TSTP Solution File: SEV239^5 by cocATP---0.2.0
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- Process Solution
%------------------------------------------------------------------------------
% File : cocATP---0.2.0
% Problem : SEV239^5 : TPTP v6.1.0. Released v4.0.0.
% Transfm : none
% Format : tptp:raw
% Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% Computer : n188.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 13:33:55 EDT 2014
% Result : Timeout 300.08s
% Output : None
% Verified :
% SZS Type : None (Parsing solution fails)
% Syntax : Number of formulae : 0
% Comments :
%------------------------------------------------------------------------------
%----NO SOLUTION OUTPUT BY SYSTEM
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% % Problem : SEV239^5 : TPTP v6.1.0. Released v4.0.0.
% % Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n188.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 08:35:31 CDT 2014
% % CPUTime : 300.08
% Python 2.7.5
% Using paths ['/home/cristobal/cocATP/CASC/TPTP/', '/export/starexec/sandbox/benchmark/', '/export/starexec/sandbox/benchmark/']
% FOF formula (<kernel.Constant object at 0x11d3b48>, <kernel.Type object at 0x11d3560>) of role type named a_type
% Using role type
% Declaring a:Type
% FOF formula (<kernel.Constant object at 0x13c1488>, <kernel.DependentProduct object at 0x11d3518>) of role type named y
% Using role type
% Declaring y:(a->Prop)
% FOF formula (((eq (a->Prop)) y) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) ((fun (Xx:a) (Xy:a)=> (((eq a) Xx) Xy)) Xx0)))))) (S Xx)))))) of role conjecture named cX5211_pme
% Conjecture to prove = (((eq (a->Prop)) y) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) ((fun (Xx:a) (Xy:a)=> (((eq a) Xx) Xy)) Xx0)))))) (S Xx)))))):Prop
% Parameter a_DUMMY:a.
% We need to prove ['(((eq (a->Prop)) y) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) ((fun (Xx:a) (Xy:a)=> (((eq a) Xx) Xy)) Xx0)))))) (S Xx))))))']
% Parameter a:Type.
% Parameter y:(a->Prop).
% Trying to prove (((eq (a->Prop)) y) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) ((fun (Xx:a) (Xy:a)=> (((eq a) Xx) Xy)) Xx0)))))) (S Xx))))))
% Found eta_expansion000:=(eta_expansion00 y):(((eq (a->Prop)) y) (fun (x:a)=> (y x)))
% Found (eta_expansion00 y) as proof of (((eq (a->Prop)) y) b)
% Found ((eta_expansion0 Prop) y) as proof of (((eq (a->Prop)) y) b)
% Found (((eta_expansion a) Prop) y) as proof of (((eq (a->Prop)) y) b)
% Found (((eta_expansion a) Prop) y) as proof of (((eq (a->Prop)) y) b)
% Found (((eta_expansion a) Prop) y) as proof of (((eq (a->Prop)) y) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx))))))
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx))))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx))))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx))))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx))))))
% Found x2:(P y)
% Found (fun (x2:(P y))=> x2) as proof of (P y)
% Found (fun (x2:(P y))=> x2) as proof of (P0 y)
% Found x2:(P y)
% Found (fun (x2:(P y))=> x2) as proof of (P y)
% Found (fun (x2:(P y))=> x2) as proof of (P0 y)
% Found x2:(P y)
% Found (fun (x2:(P y))=> x2) as proof of (P y)
% Found (fun (x2:(P y))=> x2) as proof of (P0 y)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) y)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) y)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) y)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) y)
% Found eta_expansion000:=(eta_expansion00 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx)))))):(((eq (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx)))))) (fun (x:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))))
% Found (eta_expansion00 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx)))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx)))))) b)
% Found ((eta_expansion0 Prop) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx)))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx)))))) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx)))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx)))))) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx)))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx)))))) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx)))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx)))))) b)
% Found eq_ref00:=(eq_ref0 (y x)):(((eq Prop) (y x)) (y x))
% Found (eq_ref0 (y x)) as proof of (((eq Prop) (y x)) b)
% Found ((eq_ref Prop) (y x)) as proof of (((eq Prop) (y x)) b)
% Found ((eq_ref Prop) (y x)) as proof of (((eq Prop) (y x)) b)
% Found ((eq_ref Prop) (y x)) as proof of (((eq Prop) (y x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found eq_ref00:=(eq_ref0 (y x)):(((eq Prop) (y x)) (y x))
% Found (eq_ref0 (y x)) as proof of (((eq Prop) (y x)) b)
% Found ((eq_ref Prop) (y x)) as proof of (((eq Prop) (y x)) b)
% Found ((eq_ref Prop) (y x)) as proof of (((eq Prop) (y x)) b)
% Found ((eq_ref Prop) (y x)) as proof of (((eq Prop) (y x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found x01:(P (y x))
% Found (fun (x01:(P (y x)))=> x01) as proof of (P (y x))
% Found (fun (x01:(P (y x)))=> x01) as proof of (P0 (y x))
% Found x01:(P (y x))
% Found (fun (x01:(P (y x)))=> x01) as proof of (P (y x))
% Found (fun (x01:(P (y x)))=> x01) as proof of (P0 (y x))
% Found eq_ref000:=(eq_ref00 P):((P ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))->(P ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))))
% Found (eq_ref00 P) as proof of (P0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found ((eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) P) as proof of (P0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found (((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) P) as proof of (P0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found (((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) P) as proof of (P0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found eq_ref000:=(eq_ref00 P):((P ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))->(P ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))))
% Found (eq_ref00 P) as proof of (P0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found ((eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) P) as proof of (P0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found (((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) P) as proof of (P0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found (((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) P) as proof of (P0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (y x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (y x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (y x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (y x))
% Found eq_ref00:=(eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))):(((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found (eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (y x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (y x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (y x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (y x))
% Found eq_ref00:=(eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))):(((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found (eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b)
% Found eq_ref00:=(eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))):(((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found (eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (y x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (y x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (y x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (y x))
% Found eq_ref00:=(eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))):(((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found (eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (y x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (y x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (y x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (y x))
% Found x:(P y)
% Instantiate: b:=y:(a->Prop)
% Found x as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx)))))):(((eq (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx)))))) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx))))))
% Found (eq_ref0 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx)))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx)))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx)))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx)))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx)))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx)))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx)))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx)))))) b)
% Found x:(P y)
% Instantiate: f:=y:(a->Prop)
% Found x as proof of (P0 f)
% 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)) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x0)))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x0)))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x0)))))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x0)))))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:a), (((eq Prop) (f x)) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))))
% Found x:(P y)
% Instantiate: f:=y:(a->Prop)
% Found x as proof of (P0 f)
% 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)) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x0)))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x0)))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x0)))))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x0)))))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:a), (((eq Prop) (f x)) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))))
% Found eq_ref00:=(eq_ref0 (y x)):(((eq Prop) (y x)) (y x))
% Found (eq_ref0 (y x)) as proof of (((eq Prop) (y x)) b)
% Found ((eq_ref Prop) (y x)) as proof of (((eq Prop) (y x)) b)
% Found ((eq_ref Prop) (y x)) as proof of (((eq Prop) (y x)) b)
% Found ((eq_ref Prop) (y x)) as proof of (((eq Prop) (y x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found eq_ref00:=(eq_ref0 (y x)):(((eq Prop) (y x)) (y x))
% Found (eq_ref0 (y x)) as proof of (((eq Prop) (y x)) b)
% Found ((eq_ref Prop) (y x)) as proof of (((eq Prop) (y x)) b)
% Found ((eq_ref Prop) (y x)) as proof of (((eq Prop) (y x)) b)
% Found ((eq_ref Prop) (y x)) as proof of (((eq Prop) (y x)) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 y):(((eq (a->Prop)) y) (fun (x:a)=> (y x)))
% Found (eta_expansion_dep00 y) as proof of (((eq (a->Prop)) y) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) y) as proof of (((eq (a->Prop)) y) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) y) as proof of (((eq (a->Prop)) y) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) y) as proof of (((eq (a->Prop)) y) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) y) as proof of (((eq (a->Prop)) y) b)
