%------------------------------------------------------------------------------
% File     : cocATP---0.2.0
% Problem  : SEV123^5 : TPTP v6.1.0. Released v4.0.0.
% Transfm  : none
% Format   : tptp:raw
% Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p

% Computer : n109.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 300.04s
% Output   : None 
% Verified : 
% SZS Type : None (Parsing solution fails)
% Syntax   : Number of formulae    : 0

% Comments : 
%------------------------------------------------------------------------------
%----NO SOLUTION OUTPUT BY SYSTEM
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% % Problem  : SEV123^5 : TPTP v6.1.0. Released v4.0.0.
% % Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n109.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 08:09:41 CDT 2014
% % CPUTime  : 300.04 
% 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 0xca6ab8>, <kernel.Type object at 0xca6098>) 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)) (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)))->(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))))) of role conjecture named cTHM254_B_pme
% Conjecture to prove = (forall (PROP:((a->(a->Prop))->Prop)) (S:((a->(a->Prop))->Prop)) (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)))->(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))))):Prop
% Parameter a_DUMMY:a.
% We need to prove ['(forall (PROP:((a->(a->Prop))->Prop)) (S:((a->(a->Prop))->Prop)) (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)))->(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)))))']
% Parameter a:Type.
% Trying to prove (forall (PROP:((a->(a->Prop))->Prop)) (S:((a->(a->Prop))->Prop)) (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)))->(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:(PROP Xp)
% Found x2 as proof of (PROP Xp)
% Found x3:(PROP Xp)
% Found x3 as proof of (PROP Xp)
% Found eq_ref000:=(eq_ref00 PROP):((PROP Xp)->(PROP Xp))
% Found (eq_ref00 PROP) as proof of ((PROP Xp)->(PROP Xp))
% Found ((eq_ref0 Xp) PROP) as proof of ((PROP Xp)->(PROP Xp))
% Found (((eq_ref (a->(a->Prop))) Xp) PROP) as proof of ((PROP Xp)->(PROP Xp))
% Found (fun (x2:(forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0))))=> (((eq_ref (a->(a->Prop))) Xp) PROP)) as proof of ((PROP Xp)->(PROP Xp))
% Found (fun (x2:(forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0))))=> (((eq_ref (a->(a->Prop))) Xp) PROP)) as proof of ((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)->(PROP Xp)))
% Found (and_rect00 (fun (x2:(forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0))))=> (((eq_ref (a->(a->Prop))) Xp) PROP))) as proof of (PROP Xp)
% Found ((and_rect0 (PROP Xp)) (fun (x2:(forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0))))=> (((eq_ref (a->(a->Prop))) Xp) PROP))) as proof of (PROP Xp)
% Found (((fun (P:Type) (x2:((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)->P)))=> (((((and_rect (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)) P) x2) x0)) (PROP Xp)) (fun (x2:(forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0))))=> (((eq_ref (a->(a->Prop))) Xp) PROP))) as proof of (PROP Xp)
% Found (((fun (P:Type) (x2:((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)->P)))=> (((((and_rect (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)) P) x2) x0)) (PROP Xp)) (fun (x2:(forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0))))=> (((eq_ref (a->(a->Prop))) Xp) PROP))) as proof of (PROP Xp)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found eq_ref00:=(eq_ref0 (PROP (fun (x2:a) (x10:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp0 Xx0) Xy0)))) (PROP Xp0))->((Xp0 x2) x10)))))):(((eq Prop) (PROP (fun (x2:a) (x10:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp0 Xx0) Xy0)))) (PROP Xp0))->((Xp0 x2) x10)))))) (PROP (fun (x2:a) (x10:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp0 Xx0) Xy0)))) (PROP Xp0))->((Xp0 x2) x10))))))
% Found (eq_ref0 (PROP (fun (x2:a) (x10:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp0 Xx0) Xy0)))) (PROP Xp0))->((Xp0 x2) x10)))))) as proof of (((eq Prop) (PROP (fun (x2:a) (x10:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp0 Xx0) Xy0)))) (PROP Xp0))->((Xp0 x2) x10)))))) b)
% Found ((eq_ref Prop) (PROP (fun (x2:a) (x10:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp0 Xx0) Xy0)))) (PROP Xp0))->((Xp0 x2) x10)))))) as proof of (((eq Prop) (PROP (fun (x2:a) (x10:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp0 Xx0) Xy0)))) (PROP Xp0))->((Xp0 x2) x10)))))) b)
% Found ((eq_ref Prop) (PROP (fun (x2:a) (x10:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp0 Xx0) Xy0)))) (PROP Xp0))->((Xp0 x2) x10)))))) as proof of (((eq Prop) (PROP (fun (x2:a) (x10:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp0 Xx0) Xy0)))) (PROP Xp0))->((Xp0 x2) x10)))))) b)
% Found ((eq_ref Prop) (PROP (fun (x2:a) (x10:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp0 Xx0) Xy0)))) (PROP Xp0))->((Xp0 x2) x10)))))) as proof of (((eq Prop) (PROP (fun (x2:a) (x10:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp0 Xx0) Xy0)))) (PROP Xp0))->((Xp0 x2) x10)))))) b)
% Found eq_ref00:=(eq_ref0 (PROP (fun (x2:a) (x10:a)=> (((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 x2) x10))))):(((eq Prop) (PROP (fun (x2:a) (x10:a)=> (((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 x2) x10))))) (PROP (fun (x2:a) (x10:a)=> (((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 x2) x10)))))
% Found (eq_ref0 (PROP (fun (x2:a) (x10:a)=> (((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 x2) x10))))) as proof of (((eq Prop) (PROP (fun (x2:a) (x10:a)=> (((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 x2) x10))))) b)
% Found ((eq_ref Prop) (PROP (fun (x2:a) (x10:a)=> (((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 x2) x10))))) as proof of (((eq Prop) (PROP (fun (x2:a) (x10:a)=> (((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 x2) x10))))) b)
% Found ((eq_ref Prop) (PROP (fun (x2:a) (x10:a)=> (((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 x2) x10))))) as proof of (((eq Prop) (PROP (fun (x2:a) (x10:a)=> (((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 x2) x10))))) b)
% Found ((eq_ref Prop) (PROP (fun (x2:a) (x10:a)=> (((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 x2) x10))))) as proof of (((eq Prop) (PROP (fun (x2:a) (x10:a)=> (((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 x2) x10))))) b)
% Found eq_ref00:=(eq_ref0 (PROP Xp)):(((eq Prop) (PROP Xp)) (PROP Xp))
% Found (eq_ref0 (PROP Xp)) as proof of (((eq Prop) (PROP Xp)) b)
% Found ((eq_ref Prop) (PROP Xp)) as proof of (((eq Prop) (PROP Xp)) b)
% Found ((eq_ref Prop) (PROP Xp)) as proof of (((eq Prop) (PROP Xp)) b)
% Found ((eq_ref Prop) (PROP Xp)) as proof of (((eq Prop) (PROP Xp)) b)
% Found eq_ref00:=(eq_ref0 a0):(((eq (a->(a->Prop))) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq (a->(a->Prop))) a0) (fun (x2:a) (x10:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp0 Xx0) Xy0)))) (PROP Xp0))->((Xp0 x2) x10)))))
% Found ((eq_ref (a->(a->Prop))) a0) as proof of (((eq (a->(a->Prop))) a0) (fun (x2:a) (x10:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp0 Xx0) Xy0)))) (PROP Xp0))->((Xp0 x2) x10)))))
% Found ((eq_ref (a->(a->Prop))) a0) as proof of (((eq (a->(a->Prop))) a0) (fun (x2:a) (x10:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp0 Xx0) Xy0)))) (PROP Xp0))->((Xp0 x2) x10)))))
% Found ((eq_ref (a->(a->Prop))) a0) as proof of (((eq (a->(a->Prop))) a0) (fun (x2:a) (x10:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp0 Xx0) Xy0)))) (PROP Xp0))->((Xp0 x2) x10)))))
% Found eq_ref00:=(eq_ref0 a0):(((eq (a->(a->Prop))) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq (a->(a->Prop))) a0) (fun (x2:a) (x10:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp0 Xx0) Xy0)))) (PROP Xp0))->((Xp0 x2) x10)))))
% Found ((eq_ref (a->(a->Prop))) a0) as proof of (((eq (a->(a->Prop))) a0) (fun (x2:a) (x10:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp0 Xx0) Xy0)))) (PROP Xp0))->((Xp0 x2) x10)))))
% Found ((eq_ref (a->(a->Prop))) a0) as proof of (((eq (a->(a->Prop))) a0) (fun (x2:a) (x10:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp0 Xx0) Xy0)))) (PROP Xp0))->((Xp0 x2) x10)))))
% Found ((eq_ref (a->(a->Prop))) a0) as proof of (((eq (a->(a->Prop))) a0) (fun (x2:a) (x10:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp0 Xx0) Xy0)))) (PROP Xp0))->((Xp0 x2) x10)))))
% Found iff_sym:=(fun (A:Prop) (B:Prop) (H:((iff A) B))=> ((((conj (B->A)) (A->B)) (((proj2 (A->B)) (B->A)) H)) (((proj1 (A->B)) (B->A)) H))):(forall (A:Prop) (B:Prop), (((iff A) B)->((iff B) A)))
% Instantiate: b:=(forall (A:Prop) (B:Prop), (((iff A) B)->((iff B) A))):Prop
% Found iff_sym as proof of b
% Found eq_ref00:=(eq_ref0 a0):(((eq (a->(a->Prop))) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq (a->(a->Prop))) a0) (fun (x2:a) (x10:a)=> (((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 x2) x10))))
% Found ((eq_ref (a->(a->Prop))) a0) as proof of (((eq (a->(a->Prop))) a0) (fun (x2:a) (x10:a)=> (((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 x2) x10))))
% Found ((eq_ref (a->(a->Prop))) a0) as proof of (((eq (a->(a->Prop))) a0) (fun (x2:a) (x10:a)=> (((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 x2) x10))))
% Found ((eq_ref (a->(a->Prop))) a0) as proof of (((eq (a->(a->Prop))) a0) (fun (x2:a) (x10:a)=> (((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 x2) x10))))
% Found eq_ref00:=(eq_ref0 a0):(((eq (a->(a->Prop))) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq (a->(a->Prop))) a0) (fun (x2:a) (x10:a)=> (((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 x2) x10))))
% Found ((eq_ref (a->(a->Prop))) a0) as proof of (((eq (a->(a->Prop))) a0) (fun (x2:a) (x10:a)=> (((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 x2) x10))))
% Found ((eq_ref (a->(a->Prop))) a0) as proof of (((eq (a->(a->Prop))) a0) (fun (x2:a) (x10:a)=> (((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 x2) x10))))
% Found ((eq_ref (a->(a->Prop))) a0) as proof of (((eq (a->(a->Prop))) a0) (fun (x2:a) (x10:a)=> (((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 x2) x10))))
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found iff_sym:=(fun (A:Prop) (B:Prop) (H:((iff A) B))=> ((((conj (B->A)) (A->B)) (((proj2 (A->B)) (B->A)) H)) (((proj1 (A->B)) (B->A)) H))):(forall (A:Prop) (B:Prop), (((iff A) B)->((iff B) A)))
% Instantiate: b:=(forall (A:Prop) (B:Prop), (((iff A) B)->((iff B) A))):Prop
% Found iff_sym as proof of b
% Found eq_ref00:=(eq_ref0 a0):(((eq (a->(a->Prop))) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq (a->(a->Prop))) a0) Xp)
% Found ((eq_ref (a->(a->Prop))) a0) as proof of (((eq (a->(a->Prop))) a0) Xp)
% Found ((eq_ref (a->(a->Prop))) a0) as proof of (((eq (a->(a->Prop))) a0) Xp)
% Found ((eq_ref (a->(a->Prop))) a0) as proof of (((eq (a->(a->Prop))) a0) Xp)
% Found eq_ref00:=(eq_ref0 a0):(((eq (a->(a->Prop))) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq (a->(a->Prop))) a0) Xp)
% Found ((eq_ref (a->(a->Prop))) a0) as proof of (((eq (a->(a->Prop))) a0) Xp)
% Found ((eq_ref (a->(a->Prop))) a0) as proof of (((eq (a->(a->Prop))) a0) Xp)
% Found ((eq_ref (a->(a->Prop))) a0) as proof of (((eq (a->(a->Prop))) a0) Xp)
% Found eq_ref00:=(eq_ref0 a0):(((eq (a->(a->Prop))) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq (a->(a->Prop))) a0) Xp)
% Found ((eq_ref (a->(a->Prop))) a0) as proof of (((eq (a->(a->Prop))) a0) Xp)
% Found ((eq_ref (a->(a->Prop))) a0) as proof of (((eq (a->(a->Prop))) a0) Xp)
% Found ((eq_ref (a->(a->Prop))) a0) as proof of (((eq (a->(a->Prop))) a0) Xp)
% Found eq_ref00:=(eq_ref0 (PROP (fun (x3:a) (x20:a)=> ((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 x3) x20)))))):(((eq Prop) (PROP (fun (x3:a) (x20:a)=> ((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 x3) x20)))))) (PROP (fun (x3:a) (x20:a)=> ((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 x3) x20))))))
% Found (eq_ref0 (PROP (fun (x3:a) (x20:a)=> ((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 x3) x20)))))) as proof of (((eq Prop) (PROP (fun (x3:a) (x20:a)=> ((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 x3) x20)))))) b)
% Found ((eq_ref Prop) (PROP (fun (x3:a) (x20:a)=> ((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 x3) x20)))))) as proof of (((eq Prop) (PROP (fun (x3:a) (x20:a)=> ((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 x3) x20)))))) b)
% Found ((eq_ref Prop) (PROP (fun (x3:a) (x20:a)=> ((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 x3) x20)))))) as proof of (((eq Prop) (PROP (fun (x3:a) (x20:a)=> ((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 x3) x20)))))) b)
% Found ((eq_ref Prop) (PROP (fun (x3:a) (x20:a)=> ((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 x3) x20)))))) as proof of (((eq Prop) (PROP (fun (x3:a) (x20:a)=> ((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 x3) x20)))))) b)
% Found iff_refl:=(fun (A:Prop)=> ((((conj (A->A)) (A->A)) (fun (H:A)=> H)) (fun (H:A)=> H))):(forall (P:Prop), ((iff P) P))
% Instantiate: b:=(forall (P:Prop), ((iff P) P)):Prop
% Found iff_refl as proof of b
% Found eq_ref00:=(eq_ref0 (PROP (fun (x4:a) (x30:a)=> ((PROP Xp)->((Xp x4) x30))))):(((eq Prop) (PROP (fun (x4:a) (x30:a)=> ((PROP Xp)->((Xp x4) x30))))) (PROP (fun (x4:a) (x30:a)=> ((PROP Xp)->((Xp x4) x30)))))
