TSTP Solution File: SEV125^5 by cocATP---0.2.0
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- Process Solution
%------------------------------------------------------------------------------
% File : cocATP---0.2.0
% Problem : SEV125^5 : TPTP v6.1.0. Released v4.0.0.
% Transfm : none
% Format : tptp:raw
% Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% Computer : n115.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 13:33:46 EDT 2014
% Result : Timeout 300.08s
% Output : None
% Verified :
% SZS Type : None (Parsing solution fails)
% Syntax : Number of formulae : 0
% Comments :
%------------------------------------------------------------------------------
%----NO SOLUTION OUTPUT BY SYSTEM
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% % Problem : SEV125^5 : TPTP v6.1.0. Released v4.0.0.
% % Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n115.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 08:10:11 CDT 2014
% % CPUTime : 300.08
% 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 0x20dc200>, <kernel.Type object at 0x20dd050>) of role type named a_type
% Using role type
% Declaring a:Type
% FOF formula (forall (PROP:((a->(a->Prop))->Prop)) (S:((a->(a->Prop))->Prop)), (((eq (a->(a->Prop))) (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy))))) (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy)))))) of role conjecture named cTHM254_pme
% Conjecture to prove = (forall (PROP:((a->(a->Prop))->Prop)) (S:((a->(a->Prop))->Prop)), (((eq (a->(a->Prop))) (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy))))) (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy)))))):Prop
% Parameter a_DUMMY:a.
% We need to prove ['(forall (PROP:((a->(a->Prop))->Prop)) (S:((a->(a->Prop))->Prop)), (((eq (a->(a->Prop))) (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy))))) (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy))))))']
% Parameter a:Type.
% Trying to prove (forall (PROP:((a->(a->Prop))->Prop)) (S:((a->(a->Prop))->Prop)), (((eq (a->(a->Prop))) (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy))))) (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy))))))
% Found eq_ref00:=(eq_ref0 b):(((eq (a->(a->Prop))) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->(a->Prop))) b) (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy)))))
% Found ((eq_ref (a->(a->Prop))) b) as proof of (((eq (a->(a->Prop))) b) (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy)))))
% Found ((eq_ref (a->(a->Prop))) b) as proof of (((eq (a->(a->Prop))) b) (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy)))))
% Found ((eq_ref (a->(a->Prop))) b) as proof of (((eq (a->(a->Prop))) b) (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy)))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy))))):(((eq (a->(a->Prop))) (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy))))) (fun (x:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy)))))
% Found (eta_expansion_dep00 (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy))))) as proof of (((eq (a->(a->Prop))) (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy))))) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> (a->Prop))) (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy))))) as proof of (((eq (a->(a->Prop))) (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> (a->Prop))) (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy))))) as proof of (((eq (a->(a->Prop))) (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> (a->Prop))) (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy))))) as proof of (((eq (a->(a->Prop))) (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> (a->Prop))) (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy))))) as proof of (((eq (a->(a->Prop))) (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy))))) b)
% Found x2:(P (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy)))))
% Found (fun (x2:(P (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy))))))=> x2) as proof of (P (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy)))))
% Found (fun (x2:(P (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy))))))=> x2) as proof of (P0 (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy)))))
% Found x2:(P (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy)))))
% Found (fun (x2:(P (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy))))))=> x2) as proof of (P (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy)))))
% Found (fun (x2:(P (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy))))))=> x2) as proof of (P0 (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy)))))
% Found x2:(P (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy)))))
% Found (fun (x2:(P (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy))))))=> x2) as proof of (P (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy)))))
% Found (fun (x2:(P (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy))))))=> x2) as proof of (P0 (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy)))))
% Found x2:(P (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy)))))
% Found (fun (x2:(P (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy))))))=> x2) as proof of (P (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy)))))
% Found (fun (x2:(P (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy))))))=> x2) as proof of (P0 (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy)))))
% Found eq_ref00:=(eq_ref0 b):(((eq (a->(a->Prop))) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->(a->Prop))) b) (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy)))))
% Found ((eq_ref (a->(a->Prop))) b) as proof of (((eq (a->(a->Prop))) b) (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy)))))
% Found ((eq_ref (a->(a->Prop))) b) as proof of (((eq (a->(a->Prop))) b) (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy)))))
% Found ((eq_ref (a->(a->Prop))) b) as proof of (((eq (a->(a->Prop))) b) (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy)))))
% Found eta_expansion000:=(eta_expansion00 (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy))))):(((eq (a->(a->Prop))) (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy))))) (fun (x:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy)))))
% Found (eta_expansion00 (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy))))) as proof of (((eq (a->(a->Prop))) (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy))))) b)
% Found ((eta_expansion0 (a->Prop)) (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy))))) as proof of (((eq (a->(a->Prop))) (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy))))) b)
