TSTP Solution File: SEV030^5 by cocATP---0.2.0
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- Process Solution
%------------------------------------------------------------------------------
% File : cocATP---0.2.0
% Problem : SEV030^5 : TPTP v6.1.0. Released v4.0.0.
% Transfm : none
% Format : tptp:raw
% Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% Computer : n110.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 13:33:36 EDT 2014
% Result : Timeout 300.03s
% Output : None
% Verified :
% SZS Type : None (Parsing solution fails)
% Syntax : Number of formulae : 0
% Comments :
%------------------------------------------------------------------------------
%----NO SOLUTION OUTPUT BY SYSTEM
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% % Problem : SEV030^5 : TPTP v6.1.0. Released v4.0.0.
% % Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n110.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 07:37:01 CDT 2014
% % CPUTime : 300.03
% 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 0x2168710>, <kernel.Type object at 0x21686c8>) of role type named a_type
% Using role type
% Declaring a:Type
% FOF formula (<kernel.Constant object at 0x202e758>, <kernel.DependentProduct object at 0x2168c68>) of role type named cP
% Using role type
% Declaring cP:((a->Prop)->Prop)
% FOF formula ((forall (Xq1:(a->Prop)) (Xq2:(a->Prop)), (((and (((eq (a->Prop)) Xq1) Xq2)) (cP Xq1))->(cP Xq2)))->(((and ((and (forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx))))))) (forall (Xx:a) (Xy:a) (Xp:(a->Prop)) (Xq:(a->Prop)), (((and ((and ((and ((and (cP Xp)) (cP Xq))) (Xp Xx))) (Xq Xx))) (Xp Xy))->(Xq Xy))))->((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))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy)))))))) cP)))))) of role conjecture named cTHM262_C_EXT2_pme
% Conjecture to prove = ((forall (Xq1:(a->Prop)) (Xq2:(a->Prop)), (((and (((eq (a->Prop)) Xq1) Xq2)) (cP Xq1))->(cP Xq2)))->(((and ((and (forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx))))))) (forall (Xx:a) (Xy:a) (Xp:(a->Prop)) (Xq:(a->Prop)), (((and ((and ((and ((and (cP Xp)) (cP Xq))) (Xp Xx))) (Xq Xx))) (Xp Xy))->(Xq Xy))))->((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))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy)))))))) cP)))))):Prop
% Parameter a_DUMMY:a.
% We need to prove ['((forall (Xq1:(a->Prop)) (Xq2:(a->Prop)), (((and (((eq (a->Prop)) Xq1) Xq2)) (cP Xq1))->(cP Xq2)))->(((and ((and (forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx))))))) (forall (Xx:a) (Xy:a) (Xp:(a->Prop)) (Xq:(a->Prop)), (((and ((and ((and ((and (cP Xp)) (cP Xq))) (Xp Xx))) (Xq Xx))) (Xp Xy))->(Xq Xy))))->((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))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy)))))))) cP))))))']
% Parameter a:Type.
% Parameter cP:((a->Prop)->Prop).
% Trying to prove ((forall (Xq1:(a->Prop)) (Xq2:(a->Prop)), (((and (((eq (a->Prop)) Xq1) Xq2)) (cP Xq1))->(cP Xq2)))->(((and ((and (forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx))))))) (forall (Xx:a) (Xy:a) (Xp:(a->Prop)) (Xq:(a->Prop)), (((and ((and ((and ((and (cP Xp)) (cP Xq))) (Xp Xx))) (Xq Xx))) (Xp Xy))->(Xq Xy))))->((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))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy)))))))) cP))))))
% 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))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy)))))))) cP)))):(((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))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy)))))))) cP)))) (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))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x Xx) Xy)))))))) cP))))
% Found (eta_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))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy)))))))) cP)))) 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))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy)))))))) cP)))) 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))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy)))))))) cP)))) 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))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy)))))))) cP)))) 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))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy)))))))) cP)))) 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))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy)))))))) cP)))) 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))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy)))))))) cP)))) 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))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy)))))))) cP)))) 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))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy)))))))) cP)))) 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))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy)))))))) cP)))) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (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))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy)))))))) cP)))):(((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))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy)))))))) cP)))) (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))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x Xx) Xy)))))))) cP))))
% Found (eta_expansion_dep00 (fun (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))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy)))))))) cP)))) 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))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy)))))))) cP)))) b)
% Found ((eta_expansion_dep0 (fun (x4:(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))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy)))))))) cP)))) 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))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy)))))))) cP)))) b)
% Found (((eta_expansion_dep (a->(a->Prop))) (fun (x4:(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))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy)))))))) cP)))) 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))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy)))))))) cP)))) b)
% Found (((eta_expansion_dep (a->(a->Prop))) (fun (x4:(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))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy)))))))) cP)))) 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))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy)))))))) cP)))) b)
% Found (((eta_expansion_dep (a->(a->Prop))) (fun (x4:(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))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy)))))))) cP)))) 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))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy)))))))) cP)))) b)
% Found choice000:=(choice00 (fun (x7:a) (x60:(a->Prop))=> ((and (cP x60)) (x60 x7)))):((forall (x:a), ((ex (a->Prop)) (fun (y:(a->Prop))=> ((and (cP y)) (y x)))))->((ex (a->(a->Prop))) (fun (f:(a->(a->Prop)))=> (forall (x:a), ((and (cP (f x))) ((f x) x))))))
% Found (choice00 (fun (x7:a) (x60:(a->Prop))=> ((and (cP x60)) (x60 x7)))) as proof of ((forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx)))))->(P b))
% Found ((choice0 (a->Prop)) (fun (x7:a) (x60:(a->Prop))=> ((and (cP x60)) (x60 x7)))) as proof of ((forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx)))))->(P b))
% Found (((choice a) (a->Prop)) (fun (x7:a) (x60:(a->Prop))=> ((and (cP x60)) (x60 x7)))) as proof of ((forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx)))))->(P b))
% Found (((choice a) (a->Prop)) (fun (x7:a) (x60:(a->Prop))=> ((and (cP x60)) (x60 x7)))) as proof of ((forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx)))))->(P b))
% Found (fun (x3:(forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))=> (((choice a) (a->Prop)) (fun (x7:a) (x60:(a->Prop))=> ((and (cP x60)) (x60 x7))))) as proof of ((forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx)))))->(P b))
% Found (fun (x3:(forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))=> (((choice a) (a->Prop)) (fun (x7:a) (x60:(a->Prop))=> ((and (cP x60)) (x60 x7))))) as proof of ((forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))->((forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx)))))->(P b)))
% Found (and_rect10 (fun (x3:(forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))=> (((choice a) (a->Prop)) (fun (x7:a) (x60:(a->Prop))=> ((and (cP x60)) (x60 x7)))))) as proof of (P b)
% Found ((and_rect1 (P b)) (fun (x3:(forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))=> (((choice a) (a->Prop)) (fun (x7:a) (x60:(a->Prop))=> ((and (cP x60)) (x60 x7)))))) as proof of (P b)
% Found (((fun (P0:Type) (x3:((forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))->((forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx)))))->P0)))=> (((((and_rect (forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx)))))) P0) x3) x1)) (P b)) (fun (x3:(forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))=> (((choice a) (a->Prop)) (fun (x7:a) (x60:(a->Prop))=> ((and (cP x60)) (x60 x7)))))) as proof of (P b)
% Found (((fun (P0:Type) (x3:((forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))->((forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx)))))->P0)))=> (((((and_rect (forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx)))))) P0) x3) x1)) (P b)) (fun (x3:(forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))=> (((choice a) (a->Prop)) (fun (x7:a) (x60:(a->Prop))=> ((and (cP x60)) (x60 x7)))))) as proof of (P b)
% Found choice000:=(choice00 (fun (x7:a) (x60:(a->Prop))=> ((and (cP x60)) (x60 x7)))):((forall (x:a), ((ex (a->Prop)) (fun (y:(a->Prop))=> ((and (cP y)) (y x)))))->((ex (a->(a->Prop))) (fun (f:(a->(a->Prop)))=> (forall (x:a), ((and (cP (f x))) ((f x) x))))))
% Found (choice00 (fun (x7:a) (x60:(a->Prop))=> ((and (cP x60)) (x60 x7)))) as proof of ((forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx)))))->(P b))
% Found ((choice0 (a->Prop)) (fun (x7:a) (x60:(a->Prop))=> ((and (cP x60)) (x60 x7)))) as proof of ((forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx)))))->(P b))
% Found (((choice a) (a->Prop)) (fun (x7:a) (x60:(a->Prop))=> ((and (cP x60)) (x60 x7)))) as proof of ((forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx)))))->(P b))
% Found (((choice a) (a->Prop)) (fun (x7:a) (x60:(a->Prop))=> ((and (cP x60)) (x60 x7)))) as proof of ((forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx)))))->(P b))
% Found (fun (x3:(forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))=> (((choice a) (a->Prop)) (fun (x7:a) (x60:(a->Prop))=> ((and (cP x60)) (x60 x7))))) as proof of ((forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx)))))->(P b))
% Found (fun (x3:(forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))=> (((choice a) (a->Prop)) (fun (x7:a) (x60:(a->Prop))=> ((and (cP x60)) (x60 x7))))) as proof of ((forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))->((forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx)))))->(P b)))
% Found (and_rect10 (fun (x3:(forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))=> (((choice a) (a->Prop)) (fun (x7:a) (x60:(a->Prop))=> ((and (cP x60)) (x60 x7)))))) as proof of (P b)
% Found ((and_rect1 (P b)) (fun (x3:(forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))=> (((choice a) (a->Prop)) (fun (x7:a) (x60:(a->Prop))=> ((and (cP x60)) (x60 x7)))))) as proof of (P b)
% Found (((fun (P0:Type) (x3:((forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))->((forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx)))))->P0)))=> (((((and_rect (forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx)))))) P0) x3) x1)) (P b)) (fun (x3:(forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))=> (((choice a) (a->Prop)) (fun (x7:a) (x60:(a->Prop))=> ((and (cP x60)) (x60 x7)))))) as proof of (P b)
% Found (fun (x2:(forall (Xx:a) (Xy:a) (Xp:(a->Prop)) (Xq:(a->Prop)), (((and ((and ((and ((and (cP Xp)) (cP Xq))) (Xp Xx))) (Xq Xx))) (Xp Xy))->(Xq Xy))))=> (((fun (P0:Type) (x3:((forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))->((forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx)))))->P0)))=> (((((and_rect (forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx)))))) P0) x3) x1)) (P b)) (fun (x3:(forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))=> (((choice a) (a->Prop)) (fun (x7:a) (x60:(a->Prop))=> ((and (cP x60)) (x60 x7))))))) as proof of (P b)
% Found (fun (x1:((and (forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx))))))) (x2:(forall (Xx:a) (Xy:a) (Xp:(a->Prop)) (Xq:(a->Prop)), (((and ((and ((and ((and (cP Xp)) (cP Xq))) (Xp Xx))) (Xq Xx))) (Xp Xy))->(Xq Xy))))=> (((fun (P0:Type) (x3:((forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))->((forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx)))))->P0)))=> (((((and_rect (forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx)))))) P0) x3) x1)) (P b)) (fun (x3:(forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))=> (((choice a) (a->Prop)) (fun (x7:a) (x60:(a->Prop))=> ((and (cP x60)) (x60 x7))))))) as proof of ((forall (Xx:a) (Xy:a) (Xp:(a->Prop)) (Xq:(a->Prop)), (((and ((and ((and ((and (cP Xp)) (cP Xq))) (Xp Xx))) (Xq Xx))) (Xp Xy))->(Xq Xy)))->(P b))
% Found (fun (x1:((and (forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx))))))) (x2:(forall (Xx:a) (Xy:a) (Xp:(a->Prop)) (Xq:(a->Prop)), (((and ((and ((and ((and (cP Xp)) (cP Xq))) (Xp Xx))) (Xq Xx))) (Xp Xy))->(Xq Xy))))=> (((fun (P0:Type) (x3:((forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))->((forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx)))))->P0)))=> (((((and_rect (forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx)))))) P0) x3) x1)) (P b)) (fun (x3:(forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))=> (((choice a) (a->Prop)) (fun (x7:a) (x60:(a->Prop))=> ((and (cP x60)) (x60 x7))))))) as proof of (((and (forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx))))))->((forall (Xx:a) (Xy:a) (Xp:(a->Prop)) (Xq:(a->Prop)), (((and ((and ((and ((and (cP Xp)) (cP Xq))) (Xp Xx))) (Xq Xx))) (Xp Xy))->(Xq Xy)))->(P b)))
% Found (and_rect00 (fun (x1:((and (forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx))))))) (x2:(forall (Xx:a) (Xy:a) (Xp:(a->Prop)) (Xq:(a->Prop)), (((and ((and ((and ((and (cP Xp)) (cP Xq))) (Xp Xx))) (Xq Xx))) (Xp Xy))->(Xq Xy))))=> (((fun (P0:Type) (x3:((forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))->((forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx)))))->P0)))=> (((((and_rect (forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx)))))) P0) x3) x1)) (P b)) (fun (x3:(forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))=> (((choice a) (a->Prop)) (fun (x7:a) (x60:(a->Prop))=> ((and (cP x60)) (x60 x7)))))))) as proof of (P b)
% Found ((and_rect0 (P b)) (fun (x1:((and (forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx))))))) (x2:(forall (Xx:a) (Xy:a) (Xp:(a->Prop)) (Xq:(a->Prop)), (((and ((and ((and ((and (cP Xp)) (cP Xq))) (Xp Xx))) (Xq Xx))) (Xp Xy))->(Xq Xy))))=> (((fun (P0:Type) (x3:((forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))->((forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx)))))->P0)))=> (((((and_rect (forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx)))))) P0) x3) x1)) (P b)) (fun (x3:(forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))=> (((choice a) (a->Prop)) (fun (x7:a) (x60:(a->Prop))=> ((and (cP x60)) (x60 x7)))))))) as proof of (P b)