% Found eta_expansion000:=(eta_expansion00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx))))))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx))))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx))))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx))))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx))))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx)))))):(((eq (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx)))))) (fun (x:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))))
% Found (eta_expansion_dep00 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx)))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx)))))) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx)))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx)))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx)))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx)))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx)))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx)))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx)))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx)))))) b)
% Found x:(P0 b)
% Instantiate: b:=y:(a->Prop)
% Found (fun (x:(P0 b))=> x) as proof of (P0 y)
% Found (fun (P0:((a->Prop)->Prop)) (x:(P0 b))=> x) as proof of ((P0 b)->(P0 y))
% Found (fun (P0:((a->Prop)->Prop)) (x:(P0 b))=> x) as proof of (P b)
% Found x:(P (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx))))))
% Instantiate: b:=(fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx))))):(a->Prop)
% Found x as proof of (P0 b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 y):(((eq (a->Prop)) y) (fun (x:a)=> (y x)))
% Found (eta_expansion_dep00 y) as proof of (((eq (a->Prop)) y) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) y) as proof of (((eq (a->Prop)) y) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) y) as proof of (((eq (a->Prop)) y) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) y) as proof of (((eq (a->Prop)) y) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) y) as proof of (((eq (a->Prop)) y) b)
% Found x3:(P y)
% Found (fun (x3:(P y))=> x3) as proof of (P y)
% Found (fun (x3:(P y))=> x3) as proof of (P0 y)
% Found eq_ref00:=(eq_ref0 y):(((eq (a->Prop)) y) y)
% Found (eq_ref0 y) as proof of (((eq (a->Prop)) y) b)
% Found ((eq_ref (a->Prop)) y) as proof of (((eq (a->Prop)) y) b)
% Found ((eq_ref (a->Prop)) y) as proof of (((eq (a->Prop)) y) b)
% Found ((eq_ref (a->Prop)) y) as proof of (((eq (a->Prop)) y) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx))))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx))))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx))))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx))))))
% Found eta_expansion000:=(eta_expansion00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq (a->Prop)) b) b0)
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) b0)
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) b0)
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) b0)
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) 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 (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx))))))
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) b0) as proof of (((eq (a->Prop)) b0) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx))))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b0) as proof of (((eq (a->Prop)) b0) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx))))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b0) as proof of (((eq (a->Prop)) b0) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx))))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b0) as proof of (((eq (a->Prop)) b0) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx))))))
% Found x:(P (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx))))))
% Instantiate: f:=(fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx))))):(a->Prop)
% Found x as proof of (P0 f)
% 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)) (y x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (y x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (y x0))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (y x0))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:a), (((eq Prop) (f x)) (y x)))
% Found x:(P (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx))))))
% Instantiate: f:=(fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx))))):(a->Prop)
% Found x as proof of (P0 f)
% 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)) (y x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (y x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (y x0))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (y x0))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:a), (((eq Prop) (f x)) (y x)))
% Found x0:(P (y x))
% Instantiate: b:=(y x):Prop
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))):(((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found (eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b)
% Found x0:(P (y x))
% Instantiate: b:=(y x):Prop
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))):(((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found (eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) y)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) y)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) y)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) y)
% Found eq_ref00:=(eq_ref0 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx)))))):(((eq (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx)))))) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx))))))
% Found (eq_ref0 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx)))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx)))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx)))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx)))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx)))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx)))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx)))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx)))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) y)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) y)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) y)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) y)
% Found eq_ref00:=(eq_ref0 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx)))))):(((eq (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx)))))) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx))))))
% Found (eq_ref0 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx)))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx)))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx)))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx)))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx)))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx)))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx)))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx)))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) y)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) y)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) y)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) y)
% Found eta_expansion000:=(eta_expansion00 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx)))))):(((eq (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx)))))) (fun (x:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))))
% Found (eta_expansion00 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx)))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx)))))) b)
% Found ((eta_expansion0 Prop) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx)))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx)))))) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx)))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx)))))) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx)))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx)))))) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx)))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx)))))) b)
% Found x2:(P b)
% Found (fun (x2:(P b))=> x2) as proof of (P b)
% Found (fun (x2:(P b))=> x2) as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (y x)):(((eq Prop) (y x)) (y x))
% Found (eq_ref0 (y x)) as proof of (((eq Prop) (y x)) b)
% Found ((eq_ref Prop) (y x)) as proof of (((eq Prop) (y x)) b)
% Found ((eq_ref Prop) (y x)) as proof of (((eq Prop) (y x)) b)
% Found ((eq_ref Prop) (y x)) as proof of (((eq Prop) (y x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found eq_ref00:=(eq_ref0 (y x)):(((eq Prop) (y x)) (y x))
% Found (eq_ref0 (y x)) as proof of (((eq Prop) (y x)) b)
% Found ((eq_ref Prop) (y x)) as proof of (((eq Prop) (y x)) b)
% Found ((eq_ref Prop) (y x)) as proof of (((eq Prop) (y x)) b)
% Found ((eq_ref Prop) (y x)) as proof of (((eq Prop) (y x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found eq_ref00:=(eq_ref0 (y x)):(((eq Prop) (y x)) (y x))
% Found (eq_ref0 (y x)) as proof of (((eq Prop) (y x)) b)
% Found ((eq_ref Prop) (y x)) as proof of (((eq Prop) (y x)) b)
% Found ((eq_ref Prop) (y x)) as proof of (((eq Prop) (y x)) b)
% Found ((eq_ref Prop) (y x)) as proof of (((eq Prop) (y x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found eq_ref00:=(eq_ref0 (y x)):(((eq Prop) (y x)) (y x))
% Found (eq_ref0 (y x)) as proof of (((eq Prop) (y x)) b)
% Found ((eq_ref Prop) (y x)) as proof of (((eq Prop) (y x)) b)
% Found ((eq_ref Prop) (y x)) as proof of (((eq Prop) (y x)) b)
% Found ((eq_ref Prop) (y x)) as proof of (((eq Prop) (y x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found eq_ref00:=(eq_ref0 (y x)):(((eq Prop) (y x)) (y x))
% Found (eq_ref0 (y x)) as proof of (((eq Prop) (y x)) b)
% Found ((eq_ref Prop) (y x)) as proof of (((eq Prop) (y x)) b)
% Found ((eq_ref Prop) (y x)) as proof of (((eq Prop) (y x)) b)
% Found ((eq_ref Prop) (y x)) as proof of (((eq Prop) (y x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found eq_ref00:=(eq_ref0 (y x)):(((eq Prop) (y x)) (y x))
% Found (eq_ref0 (y x)) as proof of (((eq Prop) (y x)) b)
% Found ((eq_ref Prop) (y x)) as proof of (((eq Prop) (y x)) b)
% Found ((eq_ref Prop) (y x)) as proof of (((eq Prop) (y x)) b)
% Found ((eq_ref Prop) (y x)) as proof of (((eq Prop) (y x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found x2:(P b)
% Found (fun (x2:(P b))=> x2) as proof of (P b)
% Found (fun (x2:(P b))=> x2) as proof of (P0 b)
% Found x2:(P b)
% Found (fun (x2:(P b))=> x2) as proof of (P b)
% Found (fun (x2:(P b))=> x2) as proof of (P0 b)
% Found x01:(P1 (y x))
% Found (fun (x01:(P1 (y x)))=> x01) as proof of (P1 (y x))
% Found (fun (x01:(P1 (y x)))=> x01) as proof of (P2 (y x))