% Found (eq_ref0 (PROP (fun (x4:a) (x30:a)=> ((PROP Xp)->((Xp x4) x30))))) as proof of (((eq Prop) (PROP (fun (x4:a) (x30:a)=> ((PROP Xp)->((Xp x4) x30))))) b)
% Found ((eq_ref Prop) (PROP (fun (x4:a) (x30:a)=> ((PROP Xp)->((Xp x4) x30))))) as proof of (((eq Prop) (PROP (fun (x4:a) (x30:a)=> ((PROP Xp)->((Xp x4) x30))))) b)
% Found ((eq_ref Prop) (PROP (fun (x4:a) (x30:a)=> ((PROP Xp)->((Xp x4) x30))))) as proof of (((eq Prop) (PROP (fun (x4:a) (x30:a)=> ((PROP Xp)->((Xp x4) x30))))) b)
% Found ((eq_ref Prop) (PROP (fun (x4:a) (x30:a)=> ((PROP Xp)->((Xp x4) x30))))) as proof of (((eq Prop) (PROP (fun (x4:a) (x30:a)=> ((PROP Xp)->((Xp x4) x30))))) b)
% Found eq_ref00:=(eq_ref0 (PROP Xp)):(((eq Prop) (PROP Xp)) (PROP Xp))
% Found (eq_ref0 (PROP Xp)) as proof of (((eq Prop) (PROP Xp)) b)
% Found ((eq_ref Prop) (PROP Xp)) as proof of (((eq Prop) (PROP Xp)) b)
% Found ((eq_ref Prop) (PROP Xp)) as proof of (((eq Prop) (PROP Xp)) b)
% Found ((eq_ref Prop) (PROP Xp)) as proof of (((eq Prop) (PROP Xp)) b)
% Found eq_ref00:=(eq_ref0 (PROP Xp)):(((eq Prop) (PROP Xp)) (PROP Xp))
% Found (eq_ref0 (PROP Xp)) as proof of (((eq Prop) (PROP Xp)) b)
% Found ((eq_ref Prop) (PROP Xp)) as proof of (((eq Prop) (PROP Xp)) b)
% Found ((eq_ref Prop) (PROP Xp)) as proof of (((eq Prop) (PROP Xp)) b)
% Found ((eq_ref Prop) (PROP Xp)) as proof of (((eq Prop) (PROP Xp)) b)
% Found eq_ref00:=(eq_ref0 (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx) Xy)))):(((eq ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx) Xy)))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx) Xy))))
% Found (eq_ref0 (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx) Xy)))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx) Xy)))) b)
% Found ((eq_ref ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx) Xy)))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx) Xy)))) b)
% Found ((eq_ref ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx) Xy)))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx) Xy)))) b)
% Found ((eq_ref ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx) Xy)))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx) Xy)))) b)
% Found eq_ref00:=(eq_ref0 a0):(((eq (a->(a->Prop))) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq (a->(a->Prop))) a0) (fun (x3:a) (x20:a)=> ((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 x3) x20)))))
% Found ((eq_ref (a->(a->Prop))) a0) as proof of (((eq (a->(a->Prop))) a0) (fun (x3:a) (x20:a)=> ((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 x3) x20)))))
% Found ((eq_ref (a->(a->Prop))) a0) as proof of (((eq (a->(a->Prop))) a0) (fun (x3:a) (x20:a)=> ((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 x3) x20)))))
% Found ((eq_ref (a->(a->Prop))) a0) as proof of (((eq (a->(a->Prop))) a0) (fun (x3:a) (x20:a)=> ((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 x3) x20)))))
% Found eq_ref00:=(eq_ref0 a0):(((eq (a->(a->Prop))) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq (a->(a->Prop))) a0) (fun (x3:a) (x20:a)=> ((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 x3) x20)))))
% Found ((eq_ref (a->(a->Prop))) a0) as proof of (((eq (a->(a->Prop))) a0) (fun (x3:a) (x20:a)=> ((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 x3) x20)))))
% Found ((eq_ref (a->(a->Prop))) a0) as proof of (((eq (a->(a->Prop))) a0) (fun (x3:a) (x20:a)=> ((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 x3) x20)))))
% Found ((eq_ref (a->(a->Prop))) a0) as proof of (((eq (a->(a->Prop))) a0) (fun (x3:a) (x20:a)=> ((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 x3) x20)))))
% Found eq_sym:=(fun (T:Type) (a:T) (b:T) (H:(((eq T) a) b))=> ((H (fun (x:T)=> (((eq T) x) a))) ((eq_ref T) a))):(forall (T:Type) (a:T) (b:T), ((((eq T) a) b)->(((eq T) b) a)))
% Instantiate: b:=(forall (T:Type) (a:T) (b:T), ((((eq T) a) b)->(((eq T) b) a))):Prop
% Found eq_sym as proof of b
% Found eq_ref00:=(eq_ref0 a0):(((eq (a->(a->Prop))) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq (a->(a->Prop))) a0) (fun (x4:a) (x30:a)=> ((PROP Xp)->((Xp x4) x30))))
% Found ((eq_ref (a->(a->Prop))) a0) as proof of (((eq (a->(a->Prop))) a0) (fun (x4:a) (x30:a)=> ((PROP Xp)->((Xp x4) x30))))
% Found ((eq_ref (a->(a->Prop))) a0) as proof of (((eq (a->(a->Prop))) a0) (fun (x4:a) (x30:a)=> ((PROP Xp)->((Xp x4) x30))))
% Found ((eq_ref (a->(a->Prop))) a0) as proof of (((eq (a->(a->Prop))) a0) (fun (x4:a) (x30:a)=> ((PROP Xp)->((Xp x4) x30))))
% Found eq_ref00:=(eq_ref0 a0):(((eq (a->(a->Prop))) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq (a->(a->Prop))) a0) (fun (x4:a) (x30:a)=> ((PROP Xp)->((Xp x4) x30))))
% Found ((eq_ref (a->(a->Prop))) a0) as proof of (((eq (a->(a->Prop))) a0) (fun (x4:a) (x30:a)=> ((PROP Xp)->((Xp x4) x30))))
% Found ((eq_ref (a->(a->Prop))) a0) as proof of (((eq (a->(a->Prop))) a0) (fun (x4:a) (x30:a)=> ((PROP Xp)->((Xp x4) x30))))
% Found ((eq_ref (a->(a->Prop))) a0) as proof of (((eq (a->(a->Prop))) a0) (fun (x4:a) (x30:a)=> ((PROP Xp)->((Xp x4) x30))))
% Found eq_ref00:=(eq_ref0 Xy0):(((eq a) Xy0) Xy0)
% Found (eq_ref0 Xy0) as proof of (((eq a) Xy0) b)
% Found ((eq_ref a) Xy0) as proof of (((eq a) Xy0) b)
% Found ((eq_ref a) Xy0) as proof of (((eq a) Xy0) b)
% Found ((eq_ref a) Xy0) as proof of (((eq a) Xy0) b)
% Found eta_expansion_dep0000:=(eta_expansion_dep000 (ex (a->(a->Prop)))):(((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))))->((ex (a->(a->Prop))) (fun (x:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) x) (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))))))))) ((x Xx0) Xy0)))))
% Found (eta_expansion_dep000 (ex (a->(a->Prop)))) as proof of (P (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))))
% Found ((eta_expansion_dep00 (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)))) (ex (a->(a->Prop)))) as proof of (P (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))))
% Found (((eta_expansion_dep0 (fun (x5:(a->(a->Prop)))=> 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)))) (ex (a->(a->Prop)))) as proof of (P (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))))
% Found ((((eta_expansion_dep (a->(a->Prop))) (fun (x5:(a->(a->Prop)))=> 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)))) (ex (a->(a->Prop)))) as proof of (P (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))))
% Found ((((eta_expansion_dep (a->(a->Prop))) (fun (x5:(a->(a->Prop)))=> 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)))) (ex (a->(a->Prop)))) as proof of (P (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))))
% Found conj:(forall (A:Prop) (B:Prop), (A->(B->((and A) B))))
% Instantiate: b:=(forall (A:Prop) (B:Prop), (A->(B->((and A) B)))):Prop
% Found conj as proof of b
% Found eq_ref00:=(eq_ref0 a0):(((eq (a->(a->Prop))) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq (a->(a->Prop))) a0) Xp)
% Found ((eq_ref (a->(a->Prop))) a0) as proof of (((eq (a->(a->Prop))) a0) Xp)
% Found ((eq_ref (a->(a->Prop))) a0) as proof of (((eq (a->(a->Prop))) a0) Xp)
% Found ((eq_ref (a->(a->Prop))) a0) as proof of (((eq (a->(a->Prop))) a0) Xp)
% Found eq_ref00:=(eq_ref0 a0):(((eq (a->(a->Prop))) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq (a->(a->Prop))) a0) Xp)
% Found ((eq_ref (a->(a->Prop))) a0) as proof of (((eq (a->(a->Prop))) a0) Xp)
% Found ((eq_ref (a->(a->Prop))) a0) as proof of (((eq (a->(a->Prop))) a0) Xp)
% Found ((eq_ref (a->(a->Prop))) a0) as proof of (((eq (a->(a->Prop))) a0) Xp)
% Found eq_ref00:=(eq_ref0 a0):(((eq (a->(a->Prop))) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq (a->(a->Prop))) a0) Xp)
% Found ((eq_ref (a->(a->Prop))) a0) as proof of (((eq (a->(a->Prop))) a0) Xp)
% Found ((eq_ref (a->(a->Prop))) a0) as proof of (((eq (a->(a->Prop))) a0) Xp)
% Found ((eq_ref (a->(a->Prop))) a0) as proof of (((eq (a->(a->Prop))) a0) Xp)
% Found x2:(PROP Xp)
% Instantiate: x3:=Xp:(a->(a->Prop))
% Found x2 as proof of (PROP x3)
% Found or_ind:(forall (A:Prop) (B:Prop) (P:Prop), ((A->P)->((B->P)->(((or A) B)->P))))
% Instantiate: b:=(forall (A:Prop) (B:Prop) (P:Prop), ((A->P)->((B->P)->(((or A) B)->P)))):Prop
% Found or_ind as proof of b
% Found eq_ref00:=(eq_ref0 (f x3)):(((eq Prop) (f x3)) (f x3))
% Found (eq_ref0 (f x3)) as proof of (((eq Prop) (f x3)) ((and (S x3)) ((x3 Xx) Xy)))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((and (S x3)) ((x3 Xx) Xy)))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((and (S x3)) ((x3 Xx) Xy)))
% Found (fun (x3:(a->(a->Prop)))=> ((eq_ref Prop) (f x3))) as proof of (((eq Prop) (f x3)) ((and (S x3)) ((x3 Xx) Xy)))
% Found (fun (x3:(a->(a->Prop)))=> ((eq_ref Prop) (f x3))) as proof of (forall (x:(a->(a->Prop))), (((eq Prop) (f x)) ((and (S x)) ((x Xx) Xy))))
% Found eq_ref00:=(eq_ref0 (f x3)):(((eq Prop) (f x3)) (f x3))
% Found (eq_ref0 (f x3)) as proof of (((eq Prop) (f x3)) ((and (S x3)) ((x3 Xx) Xy)))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((and (S x3)) ((x3 Xx) Xy)))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((and (S x3)) ((x3 Xx) Xy)))
% Found (fun (x3:(a->(a->Prop)))=> ((eq_ref Prop) (f x3))) as proof of (((eq Prop) (f x3)) ((and (S x3)) ((x3 Xx) Xy)))
% Found (fun (x3:(a->(a->Prop)))=> ((eq_ref Prop) (f x3))) as proof of (forall (x:(a->(a->Prop))), (((eq Prop) (f x)) ((and (S x)) ((x Xx) Xy))))
% Found eq_ref00:=(eq_ref0 a0):(((eq (a->(a->Prop))) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq (a->(a->Prop))) a0) Xp)
% Found ((eq_ref (a->(a->Prop))) a0) as proof of (((eq (a->(a->Prop))) a0) Xp)
% Found ((eq_ref (a->(a->Prop))) a0) as proof of (((eq (a->(a->Prop))) a0) Xp)
% Found ((eq_ref (a->(a->Prop))) a0) as proof of (((eq (a->(a->Prop))) a0) Xp)
% Found eq_ref00:=(eq_ref0 a0):(((eq (a->(a->Prop))) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq (a->(a->Prop))) a0) Xp)
% Found ((eq_ref (a->(a->Prop))) a0) as proof of (((eq (a->(a->Prop))) a0) Xp)
% Found ((eq_ref (a->(a->Prop))) a0) as proof of (((eq (a->(a->Prop))) a0) Xp)
% Found ((eq_ref (a->(a->Prop))) a0) as proof of (((eq (a->(a->Prop))) a0) Xp)
% Found eq_ref00:=(eq_ref0 a0):(((eq (a->(a->Prop))) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq (a->(a->Prop))) a0) Xp)
% Found ((eq_ref (a->(a->Prop))) a0) as proof of (((eq (a->(a->Prop))) a0) Xp)
% Found ((eq_ref (a->(a->Prop))) a0) as proof of (((eq (a->(a->Prop))) a0) Xp)
% Found ((eq_ref (a->(a->Prop))) a0) as proof of (((eq (a->(a->Prop))) a0) Xp)
% Found eq_ref00:=(eq_ref0 (PROP (fun (x5:a) (x40:a)=> ((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R x5) x40))))))):(((eq Prop) (PROP (fun (x5:a) (x40:a)=> ((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R x5) x40))))))) (PROP (fun (x5:a) (x40:a)=> ((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R x5) x40)))))))
% Found (eq_ref0 (PROP (fun (x5:a) (x40:a)=> ((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R x5) x40))))))) as proof of (((eq Prop) (PROP (fun (x5:a) (x40:a)=> ((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R x5) x40))))))) b)
% Found ((eq_ref Prop) (PROP (fun (x5:a) (x40:a)=> ((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R x5) x40))))))) as proof of (((eq Prop) (PROP (fun (x5:a) (x40:a)=> ((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R x5) x40))))))) b)
% Found ((eq_ref Prop) (PROP (fun (x5:a) (x40:a)=> ((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R x5) x40))))))) as proof of (((eq Prop) (PROP (fun (x5:a) (x40:a)=> ((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R x5) x40))))))) b)
% Found ((eq_ref Prop) (PROP (fun (x5:a) (x40:a)=> ((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R x5) x40))))))) as proof of (((eq Prop) (PROP (fun (x5:a) (x40:a)=> ((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R x5) x40))))))) b)
% Found x2:(PROP Xp)
% Found x2 as proof of b
% Found or_ind:(forall (A:Prop) (B:Prop) (P:Prop), ((A->P)->((B->P)->(((or A) B)->P))))
% Instantiate: b:=(forall (A:Prop) (B:Prop) (P:Prop), ((A->P)->((B->P)->(((or A) B)->P)))):Prop
% Found or_ind as proof of b
% Found eq_ref000:=(eq_ref00 (ex (a->(a->Prop)))):(((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))))->((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)))))
% Found (eq_ref00 (ex (a->(a->Prop)))) as proof of (P (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))))
% Found ((eq_ref0 (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)))) (ex (a->(a->Prop)))) as proof of (P (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))))
% Found (((eq_ref ((a->(a->Prop))->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)))) (ex (a->(a->Prop)))) as proof of (P (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))))
% Found (((eq_ref ((a->(a->Prop))->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)))) (ex (a->(a->Prop)))) as proof of (P (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))))
% Found eta_expansion0000:=(eta_expansion000 (ex (a->(a->Prop)))):(((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))))->((ex (a->(a->Prop))) (fun (x:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) x) (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))))))))) ((x Xx0) Xy0)))))
% Found (eta_expansion000 (ex (a->(a->Prop)))) as proof of (P (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))))
% Found ((eta_expansion00 (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)))) (ex (a->(a->Prop)))) as proof of (P (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))))
