%------------------------------------------------------------------------------
% File     : cocATP---0.2.0
% Problem  : SEV009^5 : TPTP v6.1.0. Released v4.0.0.
% Transfm  : none
% Format   : tptp:raw
% Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p

% Computer : n189.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 13:33:32 EDT 2014

% Result   : Timeout 300.01s
% Output   : None 
% Verified : 
% SZS Type : None (Parsing solution fails)
% Syntax   : Number of formulae    : 0

% Comments : 
%------------------------------------------------------------------------------
%----NO SOLUTION OUTPUT BY SYSTEM
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% % Problem  : SEV009^5 : TPTP v6.1.0. Released v4.0.0.
% % Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n189.star.cs.uiowa.edu
% % Model    : x86_64 x86_64
% % CPU      : Intel(R) Xeon(R) CPU E5-2609 0 @ 2.40GHz
% % Memory   : 32286.75MB
% % OS       : Linux 2.6.32-431.20.3.el6.x86_64
% % CPULimit : 300
% % DateTime : Thu Jul 17 07:25:26 CDT 2014
% % CPUTime  : 300.01 
% 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 0x13641b8>, <kernel.Type object at 0x111db48>) of role type named a_type
% Using role type
% Declaring a:Type
% FOF formula (forall (P:((a->Prop)->Prop)), ((forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp)))))))->((and ((and (forall (Xx:a), ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xx)))))) (forall (Xx:a) (Xy:a), (((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy))))->((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xx)))))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy))))) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz)))))->((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz))))))))) of role conjecture named cTHM261_B_pme
% Conjecture to prove = (forall (P:((a->Prop)->Prop)), ((forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp)))))))->((and ((and (forall (Xx:a), ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xx)))))) (forall (Xx:a) (Xy:a), (((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy))))->((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xx)))))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy))))) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz)))))->((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz))))))))):Prop
% Parameter a_DUMMY:a.
% We need to prove ['(forall (P:((a->Prop)->Prop)), ((forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp)))))))->((and ((and (forall (Xx:a), ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xx)))))) (forall (Xx:a) (Xy:a), (((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy))))->((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xx)))))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy))))) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz)))))->((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz)))))))))']
% Parameter a:Type.
% Trying to prove (forall (P:((a->Prop)->Prop)), ((forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp)))))))->((and ((and (forall (Xx:a), ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xx)))))) (forall (Xx:a) (Xy:a), (((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy))))->((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xx)))))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy))))) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz)))))->((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz)))))))))
% Found eq_ref00:=(eq_ref0 (forall (Xx:a) (Xy:a) (Xz:a), (((and ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy))))) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz)))))->((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz))))))):(((eq Prop) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy))))) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz)))))->((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz))))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy))))) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz)))))->((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz)))))))
% Found (eq_ref0 (forall (Xx:a) (Xy:a) (Xz:a), (((and ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy))))) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz)))))->((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz))))))) as proof of (((eq Prop) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy))))) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz)))))->((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz))))))) b)
% Found ((eq_ref Prop) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy))))) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz)))))->((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz))))))) as proof of (((eq Prop) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy))))) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz)))))->((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz))))))) b)
% Found ((eq_ref Prop) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy))))) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz)))))->((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz))))))) as proof of (((eq Prop) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy))))) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz)))))->((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz))))))) b)
% Found ((eq_ref Prop) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy))))) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz)))))->((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz))))))) as proof of (((eq Prop) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy))))) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz)))))->((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz))))))) b)
% 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_ref000:=(eq_ref00 (ex (a->Prop))):(((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz))))->((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz)))))
% Found (eq_ref00 (ex (a->Prop))) as proof of (P0 (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz))))
% Found ((eq_ref0 (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz)))) (ex (a->Prop))) as proof of (P0 (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz))))
% Found (((eq_ref ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz)))) (ex (a->Prop))) as proof of (P0 (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz))))
% Found (((eq_ref ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz)))) (ex (a->Prop))) as proof of (P0 (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 (ex (a->Prop))):(((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz))))->((ex (a->Prop)) (fun (x:(a->Prop))=> ((and ((and (P x)) (x Xy))) (x Xz)))))
% Found (eta_expansion_dep000 (ex (a->Prop))) as proof of (P0 (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz))))
% Found ((eta_expansion_dep00 (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz)))) (ex (a->Prop))) as proof of (P0 (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz))))
% Found (((eta_expansion_dep0 (fun (x3:(a->Prop))=> Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz)))) (ex (a->Prop))) as proof of (P0 (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz))))
% Found ((((eta_expansion_dep (a->Prop)) (fun (x3:(a->Prop))=> Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz)))) (ex (a->Prop))) as proof of (P0 (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz))))
% Found ((((eta_expansion_dep (a->Prop)) (fun (x3:(a->Prop))=> Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz)))) (ex (a->Prop))) as proof of (P0 (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz))))
% Found eta_expansion0000:=(eta_expansion000 (ex (a->Prop))):(((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz))))->((ex (a->Prop)) (fun (x:(a->Prop))=> ((and ((and (P x)) (x Xy))) (x Xz)))))
% Found (eta_expansion000 (ex (a->Prop))) as proof of (P0 (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz))))
% Found ((eta_expansion00 (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz)))) (ex (a->Prop))) as proof of (P0 (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz))))
% Found (((eta_expansion0 Prop) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz)))) (ex (a->Prop))) as proof of (P0 (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz))))
% Found ((((eta_expansion (a->Prop)) Prop) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz)))) (ex (a->Prop))) as proof of (P0 (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz))))
% Found ((((eta_expansion (a->Prop)) Prop) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz)))) (ex (a->Prop))) as proof of (P0 (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz))))
% Found eq_ref00:=(eq_ref0 (forall (Xx:a) (Xy:a), (((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy))))->((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xx))))))):(((eq Prop) (forall (Xx:a) (Xy:a), (((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy))))->((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xx))))))) (forall (Xx:a) (Xy:a), (((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy))))->((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xx)))))))