% Found x01:(P1 (y x))
% Found (fun (x01:(P1 (y x)))=> x01) as proof of (P1 (y x))
% Found (fun (x01:(P1 (y x)))=> x01) as proof of (P2 (y x))
% Found x01:(P1 (y x))
% Found (fun (x01:(P1 (y x)))=> x01) as proof of (P1 (y x))
% Found (fun (x01:(P1 (y x)))=> x01) as proof of (P2 (y x))
% Found x01:(P1 (y x))
% Found (fun (x01:(P1 (y x)))=> x01) as proof of (P1 (y x))
% Found (fun (x01:(P1 (y x)))=> x01) as proof of (P2 (y x))
% Found eta_expansion000:=(eta_expansion00 a0):(((eq ((a->Prop)->Prop)) a0) (fun (x:(a->Prop))=> (a0 x)))
% Found (eta_expansion00 a0) as proof of (((eq ((a->Prop)->Prop)) a0) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))
% Found ((eta_expansion0 Prop) a0) as proof of (((eq ((a->Prop)->Prop)) a0) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))
% Found (((eta_expansion (a->Prop)) Prop) a0) as proof of (((eq ((a->Prop)->Prop)) a0) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))
% Found (((eta_expansion (a->Prop)) Prop) a0) as proof of (((eq ((a->Prop)->Prop)) a0) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))
% Found (((eta_expansion (a->Prop)) Prop) a0) as proof of (((eq ((a->Prop)->Prop)) a0) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))
% Found eq_ref00:=(eq_ref0 a0):(((eq ((a->Prop)->Prop)) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq ((a->Prop)->Prop)) a0) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))
% Found ((eq_ref ((a->Prop)->Prop)) a0) as proof of (((eq ((a->Prop)->Prop)) a0) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))
% Found ((eq_ref ((a->Prop)->Prop)) a0) as proof of (((eq ((a->Prop)->Prop)) a0) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))
% Found ((eq_ref ((a->Prop)->Prop)) a0) as proof of (((eq ((a->Prop)->Prop)) a0) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))
% Found eq_ref00:=(eq_ref0 a0):(((eq ((a->Prop)->Prop)) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq ((a->Prop)->Prop)) a0) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))
% Found ((eq_ref ((a->Prop)->Prop)) a0) as proof of (((eq ((a->Prop)->Prop)) a0) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))
% Found ((eq_ref ((a->Prop)->Prop)) a0) as proof of (((eq ((a->Prop)->Prop)) a0) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))
% Found ((eq_ref ((a->Prop)->Prop)) a0) as proof of (((eq ((a->Prop)->Prop)) a0) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))
% Found eta_expansion000:=(eta_expansion00 a0):(((eq ((a->Prop)->Prop)) a0) (fun (x:(a->Prop))=> (a0 x)))
% Found (eta_expansion00 a0) as proof of (((eq ((a->Prop)->Prop)) a0) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))
% Found ((eta_expansion0 Prop) a0) as proof of (((eq ((a->Prop)->Prop)) a0) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))
% Found (((eta_expansion (a->Prop)) Prop) a0) as proof of (((eq ((a->Prop)->Prop)) a0) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))
% Found (((eta_expansion (a->Prop)) Prop) a0) as proof of (((eq ((a->Prop)->Prop)) a0) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))
% Found (((eta_expansion (a->Prop)) Prop) a0) as proof of (((eq ((a->Prop)->Prop)) a0) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))
% Found eq_ref00:=(eq_ref0 b0):(((eq (a->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (a->Prop)) b0) b)
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) b)
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) b)
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) b)
% Found eq_ref00:=(eq_ref0 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx)))))):(((eq (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx)))))) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx))))))
% Found (eq_ref0 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx)))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx)))))) b0)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx)))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx)))))) b0)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx)))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx)))))) b0)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx)))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx)))))) b0)
% Found eta_expansion0000:=(eta_expansion000 P):((P (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx))))))->(P (fun (x:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))))
% Found (eta_expansion000 P) as proof of (P0 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx))))))
% Found ((eta_expansion00 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx)))))) P) as proof of (P0 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx))))))
% Found (((eta_expansion0 Prop) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx)))))) P) as proof of (P0 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx))))))
% Found ((((eta_expansion a) Prop) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx)))))) P) as proof of (P0 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx))))))
% Found ((((eta_expansion a) Prop) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx)))))) P) as proof of (P0 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx))))))
% Found eq_ref00:=(eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))):(((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found (eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b)
% Found x0:(P0 b)
% Instantiate: b:=(y x):Prop
% Found (fun (x0:(P0 b))=> x0) as proof of (P0 (y x))
% Found (fun (P0:(Prop->Prop)) (x0:(P0 b))=> x0) as proof of ((P0 b)->(P0 (y x)))
% Found (fun (P0:(Prop->Prop)) (x0:(P0 b))=> x0) as proof of (P b)
% Found eq_ref00:=(eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))):(((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found (eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b)
% Found x0:(P0 b)
% Instantiate: b:=(y x):Prop
% Found (fun (x0:(P0 b))=> x0) as proof of (P0 (y x))
% Found (fun (P0:(Prop->Prop)) (x0:(P0 b))=> x0) as proof of ((P0 b)->(P0 (y x)))
% Found (fun (P0:(Prop->Prop)) (x0:(P0 b))=> x0) as proof of (P b)
% Found x02:(P (y x))
% Found (fun (x02:(P (y x)))=> x02) as proof of (P (y x))
% Found (fun (x02:(P (y x)))=> x02) as proof of (P0 (y x))
% Found eq_ref00:=(eq_ref0 (y x)):(((eq Prop) (y x)) (y x))
% Found (eq_ref0 (y x)) as proof of (((eq Prop) (y x)) b)
% Found ((eq_ref Prop) (y x)) as proof of (((eq Prop) (y x)) b)
% Found ((eq_ref Prop) (y x)) as proof of (((eq Prop) (y x)) b)
% Found ((eq_ref Prop) (y x)) as proof of (((eq Prop) (y x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found x02:(P (y x))
% Found (fun (x02:(P (y x)))=> x02) as proof of (P (y x))
% Found (fun (x02:(P (y x)))=> x02) as proof of (P0 (y x))
% Found eq_ref00:=(eq_ref0 (y x)):(((eq Prop) (y x)) (y x))
% Found (eq_ref0 (y x)) as proof of (((eq Prop) (y x)) b)
% Found ((eq_ref Prop) (y x)) as proof of (((eq Prop) (y x)) b)
% Found ((eq_ref Prop) (y x)) as proof of (((eq Prop) (y x)) b)
% Found ((eq_ref Prop) (y x)) as proof of (((eq Prop) (y x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found x0:(P ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Instantiate: b:=((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))):Prop
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (y x)):(((eq Prop) (y x)) (y x))
% Found (eq_ref0 (y x)) as proof of (((eq Prop) (y x)) b)
% Found ((eq_ref Prop) (y x)) as proof of (((eq Prop) (y x)) b)
% Found ((eq_ref Prop) (y x)) as proof of (((eq Prop) (y x)) b)
% Found ((eq_ref Prop) (y x)) as proof of (((eq Prop) (y x)) b)
% Found x0:(P ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Instantiate: b:=((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))):Prop
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (y x)):(((eq Prop) (y x)) (y x))
% Found (eq_ref0 (y x)) as proof of (((eq Prop) (y x)) b)
% Found ((eq_ref Prop) (y x)) as proof of (((eq Prop) (y x)) b)
% Found ((eq_ref Prop) (y x)) as proof of (((eq Prop) (y x)) b)
% Found ((eq_ref Prop) (y x)) as proof of (((eq Prop) (y x)) b)
% Found eq_ref00:=(eq_ref0 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx)))))):(((eq (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx)))))) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx))))))
% Found (eq_ref0 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx)))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx)))))) b0)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx)))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx)))))) b0)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx)))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx)))))) b0)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx)))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx)))))) 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) b)
% Found ((eta_expansion0 Prop) b0) as proof of (((eq (a->Prop)) b0) b)
% Found (((eta_expansion a) Prop) b0) as proof of (((eq (a->Prop)) b0) b)
% Found (((eta_expansion a) Prop) b0) as proof of (((eq (a->Prop)) b0) b)
% Found (((eta_expansion a) Prop) b0) as proof of (((eq (a->Prop)) b0) b)
% Found eq_ref00:=(eq_ref0 (y x)):(((eq Prop) (y x)) (y x))
% Found (eq_ref0 (y x)) as proof of (((eq Prop) (y x)) b0)
% Found ((eq_ref Prop) (y x)) as proof of (((eq Prop) (y x)) b0)
% Found ((eq_ref Prop) (y x)) as proof of (((eq Prop) (y x)) b0)
% Found ((eq_ref Prop) (y x)) as proof of (((eq Prop) (y x)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found eq_ref00:=(eq_ref0 (y x)):(((eq Prop) (y x)) (y x))
% Found (eq_ref0 (y x)) as proof of (((eq Prop) (y x)) b0)
% Found ((eq_ref Prop) (y x)) as proof of (((eq Prop) (y x)) b0)
% Found ((eq_ref Prop) (y x)) as proof of (((eq Prop) (y x)) b0)
% Found ((eq_ref Prop) (y x)) as proof of (((eq Prop) (y x)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found x0:(P ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Instantiate: b:=((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))):Prop
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (y x)):(((eq Prop) (y x)) (y x))
% Found (eq_ref0 (y x)) as proof of (((eq Prop) (y x)) b)
% Found ((eq_ref Prop) (y x)) as proof of (((eq Prop) (y x)) b)
% Found ((eq_ref Prop) (y x)) as proof of (((eq Prop) (y x)) b)
% Found ((eq_ref Prop) (y x)) as proof of (((eq Prop) (y x)) b)
% Found x0:(P ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Instantiate: b:=((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))):Prop
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (y x)):(((eq Prop) (y x)) (y x))
% Found (eq_ref0 (y x)) as proof of (((eq Prop) (y x)) b)
% Found ((eq_ref Prop) (y x)) as proof of (((eq Prop) (y x)) b)
% Found ((eq_ref Prop) (y x)) as proof of (((eq Prop) (y x)) b)
% Found ((eq_ref Prop) (y x)) as proof of (((eq Prop) (y x)) b)
% Found eta_expansion000:=(eta_expansion00 y):(((eq (a->Prop)) y) (fun (x:a)=> (y x)))