% Found (((eta_expansion0 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)))) (ex (a->(a->Prop)))) as proof of (P (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))))
% Found ((((eta_expansion (a->(a->Prop))) 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)))) (ex (a->(a->Prop)))) as proof of (P (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))))
% Found ((((eta_expansion (a->(a->Prop))) 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)))) (ex (a->(a->Prop)))) as proof of (P (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))))
% Found x3:(PROP Xp)
% Found x3 as proof of b
% Found x20:((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))))
% Instantiate: a0:=(fun (x3:a) (x21: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 x3) x21))))):(a->(a->Prop))
% Found (fun (x20:((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)))))=> x20) as proof of ((a0 Xx0) Xy0)
% Found (fun (Xy0:a) (x20:((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)))))=> x20) as proof of (((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))))->((a0 Xx0) Xy0))
% Found (fun (Xx0:a) (Xy0:a) (x20:((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)))))=> x20) as proof of (forall (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))))->((a0 Xx0) Xy0)))
% Found (fun (Xx0:a) (Xy0:a) (x20:((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)))))=> x20) as proof of (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))))->((a0 Xx0) Xy0)))
% Found x6:((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))))
% Instantiate: a0:=(fun (x3:a) (x20: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 x3) x20))))):(a->(a->Prop))
% Found (fun (x6:((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)))))=> x6) as proof of ((a0 Xx0) Xy0)
% Found (fun (Xy0:a) (x6:((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)))))=> x6) as proof of (((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))))->((a0 Xx0) Xy0))
% Found (fun (Xx0:a) (Xy0:a) (x6:((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)))))=> x6) as proof of (forall (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))))->((a0 Xx0) Xy0)))
% Found (fun (Xx0:a) (Xy0:a) (x6:((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)))))=> x6) as proof of (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))))->((a0 Xx0) Xy0)))
% Found or_ind:(forall (A:Prop) (B:Prop) (P:Prop), ((A->P)->((B->P)->(((or A) B)->P))))
% Instantiate: b:=(forall (A:Prop) (B:Prop) (P:Prop), ((A->P)->((B->P)->(((or A) B)->P)))):Prop
% Found or_ind as proof of b
% Found x5:((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))))
% Instantiate: x3:=(fun (x7:a) (x60: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 x7) x60))))):(a->(a->Prop))
% Found (fun (x5:((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)))))=> x5) as proof of ((x3 Xx0) Xy0)
% Found (fun (Xy0:a) (x5:((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)))))=> x5) as proof of (((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))))->((x3 Xx0) Xy0))
% Found (fun (Xx0:a) (Xy0:a) (x5:((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)))))=> x5) as proof of (forall (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))))->((x3 Xx0) Xy0)))
% Found (fun (Xx0:a) (Xy0:a) (x5:((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)))))=> x5) as proof of (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))))->((x3 Xx0) Xy0)))
% Found x2:(PROP Xp)
% Instantiate: a0:=Xp:(a->(a->Prop))
% Found x2 as proof of (PROP a0)
% Found x2:(PROP Xp)
% Found x2 as proof of (PROP Xp)
% Found x2:(PROP Xp)
% Instantiate: a0:=Xp:(a->(a->Prop))
% Found x2 as proof of (PROP a0)
% Found eq_ref00:=(eq_ref0 a0):(((eq (a->(a->Prop))) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq (a->(a->Prop))) a0) (fun (x5:a) (x40:a)=> ((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R x5) x40))))))
% Found ((eq_ref (a->(a->Prop))) a0) as proof of (((eq (a->(a->Prop))) a0) (fun (x5:a) (x40:a)=> ((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R x5) x40))))))
% Found ((eq_ref (a->(a->Prop))) a0) as proof of (((eq (a->(a->Prop))) a0) (fun (x5:a) (x40:a)=> ((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R x5) x40))))))
% Found ((eq_ref (a->(a->Prop))) a0) as proof of (((eq (a->(a->Prop))) a0) (fun (x5:a) (x40:a)=> ((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R x5) x40))))))
% Found eq_ref00:=(eq_ref0 a0):(((eq (a->(a->Prop))) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq (a->(a->Prop))) a0) (fun (x5:a) (x40:a)=> ((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R x5) x40))))))
% Found ((eq_ref (a->(a->Prop))) a0) as proof of (((eq (a->(a->Prop))) a0) (fun (x5:a) (x40:a)=> ((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R x5) x40))))))
% Found ((eq_ref (a->(a->Prop))) a0) as proof of (((eq (a->(a->Prop))) a0) (fun (x5:a) (x40:a)=> ((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R x5) x40))))))
% Found ((eq_ref (a->(a->Prop))) a0) as proof of (((eq (a->(a->Prop))) a0) (fun (x5:a) (x40:a)=> ((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R x5) x40))))))
% Found x3:(PROP Xp)
% Found x3 as proof of b
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (x2:a) (x10:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp0 Xx0) Xy0)))) (PROP Xp0))->((Xp0 x2) x10))))):(((eq (a->(a->Prop))) (fun (x2:a) (x10:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp0 Xx0) Xy0)))) (PROP Xp0))->((Xp0 x2) x10))))) (fun (x:a) (x10:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp0 Xx0) Xy0)))) (PROP Xp0))->((Xp0 x) x10)))))
% Found (eta_expansion_dep00 (fun (x2:a) (x10:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp0 Xx0) Xy0)))) (PROP Xp0))->((Xp0 x2) x10))))) as proof of (((eq (a->(a->Prop))) (fun (x2:a) (x10:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp0 Xx0) Xy0)))) (PROP Xp0))->((Xp0 x2) x10))))) b)
% Found ((eta_expansion_dep0 (fun (x2:a)=> (a->Prop))) (fun (x2:a) (x10:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp0 Xx0) Xy0)))) (PROP Xp0))->((Xp0 x2) x10))))) as proof of (((eq (a->(a->Prop))) (fun (x2:a) (x10:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp0 Xx0) Xy0)))) (PROP Xp0))->((Xp0 x2) x10))))) b)
% Found (((eta_expansion_dep a) (fun (x2:a)=> (a->Prop))) (fun (x2:a) (x10:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp0 Xx0) Xy0)))) (PROP Xp0))->((Xp0 x2) x10))))) as proof of (((eq (a->(a->Prop))) (fun (x2:a) (x10:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp0 Xx0) Xy0)))) (PROP Xp0))->((Xp0 x2) x10))))) b)
% Found (((eta_expansion_dep a) (fun (x2:a)=> (a->Prop))) (fun (x2:a) (x10:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp0 Xx0) Xy0)))) (PROP Xp0))->((Xp0 x2) x10))))) as proof of (((eq (a->(a->Prop))) (fun (x2:a) (x10:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp0 Xx0) Xy0)))) (PROP Xp0))->((Xp0 x2) x10))))) b)
% Found (((eta_expansion_dep a) (fun (x2:a)=> (a->Prop))) (fun (x2:a) (x10:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp0 Xx0) Xy0)))) (PROP Xp0))->((Xp0 x2) x10))))) as proof of (((eq (a->(a->Prop))) (fun (x2:a) (x10:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp0 Xx0) Xy0)))) (PROP Xp0))->((Xp0 x2) x10))))) b)
% Found x2:(PROP Xp)
% Found x2 as proof of (PROP a0)
% Found x2:(PROP Xp)
% Found x2 as proof of (PROP Xp)
% Found x2:(PROP Xp)
% Found x2 as proof of (PROP a0)
% Found x3:(PROP Xp)
% Found x3 as proof of (PROP Xp)
% Found x3:(PROP Xp)
% Instantiate: a0:=Xp:(a->(a->Prop))
% Found x3 as proof of (PROP a0)
% Found x3:(PROP Xp)
% Instantiate: a0:=Xp:(a->(a->Prop))
% Found x3 as proof of (PROP a0)
% Found classical_choice:=(fun (A:Type) (B:Type) (R:(A->(B->Prop))) (b:B)=> ((fun (C:((forall (x:A), ((ex B) (fun (y:B)=> (((fun (x0:A) (y0:B)=> (((ex B) (fun (z:B)=> ((R x0) z)))->((R x0) y0))) x) y))))->((ex (A->B)) (fun (f:(A->B))=> (forall (x:A), (((fun (x0:A) (y:B)=> (((ex B) (fun (z:B)=> ((R x0) z)))->((R x0) y))) x) (f x)))))))=> (C (fun (x:A)=> ((fun (C0:((or ((ex B) (fun (z:B)=> ((R x) z)))) (not ((ex B) (fun (z:B)=> ((R x) z))))))=> ((((((or_ind ((ex B) (fun (z:B)=> ((R x) z)))) (not ((ex B) (fun (z:B)=> ((R x) z))))) ((ex B) (fun (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y))))) ((((ex_ind B) (fun (z:B)=> ((R x) z))) ((ex B) (fun (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y))))) (fun (y:B) (H:((R x) y))=> ((((ex_intro B) (fun (y0:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y0)))) y) (fun (_:((ex B) (fun (z:B)=> ((R x) z))))=> H))))) (fun (N:(not ((ex B) (fun (z:B)=> ((R x) z)))))=> ((((ex_intro B) (fun (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y)))) b) (fun (H:((ex B) (fun (z:B)=> ((R x) z))))=> ((False_rect ((R x) b)) (N H)))))) C0)) (classic ((ex B) (fun (z:B)=> ((R x) z)))))))) (((choice A) B) (fun (x:A) (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y)))))):(forall (A:Type) (B:Type) (R:(A->(B->Prop))), (B->((ex (A->B)) (fun (f:(A->B))=> (forall (x:A), (((ex B) (fun (y:B)=> ((R x) y)))->((R x) (f x))))))))
% Instantiate: b:=(forall (A:Type) (B:Type) (R:(A->(B->Prop))), (B->((ex (A->B)) (fun (f:(A->B))=> (forall (x:A), (((ex B) (fun (y:B)=> ((R x) y)))->((R x) (f x)))))))):Prop
% Found classical_choice as proof of b
% Found eq_ref00:=(eq_ref0 Xy0):(((eq a) Xy0) Xy0)
% Found (eq_ref0 Xy0) as proof of (((eq a) Xy0) b)
% Found ((eq_ref a) Xy0) as proof of (((eq a) Xy0) b)
% Found ((eq_ref a) Xy0) as proof of (((eq a) Xy0) b)
% Found ((eq_ref a) Xy0) as proof of (((eq a) Xy0) b)
% Found eq_ref00:=(eq_ref0 (fun (x2:a) (x10:a)=> (((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 x2) x10)))):(((eq (a->(a->Prop))) (fun (x2:a) (x10:a)=> (((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 x2) x10)))) (fun (x2:a) (x10:a)=> (((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 x2) x10))))
% Found (eq_ref0 (fun (x2:a) (x10:a)=> (((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 x2) x10)))) as proof of (((eq (a->(a->Prop))) (fun (x2:a) (x10:a)=> (((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 x2) x10)))) b)
% Found ((eq_ref (a->(a->Prop))) (fun (x2:a) (x10:a)=> (((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 x2) x10)))) as proof of (((eq (a->(a->Prop))) (fun (x2:a) (x10:a)=> (((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 x2) x10)))) b)
% Found ((eq_ref (a->(a->Prop))) (fun (x2:a) (x10:a)=> (((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 x2) x10)))) as proof of (((eq (a->(a->Prop))) (fun (x2:a) (x10:a)=> (((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 x2) x10)))) b)
% Found ((eq_ref (a->(a->Prop))) (fun (x2:a) (x10:a)=> (((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 x2) x10)))) as proof of (((eq (a->(a->Prop))) (fun (x2:a) (x10:a)=> (((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 x2) x10)))) b)
% Found eq_ref00:=(eq_ref0 Xy0):(((eq a) Xy0) Xy0)
% Found (eq_ref0 Xy0) as proof of (((eq a) Xy0) b)
% Found ((eq_ref a) Xy0) as proof of (((eq a) Xy0) b)
% Found ((eq_ref a) Xy0) as proof of (((eq a) Xy0) b)
% Found ((eq_ref a) Xy0) as proof of (((eq a) Xy0) b)
% Found x8:((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))))
% Instantiate: a0:=(fun (x5:a) (x40: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 x5) x40))))):(a->(a->Prop))
% Found (fun (x8:((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)))))=> x8) as proof of ((a0 Xx0) Xy0)
% Found (fun (Xy0:a) (x8:((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)))))=> x8) as proof of (((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))))->((a0 Xx0) Xy0))
% Found (fun (Xx0:a) (Xy0:a) (x8:((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)))))=> x8) as proof of (forall (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))))->((a0 Xx0) Xy0)))
% Found (fun (Xx0:a) (Xy0:a) (x8:((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)))))=> x8) as proof of (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))))->((a0 Xx0) Xy0)))
% Found x40:((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))))
% Instantiate: a0:=(fun (x5:a) (x41: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 x5) x41))))):(a->(a->Prop))
% Found (fun (x40:((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)))))=> x40) as proof of ((a0 Xx0) Xy0)
% Found (fun (Xy0:a) (x40:((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)))))=> x40) as proof of (((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))))->((a0 Xx0) Xy0))
% Found (fun (Xx0:a) (Xy0:a) (x40:((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)))))=> x40) as proof of (forall (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))))->((a0 Xx0) Xy0)))
% Found (fun (Xx0:a) (Xy0:a) (x40:((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)))))=> x40) as proof of (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))))->((a0 Xx0) Xy0)))
% Found ((conj00 (fun (Xx0:a) (Xy0:a) (x40:((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)))))=> x40)) x2) as proof of (P (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))))->((a0 Xx0) Xy0))))
% Found (((conj0 (PROP Xp)) (fun (Xx0:a) (Xy0:a) (x40:((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)))))=> x40)) x2) as proof of (P (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))))->((a0 Xx0) Xy0))))
% Found ((((conj (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))))->((a0 Xx0) Xy0)))) (PROP Xp)) (fun (Xx0:a) (Xy0:a) (x40:((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)))))=> x40)) x2) as proof of (P (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))))->((a0 Xx0) Xy0))))