% Found (eq_ref0 (forall (Xx:a) (Xy:a), (((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy))))->((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xx))))))) as proof of (((eq Prop) (forall (Xx:a) (Xy:a), (((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy))))->((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xx))))))) b)
% Found ((eq_ref Prop) (forall (Xx:a) (Xy:a), (((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy))))->((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xx))))))) as proof of (((eq Prop) (forall (Xx:a) (Xy:a), (((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy))))->((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xx))))))) b)
% Found ((eq_ref Prop) (forall (Xx:a) (Xy:a), (((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy))))->((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xx))))))) as proof of (((eq Prop) (forall (Xx:a) (Xy:a), (((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy))))->((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xx))))))) b)
% Found ((eq_ref Prop) (forall (Xx:a) (Xy:a), (((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy))))->((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xx))))))) as proof of (((eq Prop) (forall (Xx:a) (Xy:a), (((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy))))->((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xx))))))) b)
% Found eq_ref000:=(eq_ref00 (ex (a->Prop))):(((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy))))->((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy)))))
% Found (eq_ref00 (ex (a->Prop))) as proof of (P0 (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy))))
% Found ((eq_ref0 (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy)))) (ex (a->Prop))) as proof of (P0 (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy))))
% Found (((eq_ref ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy)))) (ex (a->Prop))) as proof of (P0 (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy))))
% Found (((eq_ref ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy)))) (ex (a->Prop))) as proof of (P0 (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy))))
% Found eq_ref000:=(eq_ref00 (ex (a->Prop))):(((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy))))->((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy)))))
% Found (eq_ref00 (ex (a->Prop))) as proof of (P0 (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy))))
% Found ((eq_ref0 (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy)))) (ex (a->Prop))) as proof of (P0 (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy))))
% Found (((eq_ref ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy)))) (ex (a->Prop))) as proof of (P0 (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy))))
% Found (((eq_ref ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy)))) (ex (a->Prop))) as proof of (P0 (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy))))
% Found eq_ref000:=(eq_ref00 (ex (a->Prop))):(((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy))))->((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy)))))
% Found (eq_ref00 (ex (a->Prop))) as proof of (P0 (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy))))
% Found ((eq_ref0 (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy)))) (ex (a->Prop))) as proof of (P0 (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy))))
% Found (((eq_ref ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy)))) (ex (a->Prop))) as proof of (P0 (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy))))
% Found (((eq_ref ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy)))) (ex (a->Prop))) as proof of (P0 (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy))))
% 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 eta_expansion_dep000:=(eta_expansion_dep00 (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz)))):(((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz)))) (fun (x:(a->Prop))=> ((and ((and (P x)) (x Xx))) (x Xz))))
% Found (eta_expansion_dep00 (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz)))) as proof of (((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz)))) b)
% Found ((eta_expansion_dep0 (fun (x2:(a->Prop))=> Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz)))) as proof of (((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz)))) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x2:(a->Prop))=> Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz)))) as proof of (((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz)))) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x2:(a->Prop))=> Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz)))) as proof of (((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz)))) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x2:(a->Prop))=> Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz)))) as proof of (((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz)))) b)
% Found x3:((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz))))
% Instantiate: x1:=(fun (x4:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S x4))))):(a->Prop)
% Found (fun (x3:((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz)))))=> x3) as proof of (x1 Xz)
% Found (fun (x2:((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy))))) (x3:((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz)))))=> x3) as proof of (((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz))))->(x1 Xz))
% Found (fun (x2:((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy))))) (x3:((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz)))))=> x3) as proof of (((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy))))->(((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz))))->(x1 Xz)))
% Found (and_rect00 (fun (x2:((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy))))) (x3:((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz)))))=> x3)) as proof of (x1 Xz)
% Found ((and_rect0 (x1 Xz)) (fun (x2:((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy))))) (x3:((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz)))))=> x3)) as proof of (x1 Xz)
% Found (((fun (P0:Type) (x2:(((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy))))->(((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz))))->P0)))=> (((((and_rect ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy))))) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz))))) P0) x2) x0)) (x1 Xz)) (fun (x2:((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy))))) (x3:((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz)))))=> x3)) as proof of (x1 Xz)
% Found (((fun (P0:Type) (x2:(((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy))))->(((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz))))->P0)))=> (((((and_rect ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy))))) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz))))) P0) x2) x0)) (x1 Xz)) (fun (x2:((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy))))) (x3:((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz)))))=> x3)) as proof of (x1 Xz)
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) ((and ((and (P x1)) (x1 Xx))) (x1 Xz)))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and ((and (P x1)) (x1 Xx))) (x1 Xz)))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and ((and (P x1)) (x1 Xx))) (x1 Xz)))
% Found (fun (x1:(a->Prop))=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) ((and ((and (P x1)) (x1 Xx))) (x1 Xz)))
% Found (fun (x1:(a->Prop))=> ((eq_ref Prop) (f x1))) as proof of (forall (x:(a->Prop)), (((eq Prop) (f x)) ((and ((and (P x)) (x Xx))) (x Xz))))
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) ((and ((and (P x1)) (x1 Xx))) (x1 Xz)))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and ((and (P x1)) (x1 Xx))) (x1 Xz)))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and ((and (P x1)) (x1 Xx))) (x1 Xz)))
% Found (fun (x1:(a->Prop))=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) ((and ((and (P x1)) (x1 Xx))) (x1 Xz)))
% Found (fun (x1:(a->Prop))=> ((eq_ref Prop) (f x1))) as proof of (forall (x:(a->Prop)), (((eq Prop) (f x)) ((and ((and (P x)) (x Xx))) (x Xz))))
% Found x2:((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz))))
% Instantiate: b:=(fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz))):((a->Prop)->Prop)
% Found x2 as proof of (P0 b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz)))):(((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz)))) (fun (x:(a->Prop))=> ((and ((and (P x)) (x Xx))) (x Xz))))
% Found (eta_expansion_dep00 (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz)))) as proof of (((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz)))) b)
% Found ((eta_expansion_dep0 (fun (x4:(a->Prop))=> Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz)))) as proof of (((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz)))) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x4:(a->Prop))=> Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz)))) as proof of (((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz)))) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x4:(a->Prop))=> Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz)))) as proof of (((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz)))) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x4:(a->Prop))=> Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz)))) as proof of (((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz)))) b)