% Found (eta_expansion00 y) as proof of (((eq (a->Prop)) y) b)
% Found ((eta_expansion0 Prop) y) as proof of (((eq (a->Prop)) y) b)
% Found (((eta_expansion a) Prop) y) as proof of (((eq (a->Prop)) y) b)
% Found (((eta_expansion a) Prop) y) as proof of (((eq (a->Prop)) y) b)
% Found eta_expansion000:=(eta_expansion00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq (a->Prop)) b) b)
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) b)
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) b)
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) b)
% Found eta_expansion0000:=(eta_expansion000 P):((P (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx))))))->(P (fun (x:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))))
% Found (eta_expansion000 P) as proof of (P0 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx))))))
% Found ((eta_expansion00 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx)))))) P) as proof of (P0 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx))))))
% Found (((eta_expansion0 Prop) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx)))))) P) as proof of (P0 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx))))))
% Found ((((eta_expansion a) Prop) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx)))))) P) as proof of (P0 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx))))))
% Found ((((eta_expansion a) Prop) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx)))))) P) as proof of (P0 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx))))))
% Found eq_ref000:=(eq_ref00 P):((P (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx))))))->(P (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx)))))))
% Found (eq_ref00 P) as proof of (P0 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx))))))
% Found ((eq_ref0 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx)))))) P) as proof of (P0 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx))))))
% Found (((eq_ref (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx)))))) P) as proof of (P0 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx))))))
% Found (((eq_ref (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx)))))) P) as proof of (P0 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx))))))
% Found eta_expansion0000:=(eta_expansion000 P):((P (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx))))))->(P (fun (x:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))))
% Found (eta_expansion000 P) as proof of (P0 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx))))))
% Found ((eta_expansion00 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx)))))) P) as proof of (P0 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx))))))
% Found (((eta_expansion0 Prop) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx)))))) P) as proof of (P0 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx))))))
% Found ((((eta_expansion a) Prop) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx)))))) P) as proof of (P0 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx))))))
% Found ((((eta_expansion a) Prop) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx)))))) P) as proof of (P0 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx))))))
% Found eq_ref000:=(eq_ref00 P):((P (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx))))))->(P (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx)))))))
% Found (eq_ref00 P) as proof of (P0 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx))))))
% Found ((eq_ref0 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx)))))) P) as proof of (P0 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx))))))
% Found (((eq_ref (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx)))))) P) as proof of (P0 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx))))))
% Found (((eq_ref (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx)))))) P) as proof of (P0 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx))))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 y):(((eq (a->Prop)) y) (fun (x:a)=> (y x)))
% Found (eta_expansion_dep00 y) as proof of (((eq (a->Prop)) y) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) y) as proof of (((eq (a->Prop)) y) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) y) as proof of (((eq (a->Prop)) y) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) y) as proof of (((eq (a->Prop)) y) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) y) as proof of (((eq (a->Prop)) y) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx))))))
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx))))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx))))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx))))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx))))))
% Found x2:(P y)
% Found (fun (x2:(P y))=> x2) as proof of (P y)
% Found (fun (x2:(P y))=> x2) as proof of (P0 y)
% Found eq_ref00:=(eq_ref0 (b x)):(((eq Prop) (b x)) (b x))
% Found (eq_ref0 (b x)) as proof of (((eq Prop) (b x)) b0)
% Found ((eq_ref Prop) (b x)) as proof of (((eq Prop) (b x)) b0)
% Found ((eq_ref Prop) (b x)) as proof of (((eq Prop) (b x)) b0)
% Found ((eq_ref Prop) (b x)) as proof of (((eq Prop) (b x)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found eq_ref00:=(eq_ref0 (b x)):(((eq Prop) (b x)) (b x))
% Found (eq_ref0 (b x)) as proof of (((eq Prop) (b x)) b0)
% Found ((eq_ref Prop) (b x)) as proof of (((eq Prop) (b x)) b0)
% Found ((eq_ref Prop) (b x)) as proof of (((eq Prop) (b x)) b0)
% Found ((eq_ref Prop) (b x)) as proof of (((eq Prop) (b x)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found eq_ref000:=(eq_ref00 P1):((P1 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))->(P1 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))))
% Found (eq_ref00 P1) as proof of (P2 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found ((eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) P1) as proof of (P2 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found (((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) P1) as proof of (P2 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found (((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) P1) as proof of (P2 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found eq_ref000:=(eq_ref00 P1):((P1 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))->(P1 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))))
% Found (eq_ref00 P1) as proof of (P2 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found ((eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) P1) as proof of (P2 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found (((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) P1) as proof of (P2 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found (((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) P1) as proof of (P2 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found eq_ref000:=(eq_ref00 P1):((P1 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))->(P1 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))))
% Found (eq_ref00 P1) as proof of (P2 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found ((eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) P1) as proof of (P2 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found (((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) P1) as proof of (P2 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found (((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) P1) as proof of (P2 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found eq_ref000:=(eq_ref00 P1):((P1 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))->(P1 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))))
% Found (eq_ref00 P1) as proof of (P2 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found ((eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) P1) as proof of (P2 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found (((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) P1) as proof of (P2 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found (((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) P1) as proof of (P2 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found eq_ref000:=(eq_ref00 P1):((P1 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))->(P1 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))))
% Found (eq_ref00 P1) as proof of (P2 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found ((eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) P1) as proof of (P2 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found (((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) P1) as proof of (P2 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found (((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) P1) as proof of (P2 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found eq_ref000:=(eq_ref00 P1):((P1 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))->(P1 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))))
% Found (eq_ref00 P1) as proof of (P2 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found ((eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) P1) as proof of (P2 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found (((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) P1) as proof of (P2 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found (((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) P1) as proof of (P2 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found x2:(P y)
% Found (fun (x2:(P y))=> x2) as proof of (P y)
% Found (fun (x2:(P y))=> x2) as proof of (P0 y)
% Found x2:(P y)
% Found (fun (x2:(P y))=> x2) as proof of (P y)
% Found (fun (x2:(P y))=> x2) as proof of (P0 y)
% Found eq_ref000:=(eq_ref00 P):((P ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))->(P ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))))
% Found (eq_ref00 P) as proof of (P0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found ((eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) P) as proof of (P0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found (((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) P) as proof of (P0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found (((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) P) as proof of (P0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found eq_ref000:=(eq_ref00 P):((P ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))->(P ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))))