% Found ((((conj (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))))->((a0 Xx0) Xy0)))) (PROP Xp)) (fun (Xx0:a) (Xy0:a) (x40:((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)))))=> x40)) x2) as proof of (P (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))))->((a0 Xx0) Xy0))))
% Found eq_ref00:=(eq_ref0 Xy0):(((eq a) Xy0) Xy0)
% Found (eq_ref0 Xy0) as proof of (((eq a) Xy0) b)
% Found ((eq_ref a) Xy0) as proof of (((eq a) Xy0) b)
% Found ((eq_ref a) Xy0) as proof of (((eq a) Xy0) b)
% Found ((eq_ref a) Xy0) as proof of (((eq a) Xy0) b)
% Found x3:(PROP Xp)
% Found x3 as proof of (PROP Xp)
% Found x3:(PROP Xp)
% Found x3 as proof of (PROP a0)
% Found x3:(PROP Xp)
% Found x3 as proof of (PROP a0)
% Found x8:((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))))
% Instantiate: a0:=(fun (x5:a) (x40: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 x5) x40))))):(a->(a->Prop))
% Found (fun (x8:((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)))))=> x8) as proof of ((a0 Xx0) Xy0)
% Found (fun (Xy0:a) (x8:((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)))))=> x8) as proof of (((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))))->((a0 Xx0) Xy0))
% Found (fun (Xx0:a) (Xy0:a) (x8:((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)))))=> x8) as proof of (forall (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))))->((a0 Xx0) Xy0)))
% Found (fun (Xx0:a) (Xy0:a) (x8:((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)))))=> x8) as proof of (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))))->((a0 Xx0) Xy0)))
% Found x40:((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))))
% Instantiate: a0:=(fun (x5:a) (x41: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 x5) x41))))):(a->(a->Prop))
% Found (fun (x40:((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)))))=> x40) as proof of ((a0 Xx0) Xy0)
% Found (fun (Xy0:a) (x40:((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)))))=> x40) as proof of (((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))))->((a0 Xx0) Xy0))
% Found (fun (Xx0:a) (Xy0:a) (x40:((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)))))=> x40) as proof of (forall (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))))->((a0 Xx0) Xy0)))
% Found (fun (Xx0:a) (Xy0:a) (x40:((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)))))=> x40) as proof of (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))))->((a0 Xx0) Xy0)))
% Found ((conj00 (fun (Xx0:a) (Xy0:a) (x40:((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)))))=> x40)) x3) as proof of (P (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))))->((a0 Xx0) Xy0))))
% Found (((conj0 (PROP Xp)) (fun (Xx0:a) (Xy0:a) (x40:((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)))))=> x40)) x3) as proof of (P (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))))->((a0 Xx0) Xy0))))
% Found ((((conj (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))))->((a0 Xx0) Xy0)))) (PROP Xp)) (fun (Xx0:a) (Xy0:a) (x40:((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)))))=> x40)) x3) as proof of (P (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))))->((a0 Xx0) Xy0))))
% Found ((((conj (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))))->((a0 Xx0) Xy0)))) (PROP Xp)) (fun (Xx0:a) (Xy0:a) (x40:((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)))))=> x40)) x3) as proof of (P (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))))->((a0 Xx0) Xy0))))
% Found eq_ref00:=(eq_ref0 Xy0):(((eq a) Xy0) Xy0)
% Found (eq_ref0 Xy0) as proof of (((eq a) Xy0) b)
% Found ((eq_ref a) Xy0) as proof of (((eq a) Xy0) b)
% Found ((eq_ref a) Xy0) as proof of (((eq a) Xy0) b)
% Found ((eq_ref a) Xy0) as proof of (((eq a) Xy0) b)
% Found eq_ref000:=(eq_ref00 PROP):((PROP Xp)->(PROP Xp))
% Found (eq_ref00 PROP) as proof of ((PROP Xp)->(PROP Xp))
% Found ((eq_ref0 Xp) PROP) as proof of ((PROP Xp)->(PROP Xp))
% Found (((eq_ref (a->(a->Prop))) Xp) PROP) as proof of ((PROP Xp)->(PROP Xp))
% Found (fun (x2:(forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0))))=> (((eq_ref (a->(a->Prop))) Xp) PROP)) as proof of ((PROP Xp)->(PROP Xp))
% Found (fun (x2:(forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0))))=> (((eq_ref (a->(a->Prop))) Xp) PROP)) as proof of ((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)->(PROP Xp)))
% Found (and_rect00 (fun (x2:(forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0))))=> (((eq_ref (a->(a->Prop))) Xp) PROP))) as proof of (PROP Xp)
% Found ((and_rect0 (PROP Xp)) (fun (x2:(forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0))))=> (((eq_ref (a->(a->Prop))) Xp) PROP))) as proof of (PROP Xp)
% Found (((fun (P0:Type) (x2:((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)->P0)))=> (((((and_rect (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)) P0) x2) x0)) (PROP Xp)) (fun (x2:(forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0))))=> (((eq_ref (a->(a->Prop))) Xp) PROP))) as proof of (PROP Xp)
% Found (((fun (P0:Type) (x2:((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)->P0)))=> (((((and_rect (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)) P0) x2) x0)) (PROP Xp)) (fun (x2:(forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0))))=> (((eq_ref (a->(a->Prop))) Xp) PROP))) as proof of (PROP Xp)
% Found ((conj00 (fun (Xx0:a) (Xy0:a) (x20:((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)))))=> x20)) (((fun (P0:Type) (x2:((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)->P0)))=> (((((and_rect (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)) P0) x2) x0)) (PROP Xp)) (fun (x2:(forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0))))=> (((eq_ref (a->(a->Prop))) Xp) PROP)))) as proof of (P (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))))->((a0 Xx0) Xy0))))
% Found (((conj0 (PROP Xp)) (fun (Xx0:a) (Xy0:a) (x20:((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)))))=> x20)) (((fun (P0:Type) (x2:((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)->P0)))=> (((((and_rect (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)) P0) x2) x0)) (PROP Xp)) (fun (x2:(forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0))))=> (((eq_ref (a->(a->Prop))) Xp) PROP)))) as proof of (P (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))))->((a0 Xx0) Xy0))))
% Found ((((conj (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))))->((a0 Xx0) Xy0)))) (PROP Xp)) (fun (Xx0:a) (Xy0:a) (x20:((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)))))=> x20)) (((fun (P0:Type) (x2:((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)->P0)))=> (((((and_rect (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)) P0) x2) x0)) (PROP Xp)) (fun (x2:(forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0))))=> (((eq_ref (a->(a->Prop))) Xp) PROP)))) as proof of (P (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))))->((a0 Xx0) Xy0))))
% Found ((((conj (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))))->((a0 Xx0) Xy0)))) (PROP Xp)) (fun (Xx0:a) (Xy0:a) (x20:((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)))))=> x20)) (((fun (P0:Type) (x2:((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)->P0)))=> (((((and_rect (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)) P0) x2) x0)) (PROP Xp)) (fun (x2:(forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0))))=> (((eq_ref (a->(a->Prop))) Xp) PROP)))) as proof of (P (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))))->((a0 Xx0) Xy0))))
% Found eq_ref00:=(eq_ref0 Xy0):(((eq a) Xy0) Xy0)
% Found (eq_ref0 Xy0) as proof of (((eq a) Xy0) b)
% Found ((eq_ref a) Xy0) as proof of (((eq a) Xy0) b)
% Found ((eq_ref a) Xy0) as proof of (((eq a) Xy0) b)
% Found ((eq_ref a) Xy0) as proof of (((eq a) Xy0) b)
% Found eq_ref00:=(eq_ref0 Xy0):(((eq a) Xy0) Xy0)
% Found (eq_ref0 Xy0) as proof of (((eq a) Xy0) b)
% Found ((eq_ref a) Xy0) as proof of (((eq a) Xy0) b)
% Found ((eq_ref a) Xy0) as proof of (((eq a) Xy0) b)
% Found ((eq_ref a) Xy0) as proof of (((eq a) Xy0) b)
% Found x3:(PROP Xp)
% Instantiate: a0:=Xp:(a->(a->Prop))
% Found (fun (x3:(PROP Xp))=> x3) as proof of (PROP a0)
% Found (fun (x2:(forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (x3:(PROP Xp))=> x3) as proof of ((PROP Xp)->(PROP a0))
% Found (fun (x2:(forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (x3:(PROP Xp))=> x3) as proof of ((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)->(PROP a0)))
% Found (and_rect00 (fun (x2:(forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (x3:(PROP Xp))=> x3)) as proof of (PROP a0)
% Found ((and_rect0 (PROP a0)) (fun (x2:(forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (x3:(PROP Xp))=> x3)) as proof of (PROP a0)
% Found (((fun (P0:Type) (x2:((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)->P0)))=> (((((and_rect (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)) P0) x2) x0)) (PROP a0)) (fun (x2:(forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (x3:(PROP Xp))=> x3)) as proof of (PROP a0)
% Found (((fun (P0:Type) (x2:((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)->P0)))=> (((((and_rect (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)) P0) x2) x0)) (PROP a0)) (fun (x2:(forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0)))) (x3:(PROP Xp))=> x3)) as proof of (PROP a0)
% Found x3:((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))))
% Instantiate: b:=(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))):((a->(a->Prop))->Prop)
% Found x3 as proof of (P b)
% Found eta_expansion000:=(eta_expansion00 (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0)))):(((eq ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0)))) (fun (x:(a->(a->Prop)))=> ((and (S x)) ((x Xx0) Xy0))))
% Found (eta_expansion00 (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0)))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0)))) b)
% Found ((eta_expansion0 Prop) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0)))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0)))) b)
% Found (((eta_expansion (a->(a->Prop))) Prop) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0)))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0)))) b)
% Found (((eta_expansion (a->(a->Prop))) Prop) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0)))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0)))) b)
% Found (((eta_expansion (a->(a->Prop))) Prop) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0)))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0)))) b)
% Found x3:(PROP Xp)
% Instantiate: a0:=Xp:(a->(a->Prop))
% Found x3 as proof of (PROP a0)
% Found x3:(PROP Xp)
% Instantiate: a0:=Xp:(a->(a->Prop))
% Found x3 as proof of (PROP a0)
% Found x4:((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))))
% Instantiate: b:=(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))):((a->(a->Prop))->Prop)
% Found x4 as proof of (P b)
% Found eq_ref00:=(eq_ref0 (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0)))):(((eq ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0)))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))
% Found (eq_ref0 (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0)))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0)))) b)
% Found ((eq_ref ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0)))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0)))) b)
% Found ((eq_ref ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0)))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0)))) b)
% Found ((eq_ref ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0)))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0)))) b)
% Found eq_ref00:=(eq_ref0 Xy0):(((eq a) Xy0) Xy0)
% Found (eq_ref0 Xy0) as proof of (((eq a) Xy0) b)
% Found ((eq_ref a) Xy0) as proof of (((eq a) Xy0) b)
% Found ((eq_ref a) Xy0) as proof of (((eq a) Xy0) b)
% Found ((eq_ref a) Xy0) as proof of (((eq a) Xy0) b)
% Found x3:(PROP Xp)
% Found x3 as proof of (PROP Xp)
% Found x3:(PROP Xp)
% Found x3 as proof of (PROP a0)
% Found x3:(PROP Xp)
% Found x3 as proof of (PROP a0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (f x1)):(((eq (a->Prop)) (f x1)) (fun (x:a)=> ((f x1) x)))
% Found (eta_expansion_dep00 (f x1)) as proof of (((eq (a->Prop)) (f x1)) (fun (x10:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp0 Xx0) Xy0)))) (PROP Xp0))->((Xp0 x1) x10)))))
% Found ((eta_expansion_dep0 (fun (x3:a)=> Prop)) (f x1)) as proof of (((eq (a->Prop)) (f x1)) (fun (x10:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp0 Xx0) Xy0)))) (PROP Xp0))->((Xp0 x1) x10)))))
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) (f x1)) as proof of (((eq (a->Prop)) (f x1)) (fun (x10:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp0 Xx0) Xy0)))) (PROP Xp0))->((Xp0 x1) x10)))))
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) (f x1)) as proof of (((eq (a->Prop)) (f x1)) (fun (x10:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp0 Xx0) Xy0)))) (PROP Xp0))->((Xp0 x1) x10)))))
% Found (fun (x1:a)=> (((eta_expansion_dep a) (fun (x3:a)=> Prop)) (f x1))) as proof of (((eq (a->Prop)) (f x1)) (fun (x10:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp0 Xx0) Xy0)))) (PROP Xp0))->((Xp0 x1) x10)))))
% Found (fun (x1:a)=> (((eta_expansion_dep a) (fun (x3:a)=> Prop)) (f x1))) as proof of (forall (x:a), (((eq (a->Prop)) (f x)) (fun (x10:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp0 Xx0) Xy0)))) (PROP Xp0))->((Xp0 x) x10))))))
