TSTP Solution File: SEV227^5 by cocATP---0.2.0
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- Process Solution
%------------------------------------------------------------------------------
% File : cocATP---0.2.0
% Problem : SEV227^5 : TPTP v6.1.0. Released v4.0.0.
% Transfm : none
% Format : tptp:raw
% Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% Computer : n186.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 13:33:54 EDT 2014
% Result : Timeout 300.01s
% Output : None
% Verified :
% SZS Type : None (Parsing solution fails)
% Syntax : Number of formulae : 0
% Comments :
%------------------------------------------------------------------------------
%----NO SOLUTION OUTPUT BY SYSTEM
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% % Problem : SEV227^5 : TPTP v6.1.0. Released v4.0.0.
% % Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n186.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 08:33:21 CDT 2014
% % CPUTime : 300.01
% Python 2.7.5
% Using paths ['/home/cristobal/cocATP/CASC/TPTP/', '/export/starexec/sandbox/benchmark/', '/export/starexec/sandbox/benchmark/']
% FOF formula (<kernel.Constant object at 0x1e678c0>, <kernel.Type object at 0x1edb3b0>) of role type named a_type
% Using role type
% Declaring a:Type
% FOF formula (<kernel.Constant object at 0x1e675f0>, <kernel.DependentProduct object at 0x1edb128>) of role type named y
% Using role type
% Declaring y:(a->Prop)
% FOF formula (<kernel.Constant object at 0x1e67320>, <kernel.DependentProduct object at 0x1edb170>) of role type named x
% Using role type
% Declaring x:(a->Prop)
% FOF formula (((eq (a->Prop)) (fun (Xz:a)=> ((or (x Xz)) (y Xz)))) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))) of role conjecture named cX5200_pme
% Conjecture to prove = (((eq (a->Prop)) (fun (Xz:a)=> ((or (x Xz)) (y Xz)))) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))):Prop
% Parameter a_DUMMY:a.
% We need to prove ['(((eq (a->Prop)) (fun (Xz:a)=> ((or (x Xz)) (y Xz)))) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0))))))']
% Parameter a:Type.
% Parameter y:(a->Prop).
% Parameter x:(a->Prop).
% Trying to prove (((eq (a->Prop)) (fun (Xz:a)=> ((or (x Xz)) (y Xz)))) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0))))))
% Found eta_expansion000:=(eta_expansion00 (fun (Xz:a)=> ((or (x Xz)) (y Xz)))):(((eq (a->Prop)) (fun (Xz:a)=> ((or (x Xz)) (y Xz)))) (fun (x0:a)=> ((or (x x0)) (y x0))))
% Found (eta_expansion00 (fun (Xz:a)=> ((or (x Xz)) (y Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (x Xz)) (y Xz)))) b)
% Found ((eta_expansion0 Prop) (fun (Xz:a)=> ((or (x Xz)) (y Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (x Xz)) (y Xz)))) b)
% Found (((eta_expansion a) Prop) (fun (Xz:a)=> ((or (x Xz)) (y Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (x Xz)) (y Xz)))) b)
% Found (((eta_expansion a) Prop) (fun (Xz:a)=> ((or (x Xz)) (y Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (x Xz)) (y Xz)))) b)
% Found (((eta_expansion a) Prop) (fun (Xz:a)=> ((or (x Xz)) (y Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (x Xz)) (y Xz)))) 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 (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0))))))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0))))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0))))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0))))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0))))))
% Found x01:(P (fun (Xz:a)=> ((or (x Xz)) (y Xz))))
% Found (fun (x01:(P (fun (Xz:a)=> ((or (x Xz)) (y Xz)))))=> x01) as proof of (P (fun (Xz:a)=> ((or (x Xz)) (y Xz))))
% Found (fun (x01:(P (fun (Xz:a)=> ((or (x Xz)) (y Xz)))))=> x01) as proof of (P0 (fun (Xz:a)=> ((or (x Xz)) (y Xz))))
% Found x01:(P (fun (Xz:a)=> ((or (x Xz)) (y Xz))))
% Found (fun (x01:(P (fun (Xz:a)=> ((or (x Xz)) (y Xz)))))=> x01) as proof of (P (fun (Xz:a)=> ((or (x Xz)) (y Xz))))
% Found (fun (x01:(P (fun (Xz:a)=> ((or (x Xz)) (y Xz)))))=> x01) as proof of (P0 (fun (Xz:a)=> ((or (x Xz)) (y Xz))))
% Found x01:(P (fun (Xz:a)=> ((or (x Xz)) (y Xz))))
% Found (fun (x01:(P (fun (Xz:a)=> ((or (x Xz)) (y Xz)))))=> x01) as proof of (P (fun (Xz:a)=> ((or (x Xz)) (y Xz))))
% Found (fun (x01:(P (fun (Xz:a)=> ((or (x Xz)) (y Xz)))))=> x01) as proof of (P0 (fun (Xz:a)=> ((or (x Xz)) (y Xz))))
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or (x Xz)) (y Xz))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or (x Xz)) (y Xz))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or (x Xz)) (y Xz))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or (x Xz)) (y Xz))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))):(((eq (a->Prop)) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))) (fun (x0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))))
% Found (eta_expansion_dep00 (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))) as proof of (((eq (a->Prop)) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))) as proof of (((eq (a->Prop)) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))) as proof of (((eq (a->Prop)) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))) as proof of (((eq (a->Prop)) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))) as proof of (((eq (a->Prop)) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))) b)
% Found eq_ref000:=(eq_ref00 P):((P (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0))))))->(P (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))))
% Found (eq_ref00 P) as proof of (P0 (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0))))))
% Found ((eq_ref0 (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))) P) as proof of (P0 (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0))))))
% Found (((eq_ref (a->Prop)) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))) P) as proof of (P0 (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0))))))
% Found (((eq_ref (a->Prop)) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))) P) as proof of (P0 (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0))))))
% Found eq_ref000:=(eq_ref00 P):((P (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0))))))->(P (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))))
% Found (eq_ref00 P) as proof of (P0 (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0))))))
% Found ((eq_ref0 (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))) P) as proof of (P0 (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0))))))
% Found (((eq_ref (a->Prop)) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))) P) as proof of (P0 (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0))))))
% Found (((eq_ref (a->Prop)) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))) P) as proof of (P0 (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0))))))
% Found eq_ref00:=(eq_ref0 ((or (x x0)) (y x0))):(((eq Prop) ((or (x x0)) (y x0))) ((or (x x0)) (y x0)))
% Found (eq_ref0 ((or (x x0)) (y x0))) as proof of (((eq Prop) ((or (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((or (x x0)) (y x0))) as proof of (((eq Prop) ((or (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((or (x x0)) (y x0))) as proof of (((eq Prop) ((or (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((or (x x0)) (y x0))) as proof of (((eq Prop) ((or (x x0)) (y x0))) 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 ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found eq_ref00:=(eq_ref0 ((or (x x0)) (y x0))):(((eq Prop) ((or (x x0)) (y x0))) ((or (x x0)) (y x0)))
% Found (eq_ref0 ((or (x x0)) (y x0))) as proof of (((eq Prop) ((or (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((or (x x0)) (y x0))) as proof of (((eq Prop) ((or (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((or (x x0)) (y x0))) as proof of (((eq Prop) ((or (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((or (x x0)) (y x0))) as proof of (((eq Prop) ((or (x x0)) (y x0))) 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 ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found x11:(P ((or (x x0)) (y x0)))
% Found (fun (x11:(P ((or (x x0)) (y x0))))=> x11) as proof of (P ((or (x x0)) (y x0)))
% Found (fun (x11:(P ((or (x x0)) (y x0))))=> x11) as proof of (P0 ((or (x x0)) (y x0)))
% Found x11:(P ((or (x x0)) (y x0)))
% Found (fun (x11:(P ((or (x x0)) (y x0))))=> x11) as proof of (P ((or (x x0)) (y x0)))
% Found (fun (x11:(P ((or (x x0)) (y x0))))=> x11) as proof of (P0 ((or (x x0)) (y x0)))
% Found eq_ref000:=(eq_ref00 P):((P ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))->(P ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))))
% Found (eq_ref00 P) as proof of (P0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found ((eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) P) as proof of (P0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found (((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) P) as proof of (P0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found (((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) P) as proof of (P0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found eq_ref000:=(eq_ref00 P):((P ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))->(P ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))))
% Found (eq_ref00 P) as proof of (P0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found ((eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) P) as proof of (P0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found (((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) P) as proof of (P0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found (((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) P) as proof of (P0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (x x0)) (y x0)))
% Found eq_ref00:=(eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))):(((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found (eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b)
% Found eq_ref00:=(eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))):(((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found (eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (x x0)) (y x0)))
% Found eq_ref00:=(eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))):(((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found (eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (x x0)) (y x0)))
% Found eq_ref00:=(eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))):(((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found (eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (x x0)) (y x0)))
% Found x0:(P (fun (Xz:a)=> ((or (x Xz)) (y Xz))))
% Instantiate: b:=(fun (Xz:a)=> ((or (x Xz)) (y Xz))):(a->Prop)
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))):(((eq (a->Prop)) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0))))))
% Found (eq_ref0 (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))) as proof of (((eq (a->Prop)) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))) as proof of (((eq (a->Prop)) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))) as proof of (((eq (a->Prop)) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))) as proof of (((eq (a->Prop)) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))) b)
% Found x0:(P (fun (Xz:a)=> ((or (x Xz)) (y Xz))))
% Instantiate: f:=(fun (Xz:a)=> ((or (x Xz)) (y Xz))):(a->Prop)
% Found x0 as proof of (P0 f)
% Found eq_ref00:=(eq_ref0 (f x00)):(((eq Prop) (f x00)) (f x00))
% Found (eq_ref0 (f x00)) as proof of (((eq Prop) (f x00)) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x00)))))
% Found ((eq_ref Prop) (f x00)) as proof of (((eq Prop) (f x00)) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x00)))))
% Found ((eq_ref Prop) (f x00)) as proof of (((eq Prop) (f x00)) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x00)))))
% Found (fun (x00:a)=> ((eq_ref Prop) (f x00))) as proof of (((eq Prop) (f x00)) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x00)))))