% Found eq_ref000:=(eq_ref00 (ex (a->Prop))):(((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz))))->((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz)))))
% Found (eq_ref00 (ex (a->Prop))) as proof of (((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz))))->(P0 b))
% Found ((eq_ref0 (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz)))) (ex (a->Prop))) as proof of (((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz))))->(P0 b))
% Found (((eq_ref ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz)))) (ex (a->Prop))) as proof of (((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz))))->(P0 b))
% Found (((eq_ref ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz)))) (ex (a->Prop))) as proof of (((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz))))->(P0 b))
% Found (fun (x1:((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy)))))=> (((eq_ref ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz)))) (ex (a->Prop)))) as proof of (((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz))))->(P0 b))
% Found (fun (x1:((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy)))))=> (((eq_ref ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz)))) (ex (a->Prop)))) as proof of (((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy))))->(((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz))))->(P0 b)))
% Found (and_rect00 (fun (x1:((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy)))))=> (((eq_ref ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz)))) (ex (a->Prop))))) as proof of (P0 b)
% Found ((and_rect0 (P0 b)) (fun (x1:((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy)))))=> (((eq_ref ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz)))) (ex (a->Prop))))) as proof of (P0 b)
% Found (((fun (P1:Type) (x1:(((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy))))->(((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz))))->P1)))=> (((((and_rect ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy))))) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz))))) P1) x1) x0)) (P0 b)) (fun (x1:((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy)))))=> (((eq_ref ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz)))) (ex (a->Prop))))) as proof of (P0 b)
% Found (((fun (P1:Type) (x1:(((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy))))->(((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz))))->P1)))=> (((((and_rect ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy))))) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz))))) P1) x1) x0)) (P0 b)) (fun (x1:((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy)))))=> (((eq_ref ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz)))) (ex (a->Prop))))) as proof of (P0 b)
% Found x2:((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz))))
% Instantiate: f:=(fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz))):((a->Prop)->Prop)
% Found x2 as proof of (P0 f)
% Found x2:((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz))))
% Instantiate: f:=(fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz))):((a->Prop)->Prop)
% Found x2 as proof of (P0 f)
% Found eq_ref000:=(eq_ref00 (ex (a->Prop))):(((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz))))->((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz)))))
% Found (eq_ref00 (ex (a->Prop))) as proof of (((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz))))->(P0 f))
% Found ((eq_ref0 (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz)))) (ex (a->Prop))) as proof of (((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz))))->(P0 f))
% Found (((eq_ref ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz)))) (ex (a->Prop))) as proof of (((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz))))->(P0 f))
% Found (((eq_ref ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz)))) (ex (a->Prop))) as proof of (((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz))))->(P0 f))
% Found (fun (x1:((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy)))))=> (((eq_ref ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz)))) (ex (a->Prop)))) as proof of (((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz))))->(P0 f))
% Found (fun (x1:((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy)))))=> (((eq_ref ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz)))) (ex (a->Prop)))) as proof of (((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy))))->(((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz))))->(P0 f)))
% Found (and_rect00 (fun (x1:((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy)))))=> (((eq_ref ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz)))) (ex (a->Prop))))) as proof of (P0 f)
% Found ((and_rect0 (P0 f)) (fun (x1:((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy)))))=> (((eq_ref ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz)))) (ex (a->Prop))))) as proof of (P0 f)
% Found (((fun (P1:Type) (x1:(((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy))))->(((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz))))->P1)))=> (((((and_rect ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy))))) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz))))) P1) x1) x0)) (P0 f)) (fun (x1:((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy)))))=> (((eq_ref ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz)))) (ex (a->Prop))))) as proof of (P0 f)
% Found (((fun (P1:Type) (x1:(((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy))))->(((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz))))->P1)))=> (((((and_rect ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy))))) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz))))) P1) x1) x0)) (P0 f)) (fun (x1:((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy)))))=> (((eq_ref ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz)))) (ex (a->Prop))))) as proof of (P0 f)
% Found eq_ref000:=(eq_ref00 (ex (a->Prop))):(((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz))))->((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz)))))
% Found (eq_ref00 (ex (a->Prop))) as proof of (((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz))))->(P0 f))
% Found ((eq_ref0 (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz)))) (ex (a->Prop))) as proof of (((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz))))->(P0 f))
% Found (((eq_ref ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz)))) (ex (a->Prop))) as proof of (((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz))))->(P0 f))
% Found (((eq_ref ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz)))) (ex (a->Prop))) as proof of (((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz))))->(P0 f))
% Found (fun (x1:((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy)))))=> (((eq_ref ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz)))) (ex (a->Prop)))) as proof of (((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz))))->(P0 f))
% Found (fun (x1:((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy)))))=> (((eq_ref ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz)))) (ex (a->Prop)))) as proof of (((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy))))->(((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz))))->(P0 f)))
% Found (and_rect00 (fun (x1:((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy)))))=> (((eq_ref ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz)))) (ex (a->Prop))))) as proof of (P0 f)
% Found ((and_rect0 (P0 f)) (fun (x1:((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy)))))=> (((eq_ref ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz)))) (ex (a->Prop))))) as proof of (P0 f)
% Found (((fun (P1:Type) (x1:(((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy))))->(((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz))))->P1)))=> (((((and_rect ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy))))) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz))))) P1) x1) x0)) (P0 f)) (fun (x1:((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy)))))=> (((eq_ref ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz)))) (ex (a->Prop))))) as proof of (P0 f)
% Found (((fun (P1:Type) (x1:(((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy))))->(((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz))))->P1)))=> (((((and_rect ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy))))) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz))))) P1) x1) x0)) (P0 f)) (fun (x1:((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy)))))=> (((eq_ref ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz)))) (ex (a->Prop))))) as proof of (P0 f)
% Found eta_expansion000:=(eta_expansion00 (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz)))):(((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz)))) (fun (x:(a->Prop))=> ((and ((and (P x)) (x Xy))) (x Xz))))
% Found (eta_expansion00 (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz)))) as proof of (((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz)))) b)
% Found ((eta_expansion0 Prop) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz)))) as proof of (((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz)))) b)