% Found eta_expansion000:=(eta_expansion00 (f x1)):(((eq (a->Prop)) (f x1)) (fun (x:a)=> ((f x1) x)))
% Found (eta_expansion00 (f x1)) as proof of (((eq (a->Prop)) (f x1)) (fun (x10:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp0 Xx0) Xy0)))) (PROP Xp0))->((Xp0 x1) x10)))))
% Found ((eta_expansion0 Prop) (f x1)) as proof of (((eq (a->Prop)) (f x1)) (fun (x10:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp0 Xx0) Xy0)))) (PROP Xp0))->((Xp0 x1) x10)))))
% Found (((eta_expansion a) Prop) (f x1)) as proof of (((eq (a->Prop)) (f x1)) (fun (x10:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp0 Xx0) Xy0)))) (PROP Xp0))->((Xp0 x1) x10)))))
% Found (((eta_expansion a) Prop) (f x1)) as proof of (((eq (a->Prop)) (f x1)) (fun (x10:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp0 Xx0) Xy0)))) (PROP Xp0))->((Xp0 x1) x10)))))
% Found (fun (x1:a)=> (((eta_expansion a) Prop) (f x1))) as proof of (((eq (a->Prop)) (f x1)) (fun (x10:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp0 Xx0) Xy0)))) (PROP Xp0))->((Xp0 x1) x10)))))
% Found (fun (x1:a)=> (((eta_expansion a) Prop) (f x1))) as proof of (forall (x:a), (((eq (a->Prop)) (f x)) (fun (x10:a)=> (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp0 Xx0) Xy0)))) (PROP Xp0))->((Xp0 x) x10))))))
% Found x2:(PROP Xp)
% Instantiate: a0:=Xp:(a->(a->Prop))
% Found x2 as proof of (PROP a0)
% Found x2:(PROP Xp)
% Instantiate: a0:=Xp:(a->(a->Prop))
% Found x2 as proof of (PROP a0)
% Found x4:((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))))
% Instantiate: b:=(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))):((a->(a->Prop))->Prop)
% Found x4 as proof of (P b)
% Found eq_ref00:=(eq_ref0 (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0)))):(((eq ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0)))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))
% Found (eq_ref0 (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0)))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0)))) b)
% Found ((eq_ref ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0)))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0)))) b)
% Found ((eq_ref ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0)))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0)))) b)
% Found ((eq_ref ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0)))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0)))) b)
% Found eq_ref000:=(eq_ref00 PROP):((PROP Xp)->(PROP Xp))
% Found (eq_ref00 PROP) as proof of ((PROP Xp)->b)
% Found ((eq_ref0 Xp) PROP) as proof of ((PROP Xp)->b)
% Found (((eq_ref (a->(a->Prop))) Xp) PROP) as proof of ((PROP Xp)->b)
% Found (fun (x2:(forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0))))=> (((eq_ref (a->(a->Prop))) Xp) PROP)) as proof of ((PROP Xp)->b)
% Found (fun (x2:(forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0))))=> (((eq_ref (a->(a->Prop))) Xp) PROP)) as proof of ((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)->b))
% Found (and_rect00 (fun (x2:(forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0))))=> (((eq_ref (a->(a->Prop))) Xp) PROP))) as proof of b
% Found ((and_rect0 b) (fun (x2:(forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0))))=> (((eq_ref (a->(a->Prop))) Xp) PROP))) as proof of b
% Found (((fun (P0:Type) (x2:((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)->P0)))=> (((((and_rect (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)) P0) x2) x0)) b) (fun (x2:(forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0))))=> (((eq_ref (a->(a->Prop))) Xp) PROP))) as proof of b
% Found (((fun (P0:Type) (x2:((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)->P0)))=> (((((and_rect (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)) P0) x2) x0)) b) (fun (x2:(forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0))))=> (((eq_ref (a->(a->Prop))) Xp) PROP))) as proof of b
% Found x6:((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))))
% Instantiate: a0:=(fun (x5:a) (x40: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 x5) x40))))):(a->(a->Prop))
% Found (fun (x6:((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)))))=> x6) as proof of ((a0 Xx0) Xy0)
% Found (fun (Xy0:a) (x6:((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)))))=> x6) as proof of (((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))))->((a0 Xx0) Xy0))
% Found (fun (Xx0:a) (Xy0:a) (x6:((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)))))=> x6) as proof of (forall (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))))->((a0 Xx0) Xy0)))
% Found (fun (Xx0:a) (Xy0:a) (x6:((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)))))=> x6) as proof of (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))))->((a0 Xx0) Xy0)))
% Found x3:((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))))
% Instantiate: f:=(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))):((a->(a->Prop))->Prop)
% Found x3 as proof of (P f)
% Found x3:((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))))
% Instantiate: f:=(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))):((a->(a->Prop))->Prop)
% Found x3 as proof of (P f)
% Found x4:((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))))
% Instantiate: f:=(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))):((a->(a->Prop))->Prop)
% Found x4 as proof of (P f)
% Found x4:((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))))
% Instantiate: f:=(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))):((a->(a->Prop))->Prop)
% Found x4 as proof of (P f)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (f x1)):(((eq (a->Prop)) (f x1)) (fun (x:a)=> ((f x1) x)))
% Found (eta_expansion_dep00 (f x1)) as proof of (((eq (a->Prop)) (f x1)) (fun (x10:a)=> (((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 x1) x10))))
% Found ((eta_expansion_dep0 (fun (x3:a)=> Prop)) (f x1)) as proof of (((eq (a->Prop)) (f x1)) (fun (x10:a)=> (((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 x1) x10))))
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) (f x1)) as proof of (((eq (a->Prop)) (f x1)) (fun (x10:a)=> (((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 x1) x10))))
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) (f x1)) as proof of (((eq (a->Prop)) (f x1)) (fun (x10:a)=> (((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 x1) x10))))
% Found (fun (x1:a)=> (((eta_expansion_dep a) (fun (x3:a)=> Prop)) (f x1))) as proof of (((eq (a->Prop)) (f x1)) (fun (x10:a)=> (((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 x1) x10))))
% Found (fun (x1:a)=> (((eta_expansion_dep a) (fun (x3:a)=> Prop)) (f x1))) as proof of (forall (x:a), (((eq (a->Prop)) (f x)) (fun (x10:a)=> (((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) x10)))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (f x1)):(((eq (a->Prop)) (f x1)) (fun (x:a)=> ((f x1) x)))
% Found (eta_expansion_dep00 (f x1)) as proof of (((eq (a->Prop)) (f x1)) (fun (x10:a)=> (((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 x1) x10))))
% Found ((eta_expansion_dep0 (fun (x3:a)=> Prop)) (f x1)) as proof of (((eq (a->Prop)) (f x1)) (fun (x10:a)=> (((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 x1) x10))))
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) (f x1)) as proof of (((eq (a->Prop)) (f x1)) (fun (x10:a)=> (((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 x1) x10))))
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) (f x1)) as proof of (((eq (a->Prop)) (f x1)) (fun (x10:a)=> (((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 x1) x10))))
% Found (fun (x1:a)=> (((eta_expansion_dep a) (fun (x3:a)=> Prop)) (f x1))) as proof of (((eq (a->Prop)) (f x1)) (fun (x10:a)=> (((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 x1) x10))))
% Found (fun (x1:a)=> (((eta_expansion_dep a) (fun (x3:a)=> Prop)) (f x1))) as proof of (forall (x:a), (((eq (a->Prop)) (f x)) (fun (x10:a)=> (((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) x10)))))
% Found x4:((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))))
% Instantiate: b:=(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))):((a->(a->Prop))->Prop)
% Found x4 as proof of (P b)
% Found eta_expansion000:=(eta_expansion00 (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0)))):(((eq ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0)))) (fun (x:(a->(a->Prop)))=> ((and (S x)) ((x Xx0) Xy0))))
% Found (eta_expansion00 (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0)))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0)))) b)
% Found ((eta_expansion0 Prop) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0)))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0)))) b)
% Found (((eta_expansion (a->(a->Prop))) Prop) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0)))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0)))) b)
% Found (((eta_expansion (a->(a->Prop))) Prop) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0)))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0)))) b)
% Found (((eta_expansion (a->(a->Prop))) Prop) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0)))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0)))) b)
% Found x3:((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))))
% Instantiate: x6:=(fun (x8:a) (x70: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 x8) x70))))):(a->(a->Prop))
% Found x3 as proof of ((x6 Xx0) Xy0)
% Found x3 as proof of ((x6 Xx0) Xy0)
% Found x3 as proof of ((x6 Xx0) Xy0)
% Found eq_ref00:=(eq_ref0 (f x4)):(((eq Prop) (f x4)) (f x4))
% Found (eq_ref0 (f x4)) as proof of (((eq Prop) (f x4)) ((and (S x4)) ((x4 Xx0) Xy0)))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((and (S x4)) ((x4 Xx0) Xy0)))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((and (S x4)) ((x4 Xx0) Xy0)))
% Found (fun (x4:(a->(a->Prop)))=> ((eq_ref Prop) (f x4))) as proof of (((eq Prop) (f x4)) ((and (S x4)) ((x4 Xx0) Xy0)))
% Found (fun (x4:(a->(a->Prop)))=> ((eq_ref Prop) (f x4))) as proof of (forall (x:(a->(a->Prop))), (((eq Prop) (f x)) ((and (S x)) ((x Xx0) Xy0))))
% Found eq_ref00:=(eq_ref0 (f x4)):(((eq Prop) (f x4)) (f x4))
% Found (eq_ref0 (f x4)) as proof of (((eq Prop) (f x4)) ((and (S x4)) ((x4 Xx0) Xy0)))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((and (S x4)) ((x4 Xx0) Xy0)))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((and (S x4)) ((x4 Xx0) Xy0)))
% Found (fun (x4:(a->(a->Prop)))=> ((eq_ref Prop) (f x4))) as proof of (((eq Prop) (f x4)) ((and (S x4)) ((x4 Xx0) Xy0)))
% Found (fun (x4:(a->(a->Prop)))=> ((eq_ref Prop) (f x4))) as proof of (forall (x:(a->(a->Prop))), (((eq Prop) (f x)) ((and (S x)) ((x Xx0) Xy0))))
% Found eta_expansion000:=(eta_expansion00 (fun (x3:a) (x20:a)=> ((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 x3) x20))))):(((eq (a->(a->Prop))) (fun (x3:a) (x20:a)=> ((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 x3) x20))))) (fun (x:a) (x20:a)=> ((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) x20)))))
% Found (eta_expansion00 (fun (x3:a) (x20:a)=> ((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 x3) x20))))) as proof of (((eq (a->(a->Prop))) (fun (x3:a) (x20:a)=> ((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 x3) x20))))) b)
% Found ((eta_expansion0 (a->Prop)) (fun (x3:a) (x20:a)=> ((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 x3) x20))))) as proof of (((eq (a->(a->Prop))) (fun (x3:a) (x20:a)=> ((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 x3) x20))))) b)
% Found (((eta_expansion a) (a->Prop)) (fun (x3:a) (x20:a)=> ((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 x3) x20))))) as proof of (((eq (a->(a->Prop))) (fun (x3:a) (x20:a)=> ((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 x3) x20))))) b)
% Found (((eta_expansion a) (a->Prop)) (fun (x3:a) (x20:a)=> ((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 x3) x20))))) as proof of (((eq (a->(a->Prop))) (fun (x3:a) (x20:a)=> ((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 x3) x20))))) b)
% Found (((eta_expansion a) (a->Prop)) (fun (x3:a) (x20:a)=> ((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 x3) x20))))) as proof of (((eq (a->(a->Prop))) (fun (x3:a) (x20:a)=> ((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 x3) x20))))) b)
% Found eq_ref00:=(eq_ref0 (f x5)):(((eq Prop) (f x5)) (f x5))
% Found (eq_ref0 (f x5)) as proof of (((eq Prop) (f x5)) ((and (S x5)) ((x5 Xx0) Xy0)))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) ((and (S x5)) ((x5 Xx0) Xy0)))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) ((and (S x5)) ((x5 Xx0) Xy0)))
% Found (fun (x5:(a->(a->Prop)))=> ((eq_ref Prop) (f x5))) as proof of (((eq Prop) (f x5)) ((and (S x5)) ((x5 Xx0) Xy0)))
% Found (fun (x5:(a->(a->Prop)))=> ((eq_ref Prop) (f x5))) as proof of (forall (x:(a->(a->Prop))), (((eq Prop) (f x)) ((and (S x)) ((x Xx0) Xy0))))
% Found eq_ref00:=(eq_ref0 (f x5)):(((eq Prop) (f x5)) (f x5))
% Found (eq_ref0 (f x5)) as proof of (((eq Prop) (f x5)) ((and (S x5)) ((x5 Xx0) Xy0)))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) ((and (S x5)) ((x5 Xx0) Xy0)))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) ((and (S x5)) ((x5 Xx0) Xy0)))
% Found (fun (x5:(a->(a->Prop)))=> ((eq_ref Prop) (f x5))) as proof of (((eq Prop) (f x5)) ((and (S x5)) ((x5 Xx0) Xy0)))