% Found (((fun (P0:Type) (x1:(((and (forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx))))))->((forall (Xx:a) (Xy:a) (Xp:(a->Prop)) (Xq:(a->Prop)), (((and ((and ((and ((and (cP Xp)) (cP Xq))) (Xp Xx))) (Xq Xx))) (Xp Xy))->(Xq Xy)))->P0)))=> (((((and_rect ((and (forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx))))))) (forall (Xx:a) (Xy:a) (Xp:(a->Prop)) (Xq:(a->Prop)), (((and ((and ((and ((and (cP Xp)) (cP Xq))) (Xp Xx))) (Xq Xx))) (Xp Xy))->(Xq Xy)))) P0) x1) x0)) (P b)) (fun (x1:((and (forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx))))))) (x2:(forall (Xx:a) (Xy:a) (Xp:(a->Prop)) (Xq:(a->Prop)), (((and ((and ((and ((and (cP Xp)) (cP Xq))) (Xp Xx))) (Xq Xx))) (Xp Xy))->(Xq Xy))))=> (((fun (P0:Type) (x3:((forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))->((forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx)))))->P0)))=> (((((and_rect (forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx)))))) P0) x3) x1)) (P b)) (fun (x3:(forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))=> (((choice a) (a->Prop)) (fun (x7:a) (x60:(a->Prop))=> ((and (cP x60)) (x60 x7)))))))) as proof of (P b)
% Found (((fun (P0:Type) (x1:(((and (forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx))))))->((forall (Xx:a) (Xy:a) (Xp:(a->Prop)) (Xq:(a->Prop)), (((and ((and ((and ((and (cP Xp)) (cP Xq))) (Xp Xx))) (Xq Xx))) (Xp Xy))->(Xq Xy)))->P0)))=> (((((and_rect ((and (forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx))))))) (forall (Xx:a) (Xy:a) (Xp:(a->Prop)) (Xq:(a->Prop)), (((and ((and ((and ((and (cP Xp)) (cP Xq))) (Xp Xx))) (Xq Xx))) (Xp Xy))->(Xq Xy)))) P0) x1) x0)) (P b)) (fun (x1:((and (forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx))))))) (x2:(forall (Xx:a) (Xy:a) (Xp:(a->Prop)) (Xq:(a->Prop)), (((and ((and ((and ((and (cP Xp)) (cP Xq))) (Xp Xx))) (Xq Xx))) (Xp Xy))->(Xq Xy))))=> (((fun (P0:Type) (x3:((forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))->((forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx)))))->P0)))=> (((((and_rect (forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx)))))) P0) x3) x1)) (P b)) (fun (x3:(forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))=> (((choice a) (a->Prop)) (fun (x7:a) (x60:(a->Prop))=> ((and (cP x60)) (x60 x7)))))))) as proof of (P b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (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))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy)))))))) cP)))):(((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))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy)))))))) cP)))) (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))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x Xx) Xy)))))))) cP))))
% Found (eta_expansion_dep00 (fun (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))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy)))))))) cP)))) 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))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy)))))))) cP)))) b)
% Found ((eta_expansion_dep0 (fun (x6:(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))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy)))))))) cP)))) 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))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy)))))))) cP)))) b)
% Found (((eta_expansion_dep (a->(a->Prop))) (fun (x6:(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))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy)))))))) cP)))) 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))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy)))))))) cP)))) b)
% Found (((eta_expansion_dep (a->(a->Prop))) (fun (x6:(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))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy)))))))) cP)))) 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))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy)))))))) cP)))) b)
% Found (((eta_expansion_dep (a->(a->Prop))) (fun (x6:(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))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy)))))))) cP)))) 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))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy)))))))) cP)))) b)
% Found choice000:=(choice00 (fun (x7:a) (x60:(a->Prop))=> ((and (cP x60)) (x60 x7)))):((forall (x:a), ((ex (a->Prop)) (fun (y:(a->Prop))=> ((and (cP y)) (y x)))))->((ex (a->(a->Prop))) (fun (f:(a->(a->Prop)))=> (forall (x:a), ((and (cP (f x))) ((f x) x))))))
% Found (choice00 (fun (x7:a) (x60:(a->Prop))=> ((and (cP x60)) (x60 x7)))) as proof of ((forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx)))))->(P f))
% Found ((choice0 (a->Prop)) (fun (x7:a) (x60:(a->Prop))=> ((and (cP x60)) (x60 x7)))) as proof of ((forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx)))))->(P f))
% Found (((choice a) (a->Prop)) (fun (x7:a) (x60:(a->Prop))=> ((and (cP x60)) (x60 x7)))) as proof of ((forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx)))))->(P f))
% Found (((choice a) (a->Prop)) (fun (x7:a) (x60:(a->Prop))=> ((and (cP x60)) (x60 x7)))) as proof of ((forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx)))))->(P f))
% Found (fun (x3:(forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))=> (((choice a) (a->Prop)) (fun (x7:a) (x60:(a->Prop))=> ((and (cP x60)) (x60 x7))))) as proof of ((forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx)))))->(P f))
% Found (fun (x3:(forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))=> (((choice a) (a->Prop)) (fun (x7:a) (x60:(a->Prop))=> ((and (cP x60)) (x60 x7))))) as proof of ((forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))->((forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx)))))->(P f)))
% Found (and_rect10 (fun (x3:(forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))=> (((choice a) (a->Prop)) (fun (x7:a) (x60:(a->Prop))=> ((and (cP x60)) (x60 x7)))))) as proof of (P f)
% Found ((and_rect1 (P f)) (fun (x3:(forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))=> (((choice a) (a->Prop)) (fun (x7:a) (x60:(a->Prop))=> ((and (cP x60)) (x60 x7)))))) as proof of (P f)
% Found (((fun (P0:Type) (x3:((forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))->((forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx)))))->P0)))=> (((((and_rect (forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx)))))) P0) x3) x1)) (P f)) (fun (x3:(forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))=> (((choice a) (a->Prop)) (fun (x7:a) (x60:(a->Prop))=> ((and (cP x60)) (x60 x7)))))) as proof of (P f)
% Found (((fun (P0:Type) (x3:((forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))->((forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx)))))->P0)))=> (((((and_rect (forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx)))))) P0) x3) x1)) (P f)) (fun (x3:(forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))=> (((choice a) (a->Prop)) (fun (x7:a) (x60:(a->Prop))=> ((and (cP x60)) (x60 x7)))))) as proof of (P f)
% Found choice000:=(choice00 (fun (x7:a) (x60:(a->Prop))=> ((and (cP x60)) (x60 x7)))):((forall (x:a), ((ex (a->Prop)) (fun (y:(a->Prop))=> ((and (cP y)) (y x)))))->((ex (a->(a->Prop))) (fun (f:(a->(a->Prop)))=> (forall (x:a), ((and (cP (f x))) ((f x) x))))))
% Found (choice00 (fun (x7:a) (x60:(a->Prop))=> ((and (cP x60)) (x60 x7)))) as proof of ((forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx)))))->(P f))
% Found ((choice0 (a->Prop)) (fun (x7:a) (x60:(a->Prop))=> ((and (cP x60)) (x60 x7)))) as proof of ((forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx)))))->(P f))
% Found (((choice a) (a->Prop)) (fun (x7:a) (x60:(a->Prop))=> ((and (cP x60)) (x60 x7)))) as proof of ((forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx)))))->(P f))
% Found (((choice a) (a->Prop)) (fun (x7:a) (x60:(a->Prop))=> ((and (cP x60)) (x60 x7)))) as proof of ((forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx)))))->(P f))
% Found (fun (x3:(forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))=> (((choice a) (a->Prop)) (fun (x7:a) (x60:(a->Prop))=> ((and (cP x60)) (x60 x7))))) as proof of ((forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx)))))->(P f))
% Found (fun (x3:(forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))=> (((choice a) (a->Prop)) (fun (x7:a) (x60:(a->Prop))=> ((and (cP x60)) (x60 x7))))) as proof of ((forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))->((forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx)))))->(P f)))
% Found (and_rect10 (fun (x3:(forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))=> (((choice a) (a->Prop)) (fun (x7:a) (x60:(a->Prop))=> ((and (cP x60)) (x60 x7)))))) as proof of (P f)
% Found ((and_rect1 (P f)) (fun (x3:(forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))=> (((choice a) (a->Prop)) (fun (x7:a) (x60:(a->Prop))=> ((and (cP x60)) (x60 x7)))))) as proof of (P f)
% Found (((fun (P0:Type) (x3:((forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))->((forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx)))))->P0)))=> (((((and_rect (forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx)))))) P0) x3) x1)) (P f)) (fun (x3:(forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))=> (((choice a) (a->Prop)) (fun (x7:a) (x60:(a->Prop))=> ((and (cP x60)) (x60 x7)))))) as proof of (P f)
% Found (((fun (P0:Type) (x3:((forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))->((forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx)))))->P0)))=> (((((and_rect (forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx)))))) P0) x3) x1)) (P f)) (fun (x3:(forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))=> (((choice a) (a->Prop)) (fun (x7:a) (x60:(a->Prop))=> ((and (cP x60)) (x60 x7)))))) as proof of (P 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) cP)
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) cP)
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) cP)
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) cP)
% Found eq_ref00:=(eq_ref0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))):(((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found (eq_ref0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) b)
% Found ((eq_ref ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) b)
% Found ((eq_ref ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) b)
% Found ((eq_ref ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) b)
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))->(P (fun (x:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (x Xz)))) (forall (Xx:a), ((x Xx)->(forall (Xy:a), ((iff (x Xy)) ((x1 Xx) Xy)))))))))
% Found (eta_expansion_dep000 P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found ((eta_expansion_dep00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found (((eta_expansion_dep0 (fun (x3:(a->Prop))=> Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found ((((eta_expansion_dep (a->Prop)) (fun (x3:(a->Prop))=> Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found ((((eta_expansion_dep (a->Prop)) (fun (x3:(a->Prop))=> Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found eq_ref000:=(eq_ref00 P):((P (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))->(P (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))))
% Found (eq_ref00 P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found ((eq_ref0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found (((eq_ref ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found (((eq_ref ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found eta_expansion0000:=(eta_expansion000 P):((P (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))->(P (fun (x:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (x Xz)))) (forall (Xx:a), ((x Xx)->(forall (Xy:a), ((iff (x Xy)) ((x1 Xx) Xy)))))))))
% Found (eta_expansion000 P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found ((eta_expansion00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found (((eta_expansion0 Prop) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found ((((eta_expansion (a->Prop)) Prop) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found ((((eta_expansion (a->Prop)) Prop) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found choice000:=(choice00 (fun (x7:a) (x60:(a->Prop))=> ((and (cP x60)) (x60 x7)))):((forall (x:a), ((ex (a->Prop)) (fun (y:(a->Prop))=> ((and (cP y)) (y x)))))->((ex (a->(a->Prop))) (fun (f:(a->(a->Prop)))=> (forall (x:a), ((and (cP (f x))) ((f x) x))))))
% Found (choice00 (fun (x7:a) (x60:(a->Prop))=> ((and (cP x60)) (x60 x7)))) as proof of ((forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx)))))->(P f))
% Found ((choice0 (a->Prop)) (fun (x7:a) (x60:(a->Prop))=> ((and (cP x60)) (x60 x7)))) as proof of ((forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx)))))->(P f))
% Found (((choice a) (a->Prop)) (fun (x7:a) (x60:(a->Prop))=> ((and (cP x60)) (x60 x7)))) as proof of ((forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx)))))->(P f))
% Found (((choice a) (a->Prop)) (fun (x7:a) (x60:(a->Prop))=> ((and (cP x60)) (x60 x7)))) as proof of ((forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx)))))->(P f))
% Found (fun (x3:(forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))=> (((choice a) (a->Prop)) (fun (x7:a) (x60:(a->Prop))=> ((and (cP x60)) (x60 x7))))) as proof of ((forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx)))))->(P f))
% Found (fun (x3:(forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))=> (((choice a) (a->Prop)) (fun (x7:a) (x60:(a->Prop))=> ((and (cP x60)) (x60 x7))))) as proof of ((forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))->((forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx)))))->(P f)))
% Found (and_rect10 (fun (x3:(forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))=> (((choice a) (a->Prop)) (fun (x7:a) (x60:(a->Prop))=> ((and (cP x60)) (x60 x7)))))) as proof of (P f)
% Found ((and_rect1 (P f)) (fun (x3:(forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))=> (((choice a) (a->Prop)) (fun (x7:a) (x60:(a->Prop))=> ((and (cP x60)) (x60 x7)))))) as proof of (P f)
% Found (((fun (P0:Type) (x3:((forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))->((forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx)))))->P0)))=> (((((and_rect (forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx)))))) P0) x3) x1)) (P f)) (fun (x3:(forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))=> (((choice a) (a->Prop)) (fun (x7:a) (x60:(a->Prop))=> ((and (cP x60)) (x60 x7)))))) as proof of (P f)
% Found (fun (x2:(forall (Xx:a) (Xy:a) (Xp:(a->Prop)) (Xq:(a->Prop)), (((and ((and ((and ((and (cP Xp)) (cP Xq))) (Xp Xx))) (Xq Xx))) (Xp Xy))->(Xq Xy))))=> (((fun (P0:Type) (x3:((forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))->((forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx)))))->P0)))=> (((((and_rect (forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx)))))) P0) x3) x1)) (P f)) (fun (x3:(forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))=> (((choice a) (a->Prop)) (fun (x7:a) (x60:(a->Prop))=> ((and (cP x60)) (x60 x7))))))) as proof of (P f)
