%------------------------------------------------------------------------------
% File     : cocATP---0.2.0
% Problem  : SEV279^5 : TPTP v6.1.0. Released v4.0.0.
% Transfm  : none
% Format   : tptp:raw
% Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p

% Computer : n190.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 13:33:59 EDT 2014

% Result   : Timeout 300.02s
% Output   : None 
% Verified : 
% SZS Type : None (Parsing solution fails)
% Syntax   : Number of formulae    : 0

% Comments : 
%------------------------------------------------------------------------------
%----NO SOLUTION OUTPUT BY SYSTEM
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% % Problem  : SEV279^5 : TPTP v6.1.0. Released v4.0.0.
% % Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n190.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 08:42:51 CDT 2014
% % CPUTime  : 300.02 
% 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 0x13de2d8>, <kernel.Type object at 0x13de8c0>) of role type named b_type
% Using role type
% Declaring b:Type
% FOF formula (<kernel.Constant object at 0x13dfc68>, <kernel.Type object at 0x13deb48>) of role type named a_type
% Using role type
% Declaring a:Type
% FOF formula (<kernel.Constant object at 0x13dea28>, <kernel.DependentProduct object at 0x13de5a8>) of role type named h
% Using role type
% Declaring h:((b->Prop)->a)
% FOF formula (<kernel.Constant object at 0x13de8c0>, <kernel.DependentProduct object at 0x13de950>) of role type named cW
% Using role type
% Declaring cW:((b->Prop)->Prop)
% FOF formula (((and (forall (U:((b->Prop)->Prop)), (((and ((ex (b->Prop)) (fun (Z:(b->Prop))=> (U Z)))) (forall (Xx:(b->Prop)), ((U Xx)->(cW Xx))))->(cW (fun (Xx:b)=> (forall (S:(b->Prop)), ((U S)->(S Xx)))))))) (forall (Xx:a), ((ex (b->Prop)) (fun (V:(b->Prop))=> ((and (cW V)) (((eq a) Xx) (h V)))))))->((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((R Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((R Xy) Xx))))->(((eq a) Xy) Xz))))))))))) of role conjecture named cLEM562A_pme
% Conjecture to prove = (((and (forall (U:((b->Prop)->Prop)), (((and ((ex (b->Prop)) (fun (Z:(b->Prop))=> (U Z)))) (forall (Xx:(b->Prop)), ((U Xx)->(cW Xx))))->(cW (fun (Xx:b)=> (forall (S:(b->Prop)), ((U S)->(S Xx)))))))) (forall (Xx:a), ((ex (b->Prop)) (fun (V:(b->Prop))=> ((and (cW V)) (((eq a) Xx) (h V)))))))->((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((R Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((R Xy) Xx))))->(((eq a) Xy) Xz))))))))))):Prop
% Parameter b_DUMMY:b.
% Parameter a_DUMMY:a.
% We need to prove ['(((and (forall (U:((b->Prop)->Prop)), (((and ((ex (b->Prop)) (fun (Z:(b->Prop))=> (U Z)))) (forall (Xx:(b->Prop)), ((U Xx)->(cW Xx))))->(cW (fun (Xx:b)=> (forall (S:(b->Prop)), ((U S)->(S Xx)))))))) (forall (Xx:a), ((ex (b->Prop)) (fun (V:(b->Prop))=> ((and (cW V)) (((eq a) Xx) (h V)))))))->((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((R Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((R Xy) Xx))))->(((eq a) Xy) Xz)))))))))))']
% Parameter b:Type.
% Parameter a:Type.
% Parameter h:((b->Prop)->a).
% Parameter cW:((b->Prop)->Prop).
% Trying to prove (((and (forall (U:((b->Prop)->Prop)), (((and ((ex (b->Prop)) (fun (Z:(b->Prop))=> (U Z)))) (forall (Xx:(b->Prop)), ((U Xx)->(cW Xx))))->(cW (fun (Xx:b)=> (forall (S:(b->Prop)), ((U S)->(S Xx)))))))) (forall (Xx:a), ((ex (b->Prop)) (fun (V:(b->Prop))=> ((and (cW V)) (((eq a) Xx) (h V)))))))->((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((R Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((R Xy) Xx))))->(((eq a) Xy) Xz)))))))))))
% Found eta_expansion000:=(eta_expansion00 (fun (R:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((R Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((R Xy) Xx))))->(((eq a) Xy) Xz)))))))))):(((eq ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((R Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((R Xy) Xx))))->(((eq a) Xy) Xz)))))))))) (fun (x:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x Xy) Xx))))->(((eq a) Xy) Xz))))))))))
% Found (eta_expansion00 (fun (R:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((R Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((R Xy) Xx))))->(((eq a) Xy) Xz)))))))))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((R Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((R Xy) Xx))))->(((eq a) Xy) Xz)))))))))) b0)
% Found ((eta_expansion0 Prop) (fun (R:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((R Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((R Xy) Xx))))->(((eq a) Xy) Xz)))))))))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((R Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((R Xy) Xx))))->(((eq a) Xy) Xz)))))))))) b0)
% Found (((eta_expansion (a->(a->Prop))) Prop) (fun (R:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((R Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((R Xy) Xx))))->(((eq a) Xy) Xz)))))))))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((R Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((R Xy) Xx))))->(((eq a) Xy) Xz)))))))))) b0)
% Found (((eta_expansion (a->(a->Prop))) Prop) (fun (R:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((R Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((R Xy) Xx))))->(((eq a) Xy) Xz)))))))))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((R Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((R Xy) Xx))))->(((eq a) Xy) Xz)))))))))) b0)
% Found (((eta_expansion (a->(a->Prop))) Prop) (fun (R:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((R Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((R Xy) Xx))))->(((eq a) Xy) Xz)))))))))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((R Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((R Xy) Xx))))->(((eq a) Xy) Xz)))))))))) b0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (R:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((R Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((R Xy) Xx))))->(((eq a) Xy) Xz)))))))))):(((eq ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((R Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((R Xy) Xx))))->(((eq a) Xy) Xz)))))))))) (fun (x:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x Xy) Xx))))->(((eq a) Xy) Xz))))))))))
% Found (eta_expansion_dep00 (fun (R:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((R Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((R Xy) Xx))))->(((eq a) Xy) Xz)))))))))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((R Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((R Xy) Xx))))->(((eq a) Xy) Xz)))))))))) b0)
% Found ((eta_expansion_dep0 (fun (x3:(a->(a->Prop)))=> Prop)) (fun (R:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((R Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((R Xy) Xx))))->(((eq a) Xy) Xz)))))))))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((R Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((R Xy) Xx))))->(((eq a) Xy) Xz)))))))))) b0)
% Found (((eta_expansion_dep (a->(a->Prop))) (fun (x3:(a->(a->Prop)))=> Prop)) (fun (R:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((R Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((R Xy) Xx))))->(((eq a) Xy) Xz)))))))))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((R Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((R Xy) Xx))))->(((eq a) Xy) Xz)))))))))) b0)
% Found (((eta_expansion_dep (a->(a->Prop))) (fun (x3:(a->(a->Prop)))=> Prop)) (fun (R:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((R Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((R Xy) Xx))))->(((eq a) Xy) Xz)))))))))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((R Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((R Xy) Xx))))->(((eq a) Xy) Xz)))))))))) b0)
% Found (((eta_expansion_dep (a->(a->Prop))) (fun (x3:(a->(a->Prop)))=> Prop)) (fun (R:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((R Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((R Xy) Xx))))->(((eq a) Xy) Xz)))))))))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((R Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((R Xy) Xx))))->(((eq a) Xy) Xz)))))))))) b0)
% Found eta_expansion0000:=(eta_expansion000 (ex a)):(((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (x:a)=> (X x))))
% Found (eta_expansion000 (ex a)) as proof of (P (fun (Xz:a)=> (X Xz)))
% Found ((eta_expansion00 (fun (Xz:a)=> (X Xz))) (ex a)) as proof of (P (fun (Xz:a)=> (X Xz)))
% Found (((eta_expansion0 Prop) (fun (Xz:a)=> (X Xz))) (ex a)) as proof of (P (fun (Xz:a)=> (X Xz)))
% Found ((((eta_expansion a) Prop) (fun (Xz:a)=> (X Xz))) (ex a)) as proof of (P (fun (Xz:a)=> (X Xz)))
% Found ((((eta_expansion a) Prop) (fun (Xz:a)=> (X Xz))) (ex a)) as proof of (P (fun (Xz:a)=> (X Xz)))
% Found eq_ref000:=(eq_ref00 (ex a)):(((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> (X Xz))))
% Found (eq_ref00 (ex a)) as proof of (P (fun (Xz:a)=> (X Xz)))
% Found ((eq_ref0 (fun (Xz:a)=> (X Xz))) (ex a)) as proof of (P (fun (Xz:a)=> (X Xz)))
% Found (((eq_ref (a->Prop)) (fun (Xz:a)=> (X Xz))) (ex a)) as proof of (P (fun (Xz:a)=> (X Xz)))
% Found (((eq_ref (a->Prop)) (fun (Xz:a)=> (X Xz))) (ex a)) as proof of (P (fun (Xz:a)=> (X Xz)))
% Found eta_expansion0000:=(eta_expansion000 (ex a)):(((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (x:a)=> (X x))))
% Found (eta_expansion000 (ex a)) as proof of (P (fun (Xz:a)=> (X Xz)))
% Found ((eta_expansion00 (fun (Xz:a)=> (X Xz))) (ex a)) as proof of (P (fun (Xz:a)=> (X Xz)))
% Found (((eta_expansion0 Prop) (fun (Xz:a)=> (X Xz))) (ex a)) as proof of (P (fun (Xz:a)=> (X Xz)))
% Found ((((eta_expansion a) Prop) (fun (Xz:a)=> (X Xz))) (ex a)) as proof of (P (fun (Xz:a)=> (X Xz)))
% Found ((((eta_expansion a) Prop) (fun (Xz:a)=> (X Xz))) (ex a)) as proof of (P (fun (Xz:a)=> (X Xz)))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 (ex a)):(((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (x:a)=> (X x))))
% Found (eta_expansion_dep000 (ex a)) as proof of (P (fun (Xz:a)=> (X Xz)))
% Found ((eta_expansion_dep00 (fun (Xz:a)=> (X Xz))) (ex a)) as proof of (P (fun (Xz:a)=> (X Xz)))
% Found (((eta_expansion_dep0 (fun (x4:a)=> Prop)) (fun (Xz:a)=> (X Xz))) (ex a)) as proof of (P (fun (Xz:a)=> (X Xz)))
% Found ((((eta_expansion_dep a) (fun (x4:a)=> Prop)) (fun (Xz:a)=> (X Xz))) (ex a)) as proof of (P (fun (Xz:a)=> (X Xz)))
% Found ((((eta_expansion_dep a) (fun (x4:a)=> Prop)) (fun (Xz:a)=> (X Xz))) (ex a)) as proof of (P (fun (Xz:a)=> (X Xz)))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 (ex a)):(((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (x:a)=> (X x))))
% Found (eta_expansion_dep000 (ex a)) as proof of (P (fun (Xz:a)=> (X Xz)))
% Found ((eta_expansion_dep00 (fun (Xz:a)=> (X Xz))) (ex a)) as proof of (P (fun (Xz:a)=> (X Xz)))
% Found (((eta_expansion_dep0 (fun (x4:a)=> Prop)) (fun (Xz:a)=> (X Xz))) (ex a)) as proof of (P (fun (Xz:a)=> (X Xz)))
% Found ((((eta_expansion_dep a) (fun (x4:a)=> Prop)) (fun (Xz:a)=> (X Xz))) (ex a)) as proof of (P (fun (Xz:a)=> (X Xz)))
% Found ((((eta_expansion_dep a) (fun (x4:a)=> Prop)) (fun (Xz:a)=> (X Xz))) (ex a)) as proof of (P (fun (Xz:a)=> (X Xz)))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 (ex a)):(((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (x:a)=> (X x))))
% Found (eta_expansion_dep000 (ex a)) as proof of (P (fun (Xz:a)=> (X Xz)))
% Found ((eta_expansion_dep00 (fun (Xz:a)=> (X Xz))) (ex a)) as proof of (P (fun (Xz:a)=> (X Xz)))
% Found (((eta_expansion_dep0 (fun (x4:a)=> Prop)) (fun (Xz:a)=> (X Xz))) (ex a)) as proof of (P (fun (Xz:a)=> (X Xz)))
% Found ((((eta_expansion_dep a) (fun (x4:a)=> Prop)) (fun (Xz:a)=> (X Xz))) (ex a)) as proof of (P (fun (Xz:a)=> (X Xz)))
% Found ((((eta_expansion_dep a) (fun (x4:a)=> Prop)) (fun (Xz:a)=> (X Xz))) (ex a)) as proof of (P (fun (Xz:a)=> (X Xz)))
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) x1)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) x1)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) x1)
% Found (fun (x000:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))))=> ((eq_ref a) Xy)) as proof of (((eq a) Xy) x1)
% Found eq_ref000:=(eq_ref00 P):((P Xy)->(P Xy))
% Found (eq_ref00 P) as proof of ((P Xy)->(P x1))
% Found ((eq_ref0 Xy) P) as proof of ((P Xy)->(P x1))
% Found (((eq_ref a) Xy) P) as proof of ((P Xy)->(P x1))
% Found (((eq_ref a) Xy) P) as proof of ((P Xy)->(P x1))
% Found (fun (P:(a->Prop))=> (((eq_ref a) Xy) P)) as proof of ((P Xy)->(P x1))
% Found (fun (x000:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))) (P:(a->Prop))=> (((eq_ref a) Xy) P)) as proof of (((eq a) Xy) x1)
% Found x2:(P Xy)
% Instantiate: x1:=Xy:a
% Found (fun (x2:(P Xy))=> x2) as proof of (P x1)
% Found (fun (P:(a->Prop)) (x2:(P Xy))=> x2) as proof of ((P Xy)->(P x1))
% Found (fun (x000:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))) (P:(a->Prop)) (x2:(P Xy))=> x2) as proof of (((eq a) Xy) x1)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) x3)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) x3)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) x3)
% Found (fun (x000:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))))=> ((eq_ref a) Xy)) as proof of (((eq a) Xy) x3)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) x3)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) x3)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) x3)
% Found (fun (x01:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx)))))=> ((eq_ref a) Xy)) as proof of (((eq a) Xy) x3)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) x1)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) x1)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) x1)
% Found (fun (x000:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))))=> ((eq_ref a) Xy)) as proof of (((eq a) Xy) x1)
% Found eq_ref000:=(eq_ref00 P):((P Xy)->(P Xy))
% Found (eq_ref00 P) as proof of ((P Xy)->(P x3))
% Found ((eq_ref0 Xy) P) as proof of ((P Xy)->(P x3))
% Found (((eq_ref a) Xy) P) as proof of ((P Xy)->(P x3))
% Found (((eq_ref a) Xy) P) as proof of ((P Xy)->(P x3))
% Found (fun (P:(a->Prop))=> (((eq_ref a) Xy) P)) as proof of ((P Xy)->(P x3))
% Found (fun (x01:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))) (P:(a->Prop))=> (((eq_ref a) Xy) P)) as proof of (((eq a) Xy) x3)
% Found x4:(P Xy)
% Instantiate: x3:=Xy:a
% Found (fun (x4:(P Xy))=> x4) as proof of (P x3)
% Found (fun (P:(a->Prop)) (x4:(P Xy))=> x4) as proof of ((P Xy)->(P x3))
% Found (fun (x01:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))) (P:(a->Prop)) (x4:(P Xy))=> x4) as proof of (((eq a) Xy) x3)
% Found x4:(P Xy)
% Instantiate: x3:=Xy:a
% Found (fun (x4:(P Xy))=> x4) as proof of (P x3)
% Found (fun (P:(a->Prop)) (x4:(P Xy))=> x4) as proof of ((P Xy)->(P x3))
% Found (fun (x000:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))) (P:(a->Prop)) (x4:(P Xy))=> x4) as proof of (((eq a) Xy) x3)
% Found eq_ref000:=(eq_ref00 P):((P Xy)->(P Xy))
% Found (eq_ref00 P) as proof of ((P Xy)->(P x3))
% Found ((eq_ref0 Xy) P) as proof of ((P Xy)->(P x3))
% Found (((eq_ref a) Xy) P) as proof of ((P Xy)->(P x3))
% Found (((eq_ref a) Xy) P) as proof of ((P Xy)->(P x3))
% Found (fun (P:(a->Prop))=> (((eq_ref a) Xy) P)) as proof of ((P Xy)->(P x3))
% Found (fun (x000:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))) (P:(a->Prop))=> (((eq_ref a) Xy) P)) as proof of (((eq a) Xy) x3)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) x3)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) x3)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) x3)
% Found (fun (x000:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))))=> ((eq_ref a) Xy)) as proof of (((eq a) Xy) x3)
% Found x4:(P Xy)
% Instantiate: x1:=Xy:a
% Found (fun (x4:(P Xy))=> x4) as proof of (P x1)
% Found (fun (P:(a->Prop)) (x4:(P Xy))=> x4) as proof of ((P Xy)->(P x1))
% Found (fun (x000:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))) (P:(a->Prop)) (x4:(P Xy))=> x4) as proof of (((eq a) Xy) x1)
% Found eq_ref000:=(eq_ref00 P):((P Xy)->(P Xy))
% Found (eq_ref00 P) as proof of ((P Xy)->(P x1))
% Found ((eq_ref0 Xy) P) as proof of ((P Xy)->(P x1))
% Found (((eq_ref a) Xy) P) as proof of ((P Xy)->(P x1))
% Found (((eq_ref a) Xy) P) as proof of ((P Xy)->(P x1))
% Found (fun (P:(a->Prop))=> (((eq_ref a) Xy) P)) as proof of ((P Xy)->(P x1))
% Found (fun (x000:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))) (P:(a->Prop))=> (((eq_ref a) Xy) P)) as proof of (((eq a) Xy) x1)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) x1)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) x1)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) x1)
% Found (fun (x000:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))))=> ((eq_ref a) Xy)) as proof of (((eq a) Xy) x1)
% Found x4:(P Xy)
% Instantiate: x3:=Xy:a
% Found (fun (x4:(P Xy))=> x4) as proof of (P x3)
% Found (fun (P:(a->Prop)) (x4:(P Xy))=> x4) as proof of ((P Xy)->(P x3))
% Found (fun (x000:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))) (P:(a->Prop)) (x4:(P Xy))=> x4) as proof of (((eq a) Xy) x3)
% Found eq_ref000:=(eq_ref00 P):((P Xy)->(P Xy))
% Found (eq_ref00 P) as proof of ((P Xy)->(P x3))
% Found ((eq_ref0 Xy) P) as proof of ((P Xy)->(P x3))
% Found (((eq_ref a) Xy) P) as proof of ((P Xy)->(P x3))
% Found (((eq_ref a) Xy) P) as proof of ((P Xy)->(P x3))
% Found (fun (P:(a->Prop))=> (((eq_ref a) Xy) P)) as proof of ((P Xy)->(P x3))
% Found (fun (x000:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))) (P:(a->Prop))=> (((eq_ref a) Xy) P)) as proof of (((eq a) Xy) x3)
% Found x4:(P Xy)
% Instantiate: x1:=Xy:a
% Found (fun (x4:(P Xy))=> x4) as proof of (P x1)
% Found (fun (P:(a->Prop)) (x4:(P Xy))=> x4) as proof of ((P Xy)->(P x1))
% Found (fun (x000:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))) (P:(a->Prop)) (x4:(P Xy))=> x4) as proof of (((eq a) Xy) x1)
% Found eq_ref000:=(eq_ref00 P):((P Xy)->(P Xy))
% Found (eq_ref00 P) as proof of ((P Xy)->(P x1))
% Found ((eq_ref0 Xy) P) as proof of ((P Xy)->(P x1))
% Found (((eq_ref a) Xy) P) as proof of ((P Xy)->(P x1))
% Found (((eq_ref a) Xy) P) as proof of ((P Xy)->(P x1))
% Found (fun (P:(a->Prop))=> (((eq_ref a) Xy) P)) as proof of ((P Xy)->(P x1))
% Found (fun (x000:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))) (P:(a->Prop))=> (((eq_ref a) Xy) P)) as proof of (((eq a) Xy) x1)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) x1)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) x1)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) x1)
% Found (fun (x3:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((eq_ref a) Xy)) as proof of (((eq a) Xy) x1)
% Found (fun (x2:(X Xy)) (x3:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((eq_ref a) Xy)) as proof of ((forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))->(((eq a) Xy) x1))
% Found (fun (x2:(X Xy)) (x3:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((eq_ref a) Xy)) as proof of ((X Xy)->((forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))->(((eq a) Xy) x1)))
% Found (and_rect00 (fun (x2:(X Xy)) (x3:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((eq_ref a) Xy))) as proof of (((eq a) Xy) x1)
% Found ((and_rect0 (((eq a) Xy) x1)) (fun (x2:(X Xy)) (x3:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((eq_ref a) Xy))) as proof of (((eq a) Xy) x1)