% Found (fun (x5:(a->(a->Prop)))=> ((eq_ref Prop) (f x5))) as proof of (forall (x:(a->(a->Prop))), (((eq Prop) (f x)) ((and (S x)) ((x Xx0) Xy0))))
% Found x4:((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))))
% Instantiate: f:=(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))):((a->(a->Prop))->Prop)
% Found x4 as proof of (P f)
% Found x4:((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))))
% Instantiate: f:=(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))):((a->(a->Prop))->Prop)
% Found x4 as proof of (P f)
% Found x4:((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))))
% Instantiate: b:=(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))):((a->(a->Prop))->Prop)
% Found x4 as proof of (P b)
% Found eq_ref00:=(eq_ref0 (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0)))):(((eq ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0)))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))
% Found (eq_ref0 (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0)))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0)))) b)
% Found ((eq_ref ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0)))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0)))) b)
% Found ((eq_ref ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0)))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0)))) b)
% Found ((eq_ref ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0)))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0)))) b)
% Found eq_ref00:=(eq_ref0 (fun (x4:a) (x30:a)=> ((PROP Xp)->((Xp x4) x30)))):(((eq (a->(a->Prop))) (fun (x4:a) (x30:a)=> ((PROP Xp)->((Xp x4) x30)))) (fun (x4:a) (x30:a)=> ((PROP Xp)->((Xp x4) x30))))
% Found (eq_ref0 (fun (x4:a) (x30:a)=> ((PROP Xp)->((Xp x4) x30)))) as proof of (((eq (a->(a->Prop))) (fun (x4:a) (x30:a)=> ((PROP Xp)->((Xp x4) x30)))) b)
% Found ((eq_ref (a->(a->Prop))) (fun (x4:a) (x30:a)=> ((PROP Xp)->((Xp x4) x30)))) as proof of (((eq (a->(a->Prop))) (fun (x4:a) (x30:a)=> ((PROP Xp)->((Xp x4) x30)))) b)
% Found ((eq_ref (a->(a->Prop))) (fun (x4:a) (x30:a)=> ((PROP Xp)->((Xp x4) x30)))) as proof of (((eq (a->(a->Prop))) (fun (x4:a) (x30:a)=> ((PROP Xp)->((Xp x4) x30)))) b)
% Found ((eq_ref (a->(a->Prop))) (fun (x4:a) (x30:a)=> ((PROP Xp)->((Xp x4) x30)))) as proof of (((eq (a->(a->Prop))) (fun (x4:a) (x30:a)=> ((PROP Xp)->((Xp x4) x30)))) b)
% Found x3:((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))))
% Instantiate: x6:=(fun (x8:a) (x70: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 x8) x70))))):(a->(a->Prop))
% Found x3 as proof of ((x6 Xx0) Xy0)
% Found x3 as proof of ((x6 Xx0) Xy0)
% Found x3 as proof of ((x6 Xx0) Xy0)
% Found x2:((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))))
% Instantiate: b:=(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))):((a->(a->Prop))->Prop)
% Found x2 as proof of (P b)
% Found eq_ref00:=(eq_ref0 (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0)))):(((eq ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0)))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))
% Found (eq_ref0 (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0)))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0)))) b)
% Found ((eq_ref ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0)))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0)))) b)
% Found ((eq_ref ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0)))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0)))) b)
% Found ((eq_ref ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0)))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0)))) b)
% Found x4:((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))))
% Instantiate: f:=(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))):((a->(a->Prop))->Prop)
% Found x4 as proof of (P f)
% Found x4:((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))))
% Instantiate: f:=(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))):((a->(a->Prop))->Prop)
% Found x4 as proof of (P f)
% Found eq_ref00:=(eq_ref0 (f x5)):(((eq Prop) (f x5)) (f x5))
% Found (eq_ref0 (f x5)) as proof of (((eq Prop) (f x5)) ((and (S x5)) ((x5 Xx0) Xy0)))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) ((and (S x5)) ((x5 Xx0) Xy0)))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) ((and (S x5)) ((x5 Xx0) Xy0)))
% Found (fun (x5:(a->(a->Prop)))=> ((eq_ref Prop) (f x5))) as proof of (((eq Prop) (f x5)) ((and (S x5)) ((x5 Xx0) Xy0)))
% Found (fun (x5:(a->(a->Prop)))=> ((eq_ref Prop) (f x5))) as proof of (forall (x:(a->(a->Prop))), (((eq Prop) (f x)) ((and (S x)) ((x Xx0) Xy0))))
% Found eq_ref00:=(eq_ref0 (f x5)):(((eq Prop) (f x5)) (f x5))
% Found (eq_ref0 (f x5)) as proof of (((eq Prop) (f x5)) ((and (S x5)) ((x5 Xx0) Xy0)))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) ((and (S x5)) ((x5 Xx0) Xy0)))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) ((and (S x5)) ((x5 Xx0) Xy0)))
% Found (fun (x5:(a->(a->Prop)))=> ((eq_ref Prop) (f x5))) as proof of (((eq Prop) (f x5)) ((and (S x5)) ((x5 Xx0) Xy0)))
% Found (fun (x5:(a->(a->Prop)))=> ((eq_ref Prop) (f x5))) as proof of (forall (x:(a->(a->Prop))), (((eq Prop) (f x)) ((and (S x)) ((x Xx0) Xy0))))
% Found eq_ref000:=(eq_ref00 (ex (a->(a->Prop)))):(((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))))->((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)))))
% Found (eq_ref00 (ex (a->(a->Prop)))) as proof of (((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))))->(P b))
% Found ((eq_ref0 (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)))) (ex (a->(a->Prop)))) as proof of (((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))))->(P b))
% Found (((eq_ref ((a->(a->Prop))->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)))) (ex (a->(a->Prop)))) as proof of (((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))))->(P b))
% Found (((eq_ref ((a->(a->Prop))->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)))) (ex (a->(a->Prop)))) as proof of (((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))))->(P b))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 (ex (a->(a->Prop)))):(((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))))->((ex (a->(a->Prop))) (fun (x:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) x) (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))))))))) ((x Xx0) Xy0)))))
% Found (eta_expansion_dep000 (ex (a->(a->Prop)))) as proof of (((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))))->(P b))
% Found ((eta_expansion_dep00 (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)))) (ex (a->(a->Prop)))) as proof of (((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))))->(P b))
% Found (((eta_expansion_dep0 (fun (x5:(a->(a->Prop)))=> 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)))) (ex (a->(a->Prop)))) as proof of (((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))))->(P b))
% Found ((((eta_expansion_dep (a->(a->Prop))) (fun (x5:(a->(a->Prop)))=> 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)))) (ex (a->(a->Prop)))) as proof of (((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))))->(P b))
% Found ((((eta_expansion_dep (a->(a->Prop))) (fun (x5:(a->(a->Prop)))=> 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)))) (ex (a->(a->Prop)))) as proof of (((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))))->(P b))
% Found eta_expansion0000:=(eta_expansion000 (ex (a->(a->Prop)))):(((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))))->((ex (a->(a->Prop))) (fun (x:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) x) (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))))))))) ((x Xx0) Xy0)))))
% Found (eta_expansion000 (ex (a->(a->Prop)))) as proof of (((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))))->(P b))
% Found ((eta_expansion00 (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)))) (ex (a->(a->Prop)))) as proof of (((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))))->(P b))
% Found (((eta_expansion0 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)))) (ex (a->(a->Prop)))) as proof of (((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))))->(P b))
% Found ((((eta_expansion (a->(a->Prop))) 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)))) (ex (a->(a->Prop)))) as proof of (((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))))->(P b))
% Found ((((eta_expansion (a->(a->Prop))) 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)))) (ex (a->(a->Prop)))) as proof of (((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))))->(P b))
% Found x4:((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))))
% Instantiate: b:=(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))):((a->(a->Prop))->Prop)
% Found (fun (x4:((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)))))=> x4) as proof of (P b)
% Found x3:((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))))
% Instantiate: x4:=(fun (x8:a) (x70: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 x8) x70))))):(a->(a->Prop))
% Found x3 as proof of ((x4 Xx0) Xy0)
% Found x3 as proof of ((x4 Xx0) Xy0)
% Found x3 as proof of ((x4 Xx0) Xy0)
% Found eq_ref00:=(eq_ref0 Xy0):(((eq a) Xy0) Xy0)
% Found (eq_ref0 Xy0) as proof of (((eq a) Xy0) b)
% Found ((eq_ref a) Xy0) as proof of (((eq a) Xy0) b)
% Found ((eq_ref a) Xy0) as proof of (((eq a) Xy0) b)
% Found ((eq_ref a) Xy0) as proof of (((eq a) Xy0) b)
% Found eq_ref00:=(eq_ref0 (f x5)):(((eq Prop) (f x5)) (f x5))
% Found (eq_ref0 (f x5)) as proof of (((eq Prop) (f x5)) ((and (S x5)) ((x5 Xx0) Xy0)))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) ((and (S x5)) ((x5 Xx0) Xy0)))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) ((and (S x5)) ((x5 Xx0) Xy0)))
% Found (fun (x5:(a->(a->Prop)))=> ((eq_ref Prop) (f x5))) as proof of (((eq Prop) (f x5)) ((and (S x5)) ((x5 Xx0) Xy0)))
% Found (fun (x5:(a->(a->Prop)))=> ((eq_ref Prop) (f x5))) as proof of (forall (x:(a->(a->Prop))), (((eq Prop) (f x)) ((and (S x)) ((x Xx0) Xy0))))
% Found eq_ref00:=(eq_ref0 (f x5)):(((eq Prop) (f x5)) (f x5))
% Found (eq_ref0 (f x5)) as proof of (((eq Prop) (f x5)) ((and (S x5)) ((x5 Xx0) Xy0)))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) ((and (S x5)) ((x5 Xx0) Xy0)))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) ((and (S x5)) ((x5 Xx0) Xy0)))
% Found (fun (x5:(a->(a->Prop)))=> ((eq_ref Prop) (f x5))) as proof of (((eq Prop) (f x5)) ((and (S x5)) ((x5 Xx0) Xy0)))
% Found (fun (x5:(a->(a->Prop)))=> ((eq_ref Prop) (f x5))) as proof of (forall (x:(a->(a->Prop))), (((eq Prop) (f x)) ((and (S x)) ((x Xx0) Xy0))))
% Found x6:((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) x5) (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))))))))) ((x5 Xx0) Xy0))
% Instantiate: x4:=(fun (x8:a) (x70:a)=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) x5) (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))))))))) ((x5 x8) x70))):(a->(a->Prop))
% Found (fun (x6:((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) x5) (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))))))))) ((x5 Xx0) Xy0)))=> x6) as proof of ((x4 Xx0) Xy0)
% Found eta_expansion0000:=(eta_expansion000 (ex (a->(a->Prop)))):(((ex (a->(a->Prop))) (fun (x7:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) x7) (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))))))))) ((x7 Xx0) Xy0))))->((ex (a->(a->Prop))) (fun (x:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) x) (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))))))))) ((x Xx0) Xy0)))))
% Found (eta_expansion000 (ex (a->(a->Prop)))) as proof of (((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))))->(P b))
% Found ((eta_expansion00 (fun (x7:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) x7) (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))))))))) ((x7 Xx0) Xy0)))) (ex (a->(a->Prop)))) as proof of (((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))))->(P b))
% Found (((eta_expansion0 Prop) (fun (x7:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) x7) (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))))))))) ((x7 Xx0) Xy0)))) (ex (a->(a->Prop)))) as proof of (((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))))->(P b))
% Found ((((eta_expansion (a->(a->Prop))) Prop) (fun (x7:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) x7) (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))))))))) ((x7 Xx0) Xy0)))) (ex (a->(a->Prop)))) as proof of (((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))))->(P b))
% Found ((((eta_expansion (a->(a->Prop))) Prop) (fun (x7:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) x7) (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))))))))) ((x7 Xx0) Xy0)))) (ex (a->(a->Prop)))) as proof of (((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))))->(P b))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 (ex (a->(a->Prop)))):(((ex (a->(a->Prop))) (fun (x7:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) x7) (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))))))))) ((x7 Xx0) Xy0))))->((ex (a->(a->Prop))) (fun (x:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) x) (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))))))))) ((x Xx0) Xy0)))))
% Found (eta_expansion_dep000 (ex (a->(a->Prop)))) as proof of (((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))))->(P b))