% Found (fun (x1:((and (forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx))))))) (x2:(forall (Xx:a) (Xy:a) (Xp:(a->Prop)) (Xq:(a->Prop)), (((and ((and ((and ((and (cP Xp)) (cP Xq))) (Xp Xx))) (Xq Xx))) (Xp Xy))->(Xq Xy))))=> (((fun (P0:Type) (x3:((forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))->((forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx)))))->P0)))=> (((((and_rect (forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx)))))) P0) x3) x1)) (P f)) (fun (x3:(forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))=> (((choice a) (a->Prop)) (fun (x7:a) (x60:(a->Prop))=> ((and (cP x60)) (x60 x7))))))) as proof of ((forall (Xx:a) (Xy:a) (Xp:(a->Prop)) (Xq:(a->Prop)), (((and ((and ((and ((and (cP Xp)) (cP Xq))) (Xp Xx))) (Xq Xx))) (Xp Xy))->(Xq Xy)))->(P f))
% Found (fun (x1:((and (forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx))))))) (x2:(forall (Xx:a) (Xy:a) (Xp:(a->Prop)) (Xq:(a->Prop)), (((and ((and ((and ((and (cP Xp)) (cP Xq))) (Xp Xx))) (Xq Xx))) (Xp Xy))->(Xq Xy))))=> (((fun (P0:Type) (x3:((forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))->((forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx)))))->P0)))=> (((((and_rect (forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx)))))) P0) x3) x1)) (P f)) (fun (x3:(forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))=> (((choice a) (a->Prop)) (fun (x7:a) (x60:(a->Prop))=> ((and (cP x60)) (x60 x7))))))) as proof of (((and (forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx))))))->((forall (Xx:a) (Xy:a) (Xp:(a->Prop)) (Xq:(a->Prop)), (((and ((and ((and ((and (cP Xp)) (cP Xq))) (Xp Xx))) (Xq Xx))) (Xp Xy))->(Xq Xy)))->(P f)))
% Found (and_rect00 (fun (x1:((and (forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx))))))) (x2:(forall (Xx:a) (Xy:a) (Xp:(a->Prop)) (Xq:(a->Prop)), (((and ((and ((and ((and (cP Xp)) (cP Xq))) (Xp Xx))) (Xq Xx))) (Xp Xy))->(Xq Xy))))=> (((fun (P0:Type) (x3:((forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))->((forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx)))))->P0)))=> (((((and_rect (forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx)))))) P0) x3) x1)) (P f)) (fun (x3:(forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))=> (((choice a) (a->Prop)) (fun (x7:a) (x60:(a->Prop))=> ((and (cP x60)) (x60 x7)))))))) as proof of (P f)
% Found ((and_rect0 (P f)) (fun (x1:((and (forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx))))))) (x2:(forall (Xx:a) (Xy:a) (Xp:(a->Prop)) (Xq:(a->Prop)), (((and ((and ((and ((and (cP Xp)) (cP Xq))) (Xp Xx))) (Xq Xx))) (Xp Xy))->(Xq Xy))))=> (((fun (P0:Type) (x3:((forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))->((forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx)))))->P0)))=> (((((and_rect (forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx)))))) P0) x3) x1)) (P f)) (fun (x3:(forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))=> (((choice a) (a->Prop)) (fun (x7:a) (x60:(a->Prop))=> ((and (cP x60)) (x60 x7)))))))) as proof of (P f)
% Found (((fun (P0:Type) (x1:(((and (forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx))))))->((forall (Xx:a) (Xy:a) (Xp:(a->Prop)) (Xq:(a->Prop)), (((and ((and ((and ((and (cP Xp)) (cP Xq))) (Xp Xx))) (Xq Xx))) (Xp Xy))->(Xq Xy)))->P0)))=> (((((and_rect ((and (forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx))))))) (forall (Xx:a) (Xy:a) (Xp:(a->Prop)) (Xq:(a->Prop)), (((and ((and ((and ((and (cP Xp)) (cP Xq))) (Xp Xx))) (Xq Xx))) (Xp Xy))->(Xq Xy)))) P0) x1) x0)) (P f)) (fun (x1:((and (forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx))))))) (x2:(forall (Xx:a) (Xy:a) (Xp:(a->Prop)) (Xq:(a->Prop)), (((and ((and ((and ((and (cP Xp)) (cP Xq))) (Xp Xx))) (Xq Xx))) (Xp Xy))->(Xq Xy))))=> (((fun (P0:Type) (x3:((forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))->((forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx)))))->P0)))=> (((((and_rect (forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx)))))) P0) x3) x1)) (P f)) (fun (x3:(forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))=> (((choice a) (a->Prop)) (fun (x7:a) (x60:(a->Prop))=> ((and (cP x60)) (x60 x7)))))))) as proof of (P f)
% Found (((fun (P0:Type) (x1:(((and (forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx))))))->((forall (Xx:a) (Xy:a) (Xp:(a->Prop)) (Xq:(a->Prop)), (((and ((and ((and ((and (cP Xp)) (cP Xq))) (Xp Xx))) (Xq Xx))) (Xp Xy))->(Xq Xy)))->P0)))=> (((((and_rect ((and (forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx))))))) (forall (Xx:a) (Xy:a) (Xp:(a->Prop)) (Xq:(a->Prop)), (((and ((and ((and ((and (cP Xp)) (cP Xq))) (Xp Xx))) (Xq Xx))) (Xp Xy))->(Xq Xy)))) P0) x1) x0)) (P f)) (fun (x1:((and (forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx))))))) (x2:(forall (Xx:a) (Xy:a) (Xp:(a->Prop)) (Xq:(a->Prop)), (((and ((and ((and ((and (cP Xp)) (cP Xq))) (Xp Xx))) (Xq Xx))) (Xp Xy))->(Xq Xy))))=> (((fun (P0:Type) (x3:((forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))->((forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx)))))->P0)))=> (((((and_rect (forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx)))))) P0) x3) x1)) (P f)) (fun (x3:(forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))=> (((choice a) (a->Prop)) (fun (x7:a) (x60:(a->Prop))=> ((and (cP x60)) (x60 x7)))))))) as proof of (P f)
% Found choice000:=(choice00 (fun (x7:a) (x60:(a->Prop))=> ((and (cP x60)) (x60 x7)))):((forall (x:a), ((ex (a->Prop)) (fun (y:(a->Prop))=> ((and (cP y)) (y x)))))->((ex (a->(a->Prop))) (fun (f:(a->(a->Prop)))=> (forall (x:a), ((and (cP (f x))) ((f x) x))))))
% Found (choice00 (fun (x7:a) (x60:(a->Prop))=> ((and (cP x60)) (x60 x7)))) as proof of ((forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx)))))->(P f))
% Found ((choice0 (a->Prop)) (fun (x7:a) (x60:(a->Prop))=> ((and (cP x60)) (x60 x7)))) as proof of ((forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx)))))->(P f))
% Found (((choice a) (a->Prop)) (fun (x7:a) (x60:(a->Prop))=> ((and (cP x60)) (x60 x7)))) as proof of ((forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx)))))->(P f))
% Found (((choice a) (a->Prop)) (fun (x7:a) (x60:(a->Prop))=> ((and (cP x60)) (x60 x7)))) as proof of ((forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx)))))->(P f))
% Found (fun (x3:(forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))=> (((choice a) (a->Prop)) (fun (x7:a) (x60:(a->Prop))=> ((and (cP x60)) (x60 x7))))) as proof of ((forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx)))))->(P f))
% Found (fun (x3:(forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))=> (((choice a) (a->Prop)) (fun (x7:a) (x60:(a->Prop))=> ((and (cP x60)) (x60 x7))))) as proof of ((forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))->((forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx)))))->(P f)))
% Found (and_rect10 (fun (x3:(forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))=> (((choice a) (a->Prop)) (fun (x7:a) (x60:(a->Prop))=> ((and (cP x60)) (x60 x7)))))) as proof of (P f)
% Found ((and_rect1 (P f)) (fun (x3:(forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))=> (((choice a) (a->Prop)) (fun (x7:a) (x60:(a->Prop))=> ((and (cP x60)) (x60 x7)))))) as proof of (P f)
% Found (((fun (P0:Type) (x3:((forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))->((forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx)))))->P0)))=> (((((and_rect (forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx)))))) P0) x3) x1)) (P f)) (fun (x3:(forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))=> (((choice a) (a->Prop)) (fun (x7:a) (x60:(a->Prop))=> ((and (cP x60)) (x60 x7)))))) as proof of (P f)
% Found (fun (x2:(forall (Xx:a) (Xy:a) (Xp:(a->Prop)) (Xq:(a->Prop)), (((and ((and ((and ((and (cP Xp)) (cP Xq))) (Xp Xx))) (Xq Xx))) (Xp Xy))->(Xq Xy))))=> (((fun (P0:Type) (x3:((forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))->((forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx)))))->P0)))=> (((((and_rect (forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx)))))) P0) x3) x1)) (P f)) (fun (x3:(forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))=> (((choice a) (a->Prop)) (fun (x7:a) (x60:(a->Prop))=> ((and (cP x60)) (x60 x7))))))) as proof of (P f)
% Found (fun (x1:((and (forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx))))))) (x2:(forall (Xx:a) (Xy:a) (Xp:(a->Prop)) (Xq:(a->Prop)), (((and ((and ((and ((and (cP Xp)) (cP Xq))) (Xp Xx))) (Xq Xx))) (Xp Xy))->(Xq Xy))))=> (((fun (P0:Type) (x3:((forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))->((forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx)))))->P0)))=> (((((and_rect (forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx)))))) P0) x3) x1)) (P f)) (fun (x3:(forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))=> (((choice a) (a->Prop)) (fun (x7:a) (x60:(a->Prop))=> ((and (cP x60)) (x60 x7))))))) as proof of ((forall (Xx:a) (Xy:a) (Xp:(a->Prop)) (Xq:(a->Prop)), (((and ((and ((and ((and (cP Xp)) (cP Xq))) (Xp Xx))) (Xq Xx))) (Xp Xy))->(Xq Xy)))->(P f))
% Found (fun (x1:((and (forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx))))))) (x2:(forall (Xx:a) (Xy:a) (Xp:(a->Prop)) (Xq:(a->Prop)), (((and ((and ((and ((and (cP Xp)) (cP Xq))) (Xp Xx))) (Xq Xx))) (Xp Xy))->(Xq Xy))))=> (((fun (P0:Type) (x3:((forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))->((forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx)))))->P0)))=> (((((and_rect (forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx)))))) P0) x3) x1)) (P f)) (fun (x3:(forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))=> (((choice a) (a->Prop)) (fun (x7:a) (x60:(a->Prop))=> ((and (cP x60)) (x60 x7))))))) as proof of (((and (forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx))))))->((forall (Xx:a) (Xy:a) (Xp:(a->Prop)) (Xq:(a->Prop)), (((and ((and ((and ((and (cP Xp)) (cP Xq))) (Xp Xx))) (Xq Xx))) (Xp Xy))->(Xq Xy)))->(P f)))
% Found (and_rect00 (fun (x1:((and (forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx))))))) (x2:(forall (Xx:a) (Xy:a) (Xp:(a->Prop)) (Xq:(a->Prop)), (((and ((and ((and ((and (cP Xp)) (cP Xq))) (Xp Xx))) (Xq Xx))) (Xp Xy))->(Xq Xy))))=> (((fun (P0:Type) (x3:((forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))->((forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx)))))->P0)))=> (((((and_rect (forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx)))))) P0) x3) x1)) (P f)) (fun (x3:(forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))=> (((choice a) (a->Prop)) (fun (x7:a) (x60:(a->Prop))=> ((and (cP x60)) (x60 x7)))))))) as proof of (P f)
% Found ((and_rect0 (P f)) (fun (x1:((and (forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx))))))) (x2:(forall (Xx:a) (Xy:a) (Xp:(a->Prop)) (Xq:(a->Prop)), (((and ((and ((and ((and (cP Xp)) (cP Xq))) (Xp Xx))) (Xq Xx))) (Xp Xy))->(Xq Xy))))=> (((fun (P0:Type) (x3:((forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))->((forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx)))))->P0)))=> (((((and_rect (forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx)))))) P0) x3) x1)) (P f)) (fun (x3:(forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))=> (((choice a) (a->Prop)) (fun (x7:a) (x60:(a->Prop))=> ((and (cP x60)) (x60 x7)))))))) as proof of (P f)
% Found (((fun (P0:Type) (x1:(((and (forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx))))))->((forall (Xx:a) (Xy:a) (Xp:(a->Prop)) (Xq:(a->Prop)), (((and ((and ((and ((and (cP Xp)) (cP Xq))) (Xp Xx))) (Xq Xx))) (Xp Xy))->(Xq Xy)))->P0)))=> (((((and_rect ((and (forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx))))))) (forall (Xx:a) (Xy:a) (Xp:(a->Prop)) (Xq:(a->Prop)), (((and ((and ((and ((and (cP Xp)) (cP Xq))) (Xp Xx))) (Xq Xx))) (Xp Xy))->(Xq Xy)))) P0) x1) x0)) (P f)) (fun (x1:((and (forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx))))))) (x2:(forall (Xx:a) (Xy:a) (Xp:(a->Prop)) (Xq:(a->Prop)), (((and ((and ((and ((and (cP Xp)) (cP Xq))) (Xp Xx))) (Xq Xx))) (Xp Xy))->(Xq Xy))))=> (((fun (P0:Type) (x3:((forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))->((forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx)))))->P0)))=> (((((and_rect (forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx)))))) P0) x3) x1)) (P f)) (fun (x3:(forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))=> (((choice a) (a->Prop)) (fun (x7:a) (x60:(a->Prop))=> ((and (cP x60)) (x60 x7)))))))) as proof of (P f)
% Found (((fun (P0:Type) (x1:(((and (forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx))))))->((forall (Xx:a) (Xy:a) (Xp:(a->Prop)) (Xq:(a->Prop)), (((and ((and ((and ((and (cP Xp)) (cP Xq))) (Xp Xx))) (Xq Xx))) (Xp Xy))->(Xq Xy)))->P0)))=> (((((and_rect ((and (forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx))))))) (forall (Xx:a) (Xy:a) (Xp:(a->Prop)) (Xq:(a->Prop)), (((and ((and ((and ((and (cP Xp)) (cP Xq))) (Xp Xx))) (Xq Xx))) (Xp Xy))->(Xq Xy)))) P0) x1) x0)) (P f)) (fun (x1:((and (forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx))))))) (x2:(forall (Xx:a) (Xy:a) (Xp:(a->Prop)) (Xq:(a->Prop)), (((and ((and ((and ((and (cP Xp)) (cP Xq))) (Xp Xx))) (Xq Xx))) (Xp Xy))->(Xq Xy))))=> (((fun (P0:Type) (x3:((forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))->((forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx)))))->P0)))=> (((((and_rect (forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx)))))) P0) x3) x1)) (P f)) (fun (x3:(forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))=> (((choice a) (a->Prop)) (fun (x7:a) (x60:(a->Prop))=> ((and (cP x60)) (x60 x7)))))))) as proof of (P 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) cP)
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) cP)
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) cP)
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) cP)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy)))))))):(((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy)))))))) (fun (x:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (x Xz)))) (forall (Xx:a), ((x Xx)->(forall (Xy:a), ((iff (x Xy)) ((x3 Xx) Xy))))))))
% Found (eta_expansion_dep00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy)))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy)))))))) b)
% Found ((eta_expansion_dep0 (fun (x5:(a->Prop))=> Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (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))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy)))))))) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x5:(a->Prop))=> Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (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))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy)))))))) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x5:(a->Prop))=> Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (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))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy)))))))) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x5:(a->Prop))=> Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (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))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy)))))))) b)
% Found eq_ref000:=(eq_ref00 P):((P (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy))))))))->(P (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy)))))))))
% Found (eq_ref00 P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy))))))))