% Found (((fun (P:Type) (x2:((X Xy)->((forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))->P)))=> (((((and_rect (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))) P) x2) x000)) (((eq a) Xy) x1)) (fun (x2:(X Xy)) (x3:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((eq_ref a) Xy))) as proof of (((eq a) Xy) x1)
% Found (fun (x000:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))))=> (((fun (P:Type) (x2:((X Xy)->((forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))->P)))=> (((((and_rect (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))) P) x2) x000)) (((eq a) Xy) x1)) (fun (x2:(X Xy)) (x3:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((eq_ref a) Xy)))) as proof of (((eq a) Xy) x1)
% Found x2:(X x1)
% Instantiate: x3:=x1:a
% Found x2 as proof of (X x3)
% Found eq_ref00:=(eq_ref0 x1):(((eq a) x1) x1)
% Found (eq_ref0 x1) as proof of (((eq a) x1) Xy)
% Found ((eq_ref a) x1) as proof of (((eq a) x1) Xy)
% Found ((eq_ref a) x1) as proof of (((eq a) x1) Xy)
% Found ((eq_ref a) x1) as proof of (((eq a) x1) Xy)
% Found (eq_sym000 ((eq_ref a) x1)) as proof of (((eq a) Xy) x1)
% Found ((eq_sym00 Xy) ((eq_ref a) x1)) as proof of (((eq a) Xy) x1)
% Found (((eq_sym0 x1) Xy) ((eq_ref a) x1)) as proof of (((eq a) Xy) x1)
% Found ((((eq_sym a) x1) Xy) ((eq_ref a) x1)) as proof of (((eq a) Xy) x1)
% Found (fun (x000:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))))=> ((((eq_sym a) x1) Xy) ((eq_ref a) x1))) as proof of (((eq a) Xy) x1)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) x5)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) x5)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) x5)
% Found (fun (x01:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx)))))=> ((eq_ref a) Xy)) as proof of (((eq a) Xy) x5)
% Found x3:(X x2)
% Instantiate: x1:=x2:a
% Found x3 as proof of (X x1)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) x3)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) x3)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) x3)
% Found (fun (x01:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx)))))=> ((eq_ref a) Xy)) as proof of (((eq a) Xy) x3)
% Found x6:(P Xy)
% Instantiate: x5:=Xy:a
% Found (fun (x6:(P Xy))=> x6) as proof of (P x5)
% Found (fun (P:(a->Prop)) (x6:(P Xy))=> x6) as proof of ((P Xy)->(P x5))
% Found (fun (x01:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))) (P:(a->Prop)) (x6:(P Xy))=> x6) as proof of (((eq a) Xy) x5)
% Found eq_ref000:=(eq_ref00 P):((P Xy)->(P Xy))
% Found (eq_ref00 P) as proof of ((P Xy)->(P x5))
% Found ((eq_ref0 Xy) P) as proof of ((P Xy)->(P x5))
% Found (((eq_ref a) Xy) P) as proof of ((P Xy)->(P x5))
% Found (((eq_ref a) Xy) P) as proof of ((P Xy)->(P x5))
% Found (fun (P:(a->Prop))=> (((eq_ref a) Xy) P)) as proof of ((P Xy)->(P x5))
% Found (fun (x01:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))) (P:(a->Prop))=> (((eq_ref a) Xy) P)) as proof of (((eq a) Xy) x5)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) x1)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) x1)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) x1)
% Found (fun (x3:(X x2))=> ((eq_ref a) Xy)) as proof of (((eq a) Xy) x1)
% Found (fun (x2:a) (x3:(X x2))=> ((eq_ref a) Xy)) as proof of ((X x2)->(((eq a) Xy) x1))
% Found (fun (x2:a) (x3:(X x2))=> ((eq_ref a) Xy)) as proof of (forall (x:a), ((X x)->(((eq a) Xy) x1)))
% Found (ex_ind00 (fun (x2:a) (x3:(X x2))=> ((eq_ref a) Xy))) as proof of (((eq a) Xy) x1)
% Found ((ex_ind0 (((eq a) Xy) x1)) (fun (x2:a) (x3:(X x2))=> ((eq_ref a) Xy))) as proof of (((eq a) Xy) x1)
% Found (((fun (P:Prop) (x2:(forall (x:a), ((X x)->P)))=> (((((ex_ind a) (fun (Xz:a)=> (X Xz))) P) x2) x00)) (((eq a) Xy) x1)) (fun (x2:a) (x3:(X x2))=> ((eq_ref a) Xy))) as proof of (((eq a) Xy) x1)
% Found (fun (x000:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))))=> (((fun (P:Prop) (x2:(forall (x:a), ((X x)->P)))=> (((((ex_ind a) (fun (Xz:a)=> (X Xz))) P) x2) x00)) (((eq a) Xy) x1)) (fun (x2:a) (x3:(X x2))=> ((eq_ref a) Xy)))) as proof of (((eq a) Xy) x1)
% Found eq_ref000:=(eq_ref00 P):((P Xy)->(P Xy))
% Found (eq_ref00 P) as proof of ((P Xy)->(P x3))
% Found ((eq_ref0 Xy) P) as proof of ((P Xy)->(P x3))
% Found (((eq_ref a) Xy) P) as proof of ((P Xy)->(P x3))
% Found (((eq_ref a) Xy) P) as proof of ((P Xy)->(P x3))
% Found (fun (P:(a->Prop))=> (((eq_ref a) Xy) P)) as proof of ((P Xy)->(P x3))
% Found (fun (x01:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))) (P:(a->Prop))=> (((eq_ref a) Xy) P)) as proof of (((eq a) Xy) x3)
% Found x6:(P Xy)
% Instantiate: x3:=Xy:a
% Found (fun (x6:(P Xy))=> x6) as proof of (P x3)
% Found (fun (P:(a->Prop)) (x6:(P Xy))=> x6) as proof of ((P Xy)->(P x3))
% Found (fun (x01:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))) (P:(a->Prop)) (x6:(P Xy))=> x6) as proof of (((eq a) Xy) x3)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) x5)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) x5)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) x5)
% Found (fun (x000:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))))=> ((eq_ref a) Xy)) as proof of (((eq a) Xy) x5)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) x5)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) x5)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) x5)
% Found (fun (x000:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))))=> ((eq_ref a) Xy)) as proof of (((eq a) Xy) x5)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) x1)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) x1)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) x1)
% Found (fun (x3:(forall (Xx:a), ((ex (b->Prop)) (fun (V:(b->Prop))=> ((and (cW V)) (((eq a) Xx) (h V)))))))=> ((eq_ref a) Xy)) as proof of (((eq a) Xy) x1)
% Found (fun (x2:(forall (U:((b->Prop)->Prop)), (((and ((ex (b->Prop)) (fun (Z:(b->Prop))=> (U Z)))) (forall (Xx:(b->Prop)), ((U Xx)->(cW Xx))))->(cW (fun (Xx:b)=> (forall (S:(b->Prop)), ((U S)->(S Xx)))))))) (x3:(forall (Xx:a), ((ex (b->Prop)) (fun (V:(b->Prop))=> ((and (cW V)) (((eq a) Xx) (h V)))))))=> ((eq_ref a) Xy)) as proof of ((forall (Xx:a), ((ex (b->Prop)) (fun (V:(b->Prop))=> ((and (cW V)) (((eq a) Xx) (h V))))))->(((eq a) Xy) x1))
% Found (fun (x2:(forall (U:((b->Prop)->Prop)), (((and ((ex (b->Prop)) (fun (Z:(b->Prop))=> (U Z)))) (forall (Xx:(b->Prop)), ((U Xx)->(cW Xx))))->(cW (fun (Xx:b)=> (forall (S:(b->Prop)), ((U S)->(S Xx)))))))) (x3:(forall (Xx:a), ((ex (b->Prop)) (fun (V:(b->Prop))=> ((and (cW V)) (((eq a) Xx) (h V)))))))=> ((eq_ref a) Xy)) as proof of ((forall (U:((b->Prop)->Prop)), (((and ((ex (b->Prop)) (fun (Z:(b->Prop))=> (U Z)))) (forall (Xx:(b->Prop)), ((U Xx)->(cW Xx))))->(cW (fun (Xx:b)=> (forall (S:(b->Prop)), ((U S)->(S Xx)))))))->((forall (Xx:a), ((ex (b->Prop)) (fun (V:(b->Prop))=> ((and (cW V)) (((eq a) Xx) (h V))))))->(((eq a) Xy) x1)))
% Found (and_rect00 (fun (x2:(forall (U:((b->Prop)->Prop)), (((and ((ex (b->Prop)) (fun (Z:(b->Prop))=> (U Z)))) (forall (Xx:(b->Prop)), ((U Xx)->(cW Xx))))->(cW (fun (Xx:b)=> (forall (S:(b->Prop)), ((U S)->(S Xx)))))))) (x3:(forall (Xx:a), ((ex (b->Prop)) (fun (V:(b->Prop))=> ((and (cW V)) (((eq a) Xx) (h V)))))))=> ((eq_ref a) Xy))) as proof of (((eq a) Xy) x1)
% Found ((and_rect0 (((eq a) Xy) x1)) (fun (x2:(forall (U:((b->Prop)->Prop)), (((and ((ex (b->Prop)) (fun (Z:(b->Prop))=> (U Z)))) (forall (Xx:(b->Prop)), ((U Xx)->(cW Xx))))->(cW (fun (Xx:b)=> (forall (S:(b->Prop)), ((U S)->(S Xx)))))))) (x3:(forall (Xx:a), ((ex (b->Prop)) (fun (V:(b->Prop))=> ((and (cW V)) (((eq a) Xx) (h V)))))))=> ((eq_ref a) Xy))) as proof of (((eq a) Xy) x1)
% Found (((fun (P:Type) (x2:((forall (U:((b->Prop)->Prop)), (((and ((ex (b->Prop)) (fun (Z:(b->Prop))=> (U Z)))) (forall (Xx:(b->Prop)), ((U Xx)->(cW Xx))))->(cW (fun (Xx:b)=> (forall (S:(b->Prop)), ((U S)->(S Xx)))))))->((forall (Xx:a), ((ex (b->Prop)) (fun (V:(b->Prop))=> ((and (cW V)) (((eq a) Xx) (h V))))))->P)))=> (((((and_rect (forall (U:((b->Prop)->Prop)), (((and ((ex (b->Prop)) (fun (Z:(b->Prop))=> (U Z)))) (forall (Xx:(b->Prop)), ((U Xx)->(cW Xx))))->(cW (fun (Xx:b)=> (forall (S:(b->Prop)), ((U S)->(S Xx)))))))) (forall (Xx:a), ((ex (b->Prop)) (fun (V:(b->Prop))=> ((and (cW V)) (((eq a) Xx) (h V))))))) P) x2) x)) (((eq a) Xy) x1)) (fun (x2:(forall (U:((b->Prop)->Prop)), (((and ((ex (b->Prop)) (fun (Z:(b->Prop))=> (U Z)))) (forall (Xx:(b->Prop)), ((U Xx)->(cW Xx))))->(cW (fun (Xx:b)=> (forall (S:(b->Prop)), ((U S)->(S Xx)))))))) (x3:(forall (Xx:a), ((ex (b->Prop)) (fun (V:(b->Prop))=> ((and (cW V)) (((eq a) Xx) (h V)))))))=> ((eq_ref a) Xy))) as proof of (((eq a) Xy) x1)
% Found (fun (x000:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))))=> (((fun (P:Type) (x2:((forall (U:((b->Prop)->Prop)), (((and ((ex (b->Prop)) (fun (Z:(b->Prop))=> (U Z)))) (forall (Xx:(b->Prop)), ((U Xx)->(cW Xx))))->(cW (fun (Xx:b)=> (forall (S:(b->Prop)), ((U S)->(S Xx)))))))->((forall (Xx:a), ((ex (b->Prop)) (fun (V:(b->Prop))=> ((and (cW V)) (((eq a) Xx) (h V))))))->P)))=> (((((and_rect (forall (U:((b->Prop)->Prop)), (((and ((ex (b->Prop)) (fun (Z:(b->Prop))=> (U Z)))) (forall (Xx:(b->Prop)), ((U Xx)->(cW Xx))))->(cW (fun (Xx:b)=> (forall (S:(b->Prop)), ((U S)->(S Xx)))))))) (forall (Xx:a), ((ex (b->Prop)) (fun (V:(b->Prop))=> ((and (cW V)) (((eq a) Xx) (h V))))))) P) x2) x)) (((eq a) Xy) x1)) (fun (x2:(forall (U:((b->Prop)->Prop)), (((and ((ex (b->Prop)) (fun (Z:(b->Prop))=> (U Z)))) (forall (Xx:(b->Prop)), ((U Xx)->(cW Xx))))->(cW (fun (Xx:b)=> (forall (S:(b->Prop)), ((U S)->(S Xx)))))))) (x3:(forall (Xx:a), ((ex (b->Prop)) (fun (V:(b->Prop))=> ((and (cW V)) (((eq a) Xx) (h V)))))))=> ((eq_ref a) Xy)))) as proof of (((eq a) Xy) x1)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) x3)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) x3)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) x3)
% Found (fun (x000:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))))=> ((eq_ref a) Xy)) as proof of (((eq a) Xy) x3)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) x3)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) x3)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) x3)
% Found (fun (x000:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))))=> ((eq_ref a) Xy)) as proof of (((eq a) Xy) x3)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) x3)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) x3)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) x3)
% Found (fun (x5:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((eq_ref a) Xy)) as proof of (((eq a) Xy) x3)
% Found (fun (x4:(X Xy)) (x5:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((eq_ref a) Xy)) as proof of ((forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))->(((eq a) Xy) x3))
% Found (fun (x4:(X Xy)) (x5:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((eq_ref a) Xy)) as proof of ((X Xy)->((forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))->(((eq a) Xy) x3)))
% Found (and_rect00 (fun (x4:(X Xy)) (x5:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((eq_ref a) Xy))) as proof of (((eq a) Xy) x3)
% Found ((and_rect0 (((eq a) Xy) x3)) (fun (x4:(X Xy)) (x5:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((eq_ref a) Xy))) as proof of (((eq a) Xy) x3)
% Found (((fun (P:Type) (x4:((X Xy)->((forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))->P)))=> (((((and_rect (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))) P) x4) x000)) (((eq a) Xy) x3)) (fun (x4:(X Xy)) (x5:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((eq_ref a) Xy))) as proof of (((eq a) Xy) x3)
% Found (fun (x000:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))))=> (((fun (P:Type) (x4:((X Xy)->((forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))->P)))=> (((((and_rect (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))) P) x4) x000)) (((eq a) Xy) x3)) (fun (x4:(X Xy)) (x5:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((eq_ref a) Xy)))) as proof of (((eq a) Xy) x3)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) x3)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) x3)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) x3)
% Found (fun (x5:(forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))=> ((eq_ref a) Xy)) as proof of (((eq a) Xy) x3)
% Found (fun (x4:(X Xy)) (x5:(forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))=> ((eq_ref a) Xy)) as proof of ((forall (Xx:a), ((X Xx)->((x2 Xy) Xx)))->(((eq a) Xy) x3))
% Found (fun (x4:(X Xy)) (x5:(forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))=> ((eq_ref a) Xy)) as proof of ((X Xy)->((forall (Xx:a), ((X Xx)->((x2 Xy) Xx)))->(((eq a) Xy) x3)))
% Found (and_rect10 (fun (x4:(X Xy)) (x5:(forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))=> ((eq_ref a) Xy))) as proof of (((eq a) Xy) x3)
% Found ((and_rect1 (((eq a) Xy) x3)) (fun (x4:(X Xy)) (x5:(forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))=> ((eq_ref a) Xy))) as proof of (((eq a) Xy) x3)
% Found (((fun (P:Type) (x4:((X Xy)->((forall (Xx:a), ((X Xx)->((x2 Xy) Xx)))->P)))=> (((((and_rect (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx)))) P) x4) x01)) (((eq a) Xy) x3)) (fun (x4:(X Xy)) (x5:(forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))=> ((eq_ref a) Xy))) as proof of (((eq a) Xy) x3)
% Found (fun (x01:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx)))))=> (((fun (P:Type) (x4:((X Xy)->((forall (Xx:a), ((X Xx)->((x2 Xy) Xx)))->P)))=> (((((and_rect (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx)))) P) x4) x01)) (((eq a) Xy) x3)) (fun (x4:(X Xy)) (x5:(forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))=> ((eq_ref a) Xy)))) as proof of (((eq a) Xy) x3)
% Found eq_ref00:=(eq_ref0 x3):(((eq a) x3) x3)
% Found (eq_ref0 x3) as proof of (((eq a) x3) Xy)
% Found ((eq_ref a) x3) as proof of (((eq a) x3) Xy)
% Found ((eq_ref a) x3) as proof of (((eq a) x3) Xy)
% Found ((eq_ref a) x3) as proof of (((eq a) x3) Xy)
% Found (eq_sym000 ((eq_ref a) x3)) as proof of (((eq a) Xy) x3)
% Found ((eq_sym00 Xy) ((eq_ref a) x3)) as proof of (((eq a) Xy) x3)
% Found (((eq_sym0 x3) Xy) ((eq_ref a) x3)) as proof of (((eq a) Xy) x3)
% Found ((((eq_sym a) x3) Xy) ((eq_ref a) x3)) as proof of (((eq a) Xy) x3)
% Found (fun (x000:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))))=> ((((eq_sym a) x3) Xy) ((eq_ref a) x3))) as proof of (((eq a) Xy) x3)
% Found eq_ref00:=(eq_ref0 x3):(((eq a) x3) x3)
% Found (eq_ref0 x3) as proof of (((eq a) x3) Xy)
% Found ((eq_ref a) x3) as proof of (((eq a) x3) Xy)
% Found ((eq_ref a) x3) as proof of (((eq a) x3) Xy)
% Found ((eq_ref a) x3) as proof of (((eq a) x3) Xy)
% Found (eq_sym000 ((eq_ref a) x3)) as proof of (((eq a) Xy) x3)
% Found ((eq_sym00 Xy) ((eq_ref a) x3)) as proof of (((eq a) Xy) x3)
% Found (((eq_sym0 x3) Xy) ((eq_ref a) x3)) as proof of (((eq a) Xy) x3)
% Found ((((eq_sym a) x3) Xy) ((eq_ref a) x3)) as proof of (((eq a) Xy) x3)
% Found (fun (x01:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx)))))=> ((((eq_sym a) x3) Xy) ((eq_ref a) x3))) as proof of (((eq a) Xy) x3)
% Found eq_ref000:=(eq_ref00 P):((P Xy)->(P Xy))
% Found (eq_ref00 P) as proof of ((P Xy)->(P x5))
% Found ((eq_ref0 Xy) P) as proof of ((P Xy)->(P x5))
% Found (((eq_ref a) Xy) P) as proof of ((P Xy)->(P x5))
% Found (((eq_ref a) Xy) P) as proof of ((P Xy)->(P x5))
% Found (fun (P:(a->Prop))=> (((eq_ref a) Xy) P)) as proof of ((P Xy)->(P x5))
% Found (fun (x000:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))) (P:(a->Prop))=> (((eq_ref a) Xy) P)) as proof of (((eq a) Xy) x5)
% Found x6:(P Xy)
% Instantiate: x5:=Xy:a
% Found (fun (x6:(P Xy))=> x6) as proof of (P x5)
% Found (fun (P:(a->Prop)) (x6:(P Xy))=> x6) as proof of ((P Xy)->(P x5))
% Found (fun (x000:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))) (P:(a->Prop)) (x6:(P Xy))=> x6) as proof of (((eq a) Xy) x5)
% Found x6:(P Xy)
% Instantiate: x5:=Xy:a
% Found (fun (x6:(P Xy))=> x6) as proof of (P x5)
% Found (fun (P:(a->Prop)) (x6:(P Xy))=> x6) as proof of ((P Xy)->(P x5))
% Found (fun (x000:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))) (P:(a->Prop)) (x6:(P Xy))=> x6) as proof of (((eq a) Xy) x5)
% Found eq_ref000:=(eq_ref00 P):((P Xy)->(P Xy))
% Found (eq_ref00 P) as proof of ((P Xy)->(P x5))
% Found ((eq_ref0 Xy) P) as proof of ((P Xy)->(P x5))
% Found (((eq_ref a) Xy) P) as proof of ((P Xy)->(P x5))
% Found (((eq_ref a) Xy) P) as proof of ((P Xy)->(P x5))
% Found (fun (P:(a->Prop))=> (((eq_ref a) Xy) P)) as proof of ((P Xy)->(P x5))
% Found (fun (x000:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))) (P:(a->Prop))=> (((eq_ref a) Xy) P)) as proof of (((eq a) Xy) x5)
% Found x4:(X x3)
% Instantiate: x5:=x3:a
% Found x4 as proof of (X x5)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) x1)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) x1)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) x1)
% Found (fun (x000:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))))=> ((eq_ref a) Xy)) as proof of (((eq a) Xy) x1)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) x1)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) x1)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) x1)
% Found (fun (x000:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))))=> ((eq_ref a) Xy)) as proof of (((eq a) Xy) x1)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) x1)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) x1)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) x1)
% Found (fun (x5:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((eq_ref a) Xy)) as proof of (((eq a) Xy) x1)
% Found (fun (x4:(X Xy)) (x5:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((eq_ref a) Xy)) as proof of ((forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))->(((eq a) Xy) x1))
% Found (fun (x4:(X Xy)) (x5:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((eq_ref a) Xy)) as proof of ((X Xy)->((forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))->(((eq a) Xy) x1)))
% Found (and_rect00 (fun (x4:(X Xy)) (x5:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((eq_ref a) Xy))) as proof of (((eq a) Xy) x1)
% Found ((and_rect0 (((eq a) Xy) x1)) (fun (x4:(X Xy)) (x5:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((eq_ref a) Xy))) as proof of (((eq a) Xy) x1)
% Found (((fun (P:Type) (x4:((X Xy)->((forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))->P)))=> (((((and_rect (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))) P) x4) x000)) (((eq a) Xy) x1)) (fun (x4:(X Xy)) (x5:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((eq_ref a) Xy))) as proof of (((eq a) Xy) x1)
% Found (fun (x000:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))))=> (((fun (P:Type) (x4:((X Xy)->((forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))->P)))=> (((((and_rect (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))) P) x4) x000)) (((eq a) Xy) x1)) (fun (x4:(X Xy)) (x5:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((eq_ref a) Xy)))) as proof of (((eq a) Xy) x1)
% Found eq_ref00:=(eq_ref0 x1):(((eq a) x1) x1)
% Found (eq_ref0 x1) as proof of (((eq a) x1) Xy)
% Found ((eq_ref a) x1) as proof of (((eq a) x1) Xy)
% Found ((eq_ref a) x1) as proof of (((eq a) x1) Xy)
% Found ((eq_ref a) x1) as proof of (((eq a) x1) Xy)