% Found ((eta_expansion_dep00 (fun (x7:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) x7) (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))))))))) ((x7 Xx0) Xy0)))) (ex (a->(a->Prop)))) as proof of (((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))))->(P b))
% Found (((eta_expansion_dep0 (fun (x5:(a->(a->Prop)))=> Prop)) (fun (x7:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) x7) (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))))))))) ((x7 Xx0) Xy0)))) (ex (a->(a->Prop)))) as proof of (((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))))->(P b))
% Found ((((eta_expansion_dep (a->(a->Prop))) (fun (x5:(a->(a->Prop)))=> Prop)) (fun (x7:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) x7) (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))))))))) ((x7 Xx0) Xy0)))) (ex (a->(a->Prop)))) as proof of (((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))))->(P b))
% Found ((((eta_expansion_dep (a->(a->Prop))) (fun (x5:(a->(a->Prop)))=> Prop)) (fun (x7:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) x7) (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))))))))) ((x7 Xx0) Xy0)))) (ex (a->(a->Prop)))) as proof of (((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))))->(P b))
% Found x4:((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))))
% Instantiate: f:=(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))):((a->(a->Prop))->Prop)
% Found x4 as proof of (P f)
% Found x4:((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))))
% Instantiate: f:=(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))):((a->(a->Prop))->Prop)
% Found x4 as proof of (P f)
% Found eq_ref00:=(eq_ref0 Xy0):(((eq a) Xy0) Xy0)
% Found (eq_ref0 Xy0) as proof of (((eq a) Xy0) b)
% Found ((eq_ref a) Xy0) as proof of (((eq a) Xy0) b)
% Found ((eq_ref a) Xy0) as proof of (((eq a) Xy0) b)
% Found ((eq_ref a) Xy0) as proof of (((eq a) Xy0) b)
% Found eq_ref00:=(eq_ref0 Xy0):(((eq a) Xy0) Xy0)
% Found (eq_ref0 Xy0) as proof of (((eq a) Xy0) b)
% Found ((eq_ref a) Xy0) as proof of (((eq a) Xy0) b)
% Found ((eq_ref a) Xy0) as proof of (((eq a) Xy0) b)
% Found ((eq_ref a) Xy0) as proof of (((eq a) Xy0) b)
% Found x7:((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) x6) (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))))))))) ((x6 Xx0) Xy0))
% Instantiate: x5:=(fun (x9:a) (x80:a)=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) x6) (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))))))))) ((x6 x9) x80))):(a->(a->Prop))
% Found (fun (x7:((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) x6) (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))))))))) ((x6 Xx0) Xy0)))=> x7) as proof of ((x5 Xx0) Xy0)
% Found x2:((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))))
% Instantiate: f:=(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))):((a->(a->Prop))->Prop)
% Found x2 as proof of (P f)
% Found x2:((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))))
% Instantiate: f:=(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))):((a->(a->Prop))->Prop)
% Found x2 as proof of (P f)
% Found eq_ref00:=(eq_ref0 Xy0):(((eq a) Xy0) Xy0)
% Found (eq_ref0 Xy0) as proof of (((eq a) Xy0) b)
% Found ((eq_ref a) Xy0) as proof of (((eq a) Xy0) b)
% Found ((eq_ref a) Xy0) as proof of (((eq a) Xy0) b)
% Found ((eq_ref a) Xy0) as proof of (((eq a) Xy0) b)
% Found eq_ref00:=(eq_ref0 Xy0):(((eq a) Xy0) Xy0)
% Found (eq_ref0 Xy0) as proof of (((eq a) Xy0) b)
% Found ((eq_ref a) Xy0) as proof of (((eq a) Xy0) b)
% Found ((eq_ref a) Xy0) as proof of (((eq a) Xy0) b)
% Found ((eq_ref a) Xy0) as proof of (((eq a) Xy0) b)
% Found eq_ref00:=(eq_ref0 (f x5)):(((eq Prop) (f x5)) (f x5))
% Found (eq_ref0 (f x5)) as proof of (((eq Prop) (f x5)) ((and (S x5)) ((x5 Xx0) Xy0)))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) ((and (S x5)) ((x5 Xx0) Xy0)))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) ((and (S x5)) ((x5 Xx0) Xy0)))
% Found (fun (x5:(a->(a->Prop)))=> ((eq_ref Prop) (f x5))) as proof of (((eq Prop) (f x5)) ((and (S x5)) ((x5 Xx0) Xy0)))
% Found (fun (x5:(a->(a->Prop)))=> ((eq_ref Prop) (f x5))) as proof of (forall (x:(a->(a->Prop))), (((eq Prop) (f x)) ((and (S x)) ((x Xx0) Xy0))))
% Found eq_ref00:=(eq_ref0 (f x5)):(((eq Prop) (f x5)) (f x5))
% Found (eq_ref0 (f x5)) as proof of (((eq Prop) (f x5)) ((and (S x5)) ((x5 Xx0) Xy0)))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) ((and (S x5)) ((x5 Xx0) Xy0)))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) ((and (S x5)) ((x5 Xx0) Xy0)))
% Found (fun (x5:(a->(a->Prop)))=> ((eq_ref Prop) (f x5))) as proof of (((eq Prop) (f x5)) ((and (S x5)) ((x5 Xx0) Xy0)))
% Found (fun (x5:(a->(a->Prop)))=> ((eq_ref Prop) (f x5))) as proof of (forall (x:(a->(a->Prop))), (((eq Prop) (f x)) ((and (S x)) ((x Xx0) Xy0))))
% Found eq_ref000:=(eq_ref00 (ex (a->(a->Prop)))):(((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))))->((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)))))
% Found (eq_ref00 (ex (a->(a->Prop)))) as proof of (((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))))->(P f))
% Found ((eq_ref0 (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)))) (ex (a->(a->Prop)))) as proof of (((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))))->(P f))
% Found (((eq_ref ((a->(a->Prop))->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)))) (ex (a->(a->Prop)))) as proof of (((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))))->(P f))
% Found (((eq_ref ((a->(a->Prop))->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)))) (ex (a->(a->Prop)))) as proof of (((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))))->(P f))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 (ex (a->(a->Prop)))):(((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))))->((ex (a->(a->Prop))) (fun (x:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) x) (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))))))))) ((x Xx0) Xy0)))))
% Found (eta_expansion_dep000 (ex (a->(a->Prop)))) as proof of (((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))))->(P f))
% Found ((eta_expansion_dep00 (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)))) (ex (a->(a->Prop)))) as proof of (((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))))->(P f))
% Found (((eta_expansion_dep0 (fun (x5:(a->(a->Prop)))=> 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)))) (ex (a->(a->Prop)))) as proof of (((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))))->(P f))
% Found ((((eta_expansion_dep (a->(a->Prop))) (fun (x5:(a->(a->Prop)))=> 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)))) (ex (a->(a->Prop)))) as proof of (((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))))->(P f))
% Found ((((eta_expansion_dep (a->(a->Prop))) (fun (x5:(a->(a->Prop)))=> 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)))) (ex (a->(a->Prop)))) as proof of (((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))))->(P f))
% Found eta_expansion0000:=(eta_expansion000 (ex (a->(a->Prop)))):(((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))))->((ex (a->(a->Prop))) (fun (x:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) x) (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))))))))) ((x Xx0) Xy0)))))
% Found (eta_expansion000 (ex (a->(a->Prop)))) as proof of (((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))))->(P f))
% Found ((eta_expansion00 (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)))) (ex (a->(a->Prop)))) as proof of (((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))))->(P f))
% Found (((eta_expansion0 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)))) (ex (a->(a->Prop)))) as proof of (((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))))->(P f))
% Found ((((eta_expansion (a->(a->Prop))) 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)))) (ex (a->(a->Prop)))) as proof of (((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))))->(P f))
% Found ((((eta_expansion (a->(a->Prop))) 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)))) (ex (a->(a->Prop)))) as proof of (((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))))->(P f))
% Found x4:((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))))
% Instantiate: f:=(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))):((a->(a->Prop))->Prop)
% Found (fun (x4:((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)))))=> x4) as proof of (P f)
% Found eta_expansion0000:=(eta_expansion000 (ex (a->(a->Prop)))):(((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))))->((ex (a->(a->Prop))) (fun (x:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) x) (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))))))))) ((x Xx0) Xy0)))))
% Found (eta_expansion000 (ex (a->(a->Prop)))) as proof of (((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))))->(P f))
% Found ((eta_expansion00 (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)))) (ex (a->(a->Prop)))) as proof of (((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))))->(P f))
% Found (((eta_expansion0 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)))) (ex (a->(a->Prop)))) as proof of (((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))))->(P f))
% Found ((((eta_expansion (a->(a->Prop))) 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)))) (ex (a->(a->Prop)))) as proof of (((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))))->(P f))
% Found ((((eta_expansion (a->(a->Prop))) 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)))) (ex (a->(a->Prop)))) as proof of (((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))))->(P f))
% Found eq_ref000:=(eq_ref00 (ex (a->(a->Prop)))):(((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))))->((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)))))
% Found (eq_ref00 (ex (a->(a->Prop)))) as proof of (((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))))->(P f))
% Found ((eq_ref0 (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)))) (ex (a->(a->Prop)))) as proof of (((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))))->(P f))
% Found (((eq_ref ((a->(a->Prop))->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)))) (ex (a->(a->Prop)))) as proof of (((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))))->(P f))
% Found (((eq_ref ((a->(a->Prop))->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)))) (ex (a->(a->Prop)))) as proof of (((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))))->(P f))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 (ex (a->(a->Prop)))):(((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))))->((ex (a->(a->Prop))) (fun (x:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) x) (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))))))))) ((x Xx0) Xy0)))))
% Found (eta_expansion_dep000 (ex (a->(a->Prop)))) as proof of (((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))))->(P f))
% Found ((eta_expansion_dep00 (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)))) (ex (a->(a->Prop)))) as proof of (((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))))->(P f))
% Found (((eta_expansion_dep0 (fun (x5:(a->(a->Prop)))=> 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)))) (ex (a->(a->Prop)))) as proof of (((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))))->(P f))
% Found ((((eta_expansion_dep (a->(a->Prop))) (fun (x5:(a->(a->Prop)))=> 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)))) (ex (a->(a->Prop)))) as proof of (((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))))->(P f))
% Found ((((eta_expansion_dep (a->(a->Prop))) (fun (x5:(a->(a->Prop)))=> 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)))) (ex (a->(a->Prop)))) as proof of (((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))))->(P f))
% Found x4:((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))))
% Instantiate: f:=(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))):((a->(a->Prop))->Prop)
% Found (fun (x4:((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)))))=> x4) as proof of (P f)
% Found eq_ref000:=(eq_ref00 PROP):((PROP Xp)->(PROP Xp))
% Found (eq_ref00 PROP) as proof of ((PROP Xp)->(PROP Xp))
% Found ((eq_ref0 Xp) PROP) as proof of ((PROP Xp)->(PROP Xp))
% Found (((eq_ref (a->(a->Prop))) Xp) PROP) as proof of ((PROP Xp)->(PROP Xp))
% Found (fun (x2:(forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0))))=> (((eq_ref (a->(a->Prop))) Xp) PROP)) as proof of ((PROP Xp)->(PROP Xp))
% Found (fun (x2:(forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0))))=> (((eq_ref (a->(a->Prop))) Xp) PROP)) as proof of ((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)->(PROP Xp)))
% Found (and_rect00 (fun (x2:(forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0))))=> (((eq_ref (a->(a->Prop))) Xp) PROP))) as proof of (PROP Xp)
% Found ((and_rect0 (PROP Xp)) (fun (x2:(forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0))))=> (((eq_ref (a->(a->Prop))) Xp) PROP))) as proof of (PROP Xp)
% Found (((fun (P0:Type) (x2:((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)->P0)))=> (((((and_rect (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)) P0) x2) x0)) (PROP Xp)) (fun (x2:(forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0))))=> (((eq_ref (a->(a->Prop))) Xp) PROP))) as proof of (PROP Xp)