% Found (eq_ref00 P) as proof of (P0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found ((eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) P) as proof of (P0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found (((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) P) as proof of (P0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found (((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) P) as proof of (P0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (y x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (y x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (y x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (y x))
% Found eq_ref00:=(eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))):(((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found (eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b)
% Found eq_ref00:=(eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))):(((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found (eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (y x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (y x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (y x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (y x))
% Found eq_ref00:=(eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))):(((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found (eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (y x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (y x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (y x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (y x))
% Found eq_ref00:=(eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))):(((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found (eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (y x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (y x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (y x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (y x))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (y x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (y x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (y x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (y x))
% Found eq_ref00:=(eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))):(((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found (eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (y x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (y x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (y x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (y x))
% Found eq_ref00:=(eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))):(((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found (eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (y x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (y x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (y x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (y x))
% Found eq_ref00:=(eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))):(((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found (eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b)
% Found eq_ref00:=(eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))):(((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found (eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (y x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (y x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (y x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (y x))
% Found eq_ref000:=(eq_ref00 P):((P ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))->(P ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))))
% Found (eq_ref00 P) as proof of (P0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found ((eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) P) as proof of (P0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found (((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) P) as proof of (P0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found (((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) P) as proof of (P0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found eq_ref000:=(eq_ref00 P):((P ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))->(P ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))))
% Found (eq_ref00 P) as proof of (P0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found ((eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) P) as proof of (P0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found (((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) P) as proof of (P0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found (((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) P) as proof of (P0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found eq_ref00:=(eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))):(((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found (eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (y x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (y x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (y x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (y x))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (y x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (y x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (y x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (y x))
% Found eq_ref00:=(eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))):(((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found (eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (y x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (y x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (y x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (y x))
% Found eq_ref00:=(eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))):(((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found (eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b)
% Found eq_ref00:=(eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))):(((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found (eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (y x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (y x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (y x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (y x))
% Found eq_ref00:=(eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))):(((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found (eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (y x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (y x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (y x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (y x))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (y x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (y x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (y x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (y x))
% Found eq_ref00:=(eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))):(((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found (eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b)
% Found eq_ref00:=(eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))):(((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found (eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (y x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (y x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (y x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (y x))
% Found eq_ref00:=(eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))):(((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found (eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (y x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (y x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (y x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (y x))
% Found eq_ref00:=(eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))):(((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found (eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (y x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (y x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (y x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (y x))
% Found eq_ref00:=(eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))):(((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found (eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (y x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (y x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (y x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (y x))
% Found eq_ref00:=(eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))):(((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found (eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (y x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (y x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (y x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (y x))
% Found eq_ref00:=(eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))):(((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found (eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (y x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (y x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (y x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (y x))
% Found eq_ref00:=(eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))):(((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found (eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (y x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (y x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (y x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (y x))
% Found eq_ref00:=(eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))):(((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found (eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (y x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (y x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (y x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (y x))
% Found x01:(P b)
% Found (fun (x01:(P b))=> x01) as proof of (P b)
% Found (fun (x01:(P b))=> x01) as proof of (P0 b)
% Found x01:(P b)
% Found (fun (x01:(P b))=> x01) as proof of (P b)
% Found (fun (x01:(P b))=> x01) as proof of (P0 b)
% Found x01:(P (b x))
% Found (fun (x01:(P (b x)))=> x01) as proof of (P (b x))
% Found (fun (x01:(P (b x)))=> x01) as proof of (P0 (b x))
% Found x01:(P (b x))
% Found (fun (x01:(P (b x)))=> x01) as proof of (P (b x))
% Found (fun (x01:(P (b x)))=> x01) as proof of (P0 (b x))
% Found eta_expansion000:=(eta_expansion00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq (a->Prop)) b) b0)
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) b0)
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) b0)
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) b0)
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) 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 (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx))))))