% Found (eq_sym000 ((eq_ref a) x1)) as proof of (((eq a) Xy) x1)
% Found ((eq_sym00 Xy) ((eq_ref a) x1)) as proof of (((eq a) Xy) x1)
% Found (((eq_sym0 x1) Xy) ((eq_ref a) x1)) as proof of (((eq a) Xy) x1)
% Found ((((eq_sym a) x1) Xy) ((eq_ref a) x1)) as proof of (((eq a) Xy) x1)
% Found (fun (x000:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))))=> ((((eq_sym a) x1) Xy) ((eq_ref a) x1))) as proof of (((eq a) Xy) x1)
% Found eq_ref000:=(eq_ref00 P):((P Xy)->(P Xy))
% Found (eq_ref00 P) as proof of ((P Xy)->(P x3))
% Found ((eq_ref0 Xy) P) as proof of ((P Xy)->(P x3))
% Found (((eq_ref a) Xy) P) as proof of ((P Xy)->(P x3))
% Found (((eq_ref a) Xy) P) as proof of ((P Xy)->(P x3))
% Found (fun (P:(a->Prop))=> (((eq_ref a) Xy) P)) as proof of ((P Xy)->(P x3))
% Found (fun (x000:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))) (P:(a->Prop))=> (((eq_ref a) Xy) P)) as proof of (((eq a) Xy) x3)
% Found x6:(P Xy)
% Instantiate: x3:=Xy:a
% Found (fun (x6:(P Xy))=> x6) as proof of (P x3)
% Found (fun (P:(a->Prop)) (x6:(P Xy))=> x6) as proof of ((P Xy)->(P x3))
% Found (fun (x000:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))) (P:(a->Prop)) (x6:(P Xy))=> x6) as proof of (((eq a) Xy) x3)
% Found x6:(P Xy)
% Instantiate: x3:=Xy:a
% Found (fun (x6:(P Xy))=> x6) as proof of (P x3)
% Found (fun (P:(a->Prop)) (x6:(P Xy))=> x6) as proof of ((P Xy)->(P x3))
% Found (fun (x000:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))) (P:(a->Prop)) (x6:(P Xy))=> x6) as proof of (((eq a) Xy) x3)
% Found eq_ref000:=(eq_ref00 P):((P Xy)->(P Xy))
% Found (eq_ref00 P) as proof of ((P Xy)->(P x3))
% Found ((eq_ref0 Xy) P) as proof of ((P Xy)->(P x3))
% Found (((eq_ref a) Xy) P) as proof of ((P Xy)->(P x3))
% Found (((eq_ref a) Xy) P) as proof of ((P Xy)->(P x3))
% Found (fun (P:(a->Prop))=> (((eq_ref a) Xy) P)) as proof of ((P Xy)->(P x3))
% Found (fun (x000:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))) (P:(a->Prop))=> (((eq_ref a) Xy) P)) as proof of (((eq a) Xy) x3)
% Found x5:(X x4)
% Instantiate: x3:=x4:a
% Found x5 as proof of (X x3)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) x3)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) x3)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) x3)
% Found (fun (x5:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((eq_ref a) Xy)) as proof of (((eq a) Xy) x3)
% Found (fun (x4:(X Xy)) (x5:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((eq_ref a) Xy)) as proof of ((forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))->(((eq a) Xy) x3))
% Found (fun (x4:(X Xy)) (x5:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((eq_ref a) Xy)) as proof of ((X Xy)->((forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))->(((eq a) Xy) x3)))
% Found (and_rect10 (fun (x4:(X Xy)) (x5:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((eq_ref a) Xy))) as proof of (((eq a) Xy) x3)
% Found ((and_rect1 (((eq a) Xy) x3)) (fun (x4:(X Xy)) (x5:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((eq_ref a) Xy))) as proof of (((eq a) Xy) x3)
% Found (((fun (P:Type) (x4:((X Xy)->((forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))->P)))=> (((((and_rect (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))) P) x4) x000)) (((eq a) Xy) x3)) (fun (x4:(X Xy)) (x5:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((eq_ref a) Xy))) as proof of (((eq a) Xy) x3)
% Found (fun (x000:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))))=> (((fun (P:Type) (x4:((X Xy)->((forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))->P)))=> (((((and_rect (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))) P) x4) x000)) (((eq a) Xy) x3)) (fun (x4:(X Xy)) (x5:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((eq_ref a) Xy)))) as proof of (((eq a) Xy) x3)
% Found x3:(X x2)
% Instantiate: x1:=x2:a
% Found x3 as proof of (X x1)
% Found eq_ref00:=(eq_ref0 x3):(((eq a) x3) x3)
% Found (eq_ref0 x3) as proof of (((eq a) x3) Xy)
% Found ((eq_ref a) x3) as proof of (((eq a) x3) Xy)
% Found ((eq_ref a) x3) as proof of (((eq a) x3) Xy)
% Found ((eq_ref a) x3) as proof of (((eq a) x3) Xy)
% Found (eq_sym000 ((eq_ref a) x3)) as proof of (((eq a) Xy) x3)
% Found ((eq_sym00 Xy) ((eq_ref a) x3)) as proof of (((eq a) Xy) x3)
% Found (((eq_sym0 x3) Xy) ((eq_ref a) x3)) as proof of (((eq a) Xy) x3)
% Found ((((eq_sym a) x3) Xy) ((eq_ref a) x3)) as proof of (((eq a) Xy) x3)
% Found (fun (x000:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))))=> ((((eq_sym a) x3) Xy) ((eq_ref a) x3))) as proof of (((eq a) Xy) x3)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) x3)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) x3)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) x3)
% Found (fun (x5:(X x4))=> ((eq_ref a) Xy)) as proof of (((eq a) Xy) x3)
% Found (fun (x4:a) (x5:(X x4))=> ((eq_ref a) Xy)) as proof of ((X x4)->(((eq a) Xy) x3))
% Found (fun (x4:a) (x5:(X x4))=> ((eq_ref a) Xy)) as proof of (forall (x:a), ((X x)->(((eq a) Xy) x3)))
% Found (ex_ind00 (fun (x4:a) (x5:(X x4))=> ((eq_ref a) Xy))) as proof of (((eq a) Xy) x3)
% Found ((ex_ind0 (((eq a) Xy) x3)) (fun (x4:a) (x5:(X x4))=> ((eq_ref a) Xy))) as proof of (((eq a) Xy) x3)
% Found (((fun (P:Prop) (x4:(forall (x:a), ((X x)->P)))=> (((((ex_ind a) (fun (Xz:a)=> (X Xz))) P) x4) x00)) (((eq a) Xy) x3)) (fun (x4:a) (x5:(X x4))=> ((eq_ref a) Xy))) as proof of (((eq a) Xy) x3)
% Found (fun (x01:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx)))))=> (((fun (P:Prop) (x4:(forall (x:a), ((X x)->P)))=> (((((ex_ind a) (fun (Xz:a)=> (X Xz))) P) x4) x00)) (((eq a) Xy) x3)) (fun (x4:a) (x5:(X x4))=> ((eq_ref a) Xy)))) as proof of (((eq a) Xy) x3)
% Found x6:(P Xy)
% Instantiate: x1:=Xy:a
% Found (fun (x6:(P Xy))=> x6) as proof of (P x1)
% Found (fun (P:(a->Prop)) (x6:(P Xy))=> x6) as proof of ((P Xy)->(P x1))
% Found (fun (x000:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))) (P:(a->Prop)) (x6:(P Xy))=> x6) as proof of (((eq a) Xy) x1)
% Found eq_ref000:=(eq_ref00 P):((P Xy)->(P Xy))
% Found (eq_ref00 P) as proof of ((P Xy)->(P x1))
% Found ((eq_ref0 Xy) P) as proof of ((P Xy)->(P x1))
% Found (((eq_ref a) Xy) P) as proof of ((P Xy)->(P x1))
% Found (((eq_ref a) Xy) P) as proof of ((P Xy)->(P x1))
% Found (fun (P:(a->Prop))=> (((eq_ref a) Xy) P)) as proof of ((P Xy)->(P x1))
% Found (fun (x000:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))) (P:(a->Prop))=> (((eq_ref a) Xy) P)) as proof of (((eq a) Xy) x1)
% Found x6:(P Xy)
% Instantiate: x1:=Xy:a
% Found (fun (x6:(P Xy))=> x6) as proof of (P x1)
% Found (fun (P:(a->Prop)) (x6:(P Xy))=> x6) as proof of ((P Xy)->(P x1))
% Found (fun (x000:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))) (P:(a->Prop)) (x6:(P Xy))=> x6) as proof of (((eq a) Xy) x1)
% Found eq_ref000:=(eq_ref00 P):((P Xy)->(P Xy))
% Found (eq_ref00 P) as proof of ((P Xy)->(P x1))
% Found ((eq_ref0 Xy) P) as proof of ((P Xy)->(P x1))
% Found (((eq_ref a) Xy) P) as proof of ((P Xy)->(P x1))
% Found (((eq_ref a) Xy) P) as proof of ((P Xy)->(P x1))
% Found (fun (P:(a->Prop))=> (((eq_ref a) Xy) P)) as proof of ((P Xy)->(P x1))
% Found (fun (x000:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))) (P:(a->Prop))=> (((eq_ref a) Xy) P)) as proof of (((eq a) Xy) x1)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) x1)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) x1)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) x1)
% Found (fun (x5:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((eq_ref a) Xy)) as proof of (((eq a) Xy) x1)
% Found (fun (x4:(X Xy)) (x5:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((eq_ref a) Xy)) as proof of ((forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))->(((eq a) Xy) x1))
% Found (fun (x4:(X Xy)) (x5:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((eq_ref a) Xy)) as proof of ((X Xy)->((forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))->(((eq a) Xy) x1)))
% Found (and_rect10 (fun (x4:(X Xy)) (x5:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((eq_ref a) Xy))) as proof of (((eq a) Xy) x1)
% Found ((and_rect1 (((eq a) Xy) x1)) (fun (x4:(X Xy)) (x5:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((eq_ref a) Xy))) as proof of (((eq a) Xy) x1)
% Found (((fun (P:Type) (x4:((X Xy)->((forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))->P)))=> (((((and_rect (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))) P) x4) x000)) (((eq a) Xy) x1)) (fun (x4:(X Xy)) (x5:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((eq_ref a) Xy))) as proof of (((eq a) Xy) x1)
% Found (fun (x000:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))))=> (((fun (P:Type) (x4:((X Xy)->((forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))->P)))=> (((((and_rect (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))) P) x4) x000)) (((eq a) Xy) x1)) (fun (x4:(X Xy)) (x5:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((eq_ref a) Xy)))) as proof of (((eq a) Xy) x1)
% Found eq_ref00:=(eq_ref0 x1):(((eq a) x1) x1)
% Found (eq_ref0 x1) as proof of (((eq a) x1) Xy)
% Found ((eq_ref a) x1) as proof of (((eq a) x1) Xy)
% Found ((eq_ref a) x1) as proof of (((eq a) x1) Xy)
% Found ((eq_ref a) x1) as proof of (((eq a) x1) Xy)
% Found (eq_sym000 ((eq_ref a) x1)) as proof of (((eq a) Xy) x1)
% Found ((eq_sym00 Xy) ((eq_ref a) x1)) as proof of (((eq a) Xy) x1)
% Found (((eq_sym0 x1) Xy) ((eq_ref a) x1)) as proof of (((eq a) Xy) x1)
% Found ((((eq_sym a) x1) Xy) ((eq_ref a) x1)) as proof of (((eq a) Xy) x1)
% Found (fun (x000:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))))=> ((((eq_sym a) x1) Xy) ((eq_ref a) x1))) as proof of (((eq a) Xy) x1)
% Found x2:(X x1)
% Instantiate: x5:=x1:a
% Found x2 as proof of (X x5)
% Found x4:(X x3)
% Instantiate: x5:=x3:a
% Found x4 as proof of (X x5)
% Found x2:(X x1)
% Instantiate: x3:=x1:a
% Found x2 as proof of (X x3)
% Found x5:(X x4)
% Instantiate: x3:=x4:a
% Found x5 as proof of (X x3)
% Found eq_ref00:=(eq_ref0 b0):(((eq ((a->(a->Prop))->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq ((a->(a->Prop))->Prop)) b0) (fun (R:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((R Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((R Xy) Xx))))->(((eq a) Xy) Xz))))))))))
% Found ((eq_ref ((a->(a->Prop))->Prop)) b0) as proof of (((eq ((a->(a->Prop))->Prop)) b0) (fun (R:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((R Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((R Xy) Xx))))->(((eq a) Xy) Xz))))))))))
% Found ((eq_ref ((a->(a->Prop))->Prop)) b0) as proof of (((eq ((a->(a->Prop))->Prop)) b0) (fun (R:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((R Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((R Xy) Xx))))->(((eq a) Xy) Xz))))))))))
% Found ((eq_ref ((a->(a->Prop))->Prop)) b0) as proof of (((eq ((a->(a->Prop))->Prop)) b0) (fun (R:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((R Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((R Xy) Xx))))->(((eq a) Xy) Xz))))))))))
% Found eq_ref00:=(eq_ref0 a0):(((eq ((a->(a->Prop))->Prop)) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq ((a->(a->Prop))->Prop)) a0) b0)
% Found ((eq_ref ((a->(a->Prop))->Prop)) a0) as proof of (((eq ((a->(a->Prop))->Prop)) a0) b0)
% Found ((eq_ref ((a->(a->Prop))->Prop)) a0) as proof of (((eq ((a->(a->Prop))->Prop)) a0) b0)
% Found ((eq_ref ((a->(a->Prop))->Prop)) a0) as proof of (((eq ((a->(a->Prop))->Prop)) a0) b0)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) x3)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) x3)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) x3)
% Found (fun (x5:(forall (Xx:a), ((ex (b->Prop)) (fun (V:(b->Prop))=> ((and (cW V)) (((eq a) Xx) (h V)))))))=> ((eq_ref a) Xy)) as proof of (((eq a) Xy) x3)
% Found (fun (x4:(forall (U:((b->Prop)->Prop)), (((and ((ex (b->Prop)) (fun (Z:(b->Prop))=> (U Z)))) (forall (Xx:(b->Prop)), ((U Xx)->(cW Xx))))->(cW (fun (Xx:b)=> (forall (S:(b->Prop)), ((U S)->(S Xx)))))))) (x5:(forall (Xx:a), ((ex (b->Prop)) (fun (V:(b->Prop))=> ((and (cW V)) (((eq a) Xx) (h V)))))))=> ((eq_ref a) Xy)) as proof of ((forall (Xx:a), ((ex (b->Prop)) (fun (V:(b->Prop))=> ((and (cW V)) (((eq a) Xx) (h V))))))->(((eq a) Xy) x3))
% Found (fun (x4:(forall (U:((b->Prop)->Prop)), (((and ((ex (b->Prop)) (fun (Z:(b->Prop))=> (U Z)))) (forall (Xx:(b->Prop)), ((U Xx)->(cW Xx))))->(cW (fun (Xx:b)=> (forall (S:(b->Prop)), ((U S)->(S Xx)))))))) (x5:(forall (Xx:a), ((ex (b->Prop)) (fun (V:(b->Prop))=> ((and (cW V)) (((eq a) Xx) (h V)))))))=> ((eq_ref a) Xy)) as proof of ((forall (U:((b->Prop)->Prop)), (((and ((ex (b->Prop)) (fun (Z:(b->Prop))=> (U Z)))) (forall (Xx:(b->Prop)), ((U Xx)->(cW Xx))))->(cW (fun (Xx:b)=> (forall (S:(b->Prop)), ((U S)->(S Xx)))))))->((forall (Xx:a), ((ex (b->Prop)) (fun (V:(b->Prop))=> ((and (cW V)) (((eq a) Xx) (h V))))))->(((eq a) Xy) x3)))
% Found (and_rect00 (fun (x4:(forall (U:((b->Prop)->Prop)), (((and ((ex (b->Prop)) (fun (Z:(b->Prop))=> (U Z)))) (forall (Xx:(b->Prop)), ((U Xx)->(cW Xx))))->(cW (fun (Xx:b)=> (forall (S:(b->Prop)), ((U S)->(S Xx)))))))) (x5:(forall (Xx:a), ((ex (b->Prop)) (fun (V:(b->Prop))=> ((and (cW V)) (((eq a) Xx) (h V)))))))=> ((eq_ref a) Xy))) as proof of (((eq a) Xy) x3)
% Found ((and_rect0 (((eq a) Xy) x3)) (fun (x4:(forall (U:((b->Prop)->Prop)), (((and ((ex (b->Prop)) (fun (Z:(b->Prop))=> (U Z)))) (forall (Xx:(b->Prop)), ((U Xx)->(cW Xx))))->(cW (fun (Xx:b)=> (forall (S:(b->Prop)), ((U S)->(S Xx)))))))) (x5:(forall (Xx:a), ((ex (b->Prop)) (fun (V:(b->Prop))=> ((and (cW V)) (((eq a) Xx) (h V)))))))=> ((eq_ref a) Xy))) as proof of (((eq a) Xy) x3)
% Found (((fun (P:Type) (x4:((forall (U:((b->Prop)->Prop)), (((and ((ex (b->Prop)) (fun (Z:(b->Prop))=> (U Z)))) (forall (Xx:(b->Prop)), ((U Xx)->(cW Xx))))->(cW (fun (Xx:b)=> (forall (S:(b->Prop)), ((U S)->(S Xx)))))))->((forall (Xx:a), ((ex (b->Prop)) (fun (V:(b->Prop))=> ((and (cW V)) (((eq a) Xx) (h V))))))->P)))=> (((((and_rect (forall (U:((b->Prop)->Prop)), (((and ((ex (b->Prop)) (fun (Z:(b->Prop))=> (U Z)))) (forall (Xx:(b->Prop)), ((U Xx)->(cW Xx))))->(cW (fun (Xx:b)=> (forall (S:(b->Prop)), ((U S)->(S Xx)))))))) (forall (Xx:a), ((ex (b->Prop)) (fun (V:(b->Prop))=> ((and (cW V)) (((eq a) Xx) (h V))))))) P) x4) x)) (((eq a) Xy) x3)) (fun (x4:(forall (U:((b->Prop)->Prop)), (((and ((ex (b->Prop)) (fun (Z:(b->Prop))=> (U Z)))) (forall (Xx:(b->Prop)), ((U Xx)->(cW Xx))))->(cW (fun (Xx:b)=> (forall (S:(b->Prop)), ((U S)->(S Xx)))))))) (x5:(forall (Xx:a), ((ex (b->Prop)) (fun (V:(b->Prop))=> ((and (cW V)) (((eq a) Xx) (h V)))))))=> ((eq_ref a) Xy))) as proof of (((eq a) Xy) x3)
% Found (fun (x000:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))))=> (((fun (P:Type) (x4:((forall (U:((b->Prop)->Prop)), (((and ((ex (b->Prop)) (fun (Z:(b->Prop))=> (U Z)))) (forall (Xx:(b->Prop)), ((U Xx)->(cW Xx))))->(cW (fun (Xx:b)=> (forall (S:(b->Prop)), ((U S)->(S Xx)))))))->((forall (Xx:a), ((ex (b->Prop)) (fun (V:(b->Prop))=> ((and (cW V)) (((eq a) Xx) (h V))))))->P)))=> (((((and_rect (forall (U:((b->Prop)->Prop)), (((and ((ex (b->Prop)) (fun (Z:(b->Prop))=> (U Z)))) (forall (Xx:(b->Prop)), ((U Xx)->(cW Xx))))->(cW (fun (Xx:b)=> (forall (S:(b->Prop)), ((U S)->(S Xx)))))))) (forall (Xx:a), ((ex (b->Prop)) (fun (V:(b->Prop))=> ((and (cW V)) (((eq a) Xx) (h V))))))) P) x4) x)) (((eq a) Xy) x3)) (fun (x4:(forall (U:((b->Prop)->Prop)), (((and ((ex (b->Prop)) (fun (Z:(b->Prop))=> (U Z)))) (forall (Xx:(b->Prop)), ((U Xx)->(cW Xx))))->(cW (fun (Xx:b)=> (forall (S:(b->Prop)), ((U S)->(S Xx)))))))) (x5:(forall (Xx:a), ((ex (b->Prop)) (fun (V:(b->Prop))=> ((and (cW V)) (((eq a) Xx) (h V)))))))=> ((eq_ref a) Xy)))) as proof of (((eq a) Xy) x3)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) x3)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) x3)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) x3)
% Found (fun (x5:(X x4))=> ((eq_ref a) Xy)) as proof of (((eq a) Xy) x3)
% Found (fun (x4:a) (x5:(X x4))=> ((eq_ref a) Xy)) as proof of ((X x4)->(((eq a) Xy) x3))
% Found (fun (x4:a) (x5:(X x4))=> ((eq_ref a) Xy)) as proof of (forall (x:a), ((X x)->(((eq a) Xy) x3)))
% Found (ex_ind00 (fun (x4:a) (x5:(X x4))=> ((eq_ref a) Xy))) as proof of (((eq a) Xy) x3)
% Found ((ex_ind0 (((eq a) Xy) x3)) (fun (x4:a) (x5:(X x4))=> ((eq_ref a) Xy))) as proof of (((eq a) Xy) x3)
% Found (((fun (P:Prop) (x4:(forall (x:a), ((X x)->P)))=> (((((ex_ind a) (fun (Xz:a)=> (X Xz))) P) x4) x00)) (((eq a) Xy) x3)) (fun (x4:a) (x5:(X x4))=> ((eq_ref a) Xy))) as proof of (((eq a) Xy) x3)
% Found (fun (x000:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))))=> (((fun (P:Prop) (x4:(forall (x:a), ((X x)->P)))=> (((((ex_ind a) (fun (Xz:a)=> (X Xz))) P) x4) x00)) (((eq a) Xy) x3)) (fun (x4:a) (x5:(X x4))=> ((eq_ref a) Xy)))) as proof of (((eq a) Xy) x3)
% Found x00:((ex a) (fun (Xz:a)=> (X Xz)))
% Instantiate: b0:=(fun (Xz:a)=> (X Xz)):(a->Prop)
% Found x00 as proof of (P b0)
% Found eta_expansion000:=(eta_expansion00 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))):(((eq (a->Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))) (fun (x:a)=> ((and ((and (X x)) (forall (Xx:a), ((X Xx)->((x0 x) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) x))))))
% Found (eta_expansion00 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))) b0)
% Found ((eta_expansion0 Prop) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))) b0)
% Found (((eta_expansion a) Prop) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))) b0)
% Found (((eta_expansion a) Prop) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))) b0)
% Found (((eta_expansion a) Prop) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))) b0)
% Found x3:(X x2)
% Instantiate: x1:=x2:a
% Found x3 as proof of (X x1)
% Found x5:(X x4)
% Instantiate: x1:=x4:a
% Found x5 as proof of (X x1)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) x5)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) x5)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) x5)
% Found (fun (x7:(forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))=> ((eq_ref a) Xy)) as proof of (((eq a) Xy) x5)
% Found (fun (x6:(X Xy)) (x7:(forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))=> ((eq_ref a) Xy)) as proof of ((forall (Xx:a), ((X Xx)->((x2 Xy) Xx)))->(((eq a) Xy) x5))
% Found (fun (x6:(X Xy)) (x7:(forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))=> ((eq_ref a) Xy)) as proof of ((X Xy)->((forall (Xx:a), ((X Xx)->((x2 Xy) Xx)))->(((eq a) Xy) x5)))
% Found (and_rect10 (fun (x6:(X Xy)) (x7:(forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))=> ((eq_ref a) Xy))) as proof of (((eq a) Xy) x5)
% Found ((and_rect1 (((eq a) Xy) x5)) (fun (x6:(X Xy)) (x7:(forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))=> ((eq_ref a) Xy))) as proof of (((eq a) Xy) x5)
% Found (((fun (P:Type) (x6:((X Xy)->((forall (Xx:a), ((X Xx)->((x2 Xy) Xx)))->P)))=> (((((and_rect (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx)))) P) x6) x01)) (((eq a) Xy) x5)) (fun (x6:(X Xy)) (x7:(forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))=> ((eq_ref a) Xy))) as proof of (((eq a) Xy) x5)
% Found (fun (x01:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx)))))=> (((fun (P:Type) (x6:((X Xy)->((forall (Xx:a), ((X Xx)->((x2 Xy) Xx)))->P)))=> (((((and_rect (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx)))) P) x6) x01)) (((eq a) Xy) x5)) (fun (x6:(X Xy)) (x7:(forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))=> ((eq_ref a) Xy)))) as proof of (((eq a) Xy) x5)