% Found (fun (x00:a)=> ((eq_ref Prop) (f x00))) as proof of (forall (x0:a), (((eq Prop) (f x0)) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))))
% Found x0:(P (fun (Xz:a)=> ((or (x Xz)) (y Xz))))
% Instantiate: f:=(fun (Xz:a)=> ((or (x Xz)) (y Xz))):(a->Prop)
% Found x0 as proof of (P0 f)
% Found eq_ref00:=(eq_ref0 (f x00)):(((eq Prop) (f x00)) (f x00))
% Found (eq_ref0 (f x00)) as proof of (((eq Prop) (f x00)) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x00)))))
% Found ((eq_ref Prop) (f x00)) as proof of (((eq Prop) (f x00)) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x00)))))
% Found ((eq_ref Prop) (f x00)) as proof of (((eq Prop) (f x00)) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x00)))))
% Found (fun (x00:a)=> ((eq_ref Prop) (f x00))) as proof of (((eq Prop) (f x00)) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x00)))))
% Found (fun (x00:a)=> ((eq_ref Prop) (f x00))) as proof of (forall (x0:a), (((eq Prop) (f x0)) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))))
% 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 ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found eq_ref00:=(eq_ref0 ((or (x x0)) (y x0))):(((eq Prop) ((or (x x0)) (y x0))) ((or (x x0)) (y x0)))
% Found (eq_ref0 ((or (x x0)) (y x0))) as proof of (((eq Prop) ((or (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((or (x x0)) (y x0))) as proof of (((eq Prop) ((or (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((or (x x0)) (y x0))) as proof of (((eq Prop) ((or (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((or (x x0)) (y x0))) as proof of (((eq Prop) ((or (x x0)) (y x0))) 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 ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found eq_ref00:=(eq_ref0 ((or (x x0)) (y x0))):(((eq Prop) ((or (x x0)) (y x0))) ((or (x x0)) (y x0)))
% Found (eq_ref0 ((or (x x0)) (y x0))) as proof of (((eq Prop) ((or (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((or (x x0)) (y x0))) as proof of (((eq Prop) ((or (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((or (x x0)) (y x0))) as proof of (((eq Prop) ((or (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((or (x x0)) (y x0))) as proof of (((eq Prop) ((or (x x0)) (y x0))) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xz:a)=> ((or (x Xz)) (y Xz)))):(((eq (a->Prop)) (fun (Xz:a)=> ((or (x Xz)) (y Xz)))) (fun (x0:a)=> ((or (x x0)) (y x0))))
% Found (eta_expansion_dep00 (fun (Xz:a)=> ((or (x Xz)) (y Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (x Xz)) (y Xz)))) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((or (x Xz)) (y Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (x Xz)) (y Xz)))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((or (x Xz)) (y Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (x Xz)) (y Xz)))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((or (x Xz)) (y Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (x Xz)) (y Xz)))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((or (x Xz)) (y Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (x Xz)) (y Xz)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0))))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0))))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0))))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0))))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))):(((eq (a->Prop)) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))) (fun (x0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))))
% Found (eta_expansion_dep00 (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))) as proof of (((eq (a->Prop)) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))) as proof of (((eq (a->Prop)) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))) as proof of (((eq (a->Prop)) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))) as proof of (((eq (a->Prop)) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))) as proof of (((eq (a->Prop)) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))) b)
% Found x0:(P0 b)
% Instantiate: b:=(fun (Xz:a)=> ((or (x Xz)) (y Xz))):(a->Prop)
% Found (fun (x0:(P0 b))=> x0) as proof of (P0 (fun (Xz:a)=> ((or (x Xz)) (y Xz))))
% Found (fun (P0:((a->Prop)->Prop)) (x0:(P0 b))=> x0) as proof of ((P0 b)->(P0 (fun (Xz:a)=> ((or (x Xz)) (y Xz)))))
% Found (fun (P0:((a->Prop)->Prop)) (x0:(P0 b))=> x0) as proof of (P b)
% Found x0:(P (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0))))))
% Instantiate: b:=(fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0))))):(a->Prop)
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (fun (Xz:a)=> ((or (x Xz)) (y Xz)))):(((eq (a->Prop)) (fun (Xz:a)=> ((or (x Xz)) (y Xz)))) (fun (Xz:a)=> ((or (x Xz)) (y Xz))))
% Found (eq_ref0 (fun (Xz:a)=> ((or (x Xz)) (y Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (x Xz)) (y Xz)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((or (x Xz)) (y Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (x Xz)) (y Xz)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((or (x Xz)) (y Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (x Xz)) (y Xz)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((or (x Xz)) (y Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (x Xz)) (y Xz)))) b)
% Found x02:(P (fun (Xz:a)=> ((or (x Xz)) (y Xz))))
% Found (fun (x02:(P (fun (Xz:a)=> ((or (x Xz)) (y Xz)))))=> x02) as proof of (P (fun (Xz:a)=> ((or (x Xz)) (y Xz))))
% Found (fun (x02:(P (fun (Xz:a)=> ((or (x Xz)) (y Xz)))))=> x02) as proof of (P0 (fun (Xz:a)=> ((or (x Xz)) (y Xz))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xz:a)=> ((or (x Xz)) (y Xz)))):(((eq (a->Prop)) (fun (Xz:a)=> ((or (x Xz)) (y Xz)))) (fun (x0:a)=> ((or (x x0)) (y x0))))
% Found (eta_expansion_dep00 (fun (Xz:a)=> ((or (x Xz)) (y Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (x Xz)) (y Xz)))) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((or (x Xz)) (y Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (x Xz)) (y Xz)))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((or (x Xz)) (y Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (x Xz)) (y Xz)))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((or (x Xz)) (y Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (x Xz)) (y Xz)))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((or (x Xz)) (y Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (x Xz)) (y Xz)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0))))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0))))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0))))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0))))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xz:a)=> ((or (x Xz)) (y Xz)))):(((eq (a->Prop)) (fun (Xz:a)=> ((or (x Xz)) (y Xz)))) (fun (x0:a)=> ((or (x x0)) (y x0))))
% Found (eta_expansion_dep00 (fun (Xz:a)=> ((or (x Xz)) (y Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (x Xz)) (y Xz)))) b0)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((or (x Xz)) (y Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (x Xz)) (y Xz)))) b0)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((or (x Xz)) (y Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (x Xz)) (y Xz)))) b0)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((or (x Xz)) (y Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (x Xz)) (y Xz)))) b0)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((or (x Xz)) (y Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (x Xz)) (y Xz)))) b0)
% Found eta_expansion000:=(eta_expansion00 b0):(((eq (a->Prop)) b0) (fun (x:a)=> (b0 x)))
% Found (eta_expansion00 b0) as proof of (((eq (a->Prop)) b0) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0))))))
% Found ((eta_expansion0 Prop) b0) as proof of (((eq (a->Prop)) b0) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0))))))
% Found (((eta_expansion a) Prop) b0) as proof of (((eq (a->Prop)) b0) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0))))))
% Found (((eta_expansion a) Prop) b0) as proof of (((eq (a->Prop)) b0) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0))))))
% Found (((eta_expansion a) Prop) b0) as proof of (((eq (a->Prop)) b0) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0))))))
% Found x0:(P (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0))))))
% Instantiate: f:=(fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0))))):(a->Prop)
% Found x0 as proof of (P0 f)
% Found eq_ref00:=(eq_ref0 (f x00)):(((eq Prop) (f x00)) (f x00))
% Found (eq_ref0 (f x00)) as proof of (((eq Prop) (f x00)) ((or (x x00)) (y x00)))
% Found ((eq_ref Prop) (f x00)) as proof of (((eq Prop) (f x00)) ((or (x x00)) (y x00)))
% Found ((eq_ref Prop) (f x00)) as proof of (((eq Prop) (f x00)) ((or (x x00)) (y x00)))
% Found (fun (x00:a)=> ((eq_ref Prop) (f x00))) as proof of (((eq Prop) (f x00)) ((or (x x00)) (y x00)))
% Found (fun (x00:a)=> ((eq_ref Prop) (f x00))) as proof of (forall (x0:a), (((eq Prop) (f x0)) ((or (x x0)) (y x0))))
% Found x0:(P (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0))))))
% Instantiate: f:=(fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0))))):(a->Prop)
% Found x0 as proof of (P0 f)
% Found eq_ref00:=(eq_ref0 (f x00)):(((eq Prop) (f x00)) (f x00))
% Found (eq_ref0 (f x00)) as proof of (((eq Prop) (f x00)) ((or (x x00)) (y x00)))
% Found ((eq_ref Prop) (f x00)) as proof of (((eq Prop) (f x00)) ((or (x x00)) (y x00)))
% Found ((eq_ref Prop) (f x00)) as proof of (((eq Prop) (f x00)) ((or (x x00)) (y x00)))
% Found (fun (x00:a)=> ((eq_ref Prop) (f x00))) as proof of (((eq Prop) (f x00)) ((or (x x00)) (y x00)))
% Found (fun (x00:a)=> ((eq_ref Prop) (f x00))) as proof of (forall (x0:a), (((eq Prop) (f x0)) ((or (x x0)) (y x0))))
% Found x1:(P ((or (x x0)) (y x0)))
% Instantiate: b:=((or (x x0)) (y x0)):Prop
% Found x1 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))):(((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found (eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b)
% Found x1:(P ((or (x x0)) (y x0)))
% Instantiate: b:=((or (x x0)) (y x0)):Prop
% Found x1 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))):(((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found (eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b)
% Found eq_ref00:=(eq_ref0 (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))):(((eq (a->Prop)) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0))))))
% Found (eq_ref0 (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))) as proof of (((eq (a->Prop)) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))) as proof of (((eq (a->Prop)) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))) as proof of (((eq (a->Prop)) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))) as proof of (((eq (a->Prop)) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))) 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 (Xz:a)=> ((or (x Xz)) (y Xz))))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or (x Xz)) (y Xz))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or (x Xz)) (y Xz))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or (x Xz)) (y Xz))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or (x Xz)) (y Xz))))
% 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 (Xz:a)=> ((or (x Xz)) (y Xz))))
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or (x Xz)) (y Xz))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or (x Xz)) (y Xz))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or (x Xz)) (y Xz))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or (x Xz)) (y Xz))))
% Found eq_ref00:=(eq_ref0 (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))):(((eq (a->Prop)) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0))))))