% Found ((eq_ref0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy)))))))) P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy))))))))
% Found (((eq_ref ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy)))))))) P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy))))))))
% Found (((eq_ref ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy)))))))) P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy))))))))
% Found eq_ref00:=(eq_ref0 b):(((eq ((a->Prop)->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq ((a->Prop)->Prop)) b) cP)
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) cP)
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) cP)
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) cP)
% Found eq_ref00:=(eq_ref0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))):(((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found (eq_ref0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) b)
% Found ((eq_ref ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) b)
% Found ((eq_ref ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) b)
% Found ((eq_ref ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) b)
% Found eq_ref000:=(eq_ref00 P):((P (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))->(P (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))))
% Found (eq_ref00 P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found ((eq_ref0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found (((eq_ref ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found (((eq_ref ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy))))))))->(P (fun (x:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (x Xz)))) (forall (Xx:a), ((x Xx)->(forall (Xy:a), ((iff (x Xy)) ((x3 Xx) Xy)))))))))
% Found (eta_expansion_dep000 P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy))))))))
% Found ((eta_expansion_dep00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy)))))))) P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy))))))))
% Found (((eta_expansion_dep0 (fun (x5:(a->Prop))=> Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy)))))))) P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy))))))))
% Found ((((eta_expansion_dep (a->Prop)) (fun (x5:(a->Prop))=> Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy)))))))) P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy))))))))
% Found ((((eta_expansion_dep (a->Prop)) (fun (x5:(a->Prop))=> Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy)))))))) P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy))))))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy))))))))->(P (fun (x:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (x Xz)))) (forall (Xx:a), ((x Xx)->(forall (Xy:a), ((iff (x Xy)) ((x3 Xx) Xy)))))))))
% Found (eta_expansion_dep000 P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy))))))))
% Found ((eta_expansion_dep00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy)))))))) P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy))))))))
% Found (((eta_expansion_dep0 (fun (x5:(a->Prop))=> Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy)))))))) P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy))))))))
% Found ((((eta_expansion_dep (a->Prop)) (fun (x5:(a->Prop))=> Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy)))))))) P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy))))))))
% Found ((((eta_expansion_dep (a->Prop)) (fun (x5:(a->Prop))=> Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy)))))))) P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy))))))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))->(P (fun (x:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (x Xz)))) (forall (Xx:a), ((x Xx)->(forall (Xy:a), ((iff (x Xy)) ((x1 Xx) Xy)))))))))
% Found (eta_expansion_dep000 P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found ((eta_expansion_dep00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found (((eta_expansion_dep0 (fun (x5:(a->Prop))=> Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found ((((eta_expansion_dep (a->Prop)) (fun (x5:(a->Prop))=> Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found ((((eta_expansion_dep (a->Prop)) (fun (x5:(a->Prop))=> Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))->(P (fun (x:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (x Xz)))) (forall (Xx:a), ((x Xx)->(forall (Xy:a), ((iff (x Xy)) ((x1 Xx) Xy)))))))))
% Found (eta_expansion_dep000 P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found ((eta_expansion_dep00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found (((eta_expansion_dep0 (fun (x5:(a->Prop))=> Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found ((((eta_expansion_dep (a->Prop)) (fun (x5:(a->Prop))=> Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found ((((eta_expansion_dep (a->Prop)) (fun (x5:(a->Prop))=> Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found eq_ref00:=(eq_ref0 b):(((eq ((a->Prop)->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq ((a->Prop)->Prop)) b) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found eq_ref00:=(eq_ref0 cP):(((eq ((a->Prop)->Prop)) cP) cP)
% Found (eq_ref0 cP) as proof of (((eq ((a->Prop)->Prop)) cP) b)
% Found ((eq_ref ((a->Prop)->Prop)) cP) as proof of (((eq ((a->Prop)->Prop)) cP) b)
% Found ((eq_ref ((a->Prop)->Prop)) cP) as proof of (((eq ((a->Prop)->Prop)) cP) b)
% Found ((eq_ref ((a->Prop)->Prop)) cP) as proof of (((eq ((a->Prop)->Prop)) cP) b)
% Found eq_ref00:=(eq_ref0 b):(((eq ((a->Prop)->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq ((a->Prop)->Prop)) b) cP)
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) cP)
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) cP)
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) cP)
% Found eq_ref00:=(eq_ref0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x5 Xx) Xy)))))))):(((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x5 Xx) Xy)))))))) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x5 Xx) Xy))))))))
% Found (eq_ref0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x5 Xx) Xy)))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x5 Xx) Xy)))))))) b)
% Found ((eq_ref ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x5 Xx) Xy)))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x5 Xx) Xy)))))))) b)
% Found ((eq_ref ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x5 Xx) Xy)))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x5 Xx) Xy)))))))) b)
% Found ((eq_ref ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x5 Xx) Xy)))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x5 Xx) Xy)))))))) b)
% Found eta_expansion0000:=(eta_expansion000 P):((P (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x5 Xx) Xy))))))))->(P (fun (x:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (x Xz)))) (forall (Xx:a), ((x Xx)->(forall (Xy:a), ((iff (x Xy)) ((x5 Xx) Xy)))))))))
% Found (eta_expansion000 P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x5 Xx) Xy))))))))
% Found ((eta_expansion00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x5 Xx) Xy)))))))) P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x5 Xx) Xy))))))))
% Found (((eta_expansion0 Prop) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x5 Xx) Xy)))))))) P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x5 Xx) Xy))))))))
% Found ((((eta_expansion (a->Prop)) Prop) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x5 Xx) Xy)))))))) P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x5 Xx) Xy))))))))
% Found ((((eta_expansion (a->Prop)) Prop) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x5 Xx) Xy)))))))) P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x5 Xx) Xy))))))))
% Found eq_ref00:=(eq_ref0 b):(((eq ((a->Prop)->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq ((a->Prop)->Prop)) b) cP)
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) cP)
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) cP)
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) cP)
% Found eta_expansion000:=(eta_expansion00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))):(((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) (fun (x:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (x Xz)))) (forall (Xx:a), ((x Xx)->(forall (Xy:a), ((iff (x Xy)) ((x1 Xx) Xy))))))))
% Found (eta_expansion00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) b)
% Found ((eta_expansion0 Prop) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) b)
% Found (((eta_expansion (a->Prop)) Prop) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) b)
% Found (((eta_expansion (a->Prop)) Prop) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) b)
% Found (((eta_expansion (a->Prop)) Prop) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq ((a->Prop)->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq ((a->Prop)->Prop)) b) cP)
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) cP)
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) cP)
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) cP)
% Found eta_expansion000:=(eta_expansion00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy)))))))):(((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy)))))))) (fun (x:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (x Xz)))) (forall (Xx:a), ((x Xx)->(forall (Xy:a), ((iff (x Xy)) ((x3 Xx) Xy))))))))
% Found (eta_expansion00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy)))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy)))))))) b)
% Found ((eta_expansion0 Prop) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy)))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy)))))))) b)
% Found (((eta_expansion (a->Prop)) Prop) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy)))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy)))))))) b)
% Found (((eta_expansion (a->Prop)) Prop) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy)))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy)))))))) b)
% Found (((eta_expansion (a->Prop)) Prop) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy)))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy)))))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq ((a->Prop)->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq ((a->Prop)->Prop)) b) cP)
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) cP)
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) cP)
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) cP)
% Found eq_ref00:=(eq_ref0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))):(((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found (eq_ref0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) b)
% Found ((eq_ref ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) b)
% Found ((eq_ref ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) b)
% Found ((eq_ref ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) b)
% Found eta_expansion0000:=(eta_expansion000 P):((P (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))->(P (fun (x:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (x Xz)))) (forall (Xx:a), ((x Xx)->(forall (Xy:a), ((iff (x Xy)) ((x1 Xx) Xy)))))))))
% Found (eta_expansion000 P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found ((eta_expansion00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found (((eta_expansion0 Prop) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found ((((eta_expansion (a->Prop)) Prop) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found ((((eta_expansion (a->Prop)) Prop) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found eq_ref000:=(eq_ref00 P):((P cP)->(P cP))
% Found (eq_ref00 P) as proof of (P0 cP)
% Found ((eq_ref0 cP) P) as proof of (P0 cP)
% Found (((eq_ref ((a->Prop)->Prop)) cP) P) as proof of (P0 cP)
% Found (((eq_ref ((a->Prop)->Prop)) cP) P) as proof of (P0 cP)
% Found eq_ref000:=(eq_ref00 P):((P cP)->(P cP))
% Found (eq_ref00 P) as proof of (P0 cP)
% Found ((eq_ref0 cP) P) as proof of (P0 cP)
% Found (((eq_ref ((a->Prop)->Prop)) cP) P) as proof of (P0 cP)
% Found (((eq_ref ((a->Prop)->Prop)) cP) P) as proof of (P0 cP)
% Found eta_expansion0000:=(eta_expansion000 P):((P (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy))))))))->(P (fun (x:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (x Xz)))) (forall (Xx:a), ((x Xx)->(forall (Xy:a), ((iff (x Xy)) ((x3 Xx) Xy)))))))))
% Found (eta_expansion000 P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy))))))))
% Found ((eta_expansion00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy)))))))) P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy))))))))
% Found (((eta_expansion0 Prop) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy)))))))) P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy))))))))
% Found ((((eta_expansion (a->Prop)) Prop) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy)))))))) P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy))))))))
% Found ((((eta_expansion (a->Prop)) Prop) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy)))))))) P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy))))))))
% Found eq_ref000:=(eq_ref00 P):((P (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))->(P (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))))
% Found (eq_ref00 P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found ((eq_ref0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found (((eq_ref ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found (((eq_ref ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found eta_expansion0000:=(eta_expansion000 P):((P (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x5 Xx) Xy))))))))->(P (fun (x:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (x Xz)))) (forall (Xx:a), ((x Xx)->(forall (Xy:a), ((iff (x Xy)) ((x5 Xx) Xy)))))))))
% Found (eta_expansion000 P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x5 Xx) Xy))))))))
% Found ((eta_expansion00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x5 Xx) Xy)))))))) P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x5 Xx) Xy))))))))
% Found (((eta_expansion0 Prop) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x5 Xx) Xy)))))))) P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x5 Xx) Xy))))))))
% Found ((((eta_expansion (a->Prop)) Prop) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x5 Xx) Xy)))))))) P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x5 Xx) Xy))))))))
% Found ((((eta_expansion (a->Prop)) Prop) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x5 Xx) Xy)))))))) P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x5 Xx) Xy))))))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x5 Xx) Xy))))))))->(P (fun (x:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (x Xz)))) (forall (Xx:a), ((x Xx)->(forall (Xy:a), ((iff (x Xy)) ((x5 Xx) Xy)))))))))
% Found (eta_expansion_dep000 P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x5 Xx) Xy))))))))
% Found ((eta_expansion_dep00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x5 Xx) Xy)))))))) P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x5 Xx) Xy))))))))
% Found (((eta_expansion_dep0 (fun (x7:(a->Prop))=> Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x5 Xx) Xy)))))))) P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x5 Xx) Xy))))))))