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) b0) as proof of (((eq (a->Prop)) b0) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx))))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b0) as proof of (((eq (a->Prop)) b0) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx))))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b0) as proof of (((eq (a->Prop)) b0) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx))))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b0) as proof of (((eq (a->Prop)) b0) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx))))))
% Found eq_ref00:=(eq_ref0 (y x)):(((eq Prop) (y x)) (y x))
% Found (eq_ref0 (y x)) as proof of (((eq Prop) (y x)) b)
% Found ((eq_ref Prop) (y x)) as proof of (((eq Prop) (y x)) b)
% Found ((eq_ref Prop) (y x)) as proof of (((eq Prop) (y x)) b)
% Found ((eq_ref Prop) (y x)) as proof of (((eq Prop) (y x)) b)
% Found x0:(P0 b)
% Instantiate: b:=((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))):Prop
% Found (fun (x0:(P0 b))=> x0) as proof of (P0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found (fun (P0:(Prop->Prop)) (x0:(P0 b))=> x0) as proof of ((P0 b)->(P0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))))
% Found (fun (P0:(Prop->Prop)) (x0:(P0 b))=> x0) as proof of (P b)
% Found eq_ref00:=(eq_ref0 (y x)):(((eq Prop) (y x)) (y x))
% Found (eq_ref0 (y x)) as proof of (((eq Prop) (y x)) b)
% Found ((eq_ref Prop) (y x)) as proof of (((eq Prop) (y x)) b)
% Found ((eq_ref Prop) (y x)) as proof of (((eq Prop) (y x)) b)
% Found ((eq_ref Prop) (y x)) as proof of (((eq Prop) (y x)) b)
% Found x0:(P0 b)
% Instantiate: b:=((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))):Prop
% Found (fun (x0:(P0 b))=> x0) as proof of (P0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found (fun (P0:(Prop->Prop)) (x0:(P0 b))=> x0) as proof of ((P0 b)->(P0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))))
% Found (fun (P0:(Prop->Prop)) (x0:(P0 b))=> x0) as proof of (P b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) y)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) y)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) y)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) y)
% Found eta_expansion000:=(eta_expansion00 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx)))))):(((eq (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx)))))) (fun (x:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))))
% Found (eta_expansion00 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx)))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx)))))) b)
% Found ((eta_expansion0 Prop) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx)))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx)))))) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx)))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx)))))) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx)))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx)))))) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx)))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx)))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) y)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) y)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) y)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) y)
% Found eta_expansion000:=(eta_expansion00 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx)))))):(((eq (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx)))))) (fun (x:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))))
% Found (eta_expansion00 (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx)))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx)))))) b)
% Found ((eta_expansion0 Prop) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx)))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx)))))) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx)))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx)))))) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx)))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx)))))) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx)))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S Xx)))))) b)
% Found x0:(P (y x))
% Instantiate: b:=(y x):Prop
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))):(((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found (eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b)
% Found x0:(P (y x))
% Instantiate: b:=(y x):Prop
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))):(((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found (eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found eq_ref00:=(eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))):(((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found (eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b0)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b0)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b0)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found eq_ref00:=(eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))):(((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found (eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b0)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b0)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b0)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b0)
% Found eq_ref000:=(eq_ref00 P):((P ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))->(P ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))))
% Found (eq_ref00 P) as proof of (P0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found ((eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) P) as proof of (P0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found (((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) P) as proof of (P0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found (((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) P) as proof of (P0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found eq_ref000:=(eq_ref00 P):((P ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))->(P ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))))
% Found (eq_ref00 P) as proof of (P0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found ((eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) P) as proof of (P0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found (((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) P) as proof of (P0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found (((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) P) as proof of (P0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found eq_ref00:=(eq_ref0 (y x)):(((eq Prop) (y x)) (y x))
% Found (eq_ref0 (y x)) as proof of (((eq Prop) (y x)) b)
% Found ((eq_ref Prop) (y x)) as proof of (((eq Prop) (y x)) b)
% Found ((eq_ref Prop) (y x)) as proof of (((eq Prop) (y x)) b)
% Found ((eq_ref Prop) (y x)) as proof of (((eq Prop) (y x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found eq_ref00:=(eq_ref0 (y x)):(((eq Prop) (y x)) (y x))
% Found (eq_ref0 (y x)) as proof of (((eq Prop) (y x)) b)
% Found ((eq_ref Prop) (y x)) as proof of (((eq Prop) (y x)) b)
% Found ((eq_ref Prop) (y x)) as proof of (((eq Prop) (y x)) b)
% Found ((eq_ref Prop) (y x)) as proof of (((eq Prop) (y x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found eq_ref00:=(eq_ref0 (y x)):(((eq Prop) (y x)) (y x))
% Found (eq_ref0 (y x)) as proof of (((eq Prop) (y x)) b)
% Found ((eq_ref Prop) (y x)) as proof of (((eq Prop) (y x)) b)
% Found ((eq_ref Prop) (y x)) as proof of (((eq Prop) (y x)) b)
% Found ((eq_ref Prop) (y x)) as proof of (((eq Prop) (y x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found eq_ref00:=(eq_ref0 (y x)):(((eq Prop) (y x)) (y x))
% Found (eq_ref0 (y x)) as proof of (((eq Prop) (y x)) b)
% Found ((eq_ref Prop) (y x)) as proof of (((eq Prop) (y x)) b)
% Found ((eq_ref Prop) (y x)) as proof of (((eq Prop) (y x)) b)
% Found ((eq_ref Prop) (y x)) as proof of (((eq Prop) (y x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) x)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) x)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) x)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) x)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) x)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) x)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) x)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) x)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) x)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) x)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) x)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) x)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) x)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) x)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) x)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) x)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) x)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) x)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) x)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) x)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) x)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) x)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) x)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) x)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) x)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) x)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) x)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) x)
% Found eq_ref00:=(eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))):(((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found (eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b0)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b0)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b0)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) x)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) x)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) x)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) x)
% Found eq_ref00:=(eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))):(((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found (eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b0)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b0)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b0)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found x01:(P (y x))
% Found (fun (x01:(P (y x)))=> x01) as proof of (P (y x))
% Found (fun (x01:(P (y x)))=> x01) as proof of (P0 (y x))
% Found x01:(P (y x))
% Found (fun (x01:(P (y x)))=> x01) as proof of (P (y x))
% Found (fun (x01:(P (y x)))=> x01) as proof of (P0 (y x))
% Found x01:(P (y x))
% Found (fun (x01:(P (y x)))=> x01) as proof of (P (y x))
% Found (fun (x01:(P (y x)))=> x01) as proof of (P0 (y x))
% Found x01:(P (y x))
% Found (fun (x01:(P (y x)))=> x01) as proof of (P (y x))