% Found eq_ref00:=(eq_ref0 x5):(((eq a) x5) x5)
% Found (eq_ref0 x5) as proof of (((eq a) x5) Xy)
% Found ((eq_ref a) x5) as proof of (((eq a) x5) Xy)
% Found ((eq_ref a) x5) as proof of (((eq a) x5) Xy)
% Found ((eq_ref a) x5) as proof of (((eq a) x5) Xy)
% Found (eq_sym000 ((eq_ref a) x5)) as proof of (((eq a) Xy) x5)
% Found ((eq_sym00 Xy) ((eq_ref a) x5)) as proof of (((eq a) Xy) x5)
% Found (((eq_sym0 x5) Xy) ((eq_ref a) x5)) as proof of (((eq a) Xy) x5)
% Found ((((eq_sym a) x5) Xy) ((eq_ref a) x5)) as proof of (((eq a) Xy) x5)
% Found (fun (x01:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx)))))=> ((((eq_sym a) x5) Xy) ((eq_ref a) x5))) as proof of (((eq a) Xy) x5)
% Found x00:((ex a) (fun (Xz:a)=> (X Xz)))
% Instantiate: f:=(fun (Xz:a)=> (X Xz)):(a->Prop)
% Found x00 as proof of (P 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 (X x1)) (forall (Xx:a), ((X Xx)->((x0 x1) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) x1)))))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and ((and (X x1)) (forall (Xx:a), ((X Xx)->((x0 x1) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) x1)))))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and ((and (X x1)) (forall (Xx:a), ((X Xx)->((x0 x1) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) x1)))))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) ((and ((and (X x1)) (forall (Xx:a), ((X Xx)->((x0 x1) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) x1)))))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:a), (((eq Prop) (f x)) ((and ((and (X x)) (forall (Xx:a), ((X Xx)->((x0 x) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) x))))))
% Found x00:((ex a) (fun (Xz:a)=> (X Xz)))
% Instantiate: f:=(fun (Xz:a)=> (X Xz)):(a->Prop)
% Found x00 as proof of (P 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 (X x1)) (forall (Xx:a), ((X Xx)->((x0 x1) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) x1)))))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and ((and (X x1)) (forall (Xx:a), ((X Xx)->((x0 x1) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) x1)))))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and ((and (X x1)) (forall (Xx:a), ((X Xx)->((x0 x1) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) x1)))))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) ((and ((and (X x1)) (forall (Xx:a), ((X Xx)->((x0 x1) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) x1)))))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:a), (((eq Prop) (f x)) ((and ((and (X x)) (forall (Xx:a), ((X Xx)->((x0 x) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) x))))))
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) x1)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) x1)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) x1)
% Found (fun (x5:(forall (Xx:a), ((ex (b->Prop)) (fun (V:(b->Prop))=> ((and (cW V)) (((eq a) Xx) (h V)))))))=> ((eq_ref a) Xy)) as proof of (((eq a) Xy) x1)
% Found (fun (x4:(forall (U:((b->Prop)->Prop)), (((and ((ex (b->Prop)) (fun (Z:(b->Prop))=> (U Z)))) (forall (Xx:(b->Prop)), ((U Xx)->(cW Xx))))->(cW (fun (Xx:b)=> (forall (S:(b->Prop)), ((U S)->(S Xx)))))))) (x5:(forall (Xx:a), ((ex (b->Prop)) (fun (V:(b->Prop))=> ((and (cW V)) (((eq a) Xx) (h V)))))))=> ((eq_ref a) Xy)) as proof of ((forall (Xx:a), ((ex (b->Prop)) (fun (V:(b->Prop))=> ((and (cW V)) (((eq a) Xx) (h V))))))->(((eq a) Xy) x1))
% Found (fun (x4:(forall (U:((b->Prop)->Prop)), (((and ((ex (b->Prop)) (fun (Z:(b->Prop))=> (U Z)))) (forall (Xx:(b->Prop)), ((U Xx)->(cW Xx))))->(cW (fun (Xx:b)=> (forall (S:(b->Prop)), ((U S)->(S Xx)))))))) (x5:(forall (Xx:a), ((ex (b->Prop)) (fun (V:(b->Prop))=> ((and (cW V)) (((eq a) Xx) (h V)))))))=> ((eq_ref a) Xy)) as proof of ((forall (U:((b->Prop)->Prop)), (((and ((ex (b->Prop)) (fun (Z:(b->Prop))=> (U Z)))) (forall (Xx:(b->Prop)), ((U Xx)->(cW Xx))))->(cW (fun (Xx:b)=> (forall (S:(b->Prop)), ((U S)->(S Xx)))))))->((forall (Xx:a), ((ex (b->Prop)) (fun (V:(b->Prop))=> ((and (cW V)) (((eq a) Xx) (h V))))))->(((eq a) Xy) x1)))
% Found (and_rect00 (fun (x4:(forall (U:((b->Prop)->Prop)), (((and ((ex (b->Prop)) (fun (Z:(b->Prop))=> (U Z)))) (forall (Xx:(b->Prop)), ((U Xx)->(cW Xx))))->(cW (fun (Xx:b)=> (forall (S:(b->Prop)), ((U S)->(S Xx)))))))) (x5:(forall (Xx:a), ((ex (b->Prop)) (fun (V:(b->Prop))=> ((and (cW V)) (((eq a) Xx) (h V)))))))=> ((eq_ref a) Xy))) as proof of (((eq a) Xy) x1)
% Found ((and_rect0 (((eq a) Xy) x1)) (fun (x4:(forall (U:((b->Prop)->Prop)), (((and ((ex (b->Prop)) (fun (Z:(b->Prop))=> (U Z)))) (forall (Xx:(b->Prop)), ((U Xx)->(cW Xx))))->(cW (fun (Xx:b)=> (forall (S:(b->Prop)), ((U S)->(S Xx)))))))) (x5:(forall (Xx:a), ((ex (b->Prop)) (fun (V:(b->Prop))=> ((and (cW V)) (((eq a) Xx) (h V)))))))=> ((eq_ref a) Xy))) as proof of (((eq a) Xy) x1)
% Found (((fun (P:Type) (x4:((forall (U:((b->Prop)->Prop)), (((and ((ex (b->Prop)) (fun (Z:(b->Prop))=> (U Z)))) (forall (Xx:(b->Prop)), ((U Xx)->(cW Xx))))->(cW (fun (Xx:b)=> (forall (S:(b->Prop)), ((U S)->(S Xx)))))))->((forall (Xx:a), ((ex (b->Prop)) (fun (V:(b->Prop))=> ((and (cW V)) (((eq a) Xx) (h V))))))->P)))=> (((((and_rect (forall (U:((b->Prop)->Prop)), (((and ((ex (b->Prop)) (fun (Z:(b->Prop))=> (U Z)))) (forall (Xx:(b->Prop)), ((U Xx)->(cW Xx))))->(cW (fun (Xx:b)=> (forall (S:(b->Prop)), ((U S)->(S Xx)))))))) (forall (Xx:a), ((ex (b->Prop)) (fun (V:(b->Prop))=> ((and (cW V)) (((eq a) Xx) (h V))))))) P) x4) x)) (((eq a) Xy) x1)) (fun (x4:(forall (U:((b->Prop)->Prop)), (((and ((ex (b->Prop)) (fun (Z:(b->Prop))=> (U Z)))) (forall (Xx:(b->Prop)), ((U Xx)->(cW Xx))))->(cW (fun (Xx:b)=> (forall (S:(b->Prop)), ((U S)->(S Xx)))))))) (x5:(forall (Xx:a), ((ex (b->Prop)) (fun (V:(b->Prop))=> ((and (cW V)) (((eq a) Xx) (h V)))))))=> ((eq_ref a) Xy))) as proof of (((eq a) Xy) x1)
% Found (fun (x000:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))))=> (((fun (P:Type) (x4:((forall (U:((b->Prop)->Prop)), (((and ((ex (b->Prop)) (fun (Z:(b->Prop))=> (U Z)))) (forall (Xx:(b->Prop)), ((U Xx)->(cW Xx))))->(cW (fun (Xx:b)=> (forall (S:(b->Prop)), ((U S)->(S Xx)))))))->((forall (Xx:a), ((ex (b->Prop)) (fun (V:(b->Prop))=> ((and (cW V)) (((eq a) Xx) (h V))))))->P)))=> (((((and_rect (forall (U:((b->Prop)->Prop)), (((and ((ex (b->Prop)) (fun (Z:(b->Prop))=> (U Z)))) (forall (Xx:(b->Prop)), ((U Xx)->(cW Xx))))->(cW (fun (Xx:b)=> (forall (S:(b->Prop)), ((U S)->(S Xx)))))))) (forall (Xx:a), ((ex (b->Prop)) (fun (V:(b->Prop))=> ((and (cW V)) (((eq a) Xx) (h V))))))) P) x4) x)) (((eq a) Xy) x1)) (fun (x4:(forall (U:((b->Prop)->Prop)), (((and ((ex (b->Prop)) (fun (Z:(b->Prop))=> (U Z)))) (forall (Xx:(b->Prop)), ((U Xx)->(cW Xx))))->(cW (fun (Xx:b)=> (forall (S:(b->Prop)), ((U S)->(S Xx)))))))) (x5:(forall (Xx:a), ((ex (b->Prop)) (fun (V:(b->Prop))=> ((and (cW V)) (((eq a) Xx) (h V)))))))=> ((eq_ref a) Xy)))) as proof of (((eq a) Xy) x1)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) x1)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) x1)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) x1)
% Found (fun (x5:(X x4))=> ((eq_ref a) Xy)) as proof of (((eq a) Xy) x1)
% Found (fun (x4:a) (x5:(X x4))=> ((eq_ref a) Xy)) as proof of ((X x4)->(((eq a) Xy) x1))
% Found (fun (x4:a) (x5:(X x4))=> ((eq_ref a) Xy)) as proof of (forall (x:a), ((X x)->(((eq a) Xy) x1)))
% Found (ex_ind00 (fun (x4:a) (x5:(X x4))=> ((eq_ref a) Xy))) as proof of (((eq a) Xy) x1)
% Found ((ex_ind0 (((eq a) Xy) x1)) (fun (x4:a) (x5:(X x4))=> ((eq_ref a) Xy))) as proof of (((eq a) Xy) x1)
% Found (((fun (P:Prop) (x4:(forall (x:a), ((X x)->P)))=> (((((ex_ind a) (fun (Xz:a)=> (X Xz))) P) x4) x00)) (((eq a) Xy) x1)) (fun (x4:a) (x5:(X x4))=> ((eq_ref a) Xy))) as proof of (((eq a) Xy) x1)
% Found (fun (x000:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))))=> (((fun (P:Prop) (x4:(forall (x:a), ((X x)->P)))=> (((((ex_ind a) (fun (Xz:a)=> (X Xz))) P) x4) x00)) (((eq a) Xy) x1)) (fun (x4:a) (x5:(X x4))=> ((eq_ref a) Xy)))) as proof of (((eq a) Xy) x1)
% Found x5:(X x4)
% Instantiate: x3:=x4:a
% Found x5 as proof of (X x3)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) x3)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) x3)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) x3)
% Found (fun (x7:(forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))=> ((eq_ref a) Xy)) as proof of (((eq a) Xy) x3)
% Found (fun (x6:(X Xy)) (x7:(forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))=> ((eq_ref a) Xy)) as proof of ((forall (Xx:a), ((X Xx)->((x2 Xy) Xx)))->(((eq a) Xy) x3))
% Found (fun (x6:(X Xy)) (x7:(forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))=> ((eq_ref a) Xy)) as proof of ((X Xy)->((forall (Xx:a), ((X Xx)->((x2 Xy) Xx)))->(((eq a) Xy) x3)))
% Found (and_rect10 (fun (x6:(X Xy)) (x7:(forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))=> ((eq_ref a) Xy))) as proof of (((eq a) Xy) x3)
% Found ((and_rect1 (((eq a) Xy) x3)) (fun (x6:(X Xy)) (x7:(forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))=> ((eq_ref a) Xy))) as proof of (((eq a) Xy) x3)
% Found (((fun (P:Type) (x6:((X Xy)->((forall (Xx:a), ((X Xx)->((x2 Xy) Xx)))->P)))=> (((((and_rect (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx)))) P) x6) x01)) (((eq a) Xy) x3)) (fun (x6:(X Xy)) (x7:(forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))=> ((eq_ref a) Xy))) as proof of (((eq a) Xy) x3)
% Found (fun (x01:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx)))))=> (((fun (P:Type) (x6:((X Xy)->((forall (Xx:a), ((X Xx)->((x2 Xy) Xx)))->P)))=> (((((and_rect (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx)))) P) x6) x01)) (((eq a) Xy) x3)) (fun (x6:(X Xy)) (x7:(forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))=> ((eq_ref a) Xy)))) as proof of (((eq a) Xy) x3)
% Found x3:(X x2)
% Instantiate: x1:=x2:a
% Found (fun (x3:(X x2))=> x3) as proof of (X x1)
% Found eq_ref000:=(eq_ref00 X):((X x2)->(X x2))
% Found (eq_ref00 X) as proof of ((X x2)->(X x1))
% Found ((eq_ref0 x2) X) as proof of ((X x2)->(X x1))
% Found (((eq_ref a) x2) X) as proof of ((X x2)->(X x1))
% Found (((eq_ref a) x2) X) as proof of ((X x2)->(X x1))
% Found eq_ref00:=(eq_ref0 x3):(((eq a) x3) x3)
% Found (eq_ref0 x3) as proof of (((eq a) x3) Xy)
% Found ((eq_ref a) x3) as proof of (((eq a) x3) Xy)
% Found ((eq_ref a) x3) as proof of (((eq a) x3) Xy)
% Found ((eq_ref a) x3) as proof of (((eq a) x3) Xy)
% Found (eq_sym000 ((eq_ref a) x3)) as proof of (((eq a) Xy) x3)
% Found ((eq_sym00 Xy) ((eq_ref a) x3)) as proof of (((eq a) Xy) x3)
% Found (((eq_sym0 x3) Xy) ((eq_ref a) x3)) as proof of (((eq a) Xy) x3)
% Found ((((eq_sym a) x3) Xy) ((eq_ref a) x3)) as proof of (((eq a) Xy) x3)
% Found (fun (x01:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx)))))=> ((((eq_sym a) x3) Xy) ((eq_ref a) x3))) as proof of (((eq a) Xy) x3)
% Found eta_expansion000:=(eta_expansion00 (fun (Xz:a)=> (X Xz))):(((eq (a->Prop)) (fun (Xz:a)=> (X Xz))) (fun (x:a)=> (X x)))
% Found (eta_expansion00 (fun (Xz:a)=> (X Xz))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> (X Xz))) b0)
% Found ((eta_expansion0 Prop) (fun (Xz:a)=> (X Xz))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> (X Xz))) b0)
% Found (((eta_expansion a) Prop) (fun (Xz:a)=> (X Xz))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> (X Xz))) b0)
% Found (((eta_expansion a) Prop) (fun (Xz:a)=> (X Xz))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> (X Xz))) b0)
% Found (((eta_expansion a) Prop) (fun (Xz:a)=> (X Xz))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> (X Xz))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq (a->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (a->Prop)) b0) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz))))))
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz))))))
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz))))))
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz))))))
% Found eq_ref000:=(eq_ref00 (ex a)):(((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> (X Xz))))
% Found (eq_ref00 (ex a)) as proof of (P (fun (Xz:a)=> (X Xz)))
% Found ((eq_ref0 (fun (Xz:a)=> (X Xz))) (ex a)) as proof of (P (fun (Xz:a)=> (X Xz)))
% Found (((eq_ref (a->Prop)) (fun (Xz:a)=> (X Xz))) (ex a)) as proof of (P (fun (Xz:a)=> (X Xz)))
% Found (((eq_ref (a->Prop)) (fun (Xz:a)=> (X Xz))) (ex a)) as proof of (P (fun (Xz:a)=> (X Xz)))
% Found eq_ref00:=(eq_ref0 (fun (Xz:a)=> (X Xz))):(((eq (a->Prop)) (fun (Xz:a)=> (X Xz))) (fun (Xz:a)=> (X Xz)))
% Found (eq_ref0 (fun (Xz:a)=> (X Xz))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> (X Xz))) b0)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> (X Xz))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> (X Xz))) b0)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> (X Xz))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> (X Xz))) b0)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> (X Xz))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> (X Xz))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq (a->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (a->Prop)) b0) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz))))))
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz))))))
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz))))))
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz))))))
% Found eq_ref00:=(eq_ref0 b0):(((eq ((a->(a->Prop))->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq ((a->(a->Prop))->Prop)) b0) (fun (R:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((R Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((R Xy) Xx))))->(((eq a) Xy) Xz))))))))))
% Found ((eq_ref ((a->(a->Prop))->Prop)) b0) as proof of (((eq ((a->(a->Prop))->Prop)) b0) (fun (R:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((R Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((R Xy) Xx))))->(((eq a) Xy) Xz))))))))))
% Found ((eq_ref ((a->(a->Prop))->Prop)) b0) as proof of (((eq ((a->(a->Prop))->Prop)) b0) (fun (R:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((R Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((R Xy) Xx))))->(((eq a) Xy) Xz))))))))))
% Found ((eq_ref ((a->(a->Prop))->Prop)) b0) as proof of (((eq ((a->(a->Prop))->Prop)) b0) (fun (R:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((R Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((R Xy) Xx))))->(((eq a) Xy) Xz))))))))))
% Found eta_expansion000:=(eta_expansion00 a0):(((eq ((a->(a->Prop))->Prop)) a0) (fun (x:(a->(a->Prop)))=> (a0 x)))
% Found (eta_expansion00 a0) as proof of (((eq ((a->(a->Prop))->Prop)) a0) b0)
% Found ((eta_expansion0 Prop) a0) as proof of (((eq ((a->(a->Prop))->Prop)) a0) b0)
% Found (((eta_expansion (a->(a->Prop))) Prop) a0) as proof of (((eq ((a->(a->Prop))->Prop)) a0) b0)
% Found (((eta_expansion (a->(a->Prop))) Prop) a0) as proof of (((eq ((a->(a->Prop))->Prop)) a0) b0)
% Found (((eta_expansion (a->(a->Prop))) Prop) a0) as proof of (((eq ((a->(a->Prop))->Prop)) a0) b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq ((a->(a->Prop))->Prop)) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq ((a->(a->Prop))->Prop)) a0) b0)
% Found ((eq_ref ((a->(a->Prop))->Prop)) a0) as proof of (((eq ((a->(a->Prop))->Prop)) a0) b0)
% Found ((eq_ref ((a->(a->Prop))->Prop)) a0) as proof of (((eq ((a->(a->Prop))->Prop)) a0) b0)
% Found ((eq_ref ((a->(a->Prop))->Prop)) a0) as proof of (((eq ((a->(a->Prop))->Prop)) a0) b0)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) x5)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) x5)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) x5)
% Found (fun (x7:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((eq_ref a) Xy)) as proof of (((eq a) Xy) x5)
% Found (fun (x6:(X Xy)) (x7:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((eq_ref a) Xy)) as proof of ((forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))->(((eq a) Xy) x5))
% Found (fun (x6:(X Xy)) (x7:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((eq_ref a) Xy)) as proof of ((X Xy)->((forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))->(((eq a) Xy) x5)))
% Found (and_rect10 (fun (x6:(X Xy)) (x7:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((eq_ref a) Xy))) as proof of (((eq a) Xy) x5)
% Found ((and_rect1 (((eq a) Xy) x5)) (fun (x6:(X Xy)) (x7:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((eq_ref a) Xy))) as proof of (((eq a) Xy) x5)
% Found (((fun (P:Type) (x6:((X Xy)->((forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))->P)))=> (((((and_rect (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))) P) x6) x000)) (((eq a) Xy) x5)) (fun (x6:(X Xy)) (x7:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((eq_ref a) Xy))) as proof of (((eq a) Xy) x5)
% Found (fun (x000:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))))=> (((fun (P:Type) (x6:((X Xy)->((forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))->P)))=> (((((and_rect (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))) P) x6) x000)) (((eq a) Xy) x5)) (fun (x6:(X Xy)) (x7:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((eq_ref a) Xy)))) as proof of (((eq a) Xy) x5)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) x5)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) x5)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) x5)
% Found (fun (x7:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((eq_ref a) Xy)) as proof of (((eq a) Xy) x5)
% Found (fun (x6:(X Xy)) (x7:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((eq_ref a) Xy)) as proof of ((forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))->(((eq a) Xy) x5))
% Found (fun (x6:(X Xy)) (x7:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((eq_ref a) Xy)) as proof of ((X Xy)->((forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))->(((eq a) Xy) x5)))
% Found (and_rect10 (fun (x6:(X Xy)) (x7:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((eq_ref a) Xy))) as proof of (((eq a) Xy) x5)
% Found ((and_rect1 (((eq a) Xy) x5)) (fun (x6:(X Xy)) (x7:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((eq_ref a) Xy))) as proof of (((eq a) Xy) x5)
% Found (((fun (P:Type) (x6:((X Xy)->((forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))->P)))=> (((((and_rect (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))) P) x6) x000)) (((eq a) Xy) x5)) (fun (x6:(X Xy)) (x7:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((eq_ref a) Xy))) as proof of (((eq a) Xy) x5)
% Found (fun (x000:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))))=> (((fun (P:Type) (x6:((X Xy)->((forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))->P)))=> (((((and_rect (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))) P) x6) x000)) (((eq a) Xy) x5)) (fun (x6:(X Xy)) (x7:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((eq_ref a) Xy)))) as proof of (((eq a) Xy) x5)
% Found eq_ref00:=(eq_ref0 x5):(((eq a) x5) x5)
% Found (eq_ref0 x5) as proof of (((eq a) x5) Xy)
% Found ((eq_ref a) x5) as proof of (((eq a) x5) Xy)
% Found ((eq_ref a) x5) as proof of (((eq a) x5) Xy)
% Found ((eq_ref a) x5) as proof of (((eq a) x5) Xy)
% Found (eq_sym000 ((eq_ref a) x5)) as proof of (((eq a) Xy) x5)
% Found ((eq_sym00 Xy) ((eq_ref a) x5)) as proof of (((eq a) Xy) x5)