% Found (((eta_expansion (a->Prop)) Prop) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz)))) as proof of (((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz)))) b)
% Found (((eta_expansion (a->Prop)) Prop) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz)))) as proof of (((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz)))) b)
% Found (((eta_expansion (a->Prop)) Prop) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz)))) as proof of (((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz)))) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq ((a->Prop)->Prop)) b) (fun (x:(a->Prop))=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq ((a->Prop)->Prop)) b) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz))))
% Found ((eta_expansion_dep0 (fun (x3:(a->Prop))=> Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz))))
% Found (((eta_expansion_dep (a->Prop)) (fun (x3:(a->Prop))=> Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz))))
% Found (((eta_expansion_dep (a->Prop)) (fun (x3:(a->Prop))=> Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz))))
% Found (((eta_expansion_dep (a->Prop)) (fun (x3:(a->Prop))=> Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz))))
% Found eq_ref000:=(eq_ref00 (ex (a->Prop))):(((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz))))->((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz)))))
% Found (eq_ref00 (ex (a->Prop))) as proof of (P0 (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz))))
% Found ((eq_ref0 (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz)))) (ex (a->Prop))) as proof of (P0 (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz))))
% Found (((eq_ref ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz)))) (ex (a->Prop))) as proof of (P0 (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz))))
% Found (((eq_ref ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz)))) (ex (a->Prop))) as proof of (P0 (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz))))
% Found eq_ref00:=(eq_ref0 (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz)))):(((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz)))) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz))))
% Found (eq_ref0 (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz)))) as proof of (((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz)))) b)
% Found ((eq_ref ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz)))) as proof of (((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz)))) b)
% Found ((eq_ref ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz)))) as proof of (((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz)))) b)
% Found ((eq_ref ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz)))) as proof of (((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz)))) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq ((a->Prop)->Prop)) b) (fun (x:(a->Prop))=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq ((a->Prop)->Prop)) b) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz))))
% Found ((eta_expansion_dep0 (fun (x3:(a->Prop))=> Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz))))
% Found (((eta_expansion_dep (a->Prop)) (fun (x3:(a->Prop))=> Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz))))
% Found (((eta_expansion_dep (a->Prop)) (fun (x3:(a->Prop))=> Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz))))
% Found (((eta_expansion_dep (a->Prop)) (fun (x3:(a->Prop))=> Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz))))
% 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 ((and (P x3)) (x3 Xx))) (x3 Xz)))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((and ((and (P x3)) (x3 Xx))) (x3 Xz)))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((and ((and (P x3)) (x3 Xx))) (x3 Xz)))
% Found (fun (x3:(a->Prop))=> ((eq_ref Prop) (f x3))) as proof of (((eq Prop) (f x3)) ((and ((and (P x3)) (x3 Xx))) (x3 Xz)))
% Found (fun (x3:(a->Prop))=> ((eq_ref Prop) (f x3))) as proof of (forall (x:(a->Prop)), (((eq Prop) (f x)) ((and ((and (P x)) (x Xx))) (x Xz))))
% 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 ((and (P x3)) (x3 Xx))) (x3 Xz)))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((and ((and (P x3)) (x3 Xx))) (x3 Xz)))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((and ((and (P x3)) (x3 Xx))) (x3 Xz)))
% Found (fun (x3:(a->Prop))=> ((eq_ref Prop) (f x3))) as proof of (((eq Prop) (f x3)) ((and ((and (P x3)) (x3 Xx))) (x3 Xz)))
% Found (fun (x3:(a->Prop))=> ((eq_ref Prop) (f x3))) as proof of (forall (x:(a->Prop)), (((eq Prop) (f x)) ((and ((and (P x)) (x Xx))) (x Xz))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 (ex (a->Prop))):(((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz))))->((ex (a->Prop)) (fun (x:(a->Prop))=> ((and ((and (P x)) (x Xy))) (x Xz)))))
% Found (eta_expansion_dep000 (ex (a->Prop))) as proof of (P0 (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz))))
% Found ((eta_expansion_dep00 (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz)))) (ex (a->Prop))) as proof of (P0 (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz))))
% Found (((eta_expansion_dep0 (fun (x6:(a->Prop))=> Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz)))) (ex (a->Prop))) as proof of (P0 (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz))))
% Found ((((eta_expansion_dep (a->Prop)) (fun (x6:(a->Prop))=> Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz)))) (ex (a->Prop))) as proof of (P0 (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz))))
% Found ((((eta_expansion_dep (a->Prop)) (fun (x6:(a->Prop))=> Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz)))) (ex (a->Prop))) as proof of (P0 (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz))))
% Found eq_ref00:=(eq_ref0 (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xx)))):(((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xx)))) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xx))))
% Found (eq_ref0 (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xx)))) as proof of (((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xx)))) b)
% Found ((eq_ref ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xx)))) as proof of (((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xx)))) b)
% Found ((eq_ref ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xx)))) as proof of (((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xx)))) b)
% Found ((eq_ref ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xx)))) as proof of (((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xx)))) b)
% Found x5:((and ((and (P x4)) (x4 Xy))) (x4 Xz))
% Instantiate: x1:=(fun (x6:a)=> ((and ((and (P x4)) (x4 Xy))) (x4 x6))):(a->Prop)
% Found (fun (x5:((and ((and (P x4)) (x4 Xy))) (x4 Xz)))=> x5) as proof of (x1 Xz)
% Found x5:((and ((and (P x4)) (x4 Xy))) (x4 Xz))
% Instantiate: x3:=(fun (x6:a)=> ((and ((and (P x4)) (x4 Xy))) (x4 x6))):(a->Prop)
% Found (fun (x5:((and ((and (P x4)) (x4 Xy))) (x4 Xz)))=> x5) as proof of (x3 Xz)
% Found x0:((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy))))
% Instantiate: b:=(fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy))):((a->Prop)->Prop)
% Found x0 as proof of (P0 b)
% Found eta_expansion0000:=(eta_expansion000 (ex (a->Prop))):(((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz))))->((ex (a->Prop)) (fun (x:(a->Prop))=> ((and ((and (P x)) (x Xy))) (x Xz)))))
% Found (eta_expansion000 (ex (a->Prop))) as proof of (P0 (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz))))
% Found ((eta_expansion00 (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz)))) (ex (a->Prop))) as proof of (P0 (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz))))
% Found (((eta_expansion0 Prop) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz)))) (ex (a->Prop))) as proof of (P0 (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz))))
% Found ((((eta_expansion (a->Prop)) Prop) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz)))) (ex (a->Prop))) as proof of (P0 (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz))))
% Found ((((eta_expansion (a->Prop)) Prop) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz)))) (ex (a->Prop))) as proof of (P0 (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 (ex (a->Prop))):(((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz))))->((ex (a->Prop)) (fun (x:(a->Prop))=> ((and ((and (P x)) (x Xy))) (x Xz)))))
% Found (eta_expansion_dep000 (ex (a->Prop))) as proof of (P0 (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz))))
% Found ((eta_expansion_dep00 (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz)))) (ex (a->Prop))) as proof of (P0 (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz))))
% Found (((eta_expansion_dep0 (fun (x6:(a->Prop))=> Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz)))) (ex (a->Prop))) as proof of (P0 (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz))))
% Found ((((eta_expansion_dep (a->Prop)) (fun (x6:(a->Prop))=> Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz)))) (ex (a->Prop))) as proof of (P0 (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz))))
% Found ((((eta_expansion_dep (a->Prop)) (fun (x6:(a->Prop))=> Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz)))) (ex (a->Prop))) as proof of (P0 (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz))))