% Found (fun (x01:(P (y x)))=> x01) as proof of (P0 (y x))
% Found eq_ref000:=(eq_ref00 P):((P ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))->(P ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))))
% Found (eq_ref00 P) as proof of (P0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found ((eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) P) as proof of (P0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found (((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) P) as proof of (P0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found (((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) P) as proof of (P0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found eq_ref000:=(eq_ref00 P):((P ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))->(P ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))))
% Found (eq_ref00 P) as proof of (P0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found ((eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) P) as proof of (P0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found (((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) P) as proof of (P0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found (((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) P) as proof of (P0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found eq_ref000:=(eq_ref00 P):((P ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))->(P ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))))
% Found (eq_ref00 P) as proof of (P0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found ((eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) P) as proof of (P0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found (((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) P) as proof of (P0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found (((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) P) as proof of (P0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found eq_ref000:=(eq_ref00 P):((P ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))->(P ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))))
% Found (eq_ref00 P) as proof of (P0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found ((eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) P) as proof of (P0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found (((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) P) as proof of (P0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found (((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) P) as proof of (P0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found x2:(P b)
% Found (fun (x2:(P b))=> x2) as proof of (P b)
% Found (fun (x2:(P b))=> x2) as proof of (P0 b)
% Found x2:(P b)
% Found (fun (x2:(P b))=> x2) as proof of (P b)
% Found (fun (x2:(P b))=> x2) as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))):(((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found (eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b0)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b0)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b0)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (b x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x))
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (b x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x))
% Found eq_ref00:=(eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))):(((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found (eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b0)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b0)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b0)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b0)
% Found eq_ref00:=(eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))):(((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found (eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b0)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b0)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b0)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (b x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x))
% Found eq_ref00:=(eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))):(((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found (eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b0)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b0)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b0)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (b x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x))
% Found eq_ref00:=(eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))):(((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found (eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b0)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b0)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b0)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (b x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x))
% Found eq_ref00:=(eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))):(((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found (eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b0)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b0)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b0)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (b x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found eq_ref00:=(eq_ref0 (y x)):(((eq Prop) (y x)) (y x))
% Found (eq_ref0 (y x)) as proof of (((eq Prop) (y x)) b)
% Found ((eq_ref Prop) (y x)) as proof of (((eq Prop) (y x)) b)
% Found ((eq_ref Prop) (y x)) as proof of (((eq Prop) (y x)) b)
% Found ((eq_ref Prop) (y x)) as proof of (((eq Prop) (y x)) b)
% Found eq_ref00:=(eq_ref0 (y x)):(((eq Prop) (y x)) (y x))
% Found (eq_ref0 (y x)) as proof of (((eq Prop) (y x)) b)
% Found ((eq_ref Prop) (y x)) as proof of (((eq Prop) (y x)) b)
% Found ((eq_ref Prop) (y x)) as proof of (((eq Prop) (y x)) b)
% Found ((eq_ref Prop) (y x)) as proof of (((eq Prop) (y x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found eq_ref00:=(eq_ref0 (y x)):(((eq Prop) (y x)) (y x))
% Found (eq_ref0 (y x)) as proof of (((eq Prop) (y x)) b)
% Found ((eq_ref Prop) (y x)) as proof of (((eq Prop) (y x)) b)
% Found ((eq_ref Prop) (y x)) as proof of (((eq Prop) (y x)) b)
% Found ((eq_ref Prop) (y x)) as proof of (((eq Prop) (y x)) b)
% Found eq_ref00:=(eq_ref0 (y x)):(((eq Prop) (y x)) (y x))
% Found (eq_ref0 (y x)) as proof of (((eq Prop) (y x)) b)
% Found ((eq_ref Prop) (y x)) as proof of (((eq Prop) (y x)) b)
% Found ((eq_ref Prop) (y x)) as proof of (((eq Prop) (y x)) b)
% Found ((eq_ref Prop) (y x)) as proof of (((eq Prop) (y x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found eq_ref00:=(eq_ref0 (y x)):(((eq Prop) (y x)) (y x))
% Found (eq_ref0 (y x)) as proof of (((eq Prop) (y x)) b)
% Found ((eq_ref Prop) (y x)) as proof of (((eq Prop) (y x)) b)
% Found ((eq_ref Prop) (y x)) as proof of (((eq Prop) (y x)) b)
% Found ((eq_ref Prop) (y x)) as proof of (((eq Prop) (y x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found eq_ref00:=(eq_ref0 (y x)):(((eq Prop) (y x)) (y x))
% Found (eq_ref0 (y x)) as proof of (((eq Prop) (y x)) b)
% Found ((eq_ref Prop) (y x)) as proof of (((eq Prop) (y x)) b)
% Found ((eq_ref Prop) (y x)) as proof of (((eq Prop) (y x)) b)
% Found ((eq_ref Prop) (y x)) as proof of (((eq Prop) (y x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found eq_ref00:=(eq_ref0 (y x)):(((eq Prop) (y x)) (y x))
% Found (eq_ref0 (y x)) as proof of (((eq Prop) (y x)) b)
% Found ((eq_ref Prop) (y x)) as proof of (((eq Prop) (y x)) b)
% Found ((eq_ref Prop) (y x)) as proof of (((eq Prop) (y x)) b)
% Found ((eq_ref Prop) (y x)) as proof of (((eq Prop) (y x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found eq_ref00:=(eq_ref0 (y x)):(((eq Prop) (y x)) (y x))
% Found (eq_ref0 (y x)) as proof of (((eq Prop) (y x)) b)
% Found ((eq_ref Prop) (y x)) as proof of (((eq Prop) (y x)) b)
% Found ((eq_ref Prop) (y x)) as proof of (((eq Prop) (y x)) b)
% Found ((eq_ref Prop) (y x)) as proof of (((eq Prop) (y x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found eq_ref00:=(eq_ref0 (y x)):(((eq Prop) (y x)) (y x))
% Found (eq_ref0 (y x)) as proof of (((eq Prop) (y x)) b)
% Found ((eq_ref Prop) (y x)) as proof of (((eq Prop) (y x)) b)
% Found ((eq_ref Prop) (y x)) as proof of (((eq Prop) (y x)) b)
% Found ((eq_ref Prop) (y x)) as proof of (((eq Prop) (y x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found eq_ref00:=(eq_ref0 (y x)):(((eq Prop) (y x)) (y x))
% Found (eq_ref0 (y x)) as proof of (((eq Prop) (y x)) b)
% Found ((eq_ref Prop) (y x)) as proof of (((eq Prop) (y x)) b)
% Found ((eq_ref Prop) (y x)) as proof of (((eq Prop) (y x)) b)
% Found ((eq_ref Prop) (y x)) as proof of (((eq Prop) (y x)) b)
% Found eq_ref00:=(eq_ref0 (y x)):(((eq Prop) (y x)) (y x))
% Found (eq_ref0 (y x)) as proof of (((eq Prop) (y x)) b)
% Found ((eq_ref Prop) (y x)) as proof of (((eq Prop) (y x)) b)
% Found ((eq_ref Prop) (y x)) as proof of (((eq Prop) (y x)) b)
% Found ((eq_ref Prop) (y x)) as proof of (((eq Prop) (y x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found eq_ref00:=(eq_ref0 (y x)):(((eq Prop) (y x)) (y x))
% Found (eq_ref0 (y x)) as proof of (((eq Prop) (y x)) b)
% Found ((eq_ref Prop) (y x)) as proof of (((eq Prop) (y x)) b)
% Found ((eq_ref Prop) (y x)) as proof of (((eq Prop) (y x)) b)
% Found ((eq_ref Prop) (y x)) as proof of (((eq Prop) (y x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found x01:(P b)
% Found (fun (x01:(P b))=> x01) as proof of (P b)
% Found (fun (x01:(P b))=> x01) as proof of (P0 b)
% Found x01:(P b)
% Found (fun (x01:(P b))=> x01) as proof of (P b)
% Found (fun (x01:(P b))=> x01) as proof of (P0 b)
% Found x02:(P (y x))
% Found (fun (x02:(P (y x)))=> x02) as proof of (P (y x))
% Found (fun (x02:(P (y x)))=> x02) as proof of (P0 (y x))
% Found eq_ref00:=(eq_ref0 (y x)):(((eq Prop) (y x)) (y x))
% Found (eq_ref0 (y x)) as proof of (((eq Prop) (y x)) b)
% Found ((eq_ref Prop) (y x)) as proof of (((eq Prop) (y x)) b)
% Found ((eq_ref Prop) (y x)) as proof of (((eq Prop) (y x)) b)
% Found ((eq_ref Prop) (y x)) as proof of (((eq Prop) (y x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found x02:(P (y x))
% Found (fun (x02:(P (y x)))=> x02) as proof of (P (y x))
% Found (fun (x02:(P (y x)))=> x02) as proof of (P0 (y x))
% Found eq_ref00:=(eq_ref0 (y x)):(((eq Prop) (y x)) (y x))
% Found (eq_ref0 (y x)) as proof of (((eq Prop) (y x)) b)
% Found ((eq_ref Prop) (y x)) as proof of (((eq Prop) (y x)) b)
% Found ((eq_ref Prop) (y x)) as proof of (((eq Prop) (y x)) b)
% Found ((eq_ref Prop) (y x)) as proof of (((eq Prop) (y x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found eq_ref00:=(eq_ref0 a0):(((eq ((a->Prop)->Prop)) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq ((a->Prop)->Prop)) a0) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))