% Found (((eq_sym0 x5) Xy) ((eq_ref a) x5)) as proof of (((eq a) Xy) x5)
% Found ((((eq_sym a) x5) Xy) ((eq_ref a) x5)) as proof of (((eq a) Xy) x5)
% Found (fun (x000:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))))=> ((((eq_sym a) x5) Xy) ((eq_ref a) x5))) as proof of (((eq a) Xy) x5)
% Found eq_ref00:=(eq_ref0 x5):(((eq a) x5) x5)
% Found (eq_ref0 x5) as proof of (((eq a) x5) Xy)
% Found ((eq_ref a) x5) as proof of (((eq a) x5) Xy)
% Found ((eq_ref a) x5) as proof of (((eq a) x5) Xy)
% Found ((eq_ref a) x5) as proof of (((eq a) x5) Xy)
% Found (eq_sym000 ((eq_ref a) x5)) as proof of (((eq a) Xy) x5)
% Found ((eq_sym00 Xy) ((eq_ref a) x5)) as proof of (((eq a) Xy) x5)
% Found (((eq_sym0 x5) Xy) ((eq_ref a) x5)) as proof of (((eq a) Xy) x5)
% Found ((((eq_sym a) x5) Xy) ((eq_ref a) x5)) as proof of (((eq a) Xy) x5)
% Found (fun (x000:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))))=> ((((eq_sym a) x5) Xy) ((eq_ref a) x5))) as proof of (((eq a) Xy) x5)
% Found x00:((ex a) (fun (Xz:a)=> (X Xz)))
% Instantiate: b0:=(fun (Xz:a)=> (X Xz)):(a->Prop)
% Found x00 as proof of (P b0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))):(((eq (a->Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))) (fun (x:a)=> ((and ((and (X x)) (forall (Xx:a), ((X Xx)->((x0 x) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) x))))))
% Found (eta_expansion_dep00 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))) b0)
% Found ((eta_expansion_dep0 (fun (x4:a)=> Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))) b0)
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))) b0)
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))) b0)
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))) b0)
% Found x00:((ex a) (fun (Xz:a)=> (X Xz)))
% Instantiate: b0:=(fun (Xz:a)=> (X Xz)):(a->Prop)
% Found x00 as proof of (P b0)
% Found eq_ref00:=(eq_ref0 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz)))))):(((eq (a->Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz)))))) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz))))))
% Found (eq_ref0 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz)))))) b0)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz)))))) b0)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz)))))) b0)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz)))))) b0)
% Found x2:(X x1)
% Instantiate: x3:=x1:a
% Found x2 as proof of (X x3)
% Found x5:(X x4)
% Instantiate: x3:=x4:a
% Found x5 as proof of (X x3)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) x3)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) x3)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) x3)
% Found (fun (x7:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((eq_ref a) Xy)) as proof of (((eq a) Xy) x3)
% Found (fun (x6:(X Xy)) (x7:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((eq_ref a) Xy)) as proof of ((forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))->(((eq a) Xy) x3))
% Found (fun (x6:(X Xy)) (x7:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((eq_ref a) Xy)) as proof of ((X Xy)->((forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))->(((eq a) Xy) x3)))
% Found (and_rect10 (fun (x6:(X Xy)) (x7:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((eq_ref a) Xy))) as proof of (((eq a) Xy) x3)
% Found ((and_rect1 (((eq a) Xy) x3)) (fun (x6:(X Xy)) (x7:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((eq_ref a) Xy))) as proof of (((eq a) Xy) x3)
% Found (((fun (P:Type) (x6:((X Xy)->((forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))->P)))=> (((((and_rect (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))) P) x6) x000)) (((eq a) Xy) x3)) (fun (x6:(X Xy)) (x7:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((eq_ref a) Xy))) as proof of (((eq a) Xy) x3)
% Found (fun (x000:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))))=> (((fun (P:Type) (x6:((X Xy)->((forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))->P)))=> (((((and_rect (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))) P) x6) x000)) (((eq a) Xy) x3)) (fun (x6:(X Xy)) (x7:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((eq_ref a) Xy)))) as proof of (((eq a) Xy) x3)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) x3)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) x3)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) x3)
% Found (fun (x7:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((eq_ref a) Xy)) as proof of (((eq a) Xy) x3)
% Found (fun (x6:(X Xy)) (x7:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((eq_ref a) Xy)) as proof of ((forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))->(((eq a) Xy) x3))
% Found (fun (x6:(X Xy)) (x7:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((eq_ref a) Xy)) as proof of ((X Xy)->((forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))->(((eq a) Xy) x3)))
% Found (and_rect10 (fun (x6:(X Xy)) (x7:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((eq_ref a) Xy))) as proof of (((eq a) Xy) x3)
% Found ((and_rect1 (((eq a) Xy) x3)) (fun (x6:(X Xy)) (x7:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((eq_ref a) Xy))) as proof of (((eq a) Xy) x3)
% Found (((fun (P:Type) (x6:((X Xy)->((forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))->P)))=> (((((and_rect (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))) P) x6) x000)) (((eq a) Xy) x3)) (fun (x6:(X Xy)) (x7:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((eq_ref a) Xy))) as proof of (((eq a) Xy) x3)
% Found (fun (x000:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))))=> (((fun (P:Type) (x6:((X Xy)->((forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))->P)))=> (((((and_rect (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))) P) x6) x000)) (((eq a) Xy) x3)) (fun (x6:(X Xy)) (x7:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((eq_ref a) Xy)))) as proof of (((eq a) Xy) x3)
% Found x00:((ex a) (fun (Xz:a)=> (X Xz)))
% Instantiate: f:=(fun (Xz:a)=> (X Xz)):(a->Prop)
% Found x00 as proof of (P f)
% 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 (X x3)) (forall (Xx:a), ((X Xx)->((x0 x3) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) x3)))))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((and ((and (X x3)) (forall (Xx:a), ((X Xx)->((x0 x3) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) x3)))))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((and ((and (X x3)) (forall (Xx:a), ((X Xx)->((x0 x3) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) x3)))))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (((eq Prop) (f x3)) ((and ((and (X x3)) (forall (Xx:a), ((X Xx)->((x0 x3) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) x3)))))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (forall (x:a), (((eq Prop) (f x)) ((and ((and (X x)) (forall (Xx:a), ((X Xx)->((x0 x) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) x))))))
% Found x00:((ex a) (fun (Xz:a)=> (X Xz)))
% Instantiate: f:=(fun (Xz:a)=> (X Xz)):(a->Prop)
% Found x00 as proof of (P f)
% 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 (X x3)) (forall (Xx:a), ((X Xx)->((x2 x3) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) x3)))))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((and ((and (X x3)) (forall (Xx:a), ((X Xx)->((x2 x3) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) x3)))))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((and ((and (X x3)) (forall (Xx:a), ((X Xx)->((x2 x3) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) x3)))))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (((eq Prop) (f x3)) ((and ((and (X x3)) (forall (Xx:a), ((X Xx)->((x2 x3) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) x3)))))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (forall (x:a), (((eq Prop) (f x)) ((and ((and (X x)) (forall (Xx:a), ((X Xx)->((x2 x) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) x))))))
% Found x00:((ex a) (fun (Xz:a)=> (X Xz)))
% Instantiate: f:=(fun (Xz:a)=> (X Xz)):(a->Prop)
% Found x00 as proof of (P f)
% 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 (X x3)) (forall (Xx:a), ((X Xx)->((x2 x3) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) x3)))))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((and ((and (X x3)) (forall (Xx:a), ((X Xx)->((x2 x3) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) x3)))))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((and ((and (X x3)) (forall (Xx:a), ((X Xx)->((x2 x3) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) x3)))))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (((eq Prop) (f x3)) ((and ((and (X x3)) (forall (Xx:a), ((X Xx)->((x2 x3) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) x3)))))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (forall (x:a), (((eq Prop) (f x)) ((and ((and (X x)) (forall (Xx:a), ((X Xx)->((x2 x) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) x))))))
% Found x00:((ex a) (fun (Xz:a)=> (X Xz)))
% Instantiate: f:=(fun (Xz:a)=> (X Xz)):(a->Prop)
% Found x00 as proof of (P f)
% 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 (X x3)) (forall (Xx:a), ((X Xx)->((x0 x3) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) x3)))))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((and ((and (X x3)) (forall (Xx:a), ((X Xx)->((x0 x3) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) x3)))))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((and ((and (X x3)) (forall (Xx:a), ((X Xx)->((x0 x3) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) x3)))))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (((eq Prop) (f x3)) ((and ((and (X x3)) (forall (Xx:a), ((X Xx)->((x0 x3) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) x3)))))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (forall (x:a), (((eq Prop) (f x)) ((and ((and (X x)) (forall (Xx:a), ((X Xx)->((x0 x) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) x))))))
% Found eq_ref00:=(eq_ref0 x3):(((eq a) x3) x3)
% Found (eq_ref0 x3) as proof of (((eq a) x3) Xy)
% Found ((eq_ref a) x3) as proof of (((eq a) x3) Xy)
% Found ((eq_ref a) x3) as proof of (((eq a) x3) Xy)
% Found ((eq_ref a) x3) as proof of (((eq a) x3) Xy)
% Found (eq_sym000 ((eq_ref a) x3)) as proof of (((eq a) Xy) x3)
% Found ((eq_sym00 Xy) ((eq_ref a) x3)) as proof of (((eq a) Xy) x3)
% Found (((eq_sym0 x3) Xy) ((eq_ref a) x3)) as proof of (((eq a) Xy) x3)
% Found ((((eq_sym a) x3) Xy) ((eq_ref a) x3)) as proof of (((eq a) Xy) x3)
% Found (fun (x000:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))))=> ((((eq_sym a) x3) Xy) ((eq_ref a) x3))) as proof of (((eq a) Xy) x3)
% Found eq_ref00:=(eq_ref0 x3):(((eq a) x3) x3)
% Found (eq_ref0 x3) as proof of (((eq a) x3) Xy)
% Found ((eq_ref a) x3) as proof of (((eq a) x3) Xy)
% Found ((eq_ref a) x3) as proof of (((eq a) x3) Xy)
% Found ((eq_ref a) x3) as proof of (((eq a) x3) Xy)
% Found (eq_sym000 ((eq_ref a) x3)) as proof of (((eq a) Xy) x3)
% Found ((eq_sym00 Xy) ((eq_ref a) x3)) as proof of (((eq a) Xy) x3)
% Found (((eq_sym0 x3) Xy) ((eq_ref a) x3)) as proof of (((eq a) Xy) x3)
% Found ((((eq_sym a) x3) Xy) ((eq_ref a) x3)) as proof of (((eq a) Xy) x3)
% Found (fun (x000:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))))=> ((((eq_sym a) x3) Xy) ((eq_ref a) x3))) as proof of (((eq a) Xy) x3)
% Found eq_ref000:=(eq_ref00 (ex a)):(((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> (X Xz))))
% Found (eq_ref00 (ex a)) as proof of (P (fun (Xz:a)=> (X Xz)))
% Found ((eq_ref0 (fun (Xz:a)=> (X Xz))) (ex a)) as proof of (P (fun (Xz:a)=> (X Xz)))
% Found (((eq_ref (a->Prop)) (fun (Xz:a)=> (X Xz))) (ex a)) as proof of (P (fun (Xz:a)=> (X Xz)))
% Found (((eq_ref (a->Prop)) (fun (Xz:a)=> (X Xz))) (ex a)) as proof of (P (fun (Xz:a)=> (X Xz)))
% Found x00:((ex a) (fun (Xz:a)=> (X Xz)))
% Instantiate: b0:=(fun (Xz:a)=> (X Xz)):(a->Prop)
% Found x00 as proof of (P b0)
% Found eq_ref00:=(eq_ref0 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))):(((eq (a->Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz))))))
% Found (eq_ref0 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))) b0)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))) b0)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))) b0)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))) b0)
% Found eq_ref00:=(eq_ref0 (fun (R:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((R Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((R Xy) Xx))))->(((eq a) Xy) Xz)))))))))):(((eq ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((R Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((R Xy) Xx))))->(((eq a) Xy) Xz)))))))))) (fun (R:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((R Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((R Xy) Xx))))->(((eq a) Xy) Xz))))))))))
% Found (eq_ref0 (fun (R:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((R Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((R Xy) Xx))))->(((eq a) Xy) Xz)))))))))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((R Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((R Xy) Xx))))->(((eq a) Xy) Xz)))))))))) b0)
% Found ((eq_ref ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((R Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((R Xy) Xx))))->(((eq a) Xy) Xz)))))))))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((R Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((R Xy) Xx))))->(((eq a) Xy) Xz)))))))))) b0)
% Found ((eq_ref ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((R Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((R Xy) Xx))))->(((eq a) Xy) Xz)))))))))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((R Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((R Xy) Xx))))->(((eq a) Xy) Xz)))))))))) b0)
% Found ((eq_ref ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((R Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((R Xy) Xx))))->(((eq a) Xy) Xz)))))))))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((R Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((R Xy) Xx))))->(((eq a) Xy) Xz)))))))))) b0)
% Found eq_ref00:=(eq_ref0 (fun (Xz:a)=> (X Xz))):(((eq (a->Prop)) (fun (Xz:a)=> (X Xz))) (fun (Xz:a)=> (X Xz)))
% Found (eq_ref0 (fun (Xz:a)=> (X Xz))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> (X Xz))) b0)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> (X Xz))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> (X Xz))) b0)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> (X Xz))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> (X Xz))) b0)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> (X Xz))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> (X Xz))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq (a->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (a->Prop)) b0) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz))))))
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz))))))
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz))))))
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz))))))
% Found eq_ref00:=(eq_ref0 (fun (Xz:a)=> (X Xz))):(((eq (a->Prop)) (fun (Xz:a)=> (X Xz))) (fun (Xz:a)=> (X Xz)))
% Found (eq_ref0 (fun (Xz:a)=> (X Xz))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> (X Xz))) b0)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> (X Xz))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> (X Xz))) b0)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> (X Xz))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> (X Xz))) b0)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> (X Xz))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> (X Xz))) b0)
% Found eta_expansion000:=(eta_expansion00 b0):(((eq (a->Prop)) b0) (fun (x:a)=> (b0 x)))
% Found (eta_expansion00 b0) as proof of (((eq (a->Prop)) b0) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz))))))
% Found ((eta_expansion0 Prop) b0) as proof of (((eq (a->Prop)) b0) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz))))))
% Found (((eta_expansion a) Prop) b0) as proof of (((eq (a->Prop)) b0) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz))))))
% Found (((eta_expansion a) Prop) b0) as proof of (((eq (a->Prop)) b0) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz))))))
% Found (((eta_expansion a) Prop) b0) as proof of (((eq (a->Prop)) b0) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz))))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P0):((P0 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz))))))->(P0 (fun (x:a)=> ((and ((and (X x)) (forall (Xx:a), ((X Xx)->((x0 x) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) x)))))))
% Found (eta_expansion_dep000 P0) as proof of (P1 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz))))))
% Found ((eta_expansion_dep00 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))) P0) as proof of (P1 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz))))))
% Found (((eta_expansion_dep0 (fun (x2:a)=> Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))) P0) as proof of (P1 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz))))))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))) P0) as proof of (P1 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz))))))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))) P0) as proof of (P1 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz))))))
% Found x3:(X x2)
% Instantiate: x1:=x2:a
% Found x3 as proof of (X x1)
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P0):((P0 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz))))))->(P0 (fun (x:a)=> ((and ((and (X x)) (forall (Xx:a), ((X Xx)->((x0 x) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) x)))))))
% Found (eta_expansion_dep000 P0) as proof of (P1 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz))))))
% Found ((eta_expansion_dep00 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))) P0) as proof of (P1 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz))))))
% Found (((eta_expansion_dep0 (fun (x2:a)=> Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))) P0) as proof of (P1 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz))))))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))) P0) as proof of (P1 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz))))))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))) P0) as proof of (P1 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz))))))
% Found x5:(X x4)
% Instantiate: x1:=x4:a
% Found x5 as proof of (X x1)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) x1)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) x1)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) x1)
% Found (fun (x7:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((eq_ref a) Xy)) as proof of (((eq a) Xy) x1)
% Found (fun (x6:(X Xy)) (x7:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((eq_ref a) Xy)) as proof of ((forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))->(((eq a) Xy) x1))
% Found (fun (x6:(X Xy)) (x7:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((eq_ref a) Xy)) as proof of ((X Xy)->((forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))->(((eq a) Xy) x1)))
% Found (and_rect10 (fun (x6:(X Xy)) (x7:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((eq_ref a) Xy))) as proof of (((eq a) Xy) x1)