% Found x3:((and (P x1)) (x1 Xx))
% Instantiate: Xx:=Xx0:a;x6:=x1:(a->Prop)
% Found x3 as proof of ((and (P x6)) (x6 Xx0))
% Found eq_ref00:=(eq_ref0 (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xx)))):(((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xx)))) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xx))))
% Found (eq_ref0 (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xx)))) as proof of (((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xx)))) b)
% Found ((eq_ref ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xx)))) as proof of (((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xx)))) b)
% Found ((eq_ref ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xx)))) as proof of (((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xx)))) b)
% Found ((eq_ref ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xx)))) as proof of (((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xx)))) b)
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy))))) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz)))))->((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy))))) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz)))))->((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy))))) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz)))))->((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy))))) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz)))))->((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz)))))))
% Found conj:(forall (A:Prop) (B:Prop), (A->(B->((and A) B))))
% Instantiate: a0:=(forall (A:Prop) (B:Prop), (A->(B->((and A) B)))):Prop
% Found conj as proof of a0
% Found x0:((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy))))
% Instantiate: f:=(fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy))):((a->Prop)->Prop)
% Found x0 as proof of (P0 f)
% Found x0:((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy))))
% Instantiate: f:=(fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy))):((a->Prop)->Prop)
% Found x0 as proof of (P0 f)
% Found eq_ref00:=(eq_ref0 (forall (Xx:a) (Xy:a) (Xz:a), (((and ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy))))) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz)))))->((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz))))))):(((eq Prop) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy))))) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz)))))->((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz))))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy))))) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz)))))->((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz)))))))
% Found (eq_ref0 (forall (Xx:a) (Xy:a) (Xz:a), (((and ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy))))) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz)))))->((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz))))))) as proof of (((eq Prop) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy))))) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz)))))->((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz))))))) b)
% Found ((eq_ref Prop) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy))))) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz)))))->((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz))))))) as proof of (((eq Prop) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy))))) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz)))))->((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz))))))) b)
% Found ((eq_ref Prop) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy))))) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz)))))->((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz))))))) as proof of (((eq Prop) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy))))) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz)))))->((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz))))))) b)
% Found ((eq_ref Prop) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy))))) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz)))))->((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz))))))) as proof of (((eq Prop) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy))))) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz)))))->((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz))))))) b)
% Found x1:((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy))))
% Instantiate: b:=(fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy))):((a->Prop)->Prop)
% Found x1 as proof of (P0 b)
% Found x1:((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy))))
% Instantiate: b:=(fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy))):((a->Prop)->Prop)
% Found x1 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz)))):(((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz)))) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz))))
% Found (eq_ref0 (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz)))) as proof of (((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz)))) b)
% Found ((eq_ref ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz)))) as proof of (((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz)))) b)
% Found ((eq_ref ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz)))) as proof of (((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz)))) b)
% Found ((eq_ref ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz)))) as proof of (((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz)))) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz)))):(((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz)))) (fun (x:(a->Prop))=> ((and ((and (P x)) (x Xx))) (x Xz))))
% Found (eta_expansion_dep00 (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz)))) as proof of (((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz)))) b)
% Found ((eta_expansion_dep0 (fun (x6:(a->Prop))=> Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz)))) as proof of (((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz)))) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x6:(a->Prop))=> Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz)))) as proof of (((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz)))) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x6:(a->Prop))=> Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz)))) as proof of (((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz)))) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x6:(a->Prop))=> Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz)))) as proof of (((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz)))) b)
% Found eq_ref00:=(eq_ref0 (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy)))):(((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy)))) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy))))
% Found (eq_ref0 (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy)))) as proof of (((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy)))) b)
% Found ((eq_ref ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy)))) as proof of (((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy)))) b)
% Found ((eq_ref ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy)))) as proof of (((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy)))) b)
% Found ((eq_ref ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy)))) as proof of (((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy)))) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq ((a->Prop)->Prop)) b) (fun (x:(a->Prop))=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq ((a->Prop)->Prop)) b) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xx))))
% Found ((eta_expansion_dep0 (fun (x1:(a->Prop))=> Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xx))))
% Found (((eta_expansion_dep (a->Prop)) (fun (x1:(a->Prop))=> Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xx))))
% Found (((eta_expansion_dep (a->Prop)) (fun (x1:(a->Prop))=> Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xx))))
% Found (((eta_expansion_dep (a->Prop)) (fun (x1:(a->Prop))=> Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xx))))
% Found x5:((and (P x3)) (x3 Xx))
% Instantiate: x7:=x3:(a->Prop)
% Found x5 as proof of ((and (P x7)) (x7 Xx))
% Found eta_expansion0000:=(eta_expansion000 (ex (a->Prop))):(((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz))))->((ex (a->Prop)) (fun (x:(a->Prop))=> ((and ((and (P x)) (x Xy))) (x Xz)))))
% Found (eta_expansion000 (ex (a->Prop))) as proof of (P0 (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz))))
% Found ((eta_expansion00 (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz)))) (ex (a->Prop))) as proof of (P0 (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz))))
% Found (((eta_expansion0 Prop) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz)))) (ex (a->Prop))) as proof of (P0 (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz))))
% Found ((((eta_expansion (a->Prop)) Prop) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz)))) (ex (a->Prop))) as proof of (P0 (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz))))
% Found ((((eta_expansion (a->Prop)) Prop) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz)))) (ex (a->Prop))) as proof of (P0 (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz))))