% Found (eq_ref0 (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))) as proof of (((eq (a->Prop)) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))) as proof of (((eq (a->Prop)) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))) as proof of (((eq (a->Prop)) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))) as proof of (((eq (a->Prop)) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))) b)
% Found eta_expansion000:=(eta_expansion00 (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))):(((eq (a->Prop)) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))) (fun (x0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))))
% Found (eta_expansion00 (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))) as proof of (((eq (a->Prop)) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))) b)
% Found ((eta_expansion0 Prop) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))) as proof of (((eq (a->Prop)) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))) b)
% Found (((eta_expansion a) Prop) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))) as proof of (((eq (a->Prop)) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))) b)
% Found (((eta_expansion a) Prop) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))) as proof of (((eq (a->Prop)) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))) b)
% Found (((eta_expansion a) Prop) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))) as proof of (((eq (a->Prop)) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or (x Xz)) (y Xz))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or (x Xz)) (y Xz))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or (x Xz)) (y Xz))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or (x Xz)) (y Xz))))
% Found eq_ref000:=(eq_ref00 P):((P (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0))))))->(P (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))))
% Found (eq_ref00 P) as proof of (P0 (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0))))))
% Found ((eq_ref0 (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))) P) as proof of (P0 (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0))))))
% Found (((eq_ref (a->Prop)) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))) P) as proof of (P0 (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0))))))
% Found (((eq_ref (a->Prop)) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))) P) as proof of (P0 (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0))))))
% Found eq_ref00:=(eq_ref0 (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))):(((eq (a->Prop)) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0))))))
% Found (eq_ref0 (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))) as proof of (((eq (a->Prop)) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))) as proof of (((eq (a->Prop)) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))) as proof of (((eq (a->Prop)) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))) as proof of (((eq (a->Prop)) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or (x Xz)) (y Xz))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or (x Xz)) (y Xz))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or (x Xz)) (y Xz))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or (x Xz)) (y Xz))))
% Found eta_expansion0000:=(eta_expansion000 P1):((P1 (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0))))))->(P1 (fun (x0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))))
% Found (eta_expansion000 P1) as proof of (P2 (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0))))))
% Found ((eta_expansion00 (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))) P1) as proof of (P2 (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0))))))
% Found (((eta_expansion0 Prop) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))) P1) as proof of (P2 (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0))))))
% Found ((((eta_expansion a) Prop) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))) P1) as proof of (P2 (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0))))))
% Found ((((eta_expansion a) Prop) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))) P1) as proof of (P2 (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0))))))
% Found eta_expansion0000:=(eta_expansion000 P1):((P1 (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0))))))->(P1 (fun (x0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))))
% Found (eta_expansion000 P1) as proof of (P2 (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0))))))
% Found ((eta_expansion00 (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))) P1) as proof of (P2 (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0))))))
% Found (((eta_expansion0 Prop) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))) P1) as proof of (P2 (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0))))))
% Found ((((eta_expansion a) Prop) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))) P1) as proof of (P2 (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0))))))
% Found ((((eta_expansion a) Prop) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))) P1) as proof of (P2 (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0))))))
% Found eta_expansion0000:=(eta_expansion000 P1):((P1 (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0))))))->(P1 (fun (x0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))))
% Found (eta_expansion000 P1) as proof of (P2 (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0))))))
% Found ((eta_expansion00 (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))) P1) as proof of (P2 (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0))))))
% Found (((eta_expansion0 Prop) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))) P1) as proof of (P2 (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0))))))
% Found ((((eta_expansion a) Prop) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))) P1) as proof of (P2 (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0))))))
% Found ((((eta_expansion a) Prop) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))) P1) as proof of (P2 (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0))))))
% Found eta_expansion0000:=(eta_expansion000 P1):((P1 (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0))))))->(P1 (fun (x0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))))
% Found (eta_expansion000 P1) as proof of (P2 (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0))))))
% Found ((eta_expansion00 (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))) P1) as proof of (P2 (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0))))))
% Found (((eta_expansion0 Prop) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))) P1) as proof of (P2 (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0))))))
% Found ((((eta_expansion a) Prop) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))) P1) as proof of (P2 (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0))))))
% Found ((((eta_expansion a) Prop) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))) P1) as proof of (P2 (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0))))))
% 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 eq_ref00:=(eq_ref0 ((or (x x0)) (y x0))):(((eq Prop) ((or (x x0)) (y x0))) ((or (x x0)) (y x0)))
% Found (eq_ref0 ((or (x x0)) (y x0))) as proof of (((eq Prop) ((or (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((or (x x0)) (y x0))) as proof of (((eq Prop) ((or (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((or (x x0)) (y x0))) as proof of (((eq Prop) ((or (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((or (x x0)) (y x0))) as proof of (((eq Prop) ((or (x x0)) (y x0))) 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 ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found eq_ref00:=(eq_ref0 ((or (x x0)) (y x0))):(((eq Prop) ((or (x x0)) (y x0))) ((or (x x0)) (y x0)))
% Found (eq_ref0 ((or (x x0)) (y x0))) as proof of (((eq Prop) ((or (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((or (x x0)) (y x0))) as proof of (((eq Prop) ((or (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((or (x x0)) (y x0))) as proof of (((eq Prop) ((or (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((or (x x0)) (y x0))) as proof of (((eq Prop) ((or (x x0)) (y x0))) 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 ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found eq_ref00:=(eq_ref0 ((or (x x0)) (y x0))):(((eq Prop) ((or (x x0)) (y x0))) ((or (x x0)) (y x0)))
% Found (eq_ref0 ((or (x x0)) (y x0))) as proof of (((eq Prop) ((or (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((or (x x0)) (y x0))) as proof of (((eq Prop) ((or (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((or (x x0)) (y x0))) as proof of (((eq Prop) ((or (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((or (x x0)) (y x0))) as proof of (((eq Prop) ((or (x x0)) (y x0))) 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 ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found eq_ref00:=(eq_ref0 ((or (x x0)) (y x0))):(((eq Prop) ((or (x x0)) (y x0))) ((or (x x0)) (y x0)))
% Found (eq_ref0 ((or (x x0)) (y x0))) as proof of (((eq Prop) ((or (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((or (x x0)) (y x0))) as proof of (((eq Prop) ((or (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((or (x x0)) (y x0))) as proof of (((eq Prop) ((or (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((or (x x0)) (y x0))) as proof of (((eq Prop) ((or (x x0)) (y x0))) 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 ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found eq_ref00:=(eq_ref0 ((or (x x0)) (y x0))):(((eq Prop) ((or (x x0)) (y x0))) ((or (x x0)) (y x0)))
% Found (eq_ref0 ((or (x x0)) (y x0))) as proof of (((eq Prop) ((or (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((or (x x0)) (y x0))) as proof of (((eq Prop) ((or (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((or (x x0)) (y x0))) as proof of (((eq Prop) ((or (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((or (x x0)) (y x0))) as proof of (((eq Prop) ((or (x x0)) (y x0))) 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 ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found eq_ref00:=(eq_ref0 ((or (x x0)) (y x0))):(((eq Prop) ((or (x x0)) (y x0))) ((or (x x0)) (y x0)))
% Found (eq_ref0 ((or (x x0)) (y x0))) as proof of (((eq Prop) ((or (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((or (x x0)) (y x0))) as proof of (((eq Prop) ((or (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((or (x x0)) (y x0))) as proof of (((eq Prop) ((or (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((or (x x0)) (y x0))) as proof of (((eq Prop) ((or (x x0)) (y x0))) 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 ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% 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 x11:(P1 ((or (x x0)) (y x0)))
% Found (fun (x11:(P1 ((or (x x0)) (y x0))))=> x11) as proof of (P1 ((or (x x0)) (y x0)))
% Found (fun (x11:(P1 ((or (x x0)) (y x0))))=> x11) as proof of (P2 ((or (x x0)) (y x0)))
% Found x11:(P1 ((or (x x0)) (y x0)))
% Found (fun (x11:(P1 ((or (x x0)) (y x0))))=> x11) as proof of (P1 ((or (x x0)) (y x0)))
% Found (fun (x11:(P1 ((or (x x0)) (y x0))))=> x11) as proof of (P2 ((or (x x0)) (y x0)))
% Found x11:(P1 ((or (x x0)) (y x0)))
% Found (fun (x11:(P1 ((or (x x0)) (y x0))))=> x11) as proof of (P1 ((or (x x0)) (y x0)))
% Found (fun (x11:(P1 ((or (x x0)) (y x0))))=> x11) as proof of (P2 ((or (x x0)) (y x0)))
% Found x11:(P1 ((or (x x0)) (y x0)))
% Found (fun (x11:(P1 ((or (x x0)) (y x0))))=> x11) as proof of (P1 ((or (x x0)) (y x0)))
% Found (fun (x11:(P1 ((or (x x0)) (y x0))))=> x11) as proof of (P2 ((or (x x0)) (y x0)))
% 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 ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))
% Found ((eta_expansion0 Prop) a0) as proof of (((eq ((a->Prop)->Prop)) a0) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))