% Found ((and_rect1 (((eq a) Xy) x1)) (fun (x6:(X Xy)) (x7:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((eq_ref a) Xy))) as proof of (((eq a) Xy) x1)
% Found (((fun (P:Type) (x6:((X Xy)->((forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))->P)))=> (((((and_rect (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))) P) x6) x000)) (((eq a) Xy) x1)) (fun (x6:(X Xy)) (x7:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((eq_ref a) Xy))) as proof of (((eq a) Xy) x1)
% Found (fun (x000:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))))=> (((fun (P:Type) (x6:((X Xy)->((forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))->P)))=> (((((and_rect (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))) P) x6) x000)) (((eq a) Xy) x1)) (fun (x6:(X Xy)) (x7:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((eq_ref a) Xy)))) as proof of (((eq a) Xy) x1)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) x1)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) x1)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) x1)
% Found (fun (x7:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((eq_ref a) Xy)) as proof of (((eq a) Xy) x1)
% Found (fun (x6:(X Xy)) (x7:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((eq_ref a) Xy)) as proof of ((forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))->(((eq a) Xy) x1))
% Found (fun (x6:(X Xy)) (x7:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((eq_ref a) Xy)) as proof of ((X Xy)->((forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))->(((eq a) Xy) x1)))
% Found (and_rect10 (fun (x6:(X Xy)) (x7:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((eq_ref a) Xy))) as proof of (((eq a) Xy) x1)
% Found ((and_rect1 (((eq a) Xy) x1)) (fun (x6:(X Xy)) (x7:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((eq_ref a) Xy))) as proof of (((eq a) Xy) x1)
% Found (((fun (P:Type) (x6:((X Xy)->((forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))->P)))=> (((((and_rect (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))) P) x6) x000)) (((eq a) Xy) x1)) (fun (x6:(X Xy)) (x7:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((eq_ref a) Xy))) as proof of (((eq a) Xy) x1)
% Found (fun (x000:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))))=> (((fun (P:Type) (x6:((X Xy)->((forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))->P)))=> (((((and_rect (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))) P) x6) x000)) (((eq a) Xy) x1)) (fun (x6:(X Xy)) (x7:(forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))=> ((eq_ref a) Xy)))) as proof of (((eq a) Xy) x1)
% Found eq_ref00:=(eq_ref0 x1):(((eq a) x1) x1)
% Found (eq_ref0 x1) as proof of (((eq a) x1) Xy)
% Found ((eq_ref a) x1) as proof of (((eq a) x1) Xy)
% Found ((eq_ref a) x1) as proof of (((eq a) x1) Xy)
% Found ((eq_ref a) x1) as proof of (((eq a) x1) Xy)
% Found (eq_sym000 ((eq_ref a) x1)) as proof of (((eq a) Xy) x1)
% Found ((eq_sym00 Xy) ((eq_ref a) x1)) as proof of (((eq a) Xy) x1)
% Found (((eq_sym0 x1) Xy) ((eq_ref a) x1)) as proof of (((eq a) Xy) x1)
% Found ((((eq_sym a) x1) Xy) ((eq_ref a) x1)) as proof of (((eq a) Xy) x1)
% Found (fun (x000:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))))=> ((((eq_sym a) x1) Xy) ((eq_ref a) x1))) as proof of (((eq a) Xy) x1)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (forall (P:(a->Prop)), ((P Xy)->(P x1)))
% Found ((eq_ref a) Xy) as proof of (forall (P:(a->Prop)), ((P Xy)->(P x1)))
% Found ((eq_ref a) Xy) as proof of (forall (P:(a->Prop)), ((P Xy)->(P x1)))
% Found ((eq_ref a) Xy) as proof of (forall (P:(a->Prop)), ((P Xy)->(P x1)))
% Found (fun (x000:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))))=> ((eq_ref a) Xy)) as proof of (((eq a) Xy) x1)
% Found eq_ref00:=(eq_ref0 x1):(((eq a) x1) x1)
% Found (eq_ref0 x1) as proof of (((eq a) x1) Xy)
% Found ((eq_ref a) x1) as proof of (((eq a) x1) Xy)
% Found ((eq_ref a) x1) as proof of (((eq a) x1) Xy)
% Found ((eq_ref a) x1) as proof of (((eq a) x1) Xy)
% Found (eq_sym000 ((eq_ref a) x1)) as proof of (((eq a) Xy) x1)
% Found ((eq_sym00 Xy) ((eq_ref a) x1)) as proof of (((eq a) Xy) x1)
% Found (((eq_sym0 x1) Xy) ((eq_ref a) x1)) as proof of (((eq a) Xy) x1)
% Found ((((eq_sym a) x1) Xy) ((eq_ref a) x1)) as proof of (((eq a) Xy) x1)
% Found (fun (x000:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))))=> ((((eq_sym a) x1) Xy) ((eq_ref a) x1))) as proof of (((eq a) Xy) x1)
% Found x00:((ex a) (fun (Xz:a)=> (X Xz)))
% Instantiate: f:=(fun (Xz:a)=> (X Xz)):(a->Prop)
% Found x00 as proof of (P f)
% 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 (X x3)) (forall (Xx:a), ((X Xx)->((x0 x3) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) x3)))))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((and ((and (X x3)) (forall (Xx:a), ((X Xx)->((x0 x3) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) x3)))))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((and ((and (X x3)) (forall (Xx:a), ((X Xx)->((x0 x3) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) x3)))))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (((eq Prop) (f x3)) ((and ((and (X x3)) (forall (Xx:a), ((X Xx)->((x0 x3) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) x3)))))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (forall (x:a), (((eq Prop) (f x)) ((and ((and (X x)) (forall (Xx:a), ((X Xx)->((x0 x) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) x))))))
% Found x00:((ex a) (fun (Xz:a)=> (X Xz)))
% Instantiate: f:=(fun (Xz:a)=> (X Xz)):(a->Prop)
% Found x00 as proof of (P f)
% 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 (X x3)) (forall (Xx:a), ((X Xx)->((x0 x3) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) x3)))))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((and ((and (X x3)) (forall (Xx:a), ((X Xx)->((x0 x3) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) x3)))))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((and ((and (X x3)) (forall (Xx:a), ((X Xx)->((x0 x3) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) x3)))))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (((eq Prop) (f x3)) ((and ((and (X x3)) (forall (Xx:a), ((X Xx)->((x0 x3) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) x3)))))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (forall (x:a), (((eq Prop) (f x)) ((and ((and (X x)) (forall (Xx:a), ((X Xx)->((x0 x) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) x))))))
% Found x5:(X x4)
% Instantiate: x3:=x4:a
% Found (fun (x5:(X x4))=> x5) as proof of (X x3)
% Found eq_ref000:=(eq_ref00 X):((X x4)->(X x4))
% Found (eq_ref00 X) as proof of ((X x4)->(X x3))
% Found ((eq_ref0 x4) X) as proof of ((X x4)->(X x3))
% Found (((eq_ref a) x4) X) as proof of ((X x4)->(X x3))
% Found (((eq_ref a) x4) X) as proof of ((X x4)->(X x3))
% Found eta_expansion000:=(eta_expansion00 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))):(((eq (a->Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))) (fun (x:a)=> ((and ((and (X x)) (forall (Xx:a), ((X Xx)->((x0 x) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) x))))))
% Found (eta_expansion00 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))) b0)
% Found ((eta_expansion0 Prop) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))) b0)
% Found (((eta_expansion a) Prop) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))) b0)
% Found (((eta_expansion a) Prop) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))) b0)
% Found (((eta_expansion a) Prop) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))) b0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b0):(((eq (a->Prop)) b0) (fun (x:a)=> (b0 x)))
% Found (eta_expansion_dep00 b0) as proof of (((eq (a->Prop)) b0) (fun (Xz:a)=> (X Xz)))
% Found ((eta_expansion_dep0 (fun (x2:a)=> Prop)) b0) as proof of (((eq (a->Prop)) b0) (fun (Xz:a)=> (X Xz)))
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) b0) as proof of (((eq (a->Prop)) b0) (fun (Xz:a)=> (X Xz)))
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) b0) as proof of (((eq (a->Prop)) b0) (fun (Xz:a)=> (X Xz)))
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) b0) as proof of (((eq (a->Prop)) b0) (fun (Xz:a)=> (X Xz)))
% Found eq_ref00:=(eq_ref0 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))):(((eq (a->Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz))))))
% Found (eq_ref0 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))) b0)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))) b0)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))) b0)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))) b0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b0):(((eq (a->Prop)) b0) (fun (x:a)=> (b0 x)))
% Found (eta_expansion_dep00 b0) as proof of (((eq (a->Prop)) b0) (fun (Xz:a)=> (X Xz)))
% Found ((eta_expansion_dep0 (fun (x2:a)=> Prop)) b0) as proof of (((eq (a->Prop)) b0) (fun (Xz:a)=> (X Xz)))
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) b0) as proof of (((eq (a->Prop)) b0) (fun (Xz:a)=> (X Xz)))
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) b0) as proof of (((eq (a->Prop)) b0) (fun (Xz:a)=> (X Xz)))
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) b0) as proof of (((eq (a->Prop)) b0) (fun (Xz:a)=> (X Xz)))
% Found x2:(P Xy)
% Instantiate: x1:=Xy:a
% Found (fun (x2:(P Xy))=> x2) as proof of (P x1)
% Found (fun (P:(a->Prop)) (x2:(P Xy))=> x2) as proof of ((P Xy)->(P x1))
% Found (fun (P:(a->Prop)) (x2:(P Xy))=> x2) as proof of (forall (P:(a->Prop)), ((P Xy)->(P x1)))
% Found (fun (x000:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))) (P:(a->Prop)) (x2:(P Xy))=> x2) as proof of (((eq a) Xy) x1)
% Found eq_ref000:=(eq_ref00 P):((P Xy)->(P Xy))
% Found (eq_ref00 P) as proof of ((P Xy)->(P x1))
% Found ((eq_ref0 Xy) P) as proof of ((P Xy)->(P x1))
% Found (((eq_ref a) Xy) P) as proof of ((P Xy)->(P x1))
% Found (((eq_ref a) Xy) P) as proof of ((P Xy)->(P x1))
% Found (fun (P:(a->Prop))=> (((eq_ref a) Xy) P)) as proof of ((P Xy)->(P x1))
% Found (fun (P:(a->Prop))=> (((eq_ref a) Xy) P)) as proof of (forall (P:(a->Prop)), ((P Xy)->(P x1)))
% Found (fun (x000:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))) (P:(a->Prop))=> (((eq_ref a) Xy) P)) as proof of (((eq a) Xy) x1)
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P0):((P0 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz))))))->(P0 (fun (x:a)=> ((and ((and (X x)) (forall (Xx:a), ((X Xx)->((x0 x) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) x)))))))
% Found (eta_expansion_dep000 P0) as proof of (P1 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz))))))
% Found ((eta_expansion_dep00 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))) P0) as proof of (P1 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz))))))
% Found (((eta_expansion_dep0 (fun (x2:a)=> Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))) P0) as proof of (P1 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz))))))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))) P0) as proof of (P1 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz))))))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))) P0) as proof of (P1 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz))))))
% Found eta_expansion0000:=(eta_expansion000 P0):((P0 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz))))))->(P0 (fun (x:a)=> ((and ((and (X x)) (forall (Xx:a), ((X Xx)->((x0 x) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) x)))))))
% Found (eta_expansion000 P0) as proof of (P1 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz))))))
% Found ((eta_expansion00 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))) P0) as proof of (P1 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz))))))
% Found (((eta_expansion0 Prop) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))) P0) as proof of (P1 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz))))))
% Found ((((eta_expansion a) Prop) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))) P0) as proof of (P1 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz))))))
% Found ((((eta_expansion a) Prop) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))) P0) as proof of (P1 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz))))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P0):((P0 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz))))))->(P0 (fun (x:a)=> ((and ((and (X x)) (forall (Xx:a), ((X Xx)->((x0 x) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) x)))))))
% Found (eta_expansion_dep000 P0) as proof of (P1 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz))))))
% Found ((eta_expansion_dep00 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))) P0) as proof of (P1 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz))))))
% Found (((eta_expansion_dep0 (fun (x2:a)=> Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))) P0) as proof of (P1 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz))))))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))) P0) as proof of (P1 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz))))))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))) P0) as proof of (P1 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz))))))
% Found eta_expansion0000:=(eta_expansion000 P0):((P0 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz))))))->(P0 (fun (x:a)=> ((and ((and (X x)) (forall (Xx:a), ((X Xx)->((x0 x) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) x)))))))
% Found (eta_expansion000 P0) as proof of (P1 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz))))))
% Found ((eta_expansion00 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))) P0) as proof of (P1 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz))))))
% Found (((eta_expansion0 Prop) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))) P0) as proof of (P1 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz))))))
% Found ((((eta_expansion a) Prop) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))) P0) as proof of (P1 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz))))))
% Found ((((eta_expansion a) Prop) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))) P0) as proof of (P1 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz))))))
% Found eq_ref000:=(eq_ref00 P0):((P0 (X x1))->(P0 (X x1)))
% Found (eq_ref00 P0) as proof of (P1 (X x1))
% Found ((eq_ref0 (X x1)) P0) as proof of (P1 (X x1))
% Found (((eq_ref Prop) (X x1)) P0) as proof of (P1 (X x1))
% Found (((eq_ref Prop) (X x1)) P0) as proof of (P1 (X x1))
% Found eq_ref000:=(eq_ref00 P0):((P0 (X x1))->(P0 (X x1)))
% Found (eq_ref00 P0) as proof of (P1 (X x1))
% Found ((eq_ref0 (X x1)) P0) as proof of (P1 (X x1))
% Found (((eq_ref Prop) (X x1)) P0) as proof of (P1 (X x1))
% Found (((eq_ref Prop) (X x1)) P0) as proof of (P1 (X x1))
% Found eq_ref000:=(eq_ref00 P0):((P0 (X x1))->(P0 (X x1)))
% Found (eq_ref00 P0) as proof of (P1 (X x1))
% Found ((eq_ref0 (X x1)) P0) as proof of (P1 (X x1))
% Found (((eq_ref Prop) (X x1)) P0) as proof of (P1 (X x1))
% Found (((eq_ref Prop) (X x1)) P0) as proof of (P1 (X x1))
% Found eq_ref000:=(eq_ref00 P0):((P0 (X x1))->(P0 (X x1)))
% Found (eq_ref00 P0) as proof of (P1 (X x1))
% Found ((eq_ref0 (X x1)) P0) as proof of (P1 (X x1))
% Found (((eq_ref Prop) (X x1)) P0) as proof of (P1 (X x1))
% Found (((eq_ref Prop) (X x1)) P0) as proof of (P1 (X x1))
% Found eq_ref00:=(eq_ref0 (X x1)):(((eq Prop) (X x1)) (X x1))
% Found (eq_ref0 (X x1)) as proof of (((eq Prop) (X x1)) b0)
% Found ((eq_ref Prop) (X x1)) as proof of (((eq Prop) (X x1)) b0)
% Found ((eq_ref Prop) (X x1)) as proof of (((eq Prop) (X x1)) b0)
% Found ((eq_ref Prop) (X x1)) as proof of (((eq Prop) (X x1)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) ((and ((and (X x1)) (forall (Xx:a), ((X Xx)->((x0 x1) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) x1)))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and ((and (X x1)) (forall (Xx:a), ((X Xx)->((x0 x1) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) x1)))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and ((and (X x1)) (forall (Xx:a), ((X Xx)->((x0 x1) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) x1)))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and ((and (X x1)) (forall (Xx:a), ((X Xx)->((x0 x1) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) x1)))))
% Found eq_ref00:=(eq_ref0 (X x1)):(((eq Prop) (X x1)) (X x1))
% Found (eq_ref0 (X x1)) as proof of (((eq Prop) (X x1)) b0)
% Found ((eq_ref Prop) (X x1)) as proof of (((eq Prop) (X x1)) b0)
% Found ((eq_ref Prop) (X x1)) as proof of (((eq Prop) (X x1)) b0)
% Found ((eq_ref Prop) (X x1)) as proof of (((eq Prop) (X x1)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) ((and ((and (X x1)) (forall (Xx:a), ((X Xx)->((x0 x1) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) x1)))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and ((and (X x1)) (forall (Xx:a), ((X Xx)->((x0 x1) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) x1)))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and ((and (X x1)) (forall (Xx:a), ((X Xx)->((x0 x1) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) x1)))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and ((and (X x1)) (forall (Xx:a), ((X Xx)->((x0 x1) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) x1)))))
% Found eq_ref00:=(eq_ref0 (X x1)):(((eq Prop) (X x1)) (X x1))
% Found (eq_ref0 (X x1)) as proof of (((eq Prop) (X x1)) b0)
% Found ((eq_ref Prop) (X x1)) as proof of (((eq Prop) (X x1)) b0)
% Found ((eq_ref Prop) (X x1)) as proof of (((eq Prop) (X x1)) b0)
% Found ((eq_ref Prop) (X x1)) as proof of (((eq Prop) (X x1)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) ((and ((and (X x1)) (forall (Xx:a), ((X Xx)->((x0 x1) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) x1)))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and ((and (X x1)) (forall (Xx:a), ((X Xx)->((x0 x1) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) x1)))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and ((and (X x1)) (forall (Xx:a), ((X Xx)->((x0 x1) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) x1)))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and ((and (X x1)) (forall (Xx:a), ((X Xx)->((x0 x1) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) x1)))))
% Found eq_ref00:=(eq_ref0 (X x1)):(((eq Prop) (X x1)) (X x1))
% Found (eq_ref0 (X x1)) as proof of (((eq Prop) (X x1)) b0)
% Found ((eq_ref Prop) (X x1)) as proof of (((eq Prop) (X x1)) b0)
% Found ((eq_ref Prop) (X x1)) as proof of (((eq Prop) (X x1)) b0)
% Found ((eq_ref Prop) (X x1)) as proof of (((eq Prop) (X x1)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) ((and ((and (X x1)) (forall (Xx:a), ((X Xx)->((x0 x1) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) x1)))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and ((and (X x1)) (forall (Xx:a), ((X Xx)->((x0 x1) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) x1)))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and ((and (X x1)) (forall (Xx:a), ((X Xx)->((x0 x1) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) x1)))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and ((and (X x1)) (forall (Xx:a), ((X Xx)->((x0 x1) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) x1)))))