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) ((and ((and (P x0)) (x0 Xx))) (x0 Xx)))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and ((and (P x0)) (x0 Xx))) (x0 Xx)))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and ((and (P x0)) (x0 Xx))) (x0 Xx)))
% Found (fun (x0:(a->Prop))=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) ((and ((and (P x0)) (x0 Xx))) (x0 Xx)))
% Found (fun (x0:(a->Prop))=> ((eq_ref Prop) (f x0))) as proof of (forall (x:(a->Prop)), (((eq Prop) (f x)) ((and ((and (P x)) (x Xx))) (x Xx))))
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) ((and ((and (P x0)) (x0 Xx))) (x0 Xx)))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and ((and (P x0)) (x0 Xx))) (x0 Xx)))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and ((and (P x0)) (x0 Xx))) (x0 Xx)))
% Found (fun (x0:(a->Prop))=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) ((and ((and (P x0)) (x0 Xx))) (x0 Xx)))
% Found (fun (x0:(a->Prop))=> ((eq_ref Prop) (f x0))) as proof of (forall (x:(a->Prop)), (((eq Prop) (f x)) ((and ((and (P x)) (x Xx))) (x Xx))))
% Found x6:((and (P x4)) (x4 Xx))
% Instantiate: x1:=x4:(a->Prop)
% Found (fun (x7:(x4 Xy))=> x6) as proof of ((and (P x1)) (x1 Xx))
% Found (fun (x6:((and (P x4)) (x4 Xx))) (x7:(x4 Xy))=> x6) as proof of ((x4 Xy)->((and (P x1)) (x1 Xx)))
% Found (fun (x6:((and (P x4)) (x4 Xx))) (x7:(x4 Xy))=> x6) as proof of (((and (P x4)) (x4 Xx))->((x4 Xy)->((and (P x1)) (x1 Xx))))
% Found (and_rect10 (fun (x6:((and (P x4)) (x4 Xx))) (x7:(x4 Xy))=> x6)) as proof of ((and (P x1)) (x1 Xx))
% Found ((and_rect1 ((and (P x1)) (x1 Xx))) (fun (x6:((and (P x4)) (x4 Xx))) (x7:(x4 Xy))=> x6)) as proof of ((and (P x1)) (x1 Xx))
% Found (((fun (P0:Type) (x6:(((and (P x4)) (x4 Xx))->((x4 Xy)->P0)))=> (((((and_rect ((and (P x4)) (x4 Xx))) (x4 Xy)) P0) x6) x5)) ((and (P x1)) (x1 Xx))) (fun (x6:((and (P x4)) (x4 Xx))) (x7:(x4 Xy))=> x6)) as proof of ((and (P x1)) (x1 Xx))
% Found (fun (x5:((and ((and (P x4)) (x4 Xx))) (x4 Xy)))=> (((fun (P0:Type) (x6:(((and (P x4)) (x4 Xx))->((x4 Xy)->P0)))=> (((((and_rect ((and (P x4)) (x4 Xx))) (x4 Xy)) P0) x6) x5)) ((and (P x1)) (x1 Xx))) (fun (x6:((and (P x4)) (x4 Xx))) (x7:(x4 Xy))=> x6))) as proof of ((and (P x1)) (x1 Xx))
% Found x1:((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy))))
% Instantiate: f:=(fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy))):((a->Prop)->Prop)
% Found x1 as proof of (P0 f)
% Found x1:((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy))))
% Instantiate: f:=(fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy))):((a->Prop)->Prop)
% Found x1 as proof of (P0 f)
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P00):((P00 (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz))))->(P00 (fun (x:(a->Prop))=> ((and ((and (P x)) (x Xx))) (x Xz)))))
% Found (eta_expansion_dep000 P00) as proof of (P01 (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz))))
% Found ((eta_expansion_dep00 (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz)))) P00) as proof of (P01 (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz))))
% Found (((eta_expansion_dep0 (fun (x3:(a->Prop))=> Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz)))) P00) as proof of (P01 (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz))))
% Found ((((eta_expansion_dep (a->Prop)) (fun (x3:(a->Prop))=> Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz)))) P00) as proof of (P01 (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz))))
% Found ((((eta_expansion_dep (a->Prop)) (fun (x3:(a->Prop))=> Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz)))) P00) as proof of (P01 (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz))))
% Found eta_expansion0000:=(eta_expansion000 P00):((P00 (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz))))->(P00 (fun (x:(a->Prop))=> ((and ((and (P x)) (x Xx))) (x Xz)))))
% Found (eta_expansion000 P00) as proof of (P01 (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz))))
% Found ((eta_expansion00 (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz)))) P00) as proof of (P01 (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz))))
% Found (((eta_expansion0 Prop) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz)))) P00) as proof of (P01 (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz))))
% Found ((((eta_expansion (a->Prop)) Prop) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz)))) P00) as proof of (P01 (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz))))
% Found ((((eta_expansion (a->Prop)) Prop) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz)))) P00) as proof of (P01 (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz))))
% Found x1:((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy))))
% Instantiate: f:=(fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy))):((a->Prop)->Prop)
% Found x1 as proof of (P0 f)
% Found x1:((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy))))
% Instantiate: f:=(fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy))):((a->Prop)->Prop)
% Found x1 as proof of (P0 f)
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) ((and ((and (P x1)) (x1 Xy))) (x1 Xx)))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and ((and (P x1)) (x1 Xy))) (x1 Xx)))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and ((and (P x1)) (x1 Xy))) (x1 Xx)))
% Found (fun (x1:(a->Prop))=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) ((and ((and (P x1)) (x1 Xy))) (x1 Xx)))
% Found (fun (x1:(a->Prop))=> ((eq_ref Prop) (f x1))) as proof of (forall (x:(a->Prop)), (((eq Prop) (f x)) ((and ((and (P x)) (x Xy))) (x Xx))))
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) ((and ((and (P x1)) (x1 Xy))) (x1 Xx)))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and ((and (P x1)) (x1 Xy))) (x1 Xx)))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and ((and (P x1)) (x1 Xy))) (x1 Xx)))
% Found (fun (x1:(a->Prop))=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) ((and ((and (P x1)) (x1 Xy))) (x1 Xx)))
% Found (fun (x1:(a->Prop))=> ((eq_ref Prop) (f x1))) as proof of (forall (x:(a->Prop)), (((eq Prop) (f x)) ((and ((and (P x)) (x Xy))) (x Xx))))
% Found eq_ref000:=(eq_ref00 (ex (a->Prop))):(((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy))))->((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy)))))
% Found (eq_ref00 (ex (a->Prop))) as proof of (P0 (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy))))
% Found ((eq_ref0 (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy)))) (ex (a->Prop))) as proof of (P0 (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy))))
% Found (((eq_ref ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy)))) (ex (a->Prop))) as proof of (P0 (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy))))
% Found (((eq_ref ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy)))) (ex (a->Prop))) as proof of (P0 (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy))))
% Found eq_ref00:=(eq_ref0 b):(((eq ((a->Prop)->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq ((a->Prop)->Prop)) b) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xx))))
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xx))))
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xx))))
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xx))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy)))):(((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy)))) (fun (x:(a->Prop))=> ((and ((and (P x)) (x Xx))) (x Xy))))
% Found (eta_expansion_dep00 (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy)))) as proof of (((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy)))) b)
% Found ((eta_expansion_dep0 (fun (x1:(a->Prop))=> Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy)))) as proof of (((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy)))) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x1:(a->Prop))=> Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy)))) as proof of (((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy)))) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x1:(a->Prop))=> Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy)))) as proof of (((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy)))) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x1:(a->Prop))=> Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy)))) as proof of (((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy)))) b)
% Found x6:((and (P x4)) (x4 Xx))
% Instantiate: x1:=x4:(a->Prop)
% Found x6 as proof of ((and (P x1)) (x1 Xx))
% Found eq_ref000:=(eq_ref00 (ex (a->Prop))):(((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx0))) (S Xy))))->((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx0))) (S Xy)))))
% Found (eq_ref00 (ex (a->Prop))) as proof of (P0 (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx0))) (S Xy))))
% Found ((eq_ref0 (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx0))) (S Xy)))) (ex (a->Prop))) as proof of (P0 (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx0))) (S Xy))))
% Found (((eq_ref ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx0))) (S Xy)))) (ex (a->Prop))) as proof of (P0 (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx0))) (S Xy))))