% Found (((eta_expansion (a->Prop)) Prop) a0) as proof of (((eq ((a->Prop)->Prop)) a0) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))
% Found (((eta_expansion (a->Prop)) Prop) a0) as proof of (((eq ((a->Prop)->Prop)) a0) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))
% Found (((eta_expansion (a->Prop)) Prop) a0) as proof of (((eq ((a->Prop)->Prop)) a0) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))
% 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 ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))
% Found ((eq_ref ((a->Prop)->Prop)) a0) as proof of (((eq ((a->Prop)->Prop)) a0) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))
% Found ((eq_ref ((a->Prop)->Prop)) a0) as proof of (((eq ((a->Prop)->Prop)) a0) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))
% Found ((eq_ref ((a->Prop)->Prop)) a0) as proof of (((eq ((a->Prop)->Prop)) a0) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))
% 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 ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))
% Found ((eta_expansion0 Prop) a0) as proof of (((eq ((a->Prop)->Prop)) a0) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))
% Found (((eta_expansion (a->Prop)) Prop) a0) as proof of (((eq ((a->Prop)->Prop)) a0) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))
% Found (((eta_expansion (a->Prop)) Prop) a0) as proof of (((eq ((a->Prop)->Prop)) a0) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))
% Found (((eta_expansion (a->Prop)) Prop) a0) as proof of (((eq ((a->Prop)->Prop)) a0) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))
% 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 ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))
% Found ((eta_expansion0 Prop) a0) as proof of (((eq ((a->Prop)->Prop)) a0) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))
% Found (((eta_expansion (a->Prop)) Prop) a0) as proof of (((eq ((a->Prop)->Prop)) a0) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))
% Found (((eta_expansion (a->Prop)) Prop) a0) as proof of (((eq ((a->Prop)->Prop)) a0) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))
% Found (((eta_expansion (a->Prop)) Prop) a0) as proof of (((eq ((a->Prop)->Prop)) a0) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))
% 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 (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))):(((eq (a->Prop)) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0))))))
% Found (eq_ref0 (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))) as proof of (((eq (a->Prop)) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))) b0)
% Found ((eq_ref (a->Prop)) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))) as proof of (((eq (a->Prop)) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))) b0)
% Found ((eq_ref (a->Prop)) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))) as proof of (((eq (a->Prop)) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))) b0)
% Found ((eq_ref (a->Prop)) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))) as proof of (((eq (a->Prop)) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))) b0)
% Found eta_expansion0000:=(eta_expansion000 P):((P (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0))))))->(P (fun (x0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))))
% Found (eta_expansion000 P) as proof of (P0 (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0))))))
% Found ((eta_expansion00 (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))) P) as proof of (P0 (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0))))))
% Found (((eta_expansion0 Prop) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))) P) as proof of (P0 (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0))))))
% Found ((((eta_expansion a) Prop) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))) P) as proof of (P0 (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0))))))
% Found ((((eta_expansion a) Prop) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))) P) as proof of (P0 (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0))))))
% Found eq_ref00:=(eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))):(((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found (eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b)
% Found x1:(P0 b)
% Instantiate: b:=((or (x x0)) (y x0)):Prop
% Found (fun (x1:(P0 b))=> x1) as proof of (P0 ((or (x x0)) (y x0)))
% Found (fun (P0:(Prop->Prop)) (x1:(P0 b))=> x1) as proof of ((P0 b)->(P0 ((or (x x0)) (y x0))))
% Found (fun (P0:(Prop->Prop)) (x1:(P0 b))=> x1) as proof of (P b)
% Found eq_ref00:=(eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))):(((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found (eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b)
% Found x1:(P0 b)
% Instantiate: b:=((or (x x0)) (y x0)):Prop
% Found (fun (x1:(P0 b))=> x1) as proof of (P0 ((or (x x0)) (y x0)))
% Found (fun (P0:(Prop->Prop)) (x1:(P0 b))=> x1) as proof of ((P0 b)->(P0 ((or (x x0)) (y x0))))
% Found (fun (P0:(Prop->Prop)) (x1:(P0 b))=> x1) as proof of (P b)
% Found x12:(P ((or (x x0)) (y x0)))
% Found (fun (x12:(P ((or (x x0)) (y x0))))=> x12) as proof of (P ((or (x x0)) (y x0)))
% Found (fun (x12:(P ((or (x x0)) (y x0))))=> x12) as proof of (P0 ((or (x x0)) (y x0)))
% Found eq_ref00:=(eq_ref0 ((or (x x0)) (y x0))):(((eq Prop) ((or (x x0)) (y x0))) ((or (x x0)) (y x0)))
% Found (eq_ref0 ((or (x x0)) (y x0))) as proof of (((eq Prop) ((or (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((or (x x0)) (y x0))) as proof of (((eq Prop) ((or (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((or (x x0)) (y x0))) as proof of (((eq Prop) ((or (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((or (x x0)) (y x0))) as proof of (((eq Prop) ((or (x x0)) (y x0))) 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 ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found x12:(P ((or (x x0)) (y x0)))
% Found (fun (x12:(P ((or (x x0)) (y x0))))=> x12) as proof of (P ((or (x x0)) (y x0)))
% Found (fun (x12:(P ((or (x x0)) (y x0))))=> x12) as proof of (P0 ((or (x x0)) (y x0)))
% Found eq_ref00:=(eq_ref0 ((or (x x0)) (y x0))):(((eq Prop) ((or (x x0)) (y x0))) ((or (x x0)) (y x0)))
% Found (eq_ref0 ((or (x x0)) (y x0))) as proof of (((eq Prop) ((or (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((or (x x0)) (y x0))) as proof of (((eq Prop) ((or (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((or (x x0)) (y x0))) as proof of (((eq Prop) ((or (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((or (x x0)) (y x0))) as proof of (((eq Prop) ((or (x x0)) (y x0))) 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 ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found x1:(P ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Instantiate: b:=((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))):Prop
% Found x1 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 ((or (x x0)) (y x0))):(((eq Prop) ((or (x x0)) (y x0))) ((or (x x0)) (y x0)))
% Found (eq_ref0 ((or (x x0)) (y x0))) as proof of (((eq Prop) ((or (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((or (x x0)) (y x0))) as proof of (((eq Prop) ((or (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((or (x x0)) (y x0))) as proof of (((eq Prop) ((or (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((or (x x0)) (y x0))) as proof of (((eq Prop) ((or (x x0)) (y x0))) b)
% Found x1:(P ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Instantiate: b:=((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))):Prop
% Found x1 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 ((or (x x0)) (y x0))):(((eq Prop) ((or (x x0)) (y x0))) ((or (x x0)) (y x0)))
% Found (eq_ref0 ((or (x x0)) (y x0))) as proof of (((eq Prop) ((or (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((or (x x0)) (y x0))) as proof of (((eq Prop) ((or (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((or (x x0)) (y x0))) as proof of (((eq Prop) ((or (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((or (x x0)) (y x0))) as proof of (((eq Prop) ((or (x x0)) (y x0))) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))):(((eq (a->Prop)) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))) (fun (x0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))))
% Found (eta_expansion_dep00 (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))) as proof of (((eq (a->Prop)) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))) b0)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))) as proof of (((eq (a->Prop)) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))) b0)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))) as proof of (((eq (a->Prop)) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))) b0)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))) as proof of (((eq (a->Prop)) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))) b0)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))) as proof of (((eq (a->Prop)) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))) 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 ((or (x x0)) (y x0))):(((eq Prop) ((or (x x0)) (y x0))) ((or (x x0)) (y x0)))
% Found (eq_ref0 ((or (x x0)) (y x0))) as proof of (((eq Prop) ((or (x x0)) (y x0))) b0)
% Found ((eq_ref Prop) ((or (x x0)) (y x0))) as proof of (((eq Prop) ((or (x x0)) (y x0))) b0)
% Found ((eq_ref Prop) ((or (x x0)) (y x0))) as proof of (((eq Prop) ((or (x x0)) (y x0))) b0)
% Found ((eq_ref Prop) ((or (x x0)) (y x0))) as proof of (((eq Prop) ((or (x x0)) (y x0))) 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 ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found eq_ref00:=(eq_ref0 ((or (x x0)) (y x0))):(((eq Prop) ((or (x x0)) (y x0))) ((or (x x0)) (y x0)))
% Found (eq_ref0 ((or (x x0)) (y x0))) as proof of (((eq Prop) ((or (x x0)) (y x0))) b0)
% Found ((eq_ref Prop) ((or (x x0)) (y x0))) as proof of (((eq Prop) ((or (x x0)) (y x0))) b0)
% Found ((eq_ref Prop) ((or (x x0)) (y x0))) as proof of (((eq Prop) ((or (x x0)) (y x0))) b0)
% Found ((eq_ref Prop) ((or (x x0)) (y x0))) as proof of (((eq Prop) ((or (x x0)) (y x0))) 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 ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found x1:(P ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Instantiate: b:=((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))):Prop
% Found x1 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 ((or (x x0)) (y x0))):(((eq Prop) ((or (x x0)) (y x0))) ((or (x x0)) (y x0)))
% Found (eq_ref0 ((or (x x0)) (y x0))) as proof of (((eq Prop) ((or (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((or (x x0)) (y x0))) as proof of (((eq Prop) ((or (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((or (x x0)) (y x0))) as proof of (((eq Prop) ((or (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((or (x x0)) (y x0))) as proof of (((eq Prop) ((or (x x0)) (y x0))) b)
% Found x1:(P ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Instantiate: b:=((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))):Prop
% Found x1 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 ((or (x x0)) (y x0))):(((eq Prop) ((or (x x0)) (y x0))) ((or (x x0)) (y x0)))
% Found (eq_ref0 ((or (x x0)) (y x0))) as proof of (((eq Prop) ((or (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((or (x x0)) (y x0))) as proof of (((eq Prop) ((or (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((or (x x0)) (y x0))) as proof of (((eq Prop) ((or (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((or (x x0)) (y x0))) as proof of (((eq Prop) ((or (x x0)) (y x0))) b)
% Found eq_ref000:=(eq_ref00 P):((P (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0))))))->(P (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))))
% Found (eq_ref00 P) as proof of (P0 (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0))))))