% Found x00:((ex a) (fun (Xz:a)=> (X Xz)))
% Instantiate: b0:=(fun (Xz:a)=> (X Xz)):(a->Prop)
% Found x00 as proof of (P b0)
% Found eta_expansion000:=(eta_expansion00 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz)))))):(((eq (a->Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz)))))) (fun (x:a)=> ((and ((and (X x)) (forall (Xx:a), ((X Xx)->((x2 x) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) x))))))
% Found (eta_expansion00 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz)))))) b0)
% Found ((eta_expansion0 Prop) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz)))))) b0)
% Found (((eta_expansion a) Prop) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz)))))) b0)
% Found (((eta_expansion a) Prop) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz)))))) b0)
% Found (((eta_expansion a) Prop) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz)))))) b0)
% Found x00:((ex a) (fun (Xz:a)=> (X Xz)))
% Instantiate: f:=(fun (Xz:a)=> (X Xz)):(a->Prop)
% Found x00 as proof of (P f)
% Found x00:((ex a) (fun (Xz:a)=> (X Xz)))
% Instantiate: f:=(fun (Xz:a)=> (X Xz)):(a->Prop)
% Found x00 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 ((and (X x5)) (forall (Xx:a), ((X Xx)->((x2 x5) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) x5)))))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) ((and ((and (X x5)) (forall (Xx:a), ((X Xx)->((x2 x5) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) x5)))))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) ((and ((and (X x5)) (forall (Xx:a), ((X Xx)->((x2 x5) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) x5)))))
% Found (fun (x5:a)=> ((eq_ref Prop) (f x5))) as proof of (((eq Prop) (f x5)) ((and ((and (X x5)) (forall (Xx:a), ((X Xx)->((x2 x5) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) x5)))))
% Found (fun (x5:a)=> ((eq_ref Prop) (f x5))) as proof of (forall (x:a), (((eq Prop) (f x)) ((and ((and (X x)) (forall (Xx:a), ((X Xx)->((x2 x) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) x))))))
% 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 (X x5)) (forall (Xx:a), ((X Xx)->((x2 x5) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) x5)))))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) ((and ((and (X x5)) (forall (Xx:a), ((X Xx)->((x2 x5) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) x5)))))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) ((and ((and (X x5)) (forall (Xx:a), ((X Xx)->((x2 x5) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) x5)))))
% Found (fun (x5:a)=> ((eq_ref Prop) (f x5))) as proof of (((eq Prop) (f x5)) ((and ((and (X x5)) (forall (Xx:a), ((X Xx)->((x2 x5) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) x5)))))
% Found (fun (x5:a)=> ((eq_ref Prop) (f x5))) as proof of (forall (x:a), (((eq Prop) (f x)) ((and ((and (X x)) (forall (Xx:a), ((X Xx)->((x2 x) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) x))))))
% 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)) b0)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b0)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b0)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))))))
% Found ((eq_trans0000 ((eq_ref Prop) (f x0))) ((eq_ref Prop) b0)) as proof of (((eq Prop) (f x0)) (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))))))
% Found (((eq_trans000 (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz))))))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) b0)) as proof of (((eq Prop) (f x0)) (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))))))
% Found ((((eq_trans00 (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz))))))))) (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz))))))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))))))) as proof of (((eq Prop) (f x0)) (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))))))
% Found (((((eq_trans0 (f x0)) (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz))))))))) (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz))))))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))))))) as proof of (((eq Prop) (f x0)) (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))))))
% Found ((((((eq_trans Prop) (f x0)) (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz))))))))) (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz))))))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))))))) as proof of (((eq Prop) (f x0)) (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) 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)) b0)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b0)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b0)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))))))
% Found ((eq_trans0000 ((eq_ref Prop) (f x0))) ((eq_ref Prop) b0)) as proof of (((eq Prop) (f x0)) (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))))))
% Found (((eq_trans000 (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz))))))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) b0)) as proof of (((eq Prop) (f x0)) (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))))))
% Found ((((eq_trans00 (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz))))))))) (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz))))))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))))))) as proof of (((eq Prop) (f x0)) (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))))))
% Found (((((eq_trans0 (f x0)) (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz))))))))) (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz))))))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))))))) as proof of (((eq Prop) (f x0)) (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))))))
% Found ((((((eq_trans Prop) (f x0)) (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz))))))))) (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz))))))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))))))) as proof of (((eq Prop) (f x0)) (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))))))
% Found x5:(X x4)
% Instantiate: x3:=x4:a
% Found (fun (x5:(X x4))=> x5) as proof of (X x3)
% Found eq_ref000:=(eq_ref00 X):((X x4)->(X x4))
% Found (eq_ref00 X) as proof of ((X x4)->(X x3))
% Found ((eq_ref0 x4) X) as proof of ((X x4)->(X x3))
% Found (((eq_ref a) x4) X) as proof of ((X x4)->(X x3))
% Found (((eq_ref a) x4) X) as proof of ((X x4)->(X x3))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (R:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((R Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((R Xy) Xx))))->(((eq a) Xy) Xz)))))))))):(((eq ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((R Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((R Xy) Xx))))->(((eq a) Xy) Xz)))))))))) (fun (x:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x Xy) Xx))))->(((eq a) Xy) Xz))))))))))
% Found (eta_expansion_dep00 (fun (R:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((R Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((R Xy) Xx))))->(((eq a) Xy) Xz)))))))))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((R Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((R Xy) Xx))))->(((eq a) Xy) Xz)))))))))) b0)
% Found ((eta_expansion_dep0 (fun (x7:(a->(a->Prop)))=> Prop)) (fun (R:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((R Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((R Xy) Xx))))->(((eq a) Xy) Xz)))))))))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((R Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((R Xy) Xx))))->(((eq a) Xy) Xz)))))))))) b0)
% Found (((eta_expansion_dep (a->(a->Prop))) (fun (x7:(a->(a->Prop)))=> Prop)) (fun (R:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((R Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((R Xy) Xx))))->(((eq a) Xy) Xz)))))))))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((R Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((R Xy) Xx))))->(((eq a) Xy) Xz)))))))))) b0)
% Found (((eta_expansion_dep (a->(a->Prop))) (fun (x7:(a->(a->Prop)))=> Prop)) (fun (R:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((R Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((R Xy) Xx))))->(((eq a) Xy) Xz)))))))))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((R Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((R Xy) Xx))))->(((eq a) Xy) Xz)))))))))) b0)
% Found (((eta_expansion_dep (a->(a->Prop))) (fun (x7:(a->(a->Prop)))=> Prop)) (fun (R:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((R Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((R Xy) Xx))))->(((eq a) Xy) Xz)))))))))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> (forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((R Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((R Xy) Xx))))->(((eq a) Xy) Xz)))))))))) b0)
% Found eta_expansion0000:=(eta_expansion000 P0):((P0 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz))))))->(P0 (fun (x:a)=> ((and ((and (X x)) (forall (Xx:a), ((X Xx)->((x2 x) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) x)))))))
% Found (eta_expansion000 P0) as proof of (P1 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz))))))
% Found ((eta_expansion00 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz)))))) P0) as proof of (P1 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz))))))
% Found (((eta_expansion0 Prop) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz)))))) P0) as proof of (P1 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz))))))
% Found ((((eta_expansion a) Prop) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz)))))) P0) as proof of (P1 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz))))))
% Found ((((eta_expansion a) Prop) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz)))))) P0) as proof of (P1 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz))))))
% Found eta_expansion0000:=(eta_expansion000 P0):((P0 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz))))))->(P0 (fun (x:a)=> ((and ((and (X x)) (forall (Xx:a), ((X Xx)->((x2 x) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) x)))))))
% Found (eta_expansion000 P0) as proof of (P1 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz))))))
% Found ((eta_expansion00 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz)))))) P0) as proof of (P1 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz))))))
% Found (((eta_expansion0 Prop) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz)))))) P0) as proof of (P1 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz))))))
% Found ((((eta_expansion a) Prop) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz)))))) P0) as proof of (P1 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz))))))
% Found ((((eta_expansion a) Prop) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz)))))) P0) as proof of (P1 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz))))))
% Found eq_ref000:=(eq_ref00 P0):((P0 (f x0))->(P0 (f x0)))
% Found (eq_ref00 P0) as proof of (P1 (f x0))
% Found ((eq_ref0 (f x0)) P0) as proof of (P1 (f x0))
% Found (((eq_ref Prop) (f x0)) P0) as proof of (P1 (f x0))
% Found (((eq_ref Prop) (f x0)) P0) as proof of (P1 (f x0))
% Found eta_expansion0000:=(eta_expansion000 (ex a)):(((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (x:a)=> (X x))))
% Found (eta_expansion000 (ex a)) as proof of (P (fun (Xz:a)=> (X Xz)))
% Found ((eta_expansion00 (fun (Xz:a)=> (X Xz))) (ex a)) as proof of (P (fun (Xz:a)=> (X Xz)))
% Found (((eta_expansion0 Prop) (fun (Xz:a)=> (X Xz))) (ex a)) as proof of (P (fun (Xz:a)=> (X Xz)))
% Found ((((eta_expansion a) Prop) (fun (Xz:a)=> (X Xz))) (ex a)) as proof of (P (fun (Xz:a)=> (X Xz)))
% Found ((((eta_expansion a) Prop) (fun (Xz:a)=> (X Xz))) (ex a)) as proof of (P (fun (Xz:a)=> (X Xz)))
% Found eq_ref000:=(eq_ref00 P0):((P0 (f x0))->(P0 (f x0)))
% Found (eq_ref00 P0) as proof of (P1 (f x0))
% Found ((eq_ref0 (f x0)) P0) as proof of (P1 (f x0))
% Found (((eq_ref Prop) (f x0)) P0) as proof of (P1 (f x0))
% Found (((eq_ref Prop) (f x0)) P0) as proof of (P1 (f x0))
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (forall (P:(a->Prop)), ((P Xy)->(P x3)))
% Found ((eq_ref a) Xy) as proof of (forall (P:(a->Prop)), ((P Xy)->(P x3)))
% Found ((eq_ref a) Xy) as proof of (forall (P:(a->Prop)), ((P Xy)->(P x3)))
% Found ((eq_ref a) Xy) as proof of (forall (P:(a->Prop)), ((P Xy)->(P x3)))
% Found (fun (x000:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))))=> ((eq_ref a) Xy)) as proof of (((eq a) Xy) x3)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (forall (P:(a->Prop)), ((P Xy)->(P x3)))
% Found ((eq_ref a) Xy) as proof of (forall (P:(a->Prop)), ((P Xy)->(P x3)))
% Found ((eq_ref a) Xy) as proof of (forall (P:(a->Prop)), ((P Xy)->(P x3)))
% Found ((eq_ref a) Xy) as proof of (forall (P:(a->Prop)), ((P Xy)->(P x3)))
% Found (fun (x01:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx)))))=> ((eq_ref a) Xy)) as proof of (((eq a) Xy) x3)
% Found x3:(X x2)
% Instantiate: x1:=x2:a
% Found x3 as proof of (X x1)
% Found x5:(X x4)
% Instantiate: x1:=x4:a
% Found x5 as proof of (X x1)
% Found eta_expansion0000:=(eta_expansion000 (ex a)):(((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (x:a)=> (X x))))
% Found (eta_expansion000 (ex a)) as proof of (P (fun (Xz:a)=> (X Xz)))
% Found ((eta_expansion00 (fun (Xz:a)=> (X Xz))) (ex a)) as proof of (P (fun (Xz:a)=> (X Xz)))
% Found (((eta_expansion0 Prop) (fun (Xz:a)=> (X Xz))) (ex a)) as proof of (P (fun (Xz:a)=> (X Xz)))
% Found ((((eta_expansion a) Prop) (fun (Xz:a)=> (X Xz))) (ex a)) as proof of (P (fun (Xz:a)=> (X Xz)))
% Found ((((eta_expansion a) Prop) (fun (Xz:a)=> (X Xz))) (ex a)) as proof of (P (fun (Xz:a)=> (X Xz)))
% Found eta_expansion0000:=(eta_expansion000 (ex a)):(((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (x:a)=> (X x))))
% Found (eta_expansion000 (ex a)) as proof of (P (fun (Xz:a)=> (X Xz)))
% Found ((eta_expansion00 (fun (Xz:a)=> (X Xz))) (ex a)) as proof of (P (fun (Xz:a)=> (X Xz)))
% Found (((eta_expansion0 Prop) (fun (Xz:a)=> (X Xz))) (ex a)) as proof of (P (fun (Xz:a)=> (X Xz)))
% Found ((((eta_expansion a) Prop) (fun (Xz:a)=> (X Xz))) (ex a)) as proof of (P (fun (Xz:a)=> (X Xz)))
% Found ((((eta_expansion a) Prop) (fun (Xz:a)=> (X Xz))) (ex a)) as proof of (P (fun (Xz:a)=> (X Xz)))
% Found x5:(X x4)
% Instantiate: x1:=x4:a
% Found (fun (x5:(X x4))=> x5) as proof of (X x1)
% Found eq_ref000:=(eq_ref00 X):((X x4)->(X x4))
% Found (eq_ref00 X) as proof of ((X x4)->(X x1))
% Found ((eq_ref0 x4) X) as proof of ((X x4)->(X x1))
% Found (((eq_ref a) x4) X) as proof of ((X x4)->(X x1))
% Found (((eq_ref a) x4) X) as proof of ((X x4)->(X x1))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz)))))):(((eq (a->Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz)))))) (fun (x:a)=> ((and ((and (X x)) (forall (Xx:a), ((X Xx)->((x2 x) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) x))))))
% Found (eta_expansion_dep00 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz)))))) b0)
% Found ((eta_expansion_dep0 (fun (x4:a)=> Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz)))))) b0)
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz)))))) b0)
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz)))))) b0)
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz)))))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq (a->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (a->Prop)) b0) (fun (Xz:a)=> (X Xz)))
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) (fun (Xz:a)=> (X Xz)))
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) (fun (Xz:a)=> (X Xz)))
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) (fun (Xz:a)=> (X Xz)))
% Found eq_ref00:=(eq_ref0 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz)))))):(((eq (a->Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz)))))) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz))))))
% Found (eq_ref0 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz)))))) b0)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz)))))) b0)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz)))))) b0)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz)))))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq (a->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (a->Prop)) b0) (fun (Xz:a)=> (X Xz)))
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) (fun (Xz:a)=> (X Xz)))
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) (fun (Xz:a)=> (X Xz)))
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) (fun (Xz:a)=> (X Xz)))
% Found x00:((ex a) (fun (Xz:a)=> (X Xz)))
% Instantiate: b0:=(fun (Xz:a)=> (X Xz)):(a->Prop)
% Found x00 as proof of (P b0)
% Found eta_expansion000:=(eta_expansion00 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))):(((eq (a->Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))) (fun (x:a)=> ((and ((and (X x)) (forall (Xx:a), ((X Xx)->((x0 x) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) x))))))
% Found (eta_expansion00 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))) b0)
% Found ((eta_expansion0 Prop) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))) b0)
% Found (((eta_expansion a) Prop) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))) b0)
% Found (((eta_expansion a) Prop) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))) b0)
% Found (((eta_expansion a) Prop) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))) b0)
% Found x00:((ex a) (fun (Xz:a)=> (X Xz)))
% Instantiate: b0:=(fun (Xz:a)=> (X Xz)):(a->Prop)
% Found x00 as proof of (P b0)
% Found eta_expansion000:=(eta_expansion00 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))):(((eq (a->Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))) (fun (x:a)=> ((and ((and (X x)) (forall (Xx:a), ((X Xx)->((x0 x) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) x))))))
% Found (eta_expansion00 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))) b0)
% Found ((eta_expansion0 Prop) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))) b0)
% Found (((eta_expansion a) Prop) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))) b0)
% Found (((eta_expansion a) Prop) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))) b0)
% Found (((eta_expansion a) Prop) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz)))))) b0)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (forall (P:(a->Prop)), ((P Xy)->(P x1)))
% Found ((eq_ref a) Xy) as proof of (forall (P:(a->Prop)), ((P Xy)->(P x1)))
% Found ((eq_ref a) Xy) as proof of (forall (P:(a->Prop)), ((P Xy)->(P x1)))
% Found ((eq_ref a) Xy) as proof of (forall (P:(a->Prop)), ((P Xy)->(P x1)))
% Found (fun (x000:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))))=> ((eq_ref a) Xy)) as proof of (((eq a) Xy) x1)
% Found eta_expansion000:=(eta_expansion00 b0):(((eq (a->Prop)) b0) (fun (x:a)=> (b0 x)))
% Found (eta_expansion00 b0) as proof of (((eq (a->Prop)) b0) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz))))))
% Found ((eta_expansion0 Prop) b0) as proof of (((eq (a->Prop)) b0) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz))))))
% Found (((eta_expansion a) Prop) b0) as proof of (((eq (a->Prop)) b0) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz))))))