% Found (((eta_expansion a) (a->Prop)) (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy))))) as proof of (((eq (a->(a->Prop))) (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy))))) b)
% Found (((eta_expansion a) (a->Prop)) (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy))))) as proof of (((eq (a->(a->Prop))) (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy))))) b)
% Found (((eta_expansion a) (a->Prop)) (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy))))) as proof of (((eq (a->(a->Prop))) (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy))))) b)
% Found eq_ref000:=(eq_ref00 P):((P (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy)))))->(P (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy))))))
% Found (eq_ref00 P) as proof of (P0 (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy)))))
% Found ((eq_ref0 (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy))))) P) as proof of (P0 (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy)))))
% Found (((eq_ref (a->(a->Prop))) (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy))))) P) as proof of (P0 (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy)))))
% Found (((eq_ref (a->(a->Prop))) (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy))))) P) as proof of (P0 (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy)))))
% Found eta_expansion0000:=(eta_expansion000 P):((P (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy)))))->(P (fun (x:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))))
% Found (eta_expansion000 P) as proof of (P0 (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy)))))
% Found ((eta_expansion00 (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy))))) P) as proof of (P0 (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy)))))
% Found (((eta_expansion0 (a->Prop)) (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy))))) P) as proof of (P0 (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy)))))
% Found ((((eta_expansion a) (a->Prop)) (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy))))) P) as proof of (P0 (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy)))))
% Found ((((eta_expansion a) (a->Prop)) (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy))))) P) as proof of (P0 (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy)))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy)))))->(P (fun (x:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))))
% Found (eta_expansion_dep000 P) as proof of (P0 (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy)))))
% Found ((eta_expansion_dep00 (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy))))) P) as proof of (P0 (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy)))))
% Found (((eta_expansion_dep0 (fun (x1:a)=> (a->Prop))) (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy))))) P) as proof of (P0 (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy)))))
% Found ((((eta_expansion_dep a) (fun (x1:a)=> (a->Prop))) (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy))))) P) as proof of (P0 (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy)))))
% Found ((((eta_expansion_dep a) (fun (x1:a)=> (a->Prop))) (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy))))) P) as proof of (P0 (fun (Xx:a) (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp Xx) Xy)))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))):(((eq (a->Prop)) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))) (fun (x0:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) x0)))))
% Found (eta_expansion_dep00 (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy)))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy)))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy)))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy)))))
% Found eq_ref00:=(eq_ref0 (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))):(((eq (a->Prop)) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy)))))
% Found (eq_ref0 (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy)))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy)))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy)))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy)))))
% Found x01:(P (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy)))))
% Found (fun (x01:(P (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))))=> x01) as proof of (P (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy)))))
% Found (fun (x01:(P (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))))=> x01) as proof of (P0 (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy)))))
% Found x01:(P (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy)))))
% Found (fun (x01:(P (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))))=> x01) as proof of (P (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy)))))
% Found (fun (x01:(P (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))))=> x01) as proof of (P0 (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy)))))
% Found x01:(P (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy)))))
% Found (fun (x01:(P (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))))=> x01) as proof of (P (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy)))))
% Found (fun (x01:(P (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))))=> x01) as proof of (P0 (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy)))))
% Found x01:(P (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy)))))
% Found (fun (x01:(P (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))))=> x01) as proof of (P (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy)))))
% Found (fun (x01:(P (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))))=> x01) as proof of (P0 (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy)))))
% Found eq_ref00:=(eq_ref0 (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) y)))):(((eq Prop) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) y)))) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) y))))
% Found (eq_ref0 (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) y)))) as proof of (((eq Prop) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) y)))) b)
% Found ((eq_ref Prop) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) y)))) as proof of (((eq Prop) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) y)))) b)
% Found ((eq_ref Prop) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) y)))) as proof of (((eq Prop) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) y)))) b)