% Found ((eq_ref0 (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))) P) as proof of (P0 (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0))))))
% Found (((eq_ref (a->Prop)) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))) P) as proof of (P0 (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0))))))
% Found (((eq_ref (a->Prop)) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))) P) as proof of (P0 (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0))))))
% Found eta_expansion0000:=(eta_expansion000 P):((P (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0))))))->(P (fun (x0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))))
% Found (eta_expansion000 P) as proof of (P0 (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0))))))
% Found ((eta_expansion00 (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))) P) as proof of (P0 (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0))))))
% Found (((eta_expansion0 Prop) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))) P) as proof of (P0 (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0))))))
% Found ((((eta_expansion a) Prop) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))) P) as proof of (P0 (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0))))))
% Found ((((eta_expansion a) Prop) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))) P) as proof of (P0 (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0))))))
% Found eta_expansion0000:=(eta_expansion000 P):((P (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0))))))->(P (fun (x0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))))
% Found (eta_expansion000 P) as proof of (P0 (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0))))))
% Found ((eta_expansion00 (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))) P) as proof of (P0 (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0))))))
% Found (((eta_expansion0 Prop) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))) P) as proof of (P0 (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0))))))
% Found ((((eta_expansion a) Prop) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))) P) as proof of (P0 (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0))))))
% Found ((((eta_expansion a) Prop) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))) P) as proof of (P0 (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0))))))
% Found eq_ref000:=(eq_ref00 P):((P (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0))))))->(P (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))))
% Found (eq_ref00 P) as proof of (P0 (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0))))))
% Found ((eq_ref0 (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))) P) as proof of (P0 (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0))))))
% Found (((eq_ref (a->Prop)) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))) P) as proof of (P0 (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0))))))
% Found (((eq_ref (a->Prop)) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))) P) as proof of (P0 (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0))))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xz:a)=> ((or (x Xz)) (y Xz)))):(((eq (a->Prop)) (fun (Xz:a)=> ((or (x Xz)) (y Xz)))) (fun (x0:a)=> ((or (x x0)) (y x0))))
% Found (eta_expansion_dep00 (fun (Xz:a)=> ((or (x Xz)) (y Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (x Xz)) (y Xz)))) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((or (x Xz)) (y Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (x Xz)) (y Xz)))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((or (x Xz)) (y Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (x Xz)) (y Xz)))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((or (x Xz)) (y Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (x Xz)) (y Xz)))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((or (x Xz)) (y Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (x Xz)) (y Xz)))) 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 (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0))))))
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0))))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0))))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0))))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0))))))
% Found eq_ref00:=(eq_ref0 (b x0)):(((eq Prop) (b x0)) (b x0))
% Found (eq_ref0 (b x0)) as proof of (((eq Prop) (b x0)) b0)
% Found ((eq_ref Prop) (b x0)) as proof of (((eq Prop) (b x0)) b0)
% Found ((eq_ref Prop) (b x0)) as proof of (((eq Prop) (b x0)) b0)
% Found ((eq_ref Prop) (b x0)) as proof of (((eq Prop) (b x0)) 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 ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found eq_ref00:=(eq_ref0 (b x0)):(((eq Prop) (b x0)) (b x0))
% Found (eq_ref0 (b x0)) as proof of (((eq Prop) (b x0)) b0)
% Found ((eq_ref Prop) (b x0)) as proof of (((eq Prop) (b x0)) b0)
% Found ((eq_ref Prop) (b x0)) as proof of (((eq Prop) (b x0)) b0)
% Found ((eq_ref Prop) (b x0)) as proof of (((eq Prop) (b x0)) 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 ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found x01:(P (fun (Xz:a)=> ((or (x Xz)) (y Xz))))
% Found (fun (x01:(P (fun (Xz:a)=> ((or (x Xz)) (y Xz)))))=> x01) as proof of (P (fun (Xz:a)=> ((or (x Xz)) (y Xz))))
% Found (fun (x01:(P (fun (Xz:a)=> ((or (x Xz)) (y Xz)))))=> x01) as proof of (P0 (fun (Xz:a)=> ((or (x Xz)) (y Xz))))
% Found eq_ref000:=(eq_ref00 P1):((P1 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))->(P1 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))))
% Found (eq_ref00 P1) as proof of (P2 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found ((eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) P1) as proof of (P2 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found (((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) P1) as proof of (P2 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found (((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) P1) as proof of (P2 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found eq_ref000:=(eq_ref00 P1):((P1 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))->(P1 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))))
% Found (eq_ref00 P1) as proof of (P2 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found ((eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) P1) as proof of (P2 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found (((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) P1) as proof of (P2 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found (((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) P1) as proof of (P2 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found eq_ref000:=(eq_ref00 P1):((P1 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))->(P1 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))))
% Found (eq_ref00 P1) as proof of (P2 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found ((eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) P1) as proof of (P2 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found (((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) P1) as proof of (P2 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found (((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) P1) as proof of (P2 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found eq_ref000:=(eq_ref00 P1):((P1 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))->(P1 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))))
% Found (eq_ref00 P1) as proof of (P2 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found ((eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) P1) as proof of (P2 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found (((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) P1) as proof of (P2 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found (((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) P1) as proof of (P2 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found eq_ref000:=(eq_ref00 P1):((P1 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))->(P1 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))))
% Found (eq_ref00 P1) as proof of (P2 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found ((eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) P1) as proof of (P2 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found (((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) P1) as proof of (P2 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found (((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) P1) as proof of (P2 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found eq_ref000:=(eq_ref00 P1):((P1 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))->(P1 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))))
% Found (eq_ref00 P1) as proof of (P2 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found ((eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) P1) as proof of (P2 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found (((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) P1) as proof of (P2 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found (((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) P1) as proof of (P2 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found eq_ref000:=(eq_ref00 P1):((P1 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))->(P1 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))))
% Found (eq_ref00 P1) as proof of (P2 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found ((eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) P1) as proof of (P2 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found (((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) P1) as proof of (P2 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found (((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) P1) as proof of (P2 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found eq_ref000:=(eq_ref00 P1):((P1 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))->(P1 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))))
% Found (eq_ref00 P1) as proof of (P2 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found ((eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) P1) as proof of (P2 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found (((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) P1) as proof of (P2 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found (((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) P1) as proof of (P2 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found x01:(P (fun (Xz:a)=> ((or (x Xz)) (y Xz))))
% Found (fun (x01:(P (fun (Xz:a)=> ((or (x Xz)) (y Xz)))))=> x01) as proof of (P (fun (Xz:a)=> ((or (x Xz)) (y Xz))))
% Found (fun (x01:(P (fun (Xz:a)=> ((or (x Xz)) (y Xz)))))=> x01) as proof of (P0 (fun (Xz:a)=> ((or (x Xz)) (y Xz))))
% Found x01:(P (fun (Xz:a)=> ((or (x Xz)) (y Xz))))
% Found (fun (x01:(P (fun (Xz:a)=> ((or (x Xz)) (y Xz)))))=> x01) as proof of (P (fun (Xz:a)=> ((or (x Xz)) (y Xz))))
% Found (fun (x01:(P (fun (Xz:a)=> ((or (x Xz)) (y Xz)))))=> x01) as proof of (P0 (fun (Xz:a)=> ((or (x Xz)) (y Xz))))
% Found eq_ref000:=(eq_ref00 P):((P ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))->(P ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))))
% Found (eq_ref00 P) as proof of (P0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found ((eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) P) as proof of (P0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found (((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) P) as proof of (P0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found (((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) P) as proof of (P0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found eq_ref000:=(eq_ref00 P):((P ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))->(P ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))))