% Found (((eta_expansion a) Prop) b0) as proof of (((eq (a->Prop)) b0) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz))))))
% Found (((eta_expansion a) Prop) b0) as proof of (((eq (a->Prop)) b0) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz))))))
% Found eta_expansion000:=(eta_expansion00 (fun (Xz:a)=> (X Xz))):(((eq (a->Prop)) (fun (Xz:a)=> (X Xz))) (fun (x:a)=> (X x)))
% Found (eta_expansion00 (fun (Xz:a)=> (X Xz))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> (X Xz))) b0)
% Found ((eta_expansion0 Prop) (fun (Xz:a)=> (X Xz))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> (X Xz))) b0)
% Found (((eta_expansion a) Prop) (fun (Xz:a)=> (X Xz))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> (X Xz))) b0)
% Found (((eta_expansion a) Prop) (fun (Xz:a)=> (X Xz))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> (X Xz))) b0)
% Found (((eta_expansion a) Prop) (fun (Xz:a)=> (X Xz))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> (X Xz))) b0)
% Found eta_expansion000:=(eta_expansion00 b0):(((eq (a->Prop)) b0) (fun (x:a)=> (b0 x)))
% Found (eta_expansion00 b0) as proof of (((eq (a->Prop)) b0) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz))))))
% Found ((eta_expansion0 Prop) b0) as proof of (((eq (a->Prop)) b0) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz))))))
% Found (((eta_expansion a) Prop) b0) as proof of (((eq (a->Prop)) b0) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz))))))
% Found (((eta_expansion a) Prop) b0) as proof of (((eq (a->Prop)) b0) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz))))))
% Found (((eta_expansion a) Prop) b0) as proof of (((eq (a->Prop)) b0) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x0 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) Xz))))))
% Found eta_expansion000:=(eta_expansion00 (fun (Xz:a)=> (X Xz))):(((eq (a->Prop)) (fun (Xz:a)=> (X Xz))) (fun (x:a)=> (X x)))
% Found (eta_expansion00 (fun (Xz:a)=> (X Xz))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> (X Xz))) b0)
% Found ((eta_expansion0 Prop) (fun (Xz:a)=> (X Xz))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> (X Xz))) b0)
% Found (((eta_expansion a) Prop) (fun (Xz:a)=> (X Xz))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> (X Xz))) b0)
% Found (((eta_expansion a) Prop) (fun (Xz:a)=> (X Xz))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> (X Xz))) b0)
% Found (((eta_expansion a) Prop) (fun (Xz:a)=> (X Xz))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> (X Xz))) b0)
% Found x4:(P Xy)
% Instantiate: x3:=Xy:a
% Found (fun (x4:(P Xy))=> x4) as proof of (P x3)
% Found (fun (P:(a->Prop)) (x4:(P Xy))=> x4) as proof of ((P Xy)->(P x3))
% Found (fun (P:(a->Prop)) (x4:(P Xy))=> x4) as proof of (forall (P:(a->Prop)), ((P Xy)->(P x3)))
% Found (fun (x000:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))) (P:(a->Prop)) (x4:(P Xy))=> x4) as proof of (((eq a) Xy) x3)
% Found eq_ref000:=(eq_ref00 P):((P Xy)->(P Xy))
% Found (eq_ref00 P) as proof of ((P Xy)->(P x3))
% Found ((eq_ref0 Xy) P) as proof of ((P Xy)->(P x3))
% Found (((eq_ref a) Xy) P) as proof of ((P Xy)->(P x3))
% Found (((eq_ref a) Xy) P) as proof of ((P Xy)->(P x3))
% Found (fun (P:(a->Prop))=> (((eq_ref a) Xy) P)) as proof of ((P Xy)->(P x3))
% Found (fun (P:(a->Prop))=> (((eq_ref a) Xy) P)) as proof of (forall (P:(a->Prop)), ((P Xy)->(P x3)))
% Found (fun (x000:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))) (P:(a->Prop))=> (((eq_ref a) Xy) P)) as proof of (((eq a) Xy) x3)
% Found eq_ref000:=(eq_ref00 P):((P Xy)->(P Xy))
% Found (eq_ref00 P) as proof of ((P Xy)->(P x3))
% Found ((eq_ref0 Xy) P) as proof of ((P Xy)->(P x3))
% Found (((eq_ref a) Xy) P) as proof of ((P Xy)->(P x3))
% Found (((eq_ref a) Xy) P) as proof of ((P Xy)->(P x3))
% Found (fun (P:(a->Prop))=> (((eq_ref a) Xy) P)) as proof of ((P Xy)->(P x3))
% Found (fun (P:(a->Prop))=> (((eq_ref a) Xy) P)) as proof of (forall (P:(a->Prop)), ((P Xy)->(P x3)))
% Found (fun (x01:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))) (P:(a->Prop))=> (((eq_ref a) Xy) P)) as proof of (((eq a) Xy) x3)
% Found x4:(P Xy)
% Instantiate: x3:=Xy:a
% Found (fun (x4:(P Xy))=> x4) as proof of (P x3)
% Found (fun (P:(a->Prop)) (x4:(P Xy))=> x4) as proof of ((P Xy)->(P x3))
% Found (fun (P:(a->Prop)) (x4:(P Xy))=> x4) as proof of (forall (P:(a->Prop)), ((P Xy)->(P x3)))
% Found (fun (x01:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))) (P:(a->Prop)) (x4:(P Xy))=> x4) as proof of (((eq a) Xy) x3)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (forall (P:(a->Prop)), ((P Xy)->(P x3)))
% Found ((eq_ref a) Xy) as proof of (forall (P:(a->Prop)), ((P Xy)->(P x3)))
% Found ((eq_ref a) Xy) as proof of (forall (P:(a->Prop)), ((P Xy)->(P x3)))
% Found ((eq_ref a) Xy) as proof of (forall (P:(a->Prop)), ((P Xy)->(P x3)))
% Found (fun (x000:((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx)))))=> ((eq_ref a) Xy)) as proof of (((eq a) Xy) x3)
% Found eta_expansion0000:=(eta_expansion000 P0):((P0 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz))))))->(P0 (fun (x:a)=> ((and ((and (X x)) (forall (Xx:a), ((X Xx)->((x2 x) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) x)))))))
% Found (eta_expansion000 P0) as proof of (P1 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz))))))
% Found ((eta_expansion00 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz)))))) P0) as proof of (P1 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz))))))
% Found (((eta_expansion0 Prop) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz)))))) P0) as proof of (P1 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz))))))
% Found ((((eta_expansion a) Prop) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz)))))) P0) as proof of (P1 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz))))))
% Found ((((eta_expansion a) Prop) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz)))))) P0) as proof of (P1 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz))))))
% Found eta_expansion0000:=(eta_expansion000 P0):((P0 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz))))))->(P0 (fun (x:a)=> ((and ((and (X x)) (forall (Xx:a), ((X Xx)->((x2 x) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) x)))))))
% Found (eta_expansion000 P0) as proof of (P1 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz))))))
% Found ((eta_expansion00 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz)))))) P0) as proof of (P1 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz))))))
% Found (((eta_expansion0 Prop) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz)))))) P0) as proof of (P1 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz))))))
% Found ((((eta_expansion a) Prop) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz)))))) P0) as proof of (P1 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz))))))
% Found ((((eta_expansion a) Prop) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz)))))) P0) as proof of (P1 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz))))))
% Found eta_expansion0000:=(eta_expansion000 P0):((P0 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz))))))->(P0 (fun (x:a)=> ((and ((and (X x)) (forall (Xx:a), ((X Xx)->((x2 x) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) x)))))))
% Found (eta_expansion000 P0) as proof of (P1 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz))))))
% Found ((eta_expansion00 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz)))))) P0) as proof of (P1 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz))))))
% Found (((eta_expansion0 Prop) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz)))))) P0) as proof of (P1 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz))))))
% Found ((((eta_expansion a) Prop) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz)))))) P0) as proof of (P1 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz))))))
% Found ((((eta_expansion a) Prop) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz)))))) P0) as proof of (P1 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz))))))
% Found eta_expansion0000:=(eta_expansion000 P0):((P0 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz))))))->(P0 (fun (x:a)=> ((and ((and (X x)) (forall (Xx:a), ((X Xx)->((x2 x) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) x)))))))
% Found (eta_expansion000 P0) as proof of (P1 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz))))))
% Found ((eta_expansion00 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz)))))) P0) as proof of (P1 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz))))))
% Found (((eta_expansion0 Prop) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz)))))) P0) as proof of (P1 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz))))))
% Found ((((eta_expansion a) Prop) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz)))))) P0) as proof of (P1 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz))))))
% Found ((((eta_expansion a) Prop) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz)))))) P0) as proof of (P1 (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((x2 Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) Xz))))))
% Found eta_expansion0000:=(eta_expansion000 P0):((P0 (fun (Xz:a)=> (X Xz)))->(P0 (fun (x:a)=> (X x))))
% Found (eta_expansion000 P0) as proof of (P1 (fun (Xz:a)=> (X Xz)))
% Found ((eta_expansion00 (fun (Xz:a)=> (X Xz))) P0) as proof of (P1 (fun (Xz:a)=> (X Xz)))
% Found (((eta_expansion0 Prop) (fun (Xz:a)=> (X Xz))) P0) as proof of (P1 (fun (Xz:a)=> (X Xz)))
% Found ((((eta_expansion a) Prop) (fun (Xz:a)=> (X Xz))) P0) as proof of (P1 (fun (Xz:a)=> (X Xz)))
% Found ((((eta_expansion a) Prop) (fun (Xz:a)=> (X Xz))) P0) as proof of (P1 (fun (Xz:a)=> (X Xz)))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P0):((P0 (fun (Xz:a)=> (X Xz)))->(P0 (fun (x:a)=> (X x))))
% Found (eta_expansion_dep000 P0) as proof of (P1 (fun (Xz:a)=> (X Xz)))
% Found ((eta_expansion_dep00 (fun (Xz:a)=> (X Xz))) P0) as proof of (P1 (fun (Xz:a)=> (X Xz)))
% Found (((eta_expansion_dep0 (fun (x2:a)=> Prop)) (fun (Xz:a)=> (X Xz))) P0) as proof of (P1 (fun (Xz:a)=> (X Xz)))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xz:a)=> (X Xz))) P0) as proof of (P1 (fun (Xz:a)=> (X Xz)))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xz:a)=> (X Xz))) P0) as proof of (P1 (fun (Xz:a)=> (X Xz)))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P0):((P0 (fun (Xz:a)=> (X Xz)))->(P0 (fun (x:a)=> (X x))))
% Found (eta_expansion_dep000 P0) as proof of (P1 (fun (Xz:a)=> (X Xz)))
% Found ((eta_expansion_dep00 (fun (Xz:a)=> (X Xz))) P0) as proof of (P1 (fun (Xz:a)=> (X Xz)))
% Found (((eta_expansion_dep0 (fun (x2:a)=> Prop)) (fun (Xz:a)=> (X Xz))) P0) as proof of (P1 (fun (Xz:a)=> (X Xz)))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xz:a)=> (X Xz))) P0) as proof of (P1 (fun (Xz:a)=> (X Xz)))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xz:a)=> (X Xz))) P0) as proof of (P1 (fun (Xz:a)=> (X Xz)))
% Found eta_expansion0000:=(eta_expansion000 P0):((P0 (fun (Xz:a)=> (X Xz)))->(P0 (fun (x:a)=> (X x))))
% Found (eta_expansion000 P0) as proof of (P1 (fun (Xz:a)=> (X Xz)))
% Found ((eta_expansion00 (fun (Xz:a)=> (X Xz))) P0) as proof of (P1 (fun (Xz:a)=> (X Xz)))
% Found (((eta_expansion0 Prop) (fun (Xz:a)=> (X Xz))) P0) as proof of (P1 (fun (Xz:a)=> (X Xz)))
% Found ((((eta_expansion a) Prop) (fun (Xz:a)=> (X Xz))) P0) as proof of (P1 (fun (Xz:a)=> (X Xz)))
% Found ((((eta_expansion a) Prop) (fun (Xz:a)=> (X Xz))) P0) as proof of (P1 (fun (Xz:a)=> (X Xz)))
% Found x00:((ex a) (fun (Xz:a)=> (X Xz)))
% Instantiate: f:=(fun (Xz:a)=> (X Xz)):(a->Prop)
% Found x00 as proof of (P f)
% Found x00:((ex a) (fun (Xz:a)=> (X Xz)))
% Instantiate: f:=(fun (Xz:a)=> (X Xz)):(a->Prop)
% Found x00 as proof of (P f)
% Found x00:((ex a) (fun (Xz:a)=> (X Xz)))
% Instantiate: f:=(fun (Xz:a)=> (X Xz)):(a->Prop)
% Found x00 as proof of (P f)
% Found x00:((ex a) (fun (Xz:a)=> (X Xz)))
% Instantiate: f:=(fun (Xz:a)=> (X Xz)):(a->Prop)
% Found x00 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 ((and (X x5)) (forall (Xx:a), ((X Xx)->((x0 x5) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) x5)))))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) ((and ((and (X x5)) (forall (Xx:a), ((X Xx)->((x0 x5) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) x5)))))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) ((and ((and (X x5)) (forall (Xx:a), ((X Xx)->((x0 x5) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) x5)))))
% Found (fun (x5:a)=> ((eq_ref Prop) (f x5))) as proof of (((eq Prop) (f x5)) ((and ((and (X x5)) (forall (Xx:a), ((X Xx)->((x0 x5) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) x5)))))
% Found (fun (x5:a)=> ((eq_ref Prop) (f x5))) as proof of (forall (x:a), (((eq Prop) (f x)) ((and ((and (X x)) (forall (Xx:a), ((X Xx)->((x0 x) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) x))))))
% 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 (X x5)) (forall (Xx:a), ((X Xx)->((x0 x5) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) x5)))))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) ((and ((and (X x5)) (forall (Xx:a), ((X Xx)->((x0 x5) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) x5)))))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) ((and ((and (X x5)) (forall (Xx:a), ((X Xx)->((x0 x5) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) x5)))))
% Found (fun (x5:a)=> ((eq_ref Prop) (f x5))) as proof of (((eq Prop) (f x5)) ((and ((and (X x5)) (forall (Xx:a), ((X Xx)->((x0 x5) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) x5)))))
% Found (fun (x5:a)=> ((eq_ref Prop) (f x5))) as proof of (forall (x:a), (((eq Prop) (f x)) ((and ((and (X x)) (forall (Xx:a), ((X Xx)->((x0 x) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) x))))))
% 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 (X x5)) (forall (Xx:a), ((X Xx)->((x0 x5) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) x5)))))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) ((and ((and (X x5)) (forall (Xx:a), ((X Xx)->((x0 x5) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) x5)))))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) ((and ((and (X x5)) (forall (Xx:a), ((X Xx)->((x0 x5) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) x5)))))
% Found (fun (x5:a)=> ((eq_ref Prop) (f x5))) as proof of (((eq Prop) (f x5)) ((and ((and (X x5)) (forall (Xx:a), ((X Xx)->((x0 x5) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) x5)))))
% Found (fun (x5:a)=> ((eq_ref Prop) (f x5))) as proof of (forall (x:a), (((eq Prop) (f x)) ((and ((and (X x)) (forall (Xx:a), ((X Xx)->((x0 x) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) x))))))
% 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 (X x5)) (forall (Xx:a), ((X Xx)->((x0 x5) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) x5)))))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) ((and ((and (X x5)) (forall (Xx:a), ((X Xx)->((x0 x5) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) x5)))))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) ((and ((and (X x5)) (forall (Xx:a), ((X Xx)->((x0 x5) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) x5)))))
% Found (fun (x5:a)=> ((eq_ref Prop) (f x5))) as proof of (((eq Prop) (f x5)) ((and ((and (X x5)) (forall (Xx:a), ((X Xx)->((x0 x5) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) x5)))))
% Found (fun (x5:a)=> ((eq_ref Prop) (f x5))) as proof of (forall (x:a), (((eq Prop) (f x)) ((and ((and (X x)) (forall (Xx:a), ((X Xx)->((x0 x) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x0 Xy) Xx))))->(((eq a) Xy) x))))))
% Found eq_ref000:=(eq_ref00 P0):((P0 (X x3))->(P0 (X x3)))
% Found (eq_ref00 P0) as proof of (P1 (X x3))
% Found ((eq_ref0 (X x3)) P0) as proof of (P1 (X x3))
% Found (((eq_ref Prop) (X x3)) P0) as proof of (P1 (X x3))
% Found (((eq_ref Prop) (X x3)) P0) as proof of (P1 (X x3))
% Found eq_ref000:=(eq_ref00 P0):((P0 (X x3))->(P0 (X x3)))
% Found (eq_ref00 P0) as proof of (P1 (X x3))
% Found ((eq_ref0 (X x3)) P0) as proof of (P1 (X x3))
% Found (((eq_ref Prop) (X x3)) P0) as proof of (P1 (X x3))
% Found (((eq_ref Prop) (X x3)) P0) as proof of (P1 (X x3))
% Found eq_ref000:=(eq_ref00 P0):((P0 (X x3))->(P0 (X x3)))
% Found (eq_ref00 P0) as proof of (P1 (X x3))
% Found ((eq_ref0 (X x3)) P0) as proof of (P1 (X x3))
% Found (((eq_ref Prop) (X x3)) P0) as proof of (P1 (X x3))
% Found (((eq_ref Prop) (X x3)) P0) as proof of (P1 (X x3))
% Found eq_ref000:=(eq_ref00 P0):((P0 (X x3))->(P0 (X x3)))
% Found (eq_ref00 P0) as proof of (P1 (X x3))
% Found ((eq_ref0 (X x3)) P0) as proof of (P1 (X x3))
% Found (((eq_ref Prop) (X x3)) P0) as proof of (P1 (X x3))
% Found (((eq_ref Prop) (X x3)) P0) as proof of (P1 (X x3))
% Found eq_ref00:=(eq_ref0 (X x3)):(((eq Prop) (X x3)) (X x3))
% Found (eq_ref0 (X x3)) as proof of (((eq Prop) (X x3)) b0)
% Found ((eq_ref Prop) (X x3)) as proof of (((eq Prop) (X x3)) b0)
% Found ((eq_ref Prop) (X x3)) as proof of (((eq Prop) (X x3)) b0)
% Found ((eq_ref Prop) (X x3)) as proof of (((eq Prop) (X x3)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) ((and ((and (X x3)) (forall (Xx:a), ((X Xx)->((x2 x3) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) x3)))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and ((and (X x3)) (forall (Xx:a), ((X Xx)->((x2 x3) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) x3)))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and ((and (X x3)) (forall (Xx:a), ((X Xx)->((x2 x3) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) x3)))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and ((and (X x3)) (forall (Xx:a), ((X Xx)->((x2 x3) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) x3)))))
% Found eq_ref00:=(eq_ref0 (X x3)):(((eq Prop) (X x3)) (X x3))
% Found (eq_ref0 (X x3)) as proof of (((eq Prop) (X x3)) b0)
% Found ((eq_ref Prop) (X x3)) as proof of (((eq Prop) (X x3)) b0)
% Found ((eq_ref Prop) (X x3)) as proof of (((eq Prop) (X x3)) b0)
% Found ((eq_ref Prop) (X x3)) as proof of (((eq Prop) (X x3)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) ((and ((and (X x3)) (forall (Xx:a), ((X Xx)->((x2 x3) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) x3)))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and ((and (X x3)) (forall (Xx:a), ((X Xx)->((x2 x3) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) x3)))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and ((and (X x3)) (forall (Xx:a), ((X Xx)->((x2 x3) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) x3)))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and ((and (X x3)) (forall (Xx:a), ((X Xx)->((x2 x3) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) x3)))))
% Found eq_ref00:=(eq_ref0 (X x3)):(((eq Prop) (X x3)) (X x3))
% Found (eq_ref0 (X x3)) as proof of (((eq Prop) (X x3)) b0)
% Found ((eq_ref Prop) (X x3)) as proof of (((eq Prop) (X x3)) b0)
% Found ((eq_ref Prop) (X x3)) as proof of (((eq Prop) (X x3)) b0)
% Found ((eq_ref Prop) (X x3)) as proof of (((eq Prop) (X x3)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) ((and ((and (X x3)) (forall (Xx:a), ((X Xx)->((x2 x3) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((x2 Xy) Xx))))->(((eq a) Xy) x3)))))
% Found ((eq_ref P
% EOF
%------------------------------------------------------------------------------