% Found ((((eta_expansion_dep (a->Prop)) (fun (x7:(a->Prop))=> Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x5 Xx) Xy)))))))) P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x5 Xx) Xy))))))))
% Found ((((eta_expansion_dep (a->Prop)) (fun (x7:(a->Prop))=> Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x5 Xx) Xy)))))))) P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x5 Xx) Xy))))))))
% Found eq_ref00:=(eq_ref0 ((and ((ex a) (fun (Xz:a)=> (x2 Xz)))) (forall (Xx:a), ((x2 Xx)->(forall (Xy:a), ((iff (x2 Xy)) ((x1 Xx) Xy))))))):(((eq Prop) ((and ((ex a) (fun (Xz:a)=> (x2 Xz)))) (forall (Xx:a), ((x2 Xx)->(forall (Xy:a), ((iff (x2 Xy)) ((x1 Xx) Xy))))))) ((and ((ex a) (fun (Xz:a)=> (x2 Xz)))) (forall (Xx:a), ((x2 Xx)->(forall (Xy:a), ((iff (x2 Xy)) ((x1 Xx) Xy)))))))
% Found (eq_ref0 ((and ((ex a) (fun (Xz:a)=> (x2 Xz)))) (forall (Xx:a), ((x2 Xx)->(forall (Xy:a), ((iff (x2 Xy)) ((x1 Xx) Xy))))))) as proof of (((eq Prop) ((and ((ex a) (fun (Xz:a)=> (x2 Xz)))) (forall (Xx:a), ((x2 Xx)->(forall (Xy:a), ((iff (x2 Xy)) ((x1 Xx) Xy))))))) b)
% Found ((eq_ref Prop) ((and ((ex a) (fun (Xz:a)=> (x2 Xz)))) (forall (Xx:a), ((x2 Xx)->(forall (Xy:a), ((iff (x2 Xy)) ((x1 Xx) Xy))))))) as proof of (((eq Prop) ((and ((ex a) (fun (Xz:a)=> (x2 Xz)))) (forall (Xx:a), ((x2 Xx)->(forall (Xy:a), ((iff (x2 Xy)) ((x1 Xx) Xy))))))) b)
% Found ((eq_ref Prop) ((and ((ex a) (fun (Xz:a)=> (x2 Xz)))) (forall (Xx:a), ((x2 Xx)->(forall (Xy:a), ((iff (x2 Xy)) ((x1 Xx) Xy))))))) as proof of (((eq Prop) ((and ((ex a) (fun (Xz:a)=> (x2 Xz)))) (forall (Xx:a), ((x2 Xx)->(forall (Xy:a), ((iff (x2 Xy)) ((x1 Xx) Xy))))))) b)
% Found ((eq_ref Prop) ((and ((ex a) (fun (Xz:a)=> (x2 Xz)))) (forall (Xx:a), ((x2 Xx)->(forall (Xy:a), ((iff (x2 Xy)) ((x1 Xx) Xy))))))) as proof of (((eq Prop) ((and ((ex a) (fun (Xz:a)=> (x2 Xz)))) (forall (Xx:a), ((x2 Xx)->(forall (Xy:a), ((iff (x2 Xy)) ((x1 Xx) Xy))))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (cP x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cP x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cP x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cP x2))
% Found eq_ref00:=(eq_ref0 ((and ((ex a) (fun (Xz:a)=> (x2 Xz)))) (forall (Xx:a), ((x2 Xx)->(forall (Xy:a), ((iff (x2 Xy)) ((x1 Xx) Xy))))))):(((eq Prop) ((and ((ex a) (fun (Xz:a)=> (x2 Xz)))) (forall (Xx:a), ((x2 Xx)->(forall (Xy:a), ((iff (x2 Xy)) ((x1 Xx) Xy))))))) ((and ((ex a) (fun (Xz:a)=> (x2 Xz)))) (forall (Xx:a), ((x2 Xx)->(forall (Xy:a), ((iff (x2 Xy)) ((x1 Xx) Xy)))))))
% Found (eq_ref0 ((and ((ex a) (fun (Xz:a)=> (x2 Xz)))) (forall (Xx:a), ((x2 Xx)->(forall (Xy:a), ((iff (x2 Xy)) ((x1 Xx) Xy))))))) as proof of (((eq Prop) ((and ((ex a) (fun (Xz:a)=> (x2 Xz)))) (forall (Xx:a), ((x2 Xx)->(forall (Xy:a), ((iff (x2 Xy)) ((x1 Xx) Xy))))))) b)
% Found ((eq_ref Prop) ((and ((ex a) (fun (Xz:a)=> (x2 Xz)))) (forall (Xx:a), ((x2 Xx)->(forall (Xy:a), ((iff (x2 Xy)) ((x1 Xx) Xy))))))) as proof of (((eq Prop) ((and ((ex a) (fun (Xz:a)=> (x2 Xz)))) (forall (Xx:a), ((x2 Xx)->(forall (Xy:a), ((iff (x2 Xy)) ((x1 Xx) Xy))))))) b)
% Found ((eq_ref Prop) ((and ((ex a) (fun (Xz:a)=> (x2 Xz)))) (forall (Xx:a), ((x2 Xx)->(forall (Xy:a), ((iff (x2 Xy)) ((x1 Xx) Xy))))))) as proof of (((eq Prop) ((and ((ex a) (fun (Xz:a)=> (x2 Xz)))) (forall (Xx:a), ((x2 Xx)->(forall (Xy:a), ((iff (x2 Xy)) ((x1 Xx) Xy))))))) b)
% Found ((eq_ref Prop) ((and ((ex a) (fun (Xz:a)=> (x2 Xz)))) (forall (Xx:a), ((x2 Xx)->(forall (Xy:a), ((iff (x2 Xy)) ((x1 Xx) Xy))))))) as proof of (((eq Prop) ((and ((ex a) (fun (Xz:a)=> (x2 Xz)))) (forall (Xx:a), ((x2 Xx)->(forall (Xy:a), ((iff (x2 Xy)) ((x1 Xx) Xy))))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (cP x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cP x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cP x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cP x2))
% Found eq_ref000:=(eq_ref00 P):((P ((and ((ex a) (fun (Xz:a)=> (x2 Xz)))) (forall (Xx:a), ((x2 Xx)->(forall (Xy:a), ((iff (x2 Xy)) ((x1 Xx) Xy)))))))->(P ((and ((ex a) (fun (Xz:a)=> (x2 Xz)))) (forall (Xx:a), ((x2 Xx)->(forall (Xy:a), ((iff (x2 Xy)) ((x1 Xx) Xy))))))))
% Found (eq_ref00 P) as proof of (P0 ((and ((ex a) (fun (Xz:a)=> (x2 Xz)))) (forall (Xx:a), ((x2 Xx)->(forall (Xy:a), ((iff (x2 Xy)) ((x1 Xx) Xy)))))))
% Found ((eq_ref0 ((and ((ex a) (fun (Xz:a)=> (x2 Xz)))) (forall (Xx:a), ((x2 Xx)->(forall (Xy:a), ((iff (x2 Xy)) ((x1 Xx) Xy))))))) P) as proof of (P0 ((and ((ex a) (fun (Xz:a)=> (x2 Xz)))) (forall (Xx:a), ((x2 Xx)->(forall (Xy:a), ((iff (x2 Xy)) ((x1 Xx) Xy)))))))
% Found (((eq_ref Prop) ((and ((ex a) (fun (Xz:a)=> (x2 Xz)))) (forall (Xx:a), ((x2 Xx)->(forall (Xy:a), ((iff (x2 Xy)) ((x1 Xx) Xy))))))) P) as proof of (P0 ((and ((ex a) (fun (Xz:a)=> (x2 Xz)))) (forall (Xx:a), ((x2 Xx)->(forall (Xy:a), ((iff (x2 Xy)) ((x1 Xx) Xy)))))))
% Found (((eq_ref Prop) ((and ((ex a) (fun (Xz:a)=> (x2 Xz)))) (forall (Xx:a), ((x2 Xx)->(forall (Xy:a), ((iff (x2 Xy)) ((x1 Xx) Xy))))))) P) as proof of (P0 ((and ((ex a) (fun (Xz:a)=> (x2 Xz)))) (forall (Xx:a), ((x2 Xx)->(forall (Xy:a), ((iff (x2 Xy)) ((x1 Xx) Xy)))))))
% Found eq_ref000:=(eq_ref00 P):((P ((and ((ex a) (fun (Xz:a)=> (x2 Xz)))) (forall (Xx:a), ((x2 Xx)->(forall (Xy:a), ((iff (x2 Xy)) ((x1 Xx) Xy)))))))->(P ((and ((ex a) (fun (Xz:a)=> (x2 Xz)))) (forall (Xx:a), ((x2 Xx)->(forall (Xy:a), ((iff (x2 Xy)) ((x1 Xx) Xy))))))))
% Found (eq_ref00 P) as proof of (P0 ((and ((ex a) (fun (Xz:a)=> (x2 Xz)))) (forall (Xx:a), ((x2 Xx)->(forall (Xy:a), ((iff (x2 Xy)) ((x1 Xx) Xy)))))))
% Found ((eq_ref0 ((and ((ex a) (fun (Xz:a)=> (x2 Xz)))) (forall (Xx:a), ((x2 Xx)->(forall (Xy:a), ((iff (x2 Xy)) ((x1 Xx) Xy))))))) P) as proof of (P0 ((and ((ex a) (fun (Xz:a)=> (x2 Xz)))) (forall (Xx:a), ((x2 Xx)->(forall (Xy:a), ((iff (x2 Xy)) ((x1 Xx) Xy)))))))
% Found (((eq_ref Prop) ((and ((ex a) (fun (Xz:a)=> (x2 Xz)))) (forall (Xx:a), ((x2 Xx)->(forall (Xy:a), ((iff (x2 Xy)) ((x1 Xx) Xy))))))) P) as proof of (P0 ((and ((ex a) (fun (Xz:a)=> (x2 Xz)))) (forall (Xx:a), ((x2 Xx)->(forall (Xy:a), ((iff (x2 Xy)) ((x1 Xx) Xy)))))))
% Found (((eq_ref Prop) ((and ((ex a) (fun (Xz:a)=> (x2 Xz)))) (forall (Xx:a), ((x2 Xx)->(forall (Xy:a), ((iff (x2 Xy)) ((x1 Xx) Xy))))))) P) as proof of (P0 ((and ((ex a) (fun (Xz:a)=> (x2 Xz)))) (forall (Xx:a), ((x2 Xx)->(forall (Xy:a), ((iff (x2 Xy)) ((x1 Xx) Xy)))))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))->(P (fun (x:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (x Xz)))) (forall (Xx:a), ((x Xx)->(forall (Xy:a), ((iff (x Xy)) ((x1 Xx) Xy)))))))))
% Found (eta_expansion_dep000 P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found ((eta_expansion_dep00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found (((eta_expansion_dep0 (fun (x5:(a->Prop))=> Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found ((((eta_expansion_dep (a->Prop)) (fun (x5:(a->Prop))=> Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found ((((eta_expansion_dep (a->Prop)) (fun (x5:(a->Prop))=> Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))->(P (fun (x:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (x Xz)))) (forall (Xx:a), ((x Xx)->(forall (Xy:a), ((iff (x Xy)) ((x1 Xx) Xy)))))))))
% Found (eta_expansion_dep000 P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found ((eta_expansion_dep00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found (((eta_expansion_dep0 (fun (x5:(a->Prop))=> Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found ((((eta_expansion_dep (a->Prop)) (fun (x5:(a->Prop))=> Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found ((((eta_expansion_dep (a->Prop)) (fun (x5:(a->Prop))=> Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))->(P (fun (x:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (x Xz)))) (forall (Xx:a), ((x Xx)->(forall (Xy:a), ((iff (x Xy)) ((x1 Xx) Xy)))))))))
% Found (eta_expansion_dep000 P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found ((eta_expansion_dep00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found (((eta_expansion_dep0 (fun (x7:(a->Prop))=> Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found ((((eta_expansion_dep (a->Prop)) (fun (x7:(a->Prop))=> Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found ((((eta_expansion_dep (a->Prop)) (fun (x7:(a->Prop))=> Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found eta_expansion0000:=(eta_expansion000 P):((P (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))->(P (fun (x:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (x Xz)))) (forall (Xx:a), ((x Xx)->(forall (Xy:a), ((iff (x Xy)) ((x1 Xx) Xy)))))))))
% Found (eta_expansion000 P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found ((eta_expansion00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found (((eta_expansion0 Prop) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found ((((eta_expansion (a->Prop)) Prop) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found ((((eta_expansion (a->Prop)) Prop) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy))))))))->(P (fun (x:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (x Xz)))) (forall (Xx:a), ((x Xx)->(forall (Xy:a), ((iff (x Xy)) ((x3 Xx) Xy)))))))))
% Found (eta_expansion_dep000 P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy))))))))
% Found ((eta_expansion_dep00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy)))))))) P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy))))))))
% Found (((eta_expansion_dep0 (fun (x7:(a->Prop))=> Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy)))))))) P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy))))))))
% Found ((((eta_expansion_dep (a->Prop)) (fun (x7:(a->Prop))=> Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy)))))))) P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy))))))))
% Found ((((eta_expansion_dep (a->Prop)) (fun (x7:(a->Prop))=> Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy)))))))) P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy))))))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy))))))))->(P (fun (x:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (x Xz)))) (forall (Xx:a), ((x Xx)->(forall (Xy:a), ((iff (x Xy)) ((x3 Xx) Xy)))))))))
% Found (eta_expansion_dep000 P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy))))))))
% Found ((eta_expansion_dep00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy)))))))) P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy))))))))
% Found (((eta_expansion_dep0 (fun (x7:(a->Prop))=> Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy)))))))) P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy))))))))
% Found ((((eta_expansion_dep (a->Prop)) (fun (x7:(a->Prop))=> Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy)))))))) P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy))))))))
% Found ((((eta_expansion_dep (a->Prop)) (fun (x7:(a->Prop))=> Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy)))))))) P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy))))))))
% Found eq_ref00:=(eq_ref0 b):(((eq ((a->Prop)->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq ((a->Prop)->Prop)) b) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy))))))))
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy))))))))
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy))))))))
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy))))))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 cP):(((eq ((a->Prop)->Prop)) cP) (fun (x:(a->Prop))=> (cP x)))
% Found (eta_expansion_dep00 cP) as proof of (((eq ((a->Prop)->Prop)) cP) b)
% Found ((eta_expansion_dep0 (fun (x5:(a->Prop))=> Prop)) cP) as proof of (((eq ((a->Prop)->Prop)) cP) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x5:(a->Prop))=> Prop)) cP) as proof of (((eq ((a->Prop)->Prop)) cP) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x5:(a->Prop))=> Prop)) cP) as proof of (((eq ((a->Prop)->Prop)) cP) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x5:(a->Prop))=> Prop)) cP) as proof of (((eq ((a->Prop)->Prop)) cP) b)
% Found eq_ref00:=(eq_ref0 b):(((eq ((a->Prop)->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq ((a->Prop)->Prop)) b) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 cP):(((eq ((a->Prop)->Prop)) cP) (fun (x:(a->Prop))=> (cP x)))
% Found (eta_expansion_dep00 cP) as proof of (((eq ((a->Prop)->Prop)) cP) b)
% Found ((eta_expansion_dep0 (fun (x5:(a->Prop))=> Prop)) cP) as proof of (((eq ((a->Prop)->Prop)) cP) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x5:(a->Prop))=> Prop)) cP) as proof of (((eq ((a->Prop)->Prop)) cP) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x5:(a->Prop))=> Prop)) cP) as proof of (((eq ((a->Prop)->Prop)) cP) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x5:(a->Prop))=> Prop)) cP) as proof of (((eq ((a->Prop)->Prop)) cP) b)
% Found eq_ref00:=(eq_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))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy)))))))) cP))))
% 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))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy)))))))) cP))))
% 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))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy)))))))) cP))))
% 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))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy)))))))) cP))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 a0):(((eq ((a->(a->Prop))->Prop)) a0) (fun (x:(a->(a->Prop)))=> (a0 x)))
% Found (eta_expansion_dep00 a0) as proof of (((eq ((a->(a->Prop))->Prop)) a0) b)
% Found ((eta_expansion_dep0 (fun (x2:(a->(a->Prop)))=> Prop)) a0) as proof of (((eq ((a->(a->Prop))->Prop)) a0) b)
% Found (((eta_expansion_dep (a->(a->Prop))) (fun (x2:(a->(a->Prop)))=> Prop)) a0) as proof of (((eq ((a->(a->Prop))->Prop)) a0) b)
% Found (((eta_expansion_dep (a->(a->Prop))) (fun (x2:(a->(a->Prop)))=> Prop)) a0) as proof of (((eq ((a->(a->Prop))->Prop)) a0) b)
% Found (((eta_expansion_dep (a->(a->Prop))) (fun (x2:(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 b):(((eq ((a->Prop)->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq ((a->Prop)->Prop)) b) cP)
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) cP)
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) cP)
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) cP)
% Found eta_expansion000:=(eta_expansion00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))):(((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) (fun (x:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (x Xz)))) (forall (Xx:a), ((x Xx)->(forall (Xy:a), ((iff (x Xy)) ((x1 Xx) Xy))))))))
% Found (eta_expansion00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) b)
% Found ((eta_expansion0 Prop) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) b)
% Found (((eta_expansion (a->Prop)) Prop) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) b)
% Found (((eta_expansion (a->Prop)) Prop) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) b)
% Found (((eta_expansion (a->Prop)) Prop) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq ((a->Prop)->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq ((a->Prop)->Prop)) b) cP)
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) cP)
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) cP)
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) cP)