% Found (eq_ref00 P) as proof of (P0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found ((eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) P) as proof of (P0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found (((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) P) as proof of (P0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found (((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) P) as proof of (P0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found eq_ref00:=(eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))):(((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found (eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (x x0)) (y x0)))
% Found eq_ref00:=(eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))):(((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found (eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (x x0)) (y x0)))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (x x0)) (y x0)))
% Found eq_ref00:=(eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))):(((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found (eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b)
% Found eq_ref00:=(eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))):(((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found (eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (x x0)) (y x0)))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (x x0)) (y x0)))
% Found eq_ref00:=(eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))):(((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found (eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (x x0)) (y x0)))
% Found eq_ref00:=(eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))):(((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found (eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b)
% Found eq_ref00:=(eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))):(((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found (eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (x x0)) (y x0)))
% Found eq_ref000:=(eq_ref00 P):((P ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))->(P ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))))
% Found (eq_ref00 P) as proof of (P0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found ((eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) P) as proof of (P0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found (((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) P) as proof of (P0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found (((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) P) as proof of (P0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found eq_ref00:=(eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))):(((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found (eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (x x0)) (y x0)))
% Found eq_ref000:=(eq_ref00 P):((P ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))->(P ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))))
% Found (eq_ref00 P) as proof of (P0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found ((eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) P) as proof of (P0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found (((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) P) as proof of (P0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found (((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) P) as proof of (P0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found eq_ref00:=(eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))):(((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found (eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (x x0)) (y x0)))
% Found eq_ref00:=(eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))):(((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found (eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (x x0)) (y x0)))
% Found eq_ref00:=(eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))):(((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found (eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (x x0)) (y x0)))
% Found eq_ref00:=(eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))):(((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found (eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (x x0)) (y x0)))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (x x0)) (y x0)))
% Found eq_ref00:=(eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))):(((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found (eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (x x0)) (y x0)))
% Found eq_ref00:=(eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))):(((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found (eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (x x0)) (y x0)))
% Found eq_ref00:=(eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))):(((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found (eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b)
% Found eq_ref00:=(eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))):(((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found (eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (x x0)) (y x0)))
% Found eq_ref00:=(eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))):(((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found (eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (x x0)) (y x0)))
% Found eq_ref00:=(eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))):(((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found (eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (x x0)) (y x0)))
% Found eq_ref00:=(eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))):(((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found (eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (x x0)) (y x0)))
% Found eq_ref00:=(eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))):(((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found (eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (x x0)) (y x0)))
% Found eq_ref00:=(eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))):(((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found (eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (x x0)) (y x0)))
% Found eq_ref00:=(eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))):(((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found (eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (x x0)) (y x0)))
% Found eq_ref00:=(eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))):(((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found (eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (x x0)) (y x0)))
% Found eq_ref00:=(eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))):(((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found (eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (x x0)) (y x0)))
% Found x11:(P b)
% Found (fun (x11:(P b))=> x11) as proof of (P b)
% Found (fun (x11:(P b))=> x11) as proof of (P0 b)
% Found x11:(P b)
% Found (fun (x11:(P b))=> x11) as proof of (P b)
% Found (fun (x11:(P b))=> x11) as proof of (P0 b)
% Found x11:(P (b x0))
% Found (fun (x11:(P (b x0)))=> x11) as proof of (P (b x0))
% Found (fun (x11:(P (b x0)))=> x11) as proof of (P0 (b x0))
% Found x11:(P (b x0))
% Found (fun (x11:(P (b x0)))=> x11) as proof of (P (b x0))
% Found (fun (x11:(P (b x0)))=> x11) as proof of (P0 (b x0))
% 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) b0)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) b0)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) b0)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) b0)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq (a->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (a->Prop)) b0) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0))))))
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0))))))
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0))))))
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0))))))
% Found eq_ref00:=(eq_ref0 ((or (x x0)) (y x0))):(((eq Prop) ((or (x x0)) (y x0))) ((or (x x0)) (y x0)))
% Found (eq_ref0 ((or (x x0)) (y x0))) as proof of (((eq Prop) ((or (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((or (x x0)) (y x0))) as proof of (((eq Prop) ((or (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((or (x x0)) (y x0))) as proof of (((eq Prop) ((or (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((or (x x0)) (y x0))) as proof of (((eq Prop) ((or (x x0)) (y x0))) b)
% Found x1:(P0 b)
% Instantiate: b:=((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))):Prop
% Found (fun (x1:(P0 b))=> x1) as proof of (P0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found (fun (P0:(Prop->Prop)) (x1:(P0 b))=> x1) as proof of ((P0 b)->(P0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))))
% Found (fun (P0:(Prop->Prop)) (x1:(P0 b))=> x1) as proof of (P b)
% Found eq_ref00:=(eq_ref0 ((or (x x0)) (y x0))):(((eq Prop) ((or (x x0)) (y x0))) ((or (x x0)) (y x0)))
% Found (eq_ref0 ((or (x x0)) (y x0))) as proof of (((eq Prop) ((or (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((or (x x0)) (y x0))) as proof of (((eq Prop) ((or (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((or (x x0)) (y x0))) as proof of (((eq Prop) ((or (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((or (x x0)) (y x0))) as proof of (((eq Prop) ((or (x x0)) (y x0))) b)
% Found x1:(P0 b)
% Instantiate: b:=((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))):Prop
% Found (fun (x1:(P0 b))=> x1) as proof of (P0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found (fun (P0:(Prop->Prop)) (x1:(P0 b))=> x1) as proof of ((P0 b)->(P0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))))
% Found (fun (P0:(Prop->Prop)) (x1:(P0 b))=> x1) as proof of (P b)
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) (y x0))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (y x0))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (y x0))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (y x0))
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) (y x0))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (y x0))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (y x0))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (y x0))
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) (y x0))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (y x0))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (y x0))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (y x0))
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) (y x0))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (y x0))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (y x0))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (y x0))
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) (y x0))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (y x0))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (y x0))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (y x0))
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) (y x0))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (y x0))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (y x0))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (y x0))
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) (y x0))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (y x0))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (y x0))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (y x0))