% Found (((eq_ref ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx0))) (S Xy)))) (ex (a->Prop))) as proof of (P0 (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx0))) (S Xy))))
% Found x6:((and (P x3)) (x3 Xx))
% Instantiate: x5:=x3:(a->Prop)
% Found x6 as proof of ((and (P x5)) (x5 Xx))
% 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 x6:((and (P x4)) (x4 Xx))
% Instantiate: x3:=x4:(a->Prop)
% Found x6 as proof of ((and (P x3)) (x3 Xx))
% Found eta_expansion000:=(eta_expansion00 (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz)))):(((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz)))) (fun (x:(a->Prop))=> ((and ((and (P x)) (x Xx))) (x Xz))))
% Found (eta_expansion00 (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz)))) as proof of (((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz)))) b)
% Found ((eta_expansion0 Prop) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz)))) as proof of (((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz)))) b)
% Found (((eta_expansion (a->Prop)) Prop) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz)))) as proof of (((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz)))) b)
% Found (((eta_expansion (a->Prop)) Prop) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz)))) as proof of (((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz)))) b)
% Found (((eta_expansion (a->Prop)) Prop) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz)))) as proof of (((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq ((a->Prop)->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq ((a->Prop)->Prop)) b) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz))))
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz))))
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz))))
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz)))):(((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz)))) (fun (x:(a->Prop))=> ((and ((and (P x)) (x Xx))) (x Xz))))
% Found (eta_expansion_dep00 (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz)))) as proof of (((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz)))) b)
% Found ((eta_expansion_dep0 (fun (x3:(a->Prop))=> Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz)))) as proof of (((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz)))) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x3:(a->Prop))=> Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz)))) as proof of (((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz)))) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x3:(a->Prop))=> Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz)))) as proof of (((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz)))) b)
% Found (((eta_expansion_dep (a->Prop)) (fun (x3:(a->Prop))=> Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz)))) as proof of (((eq ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq ((a->Prop)->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq ((a->Prop)->Prop)) b) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz))))
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz))))
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz))))
% Found ((eq_ref ((a->Prop)->Prop)) b) as proof of (((eq ((a->Prop)->Prop)) b) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz))))
% Found eq_ref000:=(eq_ref00 (ex (a->Prop))):(((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz))))->((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz)))))
% Found (eq_ref00 (ex (a->Prop))) as proof of (P0 (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz))))
% Found ((eq_ref0 (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz)))) (ex (a->Prop))) as proof of (P0 (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz))))
% Found (((eq_ref ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz)))) (ex (a->Prop))) as proof of (P0 (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz))))
% Found (((eq_ref ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz)))) (ex (a->Prop))) as proof of (P0 (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz))))
% Found eq_ref000:=(eq_ref00 (ex (a->Prop))):(((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz))))->((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz)))))
% Found (eq_ref00 (ex (a->Prop))) as proof of (P0 (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz))))
% Found ((eq_ref0 (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz)))) (ex (a->Prop))) as proof of (P0 (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz))))
% Found (((eq_ref ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz)))) (ex (a->Prop))) as proof of (P0 (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz))))
% Found (((eq_ref ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz)))) (ex (a->Prop))) as proof of (P0 (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz))))
% Found eq_ref00:=(eq_ref0 (forall (Xx:a) (Xy:a) (Xz:a), (((and ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy))))) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz)))))->((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz))))))):(((eq Prop) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy))))) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz)))))->((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz))))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy))))) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz)))))->((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz)))))))
% Found (eq_ref0 (forall (Xx:a) (Xy:a) (Xz:a), (((and ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy))))) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz)))))->((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz))))))) as proof of (((eq Prop) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy))))) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz)))))->((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz))))))) b)
% Found ((eq_ref Prop) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy))))) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz)))))->((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz))))))) as proof of (((eq Prop) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy))))) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz)))))->((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz))))))) b)
% Found ((eq_ref Prop) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy))))) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz)))))->((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz))))))) as proof of (((eq Prop) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy))))) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz)))))->((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz))))))) b)
% Found ((eq_ref Prop) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy))))) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz)))))->((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz))))))) as proof of (((eq Prop) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xy))))) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz)))))->((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz))))))) 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 ((and (P x5)) (x5 Xx))) (x5 Xz)))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) ((and ((and (P x5)) (x5 Xx))) (x5 Xz)))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) ((and ((and (P x5)) (x5 Xx))) (x5 Xz)))
% Found (fun (x5:(a->Prop))=> ((eq_ref Prop) (f x5))) as proof of (((eq Prop) (f x5)) ((and ((and (P x5)) (x5 Xx))) (x5 Xz)))
% Found (fun (x5:(a->Prop))=> ((eq_ref Prop) (f x5))) as proof of (forall (x:(a->Prop)), (((eq Prop) (f x)) ((and ((and (P x)) (x Xx))) (x Xz))))
% 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 ((and (P x5)) (x5 Xx))) (x5 Xz)))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) ((and ((and (P x5)) (x5 Xx))) (x5 Xz)))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) ((and ((and (P x5)) (x5 Xx))) (x5 Xz)))
% Found (fun (x5:(a->Prop))=> ((eq_ref Prop) (f x5))) as proof of (((eq Prop) (f x5)) ((and ((and (P x5)) (x5 Xx))) (x5 Xz)))
% Found (fun (x5:(a->Prop))=> ((eq_ref Prop) (f x5))) as proof of (forall (x:(a->Prop)), (((eq Prop) (f x)) ((and ((and (P x)) (x Xx))) (x Xz))))
% 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 ((and (P x5)) (x5 Xx))) (x5 Xz)))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) ((and ((and (P x5)) (x5 Xx))) (x5 Xz)))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) ((and ((and (P x5)) (x5 Xx))) (x5 Xz)))
% Found (fun (x5:(a->Prop))=> ((eq_ref Prop) (f x5))) as proof of (((eq Prop) (f x5)) ((and ((and (P x5)) (x5 Xx))) (x5 Xz)))
% Found (fun (x5:(a->Prop))=> ((eq_ref Prop) (f x5))) as proof of (forall (x:(a->Prop)), (((eq Prop) (f x)) ((and ((and (P x)) (x Xx))) (x Xz))))
% 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 ((and (P x5)) (x5 Xx))) (x5 Xz)))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) ((and ((and (P x5)) (x5 Xx))) (x5 Xz)))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) ((and ((and (P x5)) (x5 Xx))) (x5 Xz)))
% Found (fun (x5:(a->Prop))=> ((eq_ref Prop) (f x5))) as proof of (((eq Prop) (f x5)) ((and ((and (P x5)) (x5 Xx))) (x5 Xz)))
% Found (fun (x5:(a->Prop))=> ((eq_ref Prop) (f x5))) as proof of (forall (x:(a->Prop)), (((eq Prop) (f x)) ((and ((and (P x)) (x Xx))) (x Xz))))