% Found (((fun (P0:Type) (x2:((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)->P0)))=> (((((and_rect (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)) P0) x2) x0)) (PROP Xp)) (fun (x2:(forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0))))=> (((eq_ref (a->(a->Prop))) Xp) PROP))) as proof of (PROP Xp)
% Found eq_ref000:=(eq_ref00 PROP):((PROP Xp)->(PROP Xp))
% Found (eq_ref00 PROP) as proof of ((PROP Xp)->(PROP a0))
% Found ((eq_ref0 Xp) PROP) as proof of ((PROP Xp)->(PROP a0))
% Found (((eq_ref (a->(a->Prop))) Xp) PROP) as proof of ((PROP Xp)->(PROP a0))
% Found (fun (x2:(forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0))))=> (((eq_ref (a->(a->Prop))) Xp) PROP)) as proof of ((PROP Xp)->(PROP a0))
% Found (fun (x2:(forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0))))=> (((eq_ref (a->(a->Prop))) Xp) PROP)) as proof of ((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)->(PROP a0)))
% Found (and_rect00 (fun (x2:(forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0))))=> (((eq_ref (a->(a->Prop))) Xp) PROP))) as proof of (PROP a0)
% Found ((and_rect0 (PROP a0)) (fun (x2:(forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0))))=> (((eq_ref (a->(a->Prop))) Xp) PROP))) as proof of (PROP a0)
% Found (((fun (P0:Type) (x2:((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)->P0)))=> (((((and_rect (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)) P0) x2) x0)) (PROP a0)) (fun (x2:(forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0))))=> (((eq_ref (a->(a->Prop))) Xp) PROP))) as proof of (PROP a0)
% Found (((fun (P0:Type) (x2:((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)->P0)))=> (((((and_rect (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)) P0) x2) x0)) (PROP a0)) (fun (x2:(forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0))))=> (((eq_ref (a->(a->Prop))) Xp) PROP))) as proof of (PROP a0)
% Found x2:(PROP Xp)
% Instantiate: b:=Xp:(a->(a->Prop))
% Found x2 as proof of (P b)
% Found eq_ref00:=(eq_ref0 (f x5)):(((eq Prop) (f x5)) (f x5))
% Found (eq_ref0 (f x5)) as proof of (((eq Prop) (f x5)) ((and (S x5)) ((x5 Xx0) Xy0)))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) ((and (S x5)) ((x5 Xx0) Xy0)))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) ((and (S x5)) ((x5 Xx0) Xy0)))
% Found (fun (x5:(a->(a->Prop)))=> ((eq_ref Prop) (f x5))) as proof of (((eq Prop) (f x5)) ((and (S x5)) ((x5 Xx0) Xy0)))
% Found (fun (x5:(a->(a->Prop)))=> ((eq_ref Prop) (f x5))) as proof of (forall (x:(a->(a->Prop))), (((eq Prop) (f x)) ((and (S x)) ((x Xx0) Xy0))))
% Found eq_ref00:=(eq_ref0 (f x5)):(((eq Prop) (f x5)) (f x5))
% Found (eq_ref0 (f x5)) as proof of (((eq Prop) (f x5)) ((and (S x5)) ((x5 Xx0) Xy0)))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) ((and (S x5)) ((x5 Xx0) Xy0)))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) ((and (S x5)) ((x5 Xx0) Xy0)))
% Found (fun (x5:(a->(a->Prop)))=> ((eq_ref Prop) (f x5))) as proof of (((eq Prop) (f x5)) ((and (S x5)) ((x5 Xx0) Xy0)))
% Found (fun (x5:(a->(a->Prop)))=> ((eq_ref Prop) (f x5))) as proof of (forall (x:(a->(a->Prop))), (((eq Prop) (f x)) ((and (S x)) ((x Xx0) Xy0))))
% Found eq_ref000:=(eq_ref00 PROP):((PROP Xp)->(PROP Xp))
% Found (eq_ref00 PROP) as proof of ((PROP Xp)->(PROP a0))
% Found ((eq_ref0 Xp) PROP) as proof of ((PROP Xp)->(PROP a0))
% Found (((eq_ref (a->(a->Prop))) Xp) PROP) as proof of ((PROP Xp)->(PROP a0))
% Found (fun (x2:(forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0))))=> (((eq_ref (a->(a->Prop))) Xp) PROP)) as proof of ((PROP Xp)->(PROP a0))
% Found (fun (x2:(forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0))))=> (((eq_ref (a->(a->Prop))) Xp) PROP)) as proof of ((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)->(PROP a0)))
% Found (and_rect00 (fun (x2:(forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0))))=> (((eq_ref (a->(a->Prop))) Xp) PROP))) as proof of (PROP a0)
% Found ((and_rect0 (PROP a0)) (fun (x2:(forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0))))=> (((eq_ref (a->(a->Prop))) Xp) PROP))) as proof of (PROP a0)
% Found (((fun (P0:Type) (x2:((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)->P0)))=> (((((and_rect (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)) P0) x2) x0)) (PROP a0)) (fun (x2:(forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0))))=> (((eq_ref (a->(a->Prop))) Xp) PROP))) as proof of (PROP a0)
% Found (((fun (P0:Type) (x2:((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)->P0)))=> (((((and_rect (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)) P0) x2) x0)) (PROP a0)) (fun (x2:(forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp Xx0) Xy0))))=> (((eq_ref (a->(a->Prop))) Xp) PROP))) as proof of (PROP a0)
% Found eq_ref00:=(eq_ref0 ((f x1) y)):(((eq Prop) ((f x1) y)) ((f x1) y))
% Found (eq_ref0 ((f x1) y)) as proof of (((eq Prop) ((f x1) y)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp0 Xx0) Xy0)))) (PROP Xp0))->((Xp0 x1) y))))
% Found ((eq_ref Prop) ((f x1) y)) as proof of (((eq Prop) ((f x1) y)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp0 Xx0) Xy0)))) (PROP Xp0))->((Xp0 x1) y))))
% Found ((eq_ref Prop) ((f x1) y)) as proof of (((eq Prop) ((f x1) y)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp0 Xx0) Xy0)))) (PROP Xp0))->((Xp0 x1) y))))
% Found (fun (y:a)=> ((eq_ref Prop) ((f x1) y))) as proof of (((eq Prop) ((f x1) y)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp0 Xx0) Xy0)))) (PROP Xp0))->((Xp0 x1) y))))
% Found (fun (x1:a) (y:a)=> ((eq_ref Prop) ((f x1) y))) as proof of (forall (y:a), (((eq Prop) ((f x1) y)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp0 Xx0) Xy0)))) (PROP Xp0))->((Xp0 x1) y)))))
% Found (fun (x1:a) (y:a)=> ((eq_ref Prop) ((f x1) y))) as proof of (forall (x:a) (y:a), (((eq Prop) ((f x) y)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp0 Xx0) Xy0)))) (PROP Xp0))->((Xp0 x) y)))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 (ex (a->(a->Prop)))):(((ex (a->(a->Prop))) (fun (x7:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) x7) (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))))))))) ((x7 Xx0) Xy0))))->((ex (a->(a->Prop))) (fun (x:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) x) (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))))))))) ((x Xx0) Xy0)))))
% Found (eta_expansion_dep000 (ex (a->(a->Prop)))) as proof of (((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))))->(P f))
% Found ((eta_expansion_dep00 (fun (x7:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) x7) (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))))))))) ((x7 Xx0) Xy0)))) (ex (a->(a->Prop)))) as proof of (((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))))->(P f))
% Found (((eta_expansion_dep0 (fun (x5:(a->(a->Prop)))=> Prop)) (fun (x7:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) x7) (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))))))))) ((x7 Xx0) Xy0)))) (ex (a->(a->Prop)))) as proof of (((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))))->(P f))
% Found ((((eta_expansion_dep (a->(a->Prop))) (fun (x5:(a->(a->Prop)))=> Prop)) (fun (x7:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) x7) (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))))))))) ((x7 Xx0) Xy0)))) (ex (a->(a->Prop)))) as proof of (((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))))->(P f))
% Found ((((eta_expansion_dep (a->(a->Prop))) (fun (x5:(a->(a->Prop)))=> Prop)) (fun (x7:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) x7) (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))))))))) ((x7 Xx0) Xy0)))) (ex (a->(a->Prop)))) as proof of (((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))))->(P f))
% Found eta_expansion0000:=(eta_expansion000 (ex (a->(a->Prop)))):(((ex (a->(a->Prop))) (fun (x7:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) x7) (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))))))))) ((x7 Xx0) Xy0))))->((ex (a->(a->Prop))) (fun (x:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) x) (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))))))))) ((x Xx0) Xy0)))))
% Found (eta_expansion000 (ex (a->(a->Prop)))) as proof of (((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))))->(P f))
% Found ((eta_expansion00 (fun (x7:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) x7) (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))))))))) ((x7 Xx0) Xy0)))) (ex (a->(a->Prop)))) as proof of (((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))))->(P f))
% Found (((eta_expansion0 Prop) (fun (x7:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) x7) (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))))))))) ((x7 Xx0) Xy0)))) (ex (a->(a->Prop)))) as proof of (((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))))->(P f))
% Found ((((eta_expansion (a->(a->Prop))) Prop) (fun (x7:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) x7) (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))))))))) ((x7 Xx0) Xy0)))) (ex (a->(a->Prop)))) as proof of (((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))))->(P f))
% Found ((((eta_expansion (a->(a->Prop))) Prop) (fun (x7:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) x7) (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))))))))) ((x7 Xx0) Xy0)))) (ex (a->(a->Prop)))) as proof of (((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))))->(P f))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 (ex (a->(a->Prop)))):(((ex (a->(a->Prop))) (fun (x7:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) x7) (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))))))))) ((x7 Xx0) Xy0))))->((ex (a->(a->Prop))) (fun (x:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) x) (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))))))))) ((x Xx0) Xy0)))))
% Found (eta_expansion_dep000 (ex (a->(a->Prop)))) as proof of (((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))))->(P f))
% Found ((eta_expansion_dep00 (fun (x7:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) x7) (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))))))))) ((x7 Xx0) Xy0)))) (ex (a->(a->Prop)))) as proof of (((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))))->(P f))
% Found (((eta_expansion_dep0 (fun (x5:(a->(a->Prop)))=> Prop)) (fun (x7:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) x7) (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))))))))) ((x7 Xx0) Xy0)))) (ex (a->(a->Prop)))) as proof of (((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))))->(P f))
% Found ((((eta_expansion_dep (a->(a->Prop))) (fun (x5:(a->(a->Prop)))=> Prop)) (fun (x7:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) x7) (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))))))))) ((x7 Xx0) Xy0)))) (ex (a->(a->Prop)))) as proof of (((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))))->(P f))
% Found ((((eta_expansion_dep (a->(a->Prop))) (fun (x5:(a->(a->Prop)))=> Prop)) (fun (x7:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) x7) (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))))))))) ((x7 Xx0) Xy0)))) (ex (a->(a->Prop)))) as proof of (((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))))->(P f))
% Found eta_expansion0000:=(eta_expansion000 (ex (a->(a->Prop)))):(((ex (a->(a->Prop))) (fun (x7:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) x7) (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))))))))) ((x7 Xx0) Xy0))))->((ex (a->(a->Prop))) (fun (x:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) x) (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))))))))) ((x Xx0) Xy0)))))
% Found (eta_expansion000 (ex (a->(a->Prop)))) as proof of (((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))))->(P f))
% Found ((eta_expansion00 (fun (x7:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) x7) (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))))))))) ((x7 Xx0) Xy0)))) (ex (a->(a->Prop)))) as proof of (((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))))->(P f))
% Found (((eta_expansion0 Prop) (fun (x7:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) x7) (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))))))))) ((x7 Xx0) Xy0)))) (ex (a->(a->Prop)))) as proof of (((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))))->(P f))
% Found ((((eta_expansion (a->(a->Prop))) Prop) (fun (x7:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) x7) (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))))))))) ((x7 Xx0) Xy0)))) (ex (a->(a->Prop)))) as proof of (((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))))->(P f))
% Found ((((eta_expansion (a->(a->Prop))) Prop) (fun (x7:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) x7) (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))))))))) ((x7 Xx0) Xy0)))) (ex (a->(a->Prop)))) as proof of (((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))))->(P f))
% Found x7:((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) x6) (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))))))))) ((x6 Xx0) Xy0))
% Instantiate: x5:=(fun (x9:a) (x80:a)=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) x6) (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))))))))) ((x6 x9) x80))):(a->(a->Prop))
% Found (fun (x7:((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) x6) (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))))))))) ((x6 Xx0) Xy0)))=> x7) as proof of ((x5 Xx0) Xy0)
% Found eq_ref00:=(eq_ref0 
% EOF
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