% Found ((eq_ref ((a->Prop)->Prop)) a0) as proof of (((eq ((a->Prop)->Prop)) a0) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))
% Found ((eq_ref ((a->Prop)->Prop)) a0) as proof of (((eq ((a->Prop)->Prop)) a0) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))
% Found ((eq_ref ((a->Prop)->Prop)) a0) as proof of (((eq ((a->Prop)->Prop)) a0) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 a0):(((eq ((a->Prop)->Prop)) a0) (fun (x:(a->Prop))=> (a0 x)))
% Found (eta_expansion_dep00 a0) as proof of (((eq ((a->Prop)->Prop)) a0) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))
% Found ((eta_expansion_dep0 (fun (x2:(a->Prop))=> Prop)) a0) as proof of (((eq ((a->Prop)->Prop)) a0) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))
% Found (((eta_expansion_dep (a->Prop)) (fun (x2:(a->Prop))=> Prop)) a0) as proof of (((eq ((a->Prop)->Prop)) a0) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))
% Found (((eta_expansion_dep (a->Prop)) (fun (x2:(a->Prop))=> Prop)) a0) as proof of (((eq ((a->Prop)->Prop)) a0) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))
% Found (((eta_expansion_dep (a->Prop)) (fun (x2:(a->Prop))=> Prop)) a0) as proof of (((eq ((a->Prop)->Prop)) a0) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 a0):(((eq ((a->Prop)->Prop)) a0) (fun (x:(a->Prop))=> (a0 x)))
% Found (eta_expansion_dep00 a0) as proof of (((eq ((a->Prop)->Prop)) a0) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))
% Found ((eta_expansion_dep0 (fun (x2:(a->Prop))=> Prop)) a0) as proof of (((eq ((a->Prop)->Prop)) a0) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))
% Found (((eta_expansion_dep (a->Prop)) (fun (x2:(a->Prop))=> Prop)) a0) as proof of (((eq ((a->Prop)->Prop)) a0) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))
% Found (((eta_expansion_dep (a->Prop)) (fun (x2:(a->Prop))=> Prop)) a0) as proof of (((eq ((a->Prop)->Prop)) a0) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))
% Found (((eta_expansion_dep (a->Prop)) (fun (x2:(a->Prop))=> Prop)) a0) as proof of (((eq ((a->Prop)->Prop)) a0) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))
% Found eta_expansion000:=(eta_expansion00 a0):(((eq ((a->Prop)->Prop)) a0) (fun (x:(a->Prop))=> (a0 x)))
% Found (eta_expansion00 a0) as proof of (((eq ((a->Prop)->Prop)) a0) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))
% Found ((eta_expansion0 Prop) a0) as proof of (((eq ((a->Prop)->Prop)) a0) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))
% Found (((eta_expansion (a->Prop)) Prop) a0) as proof of (((eq ((a->Prop)->Prop)) a0) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))
% Found (((eta_expansion (a->Prop)) Prop) a0) as proof of (((eq ((a->Prop)->Prop)) a0) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))
% Found (((eta_expansion (a->Prop)) Prop) a0) as proof of (((eq ((a->Prop)->Prop)) a0) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))
% Found eq_ref000:=(eq_ref00 P):((P ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))->(P ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))))
% Found (eq_ref00 P) as proof of (P0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found ((eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) P) as proof of (P0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found (((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) P) as proof of (P0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found (((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) P) as proof of (P0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found eq_ref000:=(eq_ref00 P):((P ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))->(P ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))))
% Found (eq_ref00 P) as proof of (P0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found ((eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) P) as proof of (P0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found (((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) P) as proof of (P0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found (((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) P) as proof of (P0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found eq_ref000:=(eq_ref00 P):((P ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))->(P ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))))
% Found (eq_ref00 P) as proof of (P0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found ((eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) P) as proof of (P0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found (((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) P) as proof of (P0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found (((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) P) as proof of (P0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found eq_ref000:=(eq_ref00 P):((P ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))->(P ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))))
% Found (eq_ref00 P) as proof of (P0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found ((eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) P) as proof of (P0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found (((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) P) as proof of (P0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found (((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) P) as proof of (P0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found eq_ref000:=(eq_ref00 P):((P ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))->(P ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))))
% Found (eq_ref00 P) as proof of (P0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found ((eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) P) as proof of (P0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found (((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) P) as proof of (P0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found (((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) P) as proof of (P0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found eq_ref000:=(eq_ref00 P):((P ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))->(P ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))))
% Found (eq_ref00 P) as proof of (P0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found ((eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) P) as proof of (P0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found (((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) P) as proof of (P0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found (((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) P) as proof of (P0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (y x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (y x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (y x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (y x))
% Found eq_ref00:=(eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))):(((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found (eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (y x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (y x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (y x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (y x))
% Found eq_ref00:=(eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))):(((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found (eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (y x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (y x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (y x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (y x))
% Found eq_ref00:=(eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))):(((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found (eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (y x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (y x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (y x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (y x))
% Found eq_ref00:=(eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))):(((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found (eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (y x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (y x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (y x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (y x))
% Found eq_ref00:=(eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))):(((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found (eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b)
% Found eq_ref00:=(eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))):(((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found (eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (y x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (y x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (y x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (y x))
% Found eq_ref00:=(eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))):(((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found (eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (y x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (y x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (y x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (y x))
% Found eq_ref00:=(eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))):(((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found (eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (y x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (y x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (y x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (y x))
% Found eq_ref00:=(eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))):(((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found (eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (y x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (y x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (y x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (y x))
% Found eq_ref00:=(eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))):(((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x)))))
% Found (eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (y x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (y x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (y x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (y x))
% Found eq_ref00:=(eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))):(((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> ((and (y Xx0)) (((eq (a->Prop)) S) (fun (Xy:a)=> (((eq a) Xx0) Xy))))))) (S x))))) ((ex (a->Prop)) (fun (S
% EOF
%------------------------------------------------------------------------------