% Found eta_expansion000:=(eta_expansion00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy)))))))):(((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy)))))))) (fun (x:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (x Xz)))) (forall (Xx:a), ((x Xx)->(forall (Xy:a), ((iff (x Xy)) ((x3 Xx) Xy))))))))
% Found (eta_expansion00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy)))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy)))))))) b)
% Found ((eta_expansion0 Prop) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy)))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy)))))))) b)
% Found (((eta_expansion (a->Prop)) Prop) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy)))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy)))))))) b)
% Found (((eta_expansion (a->Prop)) Prop) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy)))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy)))))))) b)
% Found (((eta_expansion (a->Prop)) Prop) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy)))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy)))))))) b)
% Found eq_ref000:=(eq_ref00 P):((P cP)->(P cP))
% Found (eq_ref00 P) as proof of (P0 cP)
% Found ((eq_ref0 cP) P) as proof of (P0 cP)
% Found (((eq_ref ((a->Prop)->Prop)) cP) P) as proof of (P0 cP)
% Found (((eq_ref ((a->Prop)->Prop)) cP) P) as proof of (P0 cP)
% Found eq_ref000:=(eq_ref00 P):((P cP)->(P cP))
% Found (eq_ref00 P) as proof of (P0 cP)
% Found ((eq_ref0 cP) P) as proof of (P0 cP)
% Found (((eq_ref ((a->Prop)->Prop)) cP) P) as proof of (P0 cP)
% Found (((eq_ref ((a->Prop)->Prop)) cP) P) as proof of (P0 cP)
% Found eq_ref00:=(eq_ref0 (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) cP)):(((eq Prop) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) cP)) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) cP))
% Found (eq_ref0 (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) cP)) as proof of (((eq Prop) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) cP)) b)
% Found ((eq_ref Prop) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) cP)) as proof of (((eq Prop) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) cP)) b)
% Found ((eq_ref Prop) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) cP)) as proof of (((eq Prop) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) cP)) b)
% Found ((eq_ref Prop) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) cP)) as proof of (((eq Prop) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) cP)) b)
% Found eta_expansion0000:=(eta_expansion000 P):((P (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))->(P (fun (x:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (x Xz)))) (forall (Xx:a), ((x Xx)->(forall (Xy:a), ((iff (x Xy)) ((x1 Xx) Xy)))))))))
% Found (eta_expansion000 P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found ((eta_expansion00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found (((eta_expansion0 Prop) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found ((((eta_expansion (a->Prop)) Prop) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found ((((eta_expansion (a->Prop)) Prop) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found eta_expansion0000:=(eta_expansion000 P):((P (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy))))))))->(P (fun (x:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (x Xz)))) (forall (Xx:a), ((x Xx)->(forall (Xy:a), ((iff (x Xy)) ((x3 Xx) Xy)))))))))
% Found (eta_expansion000 P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy))))))))
% Found ((eta_expansion00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy)))))))) P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy))))))))
% Found (((eta_expansion0 Prop) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy)))))))) P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy))))))))
% Found ((((eta_expansion (a->Prop)) Prop) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy)))))))) P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy))))))))
% Found ((((eta_expansion (a->Prop)) Prop) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy)))))))) P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy))))))))
% Found eq_ref00:=(eq_ref0 ((and ((ex a) (fun (Xz:a)=> (x4 Xz)))) (forall (Xx:a), ((x4 Xx)->(forall (Xy:a), ((iff (x4 Xy)) ((x3 Xx) Xy))))))):(((eq Prop) ((and ((ex a) (fun (Xz:a)=> (x4 Xz)))) (forall (Xx:a), ((x4 Xx)->(forall (Xy:a), ((iff (x4 Xy)) ((x3 Xx) Xy))))))) ((and ((ex a) (fun (Xz:a)=> (x4 Xz)))) (forall (Xx:a), ((x4 Xx)->(forall (Xy:a), ((iff (x4 Xy)) ((x3 Xx) Xy)))))))
% Found (eq_ref0 ((and ((ex a) (fun (Xz:a)=> (x4 Xz)))) (forall (Xx:a), ((x4 Xx)->(forall (Xy:a), ((iff (x4 Xy)) ((x3 Xx) Xy))))))) as proof of (((eq Prop) ((and ((ex a) (fun (Xz:a)=> (x4 Xz)))) (forall (Xx:a), ((x4 Xx)->(forall (Xy:a), ((iff (x4 Xy)) ((x3 Xx) Xy))))))) b)
% Found ((eq_ref Prop) ((and ((ex a) (fun (Xz:a)=> (x4 Xz)))) (forall (Xx:a), ((x4 Xx)->(forall (Xy:a), ((iff (x4 Xy)) ((x3 Xx) Xy))))))) as proof of (((eq Prop) ((and ((ex a) (fun (Xz:a)=> (x4 Xz)))) (forall (Xx:a), ((x4 Xx)->(forall (Xy:a), ((iff (x4 Xy)) ((x3 Xx) Xy))))))) b)
% Found ((eq_ref Prop) ((and ((ex a) (fun (Xz:a)=> (x4 Xz)))) (forall (Xx:a), ((x4 Xx)->(forall (Xy:a), ((iff (x4 Xy)) ((x3 Xx) Xy))))))) as proof of (((eq Prop) ((and ((ex a) (fun (Xz:a)=> (x4 Xz)))) (forall (Xx:a), ((x4 Xx)->(forall (Xy:a), ((iff (x4 Xy)) ((x3 Xx) Xy))))))) b)
% Found ((eq_ref Prop) ((and ((ex a) (fun (Xz:a)=> (x4 Xz)))) (forall (Xx:a), ((x4 Xx)->(forall (Xy:a), ((iff (x4 Xy)) ((x3 Xx) Xy))))))) as proof of (((eq Prop) ((and ((ex a) (fun (Xz:a)=> (x4 Xz)))) (forall (Xx:a), ((x4 Xx)->(forall (Xy:a), ((iff (x4 Xy)) ((x3 Xx) Xy))))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (cP x4))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cP x4))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cP x4))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cP x4))
% Found eq_ref00:=(eq_ref0 ((and ((ex a) (fun (Xz:a)=> (x4 Xz)))) (forall (Xx:a), ((x4 Xx)->(forall (Xy:a), ((iff (x4 Xy)) ((x3 Xx) Xy))))))):(((eq Prop) ((and ((ex a) (fun (Xz:a)=> (x4 Xz)))) (forall (Xx:a), ((x4 Xx)->(forall (Xy:a), ((iff (x4 Xy)) ((x3 Xx) Xy))))))) ((and ((ex a) (fun (Xz:a)=> (x4 Xz)))) (forall (Xx:a), ((x4 Xx)->(forall (Xy:a), ((iff (x4 Xy)) ((x3 Xx) Xy)))))))
% Found (eq_ref0 ((and ((ex a) (fun (Xz:a)=> (x4 Xz)))) (forall (Xx:a), ((x4 Xx)->(forall (Xy:a), ((iff (x4 Xy)) ((x3 Xx) Xy))))))) as proof of (((eq Prop) ((and ((ex a) (fun (Xz:a)=> (x4 Xz)))) (forall (Xx:a), ((x4 Xx)->(forall (Xy:a), ((iff (x4 Xy)) ((x3 Xx) Xy))))))) b)
% Found ((eq_ref Prop) ((and ((ex a) (fun (Xz:a)=> (x4 Xz)))) (forall (Xx:a), ((x4 Xx)->(forall (Xy:a), ((iff (x4 Xy)) ((x3 Xx) Xy))))))) as proof of (((eq Prop) ((and ((ex a) (fun (Xz:a)=> (x4 Xz)))) (forall (Xx:a), ((x4 Xx)->(forall (Xy:a), ((iff (x4 Xy)) ((x3 Xx) Xy))))))) b)
% Found ((eq_ref Prop) ((and ((ex a) (fun (Xz:a)=> (x4 Xz)))) (forall (Xx:a), ((x4 Xx)->(forall (Xy:a), ((iff (x4 Xy)) ((x3 Xx) Xy))))))) as proof of (((eq Prop) ((and ((ex a) (fun (Xz:a)=> (x4 Xz)))) (forall (Xx:a), ((x4 Xx)->(forall (Xy:a), ((iff (x4 Xy)) ((x3 Xx) Xy))))))) b)
% Found ((eq_ref Prop) ((and ((ex a) (fun (Xz:a)=> (x4 Xz)))) (forall (Xx:a), ((x4 Xx)->(forall (Xy:a), ((iff (x4 Xy)) ((x3 Xx) Xy))))))) as proof of (((eq Prop) ((and ((ex a) (fun (Xz:a)=> (x4 Xz)))) (forall (Xx:a), ((x4 Xx)->(forall (Xy:a), ((iff (x4 Xy)) ((x3 Xx) Xy))))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (cP x4))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cP x4))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cP x4))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cP x4))
% Found eq_ref000:=(eq_ref00 P):((P cP)->(P cP))
% Found (eq_ref00 P) as proof of (P0 cP)
% Found ((eq_ref0 cP) P) as proof of (P0 cP)
% Found (((eq_ref ((a->Prop)->Prop)) cP) P) as proof of (P0 cP)
% Found (((eq_ref ((a->Prop)->Prop)) cP) P) as proof of (P0 cP)
% Found eq_ref000:=(eq_ref00 P):((P cP)->(P cP))
% Found (eq_ref00 P) as proof of (P0 cP)
% Found ((eq_ref0 cP) P) as proof of (P0 cP)
% Found (((eq_ref ((a->Prop)->Prop)) cP) P) as proof of (P0 cP)
% Found (((eq_ref ((a->Prop)->Prop)) cP) P) as proof of (P0 cP)
% Found eq_ref000:=(eq_ref00 P):((P ((and ((ex a) (fun (Xz:a)=> (x4 Xz)))) (forall (Xx:a), ((x4 Xx)->(forall (Xy:a), ((iff (x4 Xy)) ((x3 Xx) Xy)))))))->(P ((and ((ex a) (fun (Xz:a)=> (x4 Xz)))) (forall (Xx:a), ((x4 Xx)->(forall (Xy:a), ((iff (x4 Xy)) ((x3 Xx) Xy))))))))
% Found (eq_ref00 P) as proof of (P0 ((and ((ex a) (fun (Xz:a)=> (x4 Xz)))) (forall (Xx:a), ((x4 Xx)->(forall (Xy:a), ((iff (x4 Xy)) ((x3 Xx) Xy)))))))
% Found ((eq_ref0 ((and ((ex a) (fun (Xz:a)=> (x4 Xz)))) (forall (Xx:a), ((x4 Xx)->(forall (Xy:a), ((iff (x4 Xy)) ((x3 Xx) Xy))))))) P) as proof of (P0 ((and ((ex a) (fun (Xz:a)=> (x4 Xz)))) (forall (Xx:a), ((x4 Xx)->(forall (Xy:a), ((iff (x4 Xy)) ((x3 Xx) Xy)))))))
% Found (((eq_ref Prop) ((and ((ex a) (fun (Xz:a)=> (x4 Xz)))) (forall (Xx:a), ((x4 Xx)->(forall (Xy:a), ((iff (x4 Xy)) ((x3 Xx) Xy))))))) P) as proof of (P0 ((and ((ex a) (fun (Xz:a)=> (x4 Xz)))) (forall (Xx:a), ((x4 Xx)->(forall (Xy:a), ((iff (x4 Xy)) ((x3 Xx) Xy)))))))
% Found (((eq_ref Prop) ((and ((ex a) (fun (Xz:a)=> (x4 Xz)))) (forall (Xx:a), ((x4 Xx)->(forall (Xy:a), ((iff (x4 Xy)) ((x3 Xx) Xy))))))) P) as proof of (P0 ((and ((ex a) (fun (Xz:a)=> (x4 Xz)))) (forall (Xx:a), ((x4 Xx)->(forall (Xy:a), ((iff (x4 Xy)) ((x3 Xx) Xy)))))))
% Found eq_ref000:=(eq_ref00 P):((P ((and ((ex a) (fun (Xz:a)=> (x4 Xz)))) (forall (Xx:a), ((x4 Xx)->(forall (Xy:a), ((iff (x4 Xy)) ((x3 Xx) Xy)))))))->(P ((and ((ex a) (fun (Xz:a)=> (x4 Xz)))) (forall (Xx:a), ((x4 Xx)->(forall (Xy:a), ((iff (x4 Xy)) ((x3 Xx) Xy))))))))
% Found (eq_ref00 P) as proof of (P0 ((and ((ex a) (fun (Xz:a)=> (x4 Xz)))) (forall (Xx:a), ((x4 Xx)->(forall (Xy:a), ((iff (x4 Xy)) ((x3 Xx) Xy)))))))
% Found ((eq_ref0 ((and ((ex a) (fun (Xz:a)=> (x4 Xz)))) (forall (Xx:a), ((x4 Xx)->(forall (Xy:a), ((iff (x4 Xy)) ((x3 Xx) Xy))))))) P) as proof of (P0 ((and ((ex a) (fun (Xz:a)=> (x4 Xz)))) (forall (Xx:a), ((x4 Xx)->(forall (Xy:a), ((iff (x4 Xy)) ((x3 Xx) Xy)))))))
% Found (((eq_ref Prop) ((and ((ex a) (fun (Xz:a)=> (x4 Xz)))) (forall (Xx:a), ((x4 Xx)->(forall (Xy:a), ((iff (x4 Xy)) ((x3 Xx) Xy))))))) P) as proof of (P0 ((and ((ex a) (fun (Xz:a)=> (x4 Xz)))) (forall (Xx:a), ((x4 Xx)->(forall (Xy:a), ((iff (x4 Xy)) ((x3 Xx) Xy)))))))
% Found (((eq_ref Prop) ((and ((ex a) (fun (Xz:a)=> (x4 Xz)))) (forall (Xx:a), ((x4 Xx)->(forall (Xy:a), ((iff (x4 Xy)) ((x3 Xx) Xy))))))) P) as proof of (P0 ((and ((ex a) (fun (Xz:a)=> (x4 Xz)))) (forall (Xx:a), ((x4 Xx)->(forall (Xy:a), ((iff (x4 Xy)) ((x3 Xx) Xy)))))))
% Found eq_ref000:=(eq_ref00 P0):((P0 (f x1))->(P0 (f x1)))
% Found (eq_ref00 P0) as proof of (P1 (f x1))
% Found ((eq_ref0 (f x1)) P0) as proof of (P1 (f x1))
% Found (((eq_ref Prop) (f x1)) P0) as proof of (P1 (f x1))
% Found (((eq_ref Prop) (f x1)) P0) as proof of (P1 (f x1))
% Found eq_ref000:=(eq_ref00 P0):((P0 (f x1))->(P0 (f x1)))
% Found (eq_ref00 P0) as proof of (P1 (f x1))
% Found ((eq_ref0 (f x1)) P0) as proof of (P1 (f x1))
% Found (((eq_ref Prop) (f x1)) P0) as proof of (P1 (f x1))
% Found (((eq_ref Prop) (f x1)) P0) as proof of (P1 (f x1))
% 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))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy)))))))) cP))))
% 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))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy)))))))) cP))))
% 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))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy)))))))) cP))))
% 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))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((Q Xx) Xy)))))))) cP))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 a0):(((eq ((a->(a->Prop))->Prop)) a0) (fun (x:(a->(a->Prop)))=> (a0 x)))
% Found (eta_expansion_dep00 a0) as proof of (((eq ((a->(a->Prop))->Prop)) a0) b)
% Found ((eta_expansion_dep0 (fun (x4:(a->(a->Prop)))=> Prop)) a0) as proof of (((eq ((a->(a->Prop))->Prop)) a0) b)
% Found (((eta_expansion_dep (a->(a->Prop))) (fun (x4:(a->(a->Prop)))=> Prop)) a0) as proof of (((eq ((a->(a->Prop))->Prop)) a0) b)
% Found (((eta_expansion_dep (a->(a->Prop))) (fun (x4:(a->(a->Prop)))=> Prop)) a0) as proof of (((eq ((a->(a->Prop))->Prop)) a0) b)
% Found (((eta_expansion_dep (a->(a->Prop))) (fun (x4:(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 ((and ((ex a) (fun (Xz:a)=> (x4 Xz)))) (forall (Xx:a), ((x4 Xx)->(forall (Xy:a), ((iff (x4 Xy)) ((x1 Xx) Xy))))))):(((eq Prop) ((and ((ex a) (fun (Xz:a)=> (x4 Xz)))) (forall (Xx:a), ((x4 Xx)->(forall (Xy:a), ((iff (x4 Xy)) ((x1 Xx) Xy))))))) ((and ((ex a) (fun (Xz:a)=> (x4 Xz)))) (forall (Xx:a), ((x4 Xx)->(forall (Xy:a), ((iff (x4 Xy)) ((x1 Xx) Xy)))))))
% Found (eq_ref0 ((and ((ex a) (fun (Xz:a)=> (x4 Xz)))) (forall (Xx:a), ((x4 Xx)->(forall (Xy:a), ((iff (x4 Xy)) ((x1 Xx) Xy))))))) as proof of (((eq Prop) ((and ((ex a) (fun (Xz:a)=> (x4 Xz)))) (forall (Xx:a), ((x4 Xx)->(forall (Xy:a), ((iff (x4 Xy)) ((x1 Xx) Xy))))))) b)
% Found ((eq_ref Prop) ((and ((ex a) (fun (Xz:a)=> (x4 Xz)))) (forall (Xx:a), ((x4 Xx)->(forall (Xy:a), ((iff (x4 Xy)) ((x1 Xx) Xy))))))) as proof of (((eq Prop) ((and ((ex a) (fun (Xz:a)=> (x4 Xz)))) (forall (Xx:a), ((x4 Xx)->(forall (Xy:a), ((iff (x4 Xy)) ((x1 Xx) Xy))))))) b)
% Found ((eq_ref Prop) ((and ((ex a) (fun (Xz:a)=> (x4 Xz)))) (forall (Xx:a), ((x4 Xx)->(forall (Xy:a), ((iff (x4 Xy)) ((x1 Xx) Xy))))))) as proof of (((eq Prop) ((and ((ex a) (fun (Xz:a)=> (x4 Xz)))) (forall (Xx:a), ((x4 Xx)->(forall (Xy:a), ((iff (x4 Xy)) ((x1 Xx) Xy))))))) b)
% Found ((eq_ref Prop) ((and ((ex a) (fun (Xz:a)=> (x4 Xz)))) (forall (Xx:a), ((x4 Xx)->(forall (Xy:a), ((iff (x4 Xy)) ((x1 Xx) Xy))))))) as proof of (((eq Prop) ((and ((ex a) (fun (Xz:a)=> (x4 Xz)))) (forall (Xx:a), ((x4 Xx)->(forall (Xy:a), ((iff (x4 Xy)) ((x1 Xx) Xy))))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (cP x4))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cP x4))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cP x4))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cP x4))
% Found eq_ref00:=(eq_ref0 ((and ((ex a) (fun (Xz:a)=> (x4 Xz)))) (forall (Xx:a), ((x4 Xx)->(forall (Xy:a), ((iff (x4 Xy)) ((x1 Xx) Xy))))))):(((eq Prop) ((and ((ex a) (fun (Xz:a)=> (x4 Xz)))) (forall (Xx:a), ((x4 Xx)->(forall (Xy:a), ((iff (x4 Xy)) ((x1 Xx) Xy))))))) ((and ((ex a) (fun (Xz:a)=> (x4 Xz)))) (forall (Xx:a), ((x4 Xx)->(forall (Xy:a), ((iff (x4 Xy)) ((x1 Xx) Xy)))))))
% Found (eq_ref0 ((and ((ex a) (fun (Xz:a)=> (x4 Xz)))) (forall (Xx:a), ((x4 Xx)->(forall (Xy:a), ((iff (x4 Xy)) ((x1 Xx) Xy))))))) as proof of (((eq Prop) ((and ((ex a) (fun (Xz:a)=> (x4 Xz)))) (forall (Xx:a), ((x4 Xx)->(forall (Xy:a), ((iff (x4 Xy)) ((x1 Xx) Xy))))))) b)
% Found ((eq_ref Prop) ((and ((ex a) (fun (Xz:a)=> (x4 Xz)))) (forall (Xx:a), ((x4 Xx)->(forall (Xy:a), ((iff (x4 Xy)) ((x1 Xx) Xy))))))) as proof of (((eq Prop) ((and ((ex a) (fun (Xz:a)=> (x4 Xz)))) (forall (Xx:a), ((x4 Xx)->(forall (Xy:a), ((iff (x4 Xy)) ((x1 Xx) Xy))))))) b)
% Found ((eq_ref Prop) ((and ((ex a) (fun (Xz:a)=> (x4 Xz)))) (forall (Xx:a), ((x4 Xx)->(forall (Xy:a), ((iff (x4 Xy)) ((x1 Xx) Xy))))))) as proof of (((eq Prop) ((and ((ex a) (fun (Xz:a)=> (x4 Xz)))) (forall (Xx:a), ((x4 Xx)->(forall (Xy:a), ((iff (x4 Xy)) ((x1 Xx) Xy))))))) b)