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) (y x0))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (y x0))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (y x0))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (y x0))
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) (y x0))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (y x0))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (y x0))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (y x0))
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) (y x0))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (y x0))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (y x0))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (y x0))
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) (y x0))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (y x0))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (y x0))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (y x0))
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) (y x0))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (y x0))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (y x0))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (y x0))
% Found x1:(P ((or (x x0)) (y x0)))
% Instantiate: b:=((or (x x0)) (y x0)):Prop
% Found x1 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))):(((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found (eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b)
% Found x1:(P ((or (x x0)) (y x0)))
% Instantiate: b:=((or (x x0)) (y x0)):Prop
% Found x1 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))):(((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found (eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) 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 (Xz:a)=> ((or (x Xz)) (y Xz))))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or (x Xz)) (y Xz))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or (x Xz)) (y Xz))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or (x Xz)) (y Xz))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or (x Xz)) (y Xz))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))):(((eq (a->Prop)) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))) (fun (x0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))))
% Found (eta_expansion_dep00 (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))) as proof of (((eq (a->Prop)) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))) as proof of (((eq (a->Prop)) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))) as proof of (((eq (a->Prop)) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))) as proof of (((eq (a->Prop)) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))) as proof of (((eq (a->Prop)) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or (x Xz)) (y Xz))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or (x Xz)) (y Xz))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or (x Xz)) (y Xz))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or (x Xz)) (y Xz))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))):(((eq (a->Prop)) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))) (fun (x0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))))
% Found (eta_expansion_dep00 (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))) as proof of (((eq (a->Prop)) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))) as proof of (((eq (a->Prop)) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))) as proof of (((eq (a->Prop)) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))) as proof of (((eq (a->Prop)) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))) as proof of (((eq (a->Prop)) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))) b)
% Found eq_ref000:=(eq_ref00 P):((P (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0))))))->(P (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))))
% Found (eq_ref00 P) as proof of (P0 (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0))))))
% Found ((eq_ref0 (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))) P) as proof of (P0 (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0))))))
% Found (((eq_ref (a->Prop)) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))) P) as proof of (P0 (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0))))))
% Found (((eq_ref (a->Prop)) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))) P) as proof of (P0 (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0))))))
% Found eq_ref000:=(eq_ref00 P):((P (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0))))))->(P (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))))
% Found (eq_ref00 P) as proof of (P0 (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0))))))
% Found ((eq_ref0 (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))) P) as proof of (P0 (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0))))))
% Found (((eq_ref (a->Prop)) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))) P) as proof of (P0 (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0))))))
% Found (((eq_ref (a->Prop)) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))) P) as proof of (P0 (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0))))))
% Found eq_ref000:=(eq_ref00 P):((P (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0))))))->(P (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))))
% Found (eq_ref00 P) as proof of (P0 (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0))))))
% Found ((eq_ref0 (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))) P) as proof of (P0 (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0))))))
% Found (((eq_ref (a->Prop)) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))) P) as proof of (P0 (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0))))))
% Found (((eq_ref (a->Prop)) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))) P) as proof of (P0 (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0))))))
% Found eq_ref000:=(eq_ref00 P):((P (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0))))))->(P (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))))
% Found (eq_ref00 P) as proof of (P0 (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0))))))
% Found ((eq_ref0 (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))) P) as proof of (P0 (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0))))))
% Found (((eq_ref (a->Prop)) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))) P) as proof of (P0 (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0))))))
% Found (((eq_ref (a->Prop)) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0)))))) P) as proof of (P0 (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S Xx0))))))
% 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 ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))):(((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found (eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b0)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b0)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b0)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b0)
% Found eq_ref000:=(eq_ref00 P):((P ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))->(P ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))))
% Found (eq_ref00 P) as proof of (P0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found ((eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) P) as proof of (P0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found (((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) P) as proof of (P0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found (((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) P) as proof of (P0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found eq_ref00:=(eq_ref0 ((or (x x0)) (y x0))):(((eq Prop) ((or (x x0)) (y x0))) ((or (x x0)) (y x0)))
% Found (eq_ref0 ((or (x x0)) (y x0))) as proof of (((eq Prop) ((or (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((or (x x0)) (y x0))) as proof of (((eq Prop) ((or (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((or (x x0)) (y x0))) as proof of (((eq Prop) ((or (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((or (x x0)) (y x0))) as proof of (((eq Prop) ((or (x x0)) (y x0))) 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 ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found eq_ref00:=(eq_ref0 ((or (x x0)) (y x0))):(((eq Prop) ((or (x x0)) (y x0))) ((or (x x0)) (y x0)))
% Found (eq_ref0 ((or (x x0)) (y x0))) as proof of (((eq Prop) ((or (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((or (x x0)) (y x0))) as proof of (((eq Prop) ((or (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((or (x x0)) (y x0))) as proof of (((eq Prop) ((or (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((or (x x0)) (y x0))) as proof of (((eq Prop) ((or (x x0)) (y x0))) 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 ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found eq_ref00:=(eq_ref0 ((or (x x0)) (y x0))):(((eq Prop) ((or (x x0)) (y x0))) ((or (x x0)) (y x0)))
% Found (eq_ref0 ((or (x x0)) (y x0))) as proof of (((eq Prop) ((or (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((or (x x0)) (y x0))) as proof of (((eq Prop) ((or (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((or (x x0)) (y x0))) as proof of (((eq Prop) ((or (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((or (x x0)) (y x0))) as proof of (((eq Prop) ((or (x x0)) (y x0))) 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 ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found eq_ref00:=(eq_ref0 ((or (x x0)) (y x0))):(((eq Prop) ((or (x x0)) (y x0))) ((or (x x0)) (y x0)))
% Found (eq_ref0 ((or (x x0)) (y x0))) as proof of (((eq Prop) ((or (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((or (x x0)) (y x0))) as proof of (((eq Prop) ((or (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((or (x x0)) (y x0))) as proof of (((eq Prop) ((or (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((or (x x0)) (y x0))) as proof of (((eq Prop) ((or (x x0)) (y x0))) 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 ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found eq_ref00:=(eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))):(((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found (eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b0)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b0)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b0)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) 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 ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))):(((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found (eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b0)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b0)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b0)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) 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 x11:(P ((or (x x0)) (y x0)))
% Found (fun (x11:(P ((or (x x0)) (y x0))))=> x11) as proof of (P ((or (x x0)) (y x0)))
% Found (fun (x11:(P ((or (x x0)) (y x0))))=> x11) as proof of (P0 ((or (x x0)) (y x0)))
% Found x11:(P ((or (x x0)) (y x0)))
% Found (fun (x11:(P ((or (x x0)) (y x0))))=> x11) as proof of (P ((or (x x0)) (y x0)))
% Found (fun (x11:(P ((or (x x0)) (y x0))))=> x11) as proof of (P0 ((or (x x0)) (y x0)))
% Found x11:(P ((or (x x0)) (y x0)))
% Found (fun (x11:(P ((or (x x0)) (y x0))))=> x11) as proof of (P ((or (x x0)) (y x0)))
% Found (fun (x11:(P ((or (x x0)) (y x0))))=> x11) as proof of (P0 ((or (x x0)) (y x0)))
% Found x11:(P ((or (x x0)) (y x0)))
% Found (fun (x11:(P ((or (x x0)) (y x0))))=> x11) as proof of (P ((or (x x0)) (y x0)))
% Found (fun (x11:(P ((or (x x0)) (y x0))))=> x11) as proof of (P0 ((or (x x0)) (y x0)))
% Found eq_ref000:=(eq_ref00 P):((P ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))->(P ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))))