% Found ((eq_ref Prop) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) y)))) as proof of (((eq Prop) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) y)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) y))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) y))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) y))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) y))))
% Found x01:(P (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy)))))
% Found (fun (x01:(P (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))))=> x01) as proof of (P (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy)))))
% Found (fun (x01:(P (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))))=> x01) as proof of (P0 (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy)))))
% Found x01:(P (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy)))))
% Found (fun (x01:(P (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))))=> x01) as proof of (P (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy)))))
% Found (fun (x01:(P (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))))=> x01) as proof of (P0 (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy)))))
% Found x01:(P (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) y))))
% Found (fun (x01:(P (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) y)))))=> x01) as proof of (P (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) y))))
% Found (fun (x01:(P (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) y)))))=> x01) as proof of (P0 (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) y))))
% Found eq_ref000:=(eq_ref00 P):((P (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy)))))->(P (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))))
% Found (eq_ref00 P) as proof of (P0 (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy)))))
% Found ((eq_ref0 (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))) P) as proof of (P0 (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy)))))
% Found (((eq_ref (a->Prop)) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))) P) as proof of (P0 (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy)))))
% Found (((eq_ref (a->Prop)) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))) P) as proof of (P0 (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy)))))
% Found eq_ref000:=(eq_ref00 P):((P (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy)))))->(P (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))))
% Found (eq_ref00 P) as proof of (P0 (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy)))))
% Found ((eq_ref0 (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))) P) as proof of (P0 (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy)))))
% Found (((eq_ref (a->Prop)) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))) P) as proof of (P0 (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy)))))
% Found (((eq_ref (a->Prop)) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))) P) as proof of (P0 (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy)))))
% Found eta_expansion0000:=(eta_expansion000 P):((P (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy)))))->(P (fun (x0:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) x0))))))
% Found (eta_expansion000 P) as proof of (P0 (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy)))))
% Found ((eta_expansion00 (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))) P) as proof of (P0 (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy)))))
% Found (((eta_expansion0 Prop) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))) P) as proof of (P0 (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy)))))
% Found ((((eta_expansion a) Prop) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))) P) as proof of (P0 (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy)))))
% Found ((((eta_expansion a) Prop) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))) P) as proof of (P0 (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy)))))
% Found eq_ref000:=(eq_ref00 P):((P (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy)))))->(P (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))))
% Found (eq_ref00 P) as proof of (P0 (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy)))))
% Found ((eq_ref0 (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))) P) as proof of (P0 (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy)))))
% Found (((eq_ref (a->Prop)) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))) P) as proof of (P0 (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy)))))
% Found (((eq_ref (a->Prop)) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))) P) as proof of (P0 (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy)))))
% Found eq_ref000:=(eq_ref00 P):((P (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy)))))->(P (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))))
% Found (eq_ref00 P) as proof of (P0 (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy)))))
% Found ((eq_ref0 (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))) P) as proof of (P0 (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy)))))
% Found (((eq_ref (a->Prop)) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))) P) as proof of (P0 (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy)))))
% Found (((eq_ref (a->Prop)) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))) P) as proof of (P0 (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy)))))
% Found eta_expansion0000:=(eta_expansion000 P):((P (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy)))))->(P (fun (x0:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) x0))))))
% Found (eta_expansion000 P) as proof of (P0 (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy)))))
% Found ((eta_expansion00 (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))) P) as proof of (P0 (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy)))))
% Found (((eta_expansion0 Prop) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))) P) as proof of (P0 (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy)))))
% Found ((((eta_expansion a) Prop) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))) P) as proof of (P0 (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy)))))
% Found ((((eta_expansion a) Prop) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))) P) as proof of (P0 (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy)))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy)))))
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy)))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy)))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy)))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy)))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))):(((eq (a->Prop)) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))) (fun (x0:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) x0)))))