% Found ((eq_ref Prop) ((and ((ex a) (fun (Xz:a)=> (x4 Xz)))) (forall (Xx:a), ((x4 Xx)->(forall (Xy:a), ((iff (x4 Xy)) ((x1 Xx) Xy))))))) as proof of (((eq Prop) ((and ((ex a) (fun (Xz:a)=> (x4 Xz)))) (forall (Xx:a), ((x4 Xx)->(forall (Xy:a), ((iff (x4 Xy)) ((x1 Xx) Xy))))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (cP x4))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cP x4))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cP x4))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cP x4))
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) b)
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) b)
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) b)
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) 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), ((x1 Xx) Xx))) (forall (Xx:a) (Xy:a), (((x1 Xx) Xy)->((x1 Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x1 Xx) Xy)) ((x1 Xy) Xz))->((x1 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) cP)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and ((and (forall (Xx:a), ((x1 Xx) Xx))) (forall (Xx:a) (Xy:a), (((x1 Xx) Xy)->((x1 Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x1 Xx) Xy)) ((x1 Xy) Xz))->((x1 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) cP)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and ((and (forall (Xx:a), ((x1 Xx) Xx))) (forall (Xx:a) (Xy:a), (((x1 Xx) Xy)->((x1 Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x1 Xx) Xy)) ((x1 Xy) Xz))->((x1 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) cP)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and ((and (forall (Xx:a), ((x1 Xx) Xx))) (forall (Xx:a) (Xy:a), (((x1 Xx) Xy)->((x1 Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x1 Xx) Xy)) ((x1 Xy) Xz))->((x1 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) cP)))
% Found ((eq_trans0000 ((eq_ref Prop) (f x1))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f x1)) ((and ((and ((and (forall (Xx:a), ((x1 Xx) Xx))) (forall (Xx:a) (Xy:a), (((x1 Xx) Xy)->((x1 Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x1 Xx) Xy)) ((x1 Xy) Xz))->((x1 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) cP)))
% Found (((eq_trans000 ((and ((and ((and (forall (Xx:a), ((x1 Xx) Xx))) (forall (Xx:a) (Xy:a), (((x1 Xx) Xy)->((x1 Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x1 Xx) Xy)) ((x1 Xy) Xz))->((x1 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) cP))) ((eq_ref Prop) (f x1))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f x1)) ((and ((and ((and (forall (Xx:a), ((x1 Xx) Xx))) (forall (Xx:a) (Xy:a), (((x1 Xx) Xy)->((x1 Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x1 Xx) Xy)) ((x1 Xy) Xz))->((x1 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) cP)))
% Found ((((eq_trans00 ((and ((and ((and (forall (Xx:a), ((x1 Xx) Xx))) (forall (Xx:a) (Xy:a), (((x1 Xx) Xy)->((x1 Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x1 Xx) Xy)) ((x1 Xy) Xz))->((x1 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) cP))) ((and ((and ((and (forall (Xx:a), ((x1 Xx) Xx))) (forall (Xx:a) (Xy:a), (((x1 Xx) Xy)->((x1 Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x1 Xx) Xy)) ((x1 Xy) Xz))->((x1 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) cP))) ((eq_ref Prop) (f x1))) ((eq_ref Prop) ((and ((and ((and (forall (Xx:a), ((x1 Xx) Xx))) (forall (Xx:a) (Xy:a), (((x1 Xx) Xy)->((x1 Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x1 Xx) Xy)) ((x1 Xy) Xz))->((x1 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) cP)))) as proof of (((eq Prop) (f x1)) ((and ((and ((and (forall (Xx:a), ((x1 Xx) Xx))) (forall (Xx:a) (Xy:a), (((x1 Xx) Xy)->((x1 Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x1 Xx) Xy)) ((x1 Xy) Xz))->((x1 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) cP)))
% Found (((((eq_trans0 (f x1)) ((and ((and ((and (forall (Xx:a), ((x1 Xx) Xx))) (forall (Xx:a) (Xy:a), (((x1 Xx) Xy)->((x1 Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x1 Xx) Xy)) ((x1 Xy) Xz))->((x1 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) cP))) ((and ((and ((and (forall (Xx:a), ((x1 Xx) Xx))) (forall (Xx:a) (Xy:a), (((x1 Xx) Xy)->((x1 Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x1 Xx) Xy)) ((x1 Xy) Xz))->((x1 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) cP))) ((eq_ref Prop) (f x1))) ((eq_ref Prop) ((and ((and ((and (forall (Xx:a), ((x1 Xx) Xx))) (forall (Xx:a) (Xy:a), (((x1 Xx) Xy)->((x1 Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x1 Xx) Xy)) ((x1 Xy) Xz))->((x1 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) cP)))) as proof of (((eq Prop) (f x1)) ((and ((and ((and (forall (Xx:a), ((x1 Xx) Xx))) (forall (Xx:a) (Xy:a), (((x1 Xx) Xy)->((x1 Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x1 Xx) Xy)) ((x1 Xy) Xz))->((x1 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) cP)))
% Found ((((((eq_trans Prop) (f x1)) ((and ((and ((and (forall (Xx:a), ((x1 Xx) Xx))) (forall (Xx:a) (Xy:a), (((x1 Xx) Xy)->((x1 Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x1 Xx) Xy)) ((x1 Xy) Xz))->((x1 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) cP))) ((and ((and ((and (forall (Xx:a), ((x1 Xx) Xx))) (forall (Xx:a) (Xy:a), (((x1 Xx) Xy)->((x1 Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x1 Xx) Xy)) ((x1 Xy) Xz))->((x1 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) cP))) ((eq_ref Prop) (f x1))) ((eq_ref Prop) ((and ((and ((and (forall (Xx:a), ((x1 Xx) Xx))) (forall (Xx:a) (Xy:a), (((x1 Xx) Xy)->((x1 Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x1 Xx) Xy)) ((x1 Xy) Xz))->((x1 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) cP)))) as proof of (((eq Prop) (f x1)) ((and ((and ((and (forall (Xx:a), ((x1 Xx) Xx))) (forall (Xx:a) (Xy:a), (((x1 Xx) Xy)->((x1 Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x1 Xx) Xy)) ((x1 Xy) Xz))->((x1 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) cP)))
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) b)
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) b)
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) b)
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) 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), ((x1 Xx) Xx))) (forall (Xx:a) (Xy:a), (((x1 Xx) Xy)->((x1 Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x1 Xx) Xy)) ((x1 Xy) Xz))->((x1 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) cP)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and ((and (forall (Xx:a), ((x1 Xx) Xx))) (forall (Xx:a) (Xy:a), (((x1 Xx) Xy)->((x1 Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x1 Xx) Xy)) ((x1 Xy) Xz))->((x1 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) cP)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and ((and (forall (Xx:a), ((x1 Xx) Xx))) (forall (Xx:a) (Xy:a), (((x1 Xx) Xy)->((x1 Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x1 Xx) Xy)) ((x1 Xy) Xz))->((x1 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) cP)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and ((and (forall (Xx:a), ((x1 Xx) Xx))) (forall (Xx:a) (Xy:a), (((x1 Xx) Xy)->((x1 Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x1 Xx) Xy)) ((x1 Xy) Xz))->((x1 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) cP)))
% Found ((eq_trans0000 ((eq_ref Prop) (f x1))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f x1)) ((and ((and ((and (forall (Xx:a), ((x1 Xx) Xx))) (forall (Xx:a) (Xy:a), (((x1 Xx) Xy)->((x1 Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x1 Xx) Xy)) ((x1 Xy) Xz))->((x1 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) cP)))
% Found (((eq_trans000 ((and ((and ((and (forall (Xx:a), ((x1 Xx) Xx))) (forall (Xx:a) (Xy:a), (((x1 Xx) Xy)->((x1 Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x1 Xx) Xy)) ((x1 Xy) Xz))->((x1 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) cP))) ((eq_ref Prop) (f x1))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f x1)) ((and ((and ((and (forall (Xx:a), ((x1 Xx) Xx))) (forall (Xx:a) (Xy:a), (((x1 Xx) Xy)->((x1 Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x1 Xx) Xy)) ((x1 Xy) Xz))->((x1 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) cP)))
% Found ((((eq_trans00 ((and ((and ((and (forall (Xx:a), ((x1 Xx) Xx))) (forall (Xx:a) (Xy:a), (((x1 Xx) Xy)->((x1 Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x1 Xx) Xy)) ((x1 Xy) Xz))->((x1 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) cP))) ((and ((and ((and (forall (Xx:a), ((x1 Xx) Xx))) (forall (Xx:a) (Xy:a), (((x1 Xx) Xy)->((x1 Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x1 Xx) Xy)) ((x1 Xy) Xz))->((x1 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) cP))) ((eq_ref Prop) (f x1))) ((eq_ref Prop) ((and ((and ((and (forall (Xx:a), ((x1 Xx) Xx))) (forall (Xx:a) (Xy:a), (((x1 Xx) Xy)->((x1 Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x1 Xx) Xy)) ((x1 Xy) Xz))->((x1 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) cP)))) as proof of (((eq Prop) (f x1)) ((and ((and ((and (forall (Xx:a), ((x1 Xx) Xx))) (forall (Xx:a) (Xy:a), (((x1 Xx) Xy)->((x1 Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x1 Xx) Xy)) ((x1 Xy) Xz))->((x1 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) cP)))
% Found (((((eq_trans0 (f x1)) ((and ((and ((and (forall (Xx:a), ((x1 Xx) Xx))) (forall (Xx:a) (Xy:a), (((x1 Xx) Xy)->((x1 Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x1 Xx) Xy)) ((x1 Xy) Xz))->((x1 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) cP))) ((and ((and ((and (forall (Xx:a), ((x1 Xx) Xx))) (forall (Xx:a) (Xy:a), (((x1 Xx) Xy)->((x1 Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x1 Xx) Xy)) ((x1 Xy) Xz))->((x1 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) cP))) ((eq_ref Prop) (f x1))) ((eq_ref Prop) ((and ((and ((and (forall (Xx:a), ((x1 Xx) Xx))) (forall (Xx:a) (Xy:a), (((x1 Xx) Xy)->((x1 Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x1 Xx) Xy)) ((x1 Xy) Xz))->((x1 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) cP)))) as proof of (((eq Prop) (f x1)) ((and ((and ((and (forall (Xx:a), ((x1 Xx) Xx))) (forall (Xx:a) (Xy:a), (((x1 Xx) Xy)->((x1 Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x1 Xx) Xy)) ((x1 Xy) Xz))->((x1 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) cP)))
% Found ((((((eq_trans Prop) (f x1)) ((and ((and ((and (forall (Xx:a), ((x1 Xx) Xx))) (forall (Xx:a) (Xy:a), (((x1 Xx) Xy)->((x1 Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x1 Xx) Xy)) ((x1 Xy) Xz))->((x1 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) cP))) ((and ((and ((and (forall (Xx:a), ((x1 Xx) Xx))) (forall (Xx:a) (Xy:a), (((x1 Xx) Xy)->((x1 Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x1 Xx) Xy)) ((x1 Xy) Xz))->((x1 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) cP))) ((eq_ref Prop) (f x1))) ((eq_ref Prop) ((and ((and ((and (forall (Xx:a), ((x1 Xx) Xx))) (forall (Xx:a) (Xy:a), (((x1 Xx) Xy)->((x1 Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x1 Xx) Xy)) ((x1 Xy) Xz))->((x1 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) cP)))) as proof of (((eq Prop) (f x1)) ((and ((and ((and (forall (Xx:a), ((x1 Xx) Xx))) (forall (Xx:a) (Xy:a), (((x1 Xx) Xy)->((x1 Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x1 Xx) Xy)) ((x1 Xy) Xz))->((x1 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) cP)))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))->(P (fun (x:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (x Xz)))) (forall (Xx:a), ((x Xx)->(forall (Xy:a), ((iff (x Xy)) ((x1 Xx) Xy)))))))))
% Found (eta_expansion_dep000 P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found ((eta_expansion_dep00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found (((eta_expansion_dep0 (fun (x7:(a->Prop))=> Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found ((((eta_expansion_dep (a->Prop)) (fun (x7:(a->Prop))=> Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found ((((eta_expansion_dep (a->Prop)) (fun (x7:(a->Prop))=> Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found eq_ref000:=(eq_ref00 P):((P (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))->(P (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))))
% Found (eq_ref00 P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found ((eq_ref0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found (((eq_ref ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found (((eq_ref ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy))))))))
% Found eta_expansion0000:=(eta_expansion000 P):((P (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy))))))))->(P (fun (x:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (x Xz)))) (forall (Xx:a), ((x Xx)->(forall (Xy:a), ((iff (x Xy)) ((x3 Xx) Xy)))))))))
% Found (eta_expansion000 P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy))))))))
% Found ((eta_expansion00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy)))))))) P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy))))))))
% Found (((eta_expansion0 Prop) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy)))))))) P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy))))))))
% Found ((((eta_expansion (a->Prop)) Prop) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy)))))))) P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy))))))))
% Found ((((eta_expansion (a->Prop)) Prop) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy)))))))) P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy))))))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy))))))))->(P (fun (x:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (x Xz)))) (forall (Xx:a), ((x Xx)->(forall (Xy:a), ((iff (x Xy)) ((x3 Xx) Xy)))))))))
% Found (eta_expansion_dep000 P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy))))))))
% Found ((eta_expansion_dep00 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy)))))))) P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy))))))))
% Found (((eta_expansion_dep0 (fun (x7:(a->Prop))=> Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy)))))))) P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy))))))))
% Found ((((eta_expansion_dep (a->Prop)) (fun (x7:(a->Prop))=> Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy)))))))) P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy))))))))
% Found ((((eta_expansion_dep (a->Prop)) (fun (x7:(a->Prop))=> Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy)))))))) P) as proof of (P0 (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x3 Xx) Xy))))))))
% Found eq_ref000:=(eq_ref00 P0):((P0 (f x1))->(P0 (f x1)))
% Found (eq_ref00 P0) as proof of (P1 (f x1))
% Found ((eq_ref0 (f x1)) P0) as proof of (P1 (f x1))
% Found (((eq_ref Prop) (f x1)) P0) as proof of (P1 (f x1))
% Found (((eq_ref Prop) (f x1)) P0) as proof of (P1 (f x1))
% Found eq_ref000:=(eq_ref00 P0):((P0 (f x1))->(P0 (f x1)))
% Found (eq_ref00 P0) as proof of (P1 (f x1))
% Found ((eq_ref0 (f x1)) P0) as proof of (P1 (f x1))
% Found (((eq_ref Prop) (f x1)) P0) as proof of (P1 (f x1))
% Found (((eq_ref Prop) (f x1)) P0) as proof of (P1 (f x1))
% Found eq_ref000:=(eq_ref00 P):((P ((and ((ex a) (fun (Xz:a)=> (x4 Xz)))) (forall (Xx:a), ((x4 Xx)->(forall (Xy:a), ((iff (x4 Xy)) ((x1 Xx) Xy)))))))->(P ((and ((ex a) (fun (Xz:a)=> (x4 Xz)))) (forall (Xx:a), ((x4 Xx)->(forall (Xy:a), ((iff (x4 Xy)) ((x1 Xx) Xy))))))))
% Found (eq_ref00 P) as proof of (P0 ((and ((ex a) (fun (Xz:a)=> (x4 Xz)))) (forall (Xx:a), ((x4 Xx)->(forall (Xy:a), ((iff (x4 Xy)) ((x1 Xx) Xy)))))))
