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy))))))) P))) as proof of (((eq Prop) ((and ((and ((and (forall (Xx:a), ((x0 Xx) Xx))) (forall (Xx:a) (Xy:a), (((x0 Xx) Xy)->((x0 Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xz))->((x0 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy))))))) P))) b)
% Found ((eq_ref Prop) ((and ((and ((and (forall (Xx:a), ((x0 Xx) Xx))) (forall (Xx:a) (Xy:a), (((x0 Xx) Xy)->((x0 Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xz))->((x0 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x0 Xx) Xy))))))) P))) as proof of (((eq Prop) ((and ((and ((and (forall (Xx:a), ((x0 Xx) Xx))) (forall (Xx:a) (Xy:a), (((x0 Xx) Xy)->
% EOF
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