% Found (eta_expansion_dep00 (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))) b)
% Found eta_expansion000:=(eta_expansion00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy)))))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy)))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy)))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy)))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy)))))
% Found eta_expansion000:=(eta_expansion00 (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))):(((eq (a->Prop)) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))) (fun (x0:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) x0)))))
% Found (eta_expansion00 (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))) b)
% Found ((eta_expansion0 Prop) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))) b)
% Found (((eta_expansion a) Prop) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))) b)
% Found (((eta_expansion a) Prop) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))) b)
% Found (((eta_expansion a) Prop) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))) b)
% Found eta_expansion000:=(eta_expansion00 (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))):(((eq (a->Prop)) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))) (fun (x0:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) x0)))))
% Found (eta_expansion00 (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))) b)
% Found ((eta_expansion0 Prop) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))) b)
% Found (((eta_expansion a) Prop) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))) b)
% Found (((eta_expansion a) Prop) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))) b)
% Found (((eta_expansion a) Prop) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy)))))
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy)))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy)))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy)))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy)))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))):(((eq (a->Prop)) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))) (fun (x0:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) x0)))))
% Found (eta_expansion_dep00 (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy)))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy)))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy)))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy)))))
% Found eq_ref00:=(eq_ref0 (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) x0)))):(((eq Prop) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) x0)))) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) x0))))
% Found (eq_ref0 (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) x0)))) as proof of (((eq Prop) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) x0)))) b)
% Found ((eq_ref Prop) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) x0)))) as proof of (((eq Prop) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) x0)))) b)
% Found ((eq_ref Prop) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) x0)))) as proof of (((eq Prop) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) x0)))) b)
% Found ((eq_ref Prop) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) x0)))) as proof of (((eq Prop) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) x0)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) x0))))
% Found eq_ref00:=(eq_ref0 (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) x0)))):(((eq Prop) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) x0)))) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) x0))))
% Found (eq_ref0 (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) x0)))) as proof of (((eq Prop) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) x0)))) b)
% Found ((eq_ref Prop) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) x0)))) as proof of (((eq Prop) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) x0)))) b)
% Found ((eq_ref Prop) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) x0)))) as proof of (((eq Prop) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) x0)))) b)
% Found ((eq_ref Prop) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) x0)))) as proof of (((eq Prop) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) x0)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) x0))))
% Found eta_expansion0000:=(eta_expansion000 P):((P (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy)))))->(P (fun (x0:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) x0))))))
% Found (eta_expansion000 P) as proof of (P0 (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy)))))
% Found ((eta_expansion00 (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))) P) as proof of (P0 (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy)))))
% Found (((eta_expansion0 Prop) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))) P) as proof of (P0 (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy)))))
% Found ((((eta_expansion a) Prop) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))) P) as proof of (P0 (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy)))))
% Found ((((eta_expansion a) Prop) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))) P) as proof of (P0 (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy)))))
% Found eta_expansion0000:=(eta_expansion000 P):((P (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy)))))->(P (fun (x0:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) x0))))))
% Found (eta_expansion000 P) as proof of (P0 (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy)))))
% Found ((eta_expansion00 (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))) P) as proof of (P0 (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy)))))
% Found (((eta_expansion0 Prop) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))) P) as proof of (P0 (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy)))))
% Found ((((eta_expansion a) Prop) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))) P) as proof of (P0 (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy)))))
% Found ((((eta_expansion a) Prop) (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp0 Xx2) Xy2)))) (PROP Xp0))->((Xp0 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (PROP Xp))->((Xp x) Xy))))) P) as proof of (P0 (fun (Xy:a)=> (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun
% EOF
%------------------------------------------------------------------------------