% Found ((eq_ref0 ((and ((ex a) (fun (Xz:a)=> (x4 Xz)))) (forall (Xx:a), ((x4 Xx)->(forall (Xy:a), ((iff (x4 Xy)) ((x1 Xx) Xy))))))) P) as proof of (P0 ((and ((ex a) (fun (Xz:a)=> (x4 Xz)))) (forall (Xx:a), ((x4 Xx)->(forall (Xy:a), ((iff (x4 Xy)) ((x1 Xx) Xy)))))))
% Found (((eq_ref Prop) ((and ((ex a) (fun (Xz:a)=> (x4 Xz)))) (forall (Xx:a), ((x4 Xx)->(forall (Xy:a), ((iff (x4 Xy)) ((x1 Xx) Xy))))))) P) as proof of (P0 ((and ((ex a) (fun (Xz:a)=> (x4 Xz)))) (forall (Xx:a), ((x4 Xx)->(forall (Xy:a), ((iff (x4 Xy)) ((x1 Xx) Xy)))))))
% Found (((eq_ref Prop) ((and ((ex a) (fun (Xz:a)=> (x4 Xz)))) (forall (Xx:a), ((x4 Xx)->(forall (Xy:a), ((iff (x4 Xy)) ((x1 Xx) Xy))))))) P) as proof of (P0 ((and ((ex a) (fun (Xz:a)=> (x4 Xz)))) (forall (Xx:a), ((x4 Xx)->(forall (Xy:a), ((iff (x4 Xy)) ((x1 Xx) Xy)))))))
% Found eq_ref000:=(eq_ref00 P):((P ((and ((ex a) (fun (Xz:a)=> (x4 Xz)))) (forall (Xx:a), ((x4 Xx)->(forall (Xy:a), ((iff (x4 Xy)) ((x1 Xx) Xy)))))))->(P ((and ((ex a) (fun (Xz:a)=> (x4 Xz)))) (forall (Xx:a), ((x4 Xx)->(forall (Xy:a), ((iff (x4 Xy)) ((x1 Xx) Xy))))))))
% Found (eq_ref00 P) as proof of (P0 ((and ((ex a) (fun (Xz:a)=> (x4 Xz)))) (forall (Xx:a), ((x4 Xx)->(forall (Xy:a), ((iff (x4 Xy)) ((x1 Xx) Xy)))))))
% Found ((eq_ref0 ((and ((ex a) (fun (Xz:a)=> (x4 Xz)))) (forall (Xx:a), ((x4 Xx)->(forall (Xy:a), ((iff (x4 Xy)) ((x1 Xx) Xy))))))) P) as proof of (P0 ((and ((ex a) (fun (Xz:a)=> (x4 Xz)))) (forall (Xx:a), ((x4 Xx)->(forall (Xy:a), ((iff (x4 Xy)) ((x1 Xx) Xy)))))))
% Found (((eq_ref Prop) ((and ((ex a) (fun (Xz:a)=> (x4 Xz)))) (forall (Xx:a), ((x4 Xx)->(forall (Xy:a), ((iff (x4 Xy)) ((x1 Xx) Xy))))))) P) as proof of (P0 ((and ((ex a) (fun (Xz:a)=> (x4 Xz)))) (forall (Xx:a), ((x4 Xx)->(forall (Xy:a), ((iff (x4 Xy)) ((x1 Xx) Xy)))))))
% Found (((eq_ref Prop) ((and ((ex a) (fun (Xz:a)=> (x4 Xz)))) (forall (Xx:a), ((x4 Xx)->(forall (Xy:a), ((iff (x4 Xy)) ((x1 Xx) Xy))))))) P) as proof of (P0 ((and ((ex a) (fun (Xz:a)=> (x4 Xz)))) (forall (Xx:a), ((x4 Xx)->(forall (Xy:a), ((iff (x4 Xy)) ((x1 Xx) Xy)))))))
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) b)
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) b)
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) b)
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) 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), ((x1 Xx) Xx))) (forall (Xx:a) (Xy:a), (((x1 Xx) Xy)->((x1 Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x1 Xx) Xy)) ((x1 Xy) Xz))->((x1 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) cP)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and ((and (forall (Xx:a), ((x1 Xx) Xx))) (forall (Xx:a) (Xy:a), (((x1 Xx) Xy)->((x1 Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x1 Xx) Xy)) ((x1 Xy) Xz))->((x1 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) cP)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and ((and (forall (Xx:a), ((x1 Xx) Xx))) (forall (Xx:a) (Xy:a), (((x1 Xx) Xy)->((x1 Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x1 Xx) Xy)) ((x1 Xy) Xz))->((x1 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) cP)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and ((and (forall (Xx:a), ((x1 Xx) Xx))) (forall (Xx:a) (Xy:a), (((x1 Xx) Xy)->((x1 Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x1 Xx) Xy)) ((x1 Xy) Xz))->((x1 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) cP)))
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) b)
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) b)
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) b)
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) 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), ((x1 Xx) Xx))) (forall (Xx:a) (Xy:a), (((x1 Xx) Xy)->((x1 Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x1 Xx) Xy)) ((x1 Xy) Xz))->((x1 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) cP)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and ((and (forall (Xx:a), ((x1 Xx) Xx))) (forall (Xx:a) (Xy:a), (((x1 Xx) Xy)->((x1 Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x1 Xx) Xy)) ((x1 Xy) Xz))->((x1 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) cP)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and ((and (forall (Xx:a), ((x1 Xx) Xx))) (forall (Xx:a) (Xy:a), (((x1 Xx) Xy)->((x1 Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x1 Xx) Xy)) ((x1 Xy) Xz))->((x1 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) cP)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and ((and (forall (Xx:a), ((x1 Xx) Xx))) (forall (Xx:a) (Xy:a), (((x1 Xx) Xy)->((x1 Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((x1 Xx) Xy)) ((x1 Xy) Xz))->((x1 Xx) Xz))))) (((eq ((a->Prop)->Prop)) (fun (Xs:(a->Prop))=> ((and ((ex a) (fun (Xz:a)=> (Xs Xz)))) (forall (Xx:a), ((Xs Xx)->(forall (Xy:a), ((iff (Xs Xy)) ((x1 Xx) Xy)))))))) cP)))
% Found choice000:=(choice00 (fun (x7:a) (x60:(a->Prop))=> ((and (cP x60)) (x60 x7)))):((forall (x:a), ((ex (a->Prop)) (fun (y:(a->Prop))=> ((and (cP y)) (y x)))))->((ex (a->(a->Prop))) (fun (f:(a->(a->Prop)))=> (forall (x:a), ((and (cP (f x))) ((f x) x))))))
% Found (choice00 (fun (x7:a) (x60:(a->Prop))=> ((and (cP x60)) (x60 x7)))) as proof of ((forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx)))))->(P a0))
% Found ((choice0 (a->Prop)) (fun (x7:a) (x60:(a->Prop))=> ((and (cP x60)) (x60 x7)))) as proof of ((forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx)))))->(P a0))
% Found (((choice a) (a->Prop)) (fun (x7:a) (x60:(a->Prop))=> ((and (cP x60)) (x60 x7)))) as proof of ((forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx)))))->(P a0))
% Found (((choice a) (a->Prop)) (fun (x7:a) (x60:(a->Prop))=> ((and (cP x60)) (x60 x7)))) as proof of ((forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx)))))->(P a0))
% Found (fun (x3:(forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))=> (((choice a) (a->Prop)) (fun (x7:a) (x60:(a->Prop))=> ((and (cP x60)) (x60 x7))))) as proof of ((forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx)))))->(P a0))
% Found (fun (x3:(forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))=> (((choice a) (a->Prop)) (fun (x7:a) (x60:(a->Prop))=> ((and (cP x60)) (x60 x7))))) as proof of ((forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))->((forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx)))))->(P a0)))
% Found (and_rect10 (fun (x3:(forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))=> (((choice a) (a->Prop)) (fun (x7:a) (x60:(a->Prop))=> ((and (cP x60)) (x60 x7)))))) as proof of (P a0)
% Found ((and_rect1 (P a0)) (fun (x3:(forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))=> (((choice a) (a->Prop)) (fun (x7:a) (x60:(a->Prop))=> ((and (cP x60)) (x60 x7)))))) as proof of (P a0)
% Found (((fun (P0:Type) (x3:((forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))->((forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx)))))->P0)))=> (((((and_rect (forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx)))))) P0) x3) x1)) (P a0)) (fun (x3:(forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))=> (((choice a) (a->Prop)) (fun (x7:a) (x60:(a->Prop))=> ((and (cP x60)) (x60 x7)))))) as proof of (P a0)
% Found (((fun (P0:Type) (x3:((forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))->((forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx)))))->P0)))=> (((((and_rect (forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx)))))) P0) x3) x1)) (P a0)) (fun (x3:(forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))=> (((choice a) (a->Prop)) (fun (x7:a) (x60:(a->Prop))=> ((and (cP x60)) (x60 x7)))))) as proof of (P a0)
% 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 choice000:=(choice00 (fun (x7:a) (x60:(a->Prop))=> ((and (cP x60)) (x60 x7)))):((forall (x:a), ((ex (a->Prop)) (fun (y:(a->Prop))=> ((and (cP y)) (y x)))))->((ex (a->(a->Prop))) (fun (f:(a->(a->Prop)))=> (forall (x:a), ((and (cP (f x))) ((f x) x))))))
% Found (choice00 (fun (x7:a) (x60:(a->Prop))=> ((and (cP x60)) (x60 x7)))) as proof of ((forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx)))))->(P a0))
% Found ((choice0 (a->Prop)) (fun (x7:a) (x60:(a->Prop))=> ((and (cP x60)) (x60 x7)))) as proof of ((forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx)))))->(P a0))
% Found (((choice a) (a->Prop)) (fun (x7:a) (x60:(a->Prop))=> ((and (cP x60)) (x60 x7)))) as proof of ((forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx)))))->(P a0))
% Found (((choice a) (a->Prop)) (fun (x7:a) (x60:(a->Prop))=> ((and (cP x60)) (x60 x7)))) as proof of ((forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx)))))->(P a0))
% Found (fun (x3:(forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))=> (((choice a) (a->Prop)) (fun (x7:a) (x60:(a->Prop))=> ((and (cP x60)) (x60 x7))))) as proof of ((forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx)))))->(P a0))
% Found (fun (x3:(forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))=> (((choice a) (a->Prop)) (fun (x7:a) (x60:(a->Prop))=> ((and (cP x60)) (x60 x7))))) as proof of ((forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))->((forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx)))))->(P a0)))
% Found (and_rect10 (fun (x3:(forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))=> (((choice a) (a->Prop)) (fun (x7:a) (x60:(a->Prop))=> ((and (cP x60)) (x60 x7)))))) as proof of (P a0)
% Found ((and_rect1 (P a0)) (fun (x3:(forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))=> (((choice a) (a->Prop)) (fun (x7:a) (x60:(a->Prop))=> ((and (cP x60)) (x60 x7)))))) as proof of (P a0)
% Found (((fun (P0:Type) (x3:((forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))->((forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx)))))->P0)))=> (((((and_rect (forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx)))))) P0) x3) x1)) (P a0)) (fun (x3:(forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))=> (((choice a) (a->Prop)) (fun (x7:a) (x60:(a->Prop))=> ((and (cP x60)) (x60 x7)))))) as proof of (P a0)
% Found (fun (x2:(forall (Xx:a) (Xy:a) (Xp:(a->Prop)) (Xq:(a->Prop)), (((and ((and ((and ((and (cP Xp)) (cP Xq))) (Xp Xx))) (Xq Xx))) (Xp Xy))->(Xq Xy))))=> (((fun (P0:Type) (x3:((forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))->((forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx)))))->P0)))=> (((((and_rect (forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx)))))) P0) x3) x1)) (P a0)) (fun (x3:(forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))=> (((choice a) (a->Prop)) (fun (x7:a) (x60:(a->Prop))=> ((and (cP x60)) (x60 x7))))))) as proof of (P a0)
% Found (fun (x1:((and (forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx))))))) (x2:(forall (Xx:a) (Xy:a) (Xp:(a->Prop)) (Xq:(a->Prop)), (((and ((and ((and ((and (cP Xp)) (cP Xq))) (Xp Xx))) (Xq Xx))) (Xp Xy))->(Xq Xy))))=> (((fun (P0:Type) (x3:((forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))->((forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx)))))->P0)))=> (((((and_rect (forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx)))))) P0) x3) x1)) (P a0)) (fun (x3:(forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))=> (((choice a) (a->Prop)) (fun (x7:a) (x60:(a->Prop))=> ((and (cP x60)) (x60 x7))))))) as proof of ((forall (Xx:a) (Xy:a) (Xp:(a->Prop)) (Xq:(a->Prop)), (((and ((and ((and ((and (cP Xp)) (cP Xq))) (Xp Xx))) (Xq Xx))) (Xp Xy))->(Xq Xy)))->(P a0))
% Found (fun (x1:((and (forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx))))))) (x2:(forall (Xx:a) (Xy:a) (Xp:(a->Prop)) (Xq:(a->Prop)), (((and ((and ((and ((and (cP Xp)) (cP Xq))) (Xp Xx))) (Xq Xx))) (Xp Xy))->(Xq Xy))))=> (((fun (P0:Type) (x3:((forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))->((forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx)))))->P0)))=> (((((and_rect (forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx)))))) P0) x3) x1)) (P a0)) (fun (x3:(forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))=> (((choice a) (a->Prop)) (fun (x7:a) (x60:(a->Prop))=> ((and (cP x60)) (x60 x7))))))) as proof of (((and (forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx))))))->((forall (Xx:a) (Xy:a) (Xp:(a->Prop)) (Xq:(a->Prop)), (((and ((and ((and ((and (cP Xp)) (cP Xq))) (Xp Xx))) (Xq Xx))) (Xp Xy))->(Xq Xy)))->(P a0)))
% Found (and_rect00 (fun (x1:((and (forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx))))))) (x2:(forall (Xx:a) (Xy:a) (Xp:(a->Prop)) (Xq:(a->Prop)), (((and ((and ((and ((and (cP Xp)) (cP Xq))) (Xp Xx))) (Xq Xx))) (Xp Xy))->(Xq Xy))))=> (((fun (P0:Type) (x3:((forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))->((forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx)))))->P0)))=> (((((and_rect (forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx)))))) P0) x3) x1)) (P a0)) (fun (x3:(forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))=> (((choice a) (a->Prop)) (fun (x7:a) (x60:(a->Prop))=> ((and (cP x60)) (x60 x7)))))))) as proof of (P a0)
% Found ((and_rect0 (P a0)) (fun (x1:((and (forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx))))))) (x2:(forall (Xx:a) (Xy:a) (Xp:(a->Prop)) (Xq:(a->Prop)), (((and ((and ((and ((and (cP Xp)) (cP Xq))) (Xp Xx))) (Xq Xx))) (Xp Xy))->(Xq Xy))))=> (((fun (P0:Type) (x3:((forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))->((forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx)))))->P0)))=> (((((and_rect (forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx)))))) P0) x3) x1)) (P a0)) (fun (x3:(forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))=> (((choice a) (a->Prop)) (fun (x7:a) (x60:(a->Prop))=> ((and (cP x60)) (x60 x7)))))))) as proof of (P a0)
% Found (((fun (P0:Type) (x1:(((and (forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx))))))->((forall (Xx:a) (Xy:a) (Xp:(a->Prop)) (Xq:(a->Prop)), (((and ((and ((and ((and (cP Xp)) (cP Xq))) (Xp Xx))) (Xq Xx))) (Xp Xy))->(Xq Xy)))->P0)))=> (((((and_rect ((and (forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx))))))) (forall (Xx:a) (Xy:a) (Xp:(a->Prop)) (Xq:(a->Prop)), (((and ((and ((and ((and (cP Xp)) (cP Xq))) (Xp Xx))) (Xq Xx))) (Xp Xy))->(Xq Xy)))) P0) x1) x0)) (P a0)) (fun (x1:((and (forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx))))))) (x2:(forall (Xx:a) (Xy:a) (Xp:(a->Prop)) (Xq:(a->Prop)), (((and ((and ((and ((and (cP Xp)) (cP Xq))) (Xp Xx))) (Xq Xx))) (Xp Xy))->(Xq Xy))))=> (((fun (P0:Type) (x3:((forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))->((forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx)))))->P0)))=> (((((and_rect (forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx)))))) P0) x3) x1)) (P a0)) (fun (x3:(forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))))=> (((choice a) (a->Prop)) (fun (x7:a) (x60:(a->Prop))=> ((and (cP x60)) (x60 x7)))))))) as proof of (P a0)
% Found (((fun (P0:Type) (x1:(((and (forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx))))))->((forall (Xx:a) (Xy:a) (Xp:(a->Prop)) (Xq:(a->Prop)), (((and ((and ((and ((and (cP Xp)) (cP Xq))) (Xp Xx))) (Xq Xx))) (Xp Xy))->(Xq Xy)))->P0)))=> (((((and_rect ((and (forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and (cP Xp)) (Xp Xx))))))) (forall (Xx:a) (Xy:a) (Xp:(a->Prop)) (Xq:(a->Prop)), (((and ((and ((and ((and (cP Xp)) (cP Xq))) (Xp Xx))) (Xq Xx))) (Xp Xy))->(Xq Xy)))) P0) x1) x0)) (P a0)) (fun (x1:((and (forall (Xp:(a->Prop)), ((cP Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (
% EOF
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