% Found x3:((and (P x1)) (x1 Xx))
% Instantiate: x8:=x1:(a->Prop);Xx:=Xx0:a
% Found x3 as proof of ((and (P x8)) (x8 Xx0))
% Found x4:((and (P x1)) (x1 Xx))
% Instantiate: Xx:=Xx0:a;x6:=x1:(a->Prop)
% Found x4 as proof of ((and (P x6)) (x6 Xx0))
% 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_ref000:=(eq_ref00 (ex (a->Prop))):(((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz))))->((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz)))))
% Found (eq_ref00 (ex (a->Prop))) as proof of (P0 (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz))))
% Found ((eq_ref0 (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz)))) (ex (a->Prop))) as proof of (P0 (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz))))
% Found (((eq_ref ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz)))) (ex (a->Prop))) as proof of (P0 (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz))))
% Found (((eq_ref ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz)))) (ex (a->Prop))) as proof of (P0 (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz))))
% Found eq_ref000:=(eq_ref00 (ex (a->Prop))):(((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx0))) (S Xy))))->((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx0))) (S Xy)))))
% Found (eq_ref00 (ex (a->Prop))) as proof of (P0 (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx0))) (S Xy))))
% Found ((eq_ref0 (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx0))) (S Xy)))) (ex (a->Prop))) as proof of (P0 (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx0))) (S Xy))))
% Found (((eq_ref ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx0))) (S Xy)))) (ex (a->Prop))) as proof of (P0 (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx0))) (S Xy))))
% Found (((eq_ref ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx0))) (S Xy)))) (ex (a->Prop))) as proof of (P0 (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx0))) (S Xy))))
% Found eq_ref000:=(eq_ref00 (ex (a->Prop))):(((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx0))) (S Xy))))->((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx0))) (S Xy)))))
% Found (eq_ref00 (ex (a->Prop))) as proof of (P0 (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx0))) (S Xy))))
% Found ((eq_ref0 (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx0))) (S Xy)))) (ex (a->Prop))) as proof of (P0 (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx0))) (S Xy))))
% Found (((eq_ref ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx0))) (S Xy)))) (ex (a->Prop))) as proof of (P0 (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx0))) (S Xy))))
% Found (((eq_ref ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx0))) (S Xy)))) (ex (a->Prop))) as proof of (P0 (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx0))) (S Xy))))
% Found x5:((and (P x1)) (x1 Xx))
% Instantiate: Xx:=Xx0:a;x4:=x1:(a->Prop)
% Found x5 as proof of ((and (P x4)) (x4 Xx0))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P00):((P00 (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz))))->(P00 (fun (x:(a->Prop))=> ((and ((and (P x)) (x Xx))) (x Xz)))))
% Found (eta_expansion_dep000 P00) as proof of (P01 (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz))))
% Found ((eta_expansion_dep00 (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz)))) P00) as proof of (P01 (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz))))
% Found (((eta_expansion_dep0 (fun (x3:(a->Prop))=> Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz)))) P00) as proof of (P01 (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz))))
% Found ((((eta_expansion_dep (a->Prop)) (fun (x3:(a->Prop))=> Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz)))) P00) as proof of (P01 (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz))))
% Found ((((eta_expansion_dep (a->Prop)) (fun (x3:(a->Prop))=> Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz)))) P00) as proof of (P01 (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz))))
% Found eq_ref000:=(eq_ref00 P00):((P00 (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz))))->(P00 (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz)))))
% Found (eq_ref00 P00) as proof of (P01 (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz))))
% Found ((eq_ref0 (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz)))) P00) as proof of (P01 (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz))))
% Found (((eq_ref ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz)))) P00) as proof of (P01 (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz))))
% Found (((eq_ref ((a->Prop)->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz)))) P00) as proof of (P01 (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P00):((P00 (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz))))->(P00 (fun (x:(a->Prop))=> ((and ((and (P x)) (x Xx))) (x Xz)))))
% Found (eta_expansion_dep000 P00) as proof of (P01 (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz))))
% Found ((eta_expansion_dep00 (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz)))) P00) as proof of (P01 (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz))))
% Found (((eta_expansion_dep0 (fun (x3:(a->Prop))=> Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz)))) P00) as proof of (P01 (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz))))
% Found ((((eta_expansion_dep (a->Prop)) (fun (x3:(a->Prop))=> Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz)))) P00) as proof of (P01 (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz))))
% Found ((((eta_expansion_dep (a->Prop)) (fun (x3:(a->Prop))=> Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz)))) P00) as proof of (P01 (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P00):((P00 (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz))))->(P00 (fun (x:(a->Prop))=> ((and ((and (P x)) (x Xx))) (x Xz)))))
% Found (eta_expansion_dep000 P00) as proof of (P01 (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz))))
% Found ((eta_expansion_dep00 (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz)))) P00) as proof of (P01 (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz))))
% Found (((eta_expansion_dep0 (fun (x3:(a->Prop))=> Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz)))) P00) as proof of (P01 (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz))))
% Found ((((eta_expansion_dep (a->Prop)) (fun (x3:(a->Prop))=> Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz)))) P00) as proof of (P01 (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz))))
% Found ((((eta_expansion_dep (a->Prop)) (fun (x3:(a->Prop))=> Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz)))) P00) as proof of (P01 (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xx))) (S Xz))))
% Found x3:((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S Xz))))
% Instantiate: x1:=(fun (x8:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((and (P S)) (S Xy))) (S x8))))):(a->Prop)
% Found x3 as proof of (x1 Xz)
% Found x3 as proof of (x1 Xz)
% Found x3 as proof of (x1 Xz)
% Found eq_ref00:=(eq_ref0 ((and ((and (P x2)) (x2 Xy))) (x2 Xz))):(((eq Prop) ((and ((and (P x2)) (x2 Xy))) (x2 Xz))) ((and ((and (P x2)) (x2 Xy))) (x2 Xz)))
% Found (eq_ref0 ((and ((and (P x2)) (x2 Xy))) (x2 Xz))) as proof of (((eq Prop) ((and ((and (P x2)) (x2 Xy))) (x2 Xz))) b)
% Found ((eq_ref Prop) ((and ((and (P x2)) (x2 Xy))) (x2 Xz))) as proof of (((eq Prop) ((and ((and (P x2)) (x2 Xy))) (x2 Xz))) b)
% Found ((eq_ref Prop) ((and ((and (P x2)) (x2 Xy))) (x2 Xz))) as proof of (((eq Prop) ((and ((and (P x2)) (x2 Xy))) (x2 Xz))) b)
% Found ((eq_ref Prop) ((and ((and (P x2)) (x2 Xy))) (x2 Xz))) as proof of (((eq Prop) ((and ((and (P x2)) (x2 Xy))) (x2 Xz))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((and (P x2)) (x2 Xx))) (x2 Xz)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (P x2)) (x2 Xx))) (x2 Xz)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (P x2)) (x2 Xx))) (x2 Xz)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (P x2)) (x2 Xx))) (x2 Xz)))
% Found eq_ref00:=(eq_ref0 ((and ((and (P x2)) (x2 Xy))) (x2 Xz))):(((eq Prop) ((and ((and (P x2)) (x2 Xy))) (x2 Xz))) ((and ((and (P x2)) (x2 Xy))) (x2 Xz)))
% Found (eq_ref0 ((and ((and (P x2)) (x2 Xy))) (x2 Xz))) as proof of (((eq Prop) ((and ((and (P x2)) (x2 Xy))) (x2 Xz))) b)
% Found ((eq_ref Prop) ((and ((and (P x2)) (x2 Xy))) (x2 Xz))) as proof of (((eq Prop) ((and ((and (P x2)) (x2 Xy))) (x2 Xz))) b)
% Found ((eq_ref Prop) ((and ((and (P x2)) (x2 Xy))) (x2 Xz))) as proof of (((eq Prop) ((and ((and (P x2)) (x2 Xy))) (x2 Xz))) b)
% Found ((eq_ref Prop) ((and ((and (P x2)) (x2 Xy))) (x2 Xz))) as proof of (((eq Prop) ((and ((and (P x2)) (x2 Xy))) (x2 Xz))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((and (P x2)) (x2 Xx))) (x2 Xz)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (P x2)) (x2 Xx))) (x2 Xz)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (P x2)) (x2 Xx))) (x2 Xz)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (P x2)) (x2 Xx))) (x2 Xz)))
% Found eq_ref00:=(eq_ref0 ((and ((and (P x2)) (x2 Xy))) (x2 Xz))):(((eq Prop) ((and ((and (P x2)) (x2 Xy))) (x2 Xz))) ((and ((and (P x2)) (x2 Xy))) (x2 Xz)))
% Found (eq_ref0 ((a
% EOF
%------------------------------------------------------------------------------