% Found (eq_ref00 P) as proof of (P0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found ((eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) P) as proof of (P0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found (((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) P) as proof of (P0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found (((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) P) as proof of (P0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found eq_ref000:=(eq_ref00 P):((P ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))->(P ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))))
% Found (eq_ref00 P) as proof of (P0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found ((eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) P) as proof of (P0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found (((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) P) as proof of (P0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found (((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) P) as proof of (P0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found eq_ref000:=(eq_ref00 P):((P ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))->(P ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))))
% Found (eq_ref00 P) as proof of (P0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found ((eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) P) as proof of (P0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found (((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) P) as proof of (P0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found (((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) P) as proof of (P0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found eq_ref000:=(eq_ref00 P):((P ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))->(P ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))))
% Found (eq_ref00 P) as proof of (P0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found ((eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) P) as proof of (P0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found (((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) P) as proof of (P0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found (((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) P) as proof of (P0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% 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 eq_ref00:=(eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))):(((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found (eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b0)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b0)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b0)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (b x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x0))
% Found eq_ref00:=(eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))):(((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found (eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b0)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b0)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b0)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (b x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x0))
% Found eq_ref00:=(eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))):(((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found (eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b0)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b0)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b0)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (b x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x0))
% Found eq_ref00:=(eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))):(((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found (eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b0)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b0)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b0)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (b x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x0))
% Found eq_ref00:=(eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))):(((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found (eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b0)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b0)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b0)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (b x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x0))
% Found eq_ref00:=(eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))):(((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found (eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b0)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b0)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b0)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (b x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x0))
% Found x11:(P b)
% Found (fun (x11:(P b))=> x11) as proof of (P b)
% Found (fun (x11:(P b))=> x11) as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 ((or (x x0)) (y x0))):(((eq Prop) ((or (x x0)) (y x0))) ((or (x x0)) (y x0)))
% Found (eq_ref0 ((or (x x0)) (y x0))) as proof of (((eq Prop) ((or (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((or (x x0)) (y x0))) as proof of (((eq Prop) ((or (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((or (x x0)) (y x0))) as proof of (((eq Prop) ((or (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((or (x x0)) (y x0))) as proof of (((eq Prop) ((or (x x0)) (y x0))) 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 ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found eq_ref00:=(eq_ref0 ((or (x x0)) (y x0))):(((eq Prop) ((or (x x0)) (y x0))) ((or (x x0)) (y x0)))
% Found (eq_ref0 ((or (x x0)) (y x0))) as proof of (((eq Prop) ((or (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((or (x x0)) (y x0))) as proof of (((eq Prop) ((or (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((or (x x0)) (y x0))) as proof of (((eq Prop) ((or (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((or (x x0)) (y x0))) as proof of (((eq Prop) ((or (x x0)) (y x0))) 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 ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% 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 ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found eq_ref00:=(eq_ref0 ((or (x x0)) (y x0))):(((eq Prop) ((or (x x0)) (y x0))) ((or (x x0)) (y x0)))
% Found (eq_ref0 ((or (x x0)) (y x0))) as proof of (((eq Prop) ((or (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((or (x x0)) (y x0))) as proof of (((eq Prop) ((or (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((or (x x0)) (y x0))) as proof of (((eq Prop) ((or (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((or (x x0)) (y x0))) as proof of (((eq Prop) ((or (x x0)) (y x0))) 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 ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found eq_ref00:=(eq_ref0 ((or (x x0)) (y x0))):(((eq Prop) ((or (x x0)) (y x0))) ((or (x x0)) (y x0)))
% Found (eq_ref0 ((or (x x0)) (y x0))) as proof of (((eq Prop) ((or (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((or (x x0)) (y x0))) as proof of (((eq Prop) ((or (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((or (x x0)) (y x0))) as proof of (((eq Prop) ((or (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((or (x x0)) (y x0))) as proof of (((eq Prop) ((or (x x0)) (y x0))) 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 ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found eq_ref00:=(eq_ref0 ((or (x x0)) (y x0))):(((eq Prop) ((or (x x0)) (y x0))) ((or (x x0)) (y x0)))
% Found (eq_ref0 ((or (x x0)) (y x0))) as proof of (((eq Prop) ((or (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((or (x x0)) (y x0))) as proof of (((eq Prop) ((or (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((or (x x0)) (y x0))) as proof of (((eq Prop) ((or (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((or (x x0)) (y x0))) as proof of (((eq Prop) ((or (x x0)) (y x0))) b)
% Found eq_ref00:=(eq_ref0 ((or (x x0)) (y x0))):(((eq Prop) ((or (x x0)) (y x0))) ((or (x x0)) (y x0)))
% Found (eq_ref0 ((or (x x0)) (y x0))) as proof of (((eq Prop) ((or (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((or (x x0)) (y x0))) as proof of (((eq Prop) ((or (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((or (x x0)) (y x0))) as proof of (((eq Prop) ((or (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((or (x x0)) (y x0))) as proof of (((eq Prop) ((or (x x0)) (y x0))) 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 ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% 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 ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found eq_ref00:=(eq_ref0 ((or (x x0)) (y x0))):(((eq Prop) ((or (x x0)) (y x0))) ((or (x x0)) (y x0)))
% Found (eq_ref0 ((or (x x0)) (y x0))) as proof of (((eq Prop) ((or (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((or (x x0)) (y x0))) as proof of (((eq Prop) ((or (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((or (x x0)) (y x0))) as proof of (((eq Prop) ((or (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((or (x x0)) (y x0))) as proof of (((eq Prop) ((or (x x0)) (y x0))) 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 ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found eq_ref00:=(eq_ref0 ((or (x x0)) (y x0))):(((eq Prop) ((or (x x0)) (y x0))) ((or (x x0)) (y x0)))
% Found (eq_ref0 ((or (x x0)) (y x0))) as proof of (((eq Prop) ((or (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((or (x x0)) (y x0))) as proof of (((eq Prop) ((or (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((or (x x0)) (y x0))) as proof of (((eq Prop) ((or (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((or (x x0)) (y x0))) as proof of (((eq Prop) ((or (x x0)) (y x0))) 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 ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found eq_ref00:=(eq_ref0 ((or (x x0)) (y x0))):(((eq Prop) ((or (x x0)) (y x0))) ((or (x x0)) (y x0)))
% Found (eq_ref0 ((or (x x0)) (y x0))) as proof of (((eq Prop) ((or (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((or (x x0)) (y x0))) as proof of (((eq Prop) ((or (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((or (x x0)) (y x0))) as proof of (((eq Prop) ((or (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((or (x x0)) (y x0))) as proof of (((eq Prop) ((or (x x0)) (y x0))) b)
% Found eq_ref00:=(eq_ref0 ((or (x x0)) (y x0))):(((eq Prop) ((or (x x0)) (y x0))) ((or (x x0)) (y x0)))
% Found (eq_ref0 ((or (x x0)) (y x0))) as proof of (((eq Prop) ((or (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((or (x x0)) (y x0))) as proof of (((eq Prop) ((or (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((or (x x0)) (y x0))) as proof of (((eq Prop) ((or (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((or (x x0)) (y x0))) as proof of (((eq Prop) ((or (x x0)) (y x0))) 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 ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% 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 ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found eq_ref00:=(eq_ref0 ((or (x x0)) (y x0))):(((eq Prop) ((or (x x0)) (y x0))) ((or (x x0)) (y x0)))
% Found (eq_ref0 ((or (x x0)) (y x0))) as proof of (((eq Prop) ((or (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((or (x x0)) (y x0))) as proof of (((eq Prop) ((or (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((or (x x0)) (y x0))) as proof of (((eq Prop) ((or (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((or (x x0)) (y x0))) as proof of (((eq Prop) ((or (x x0)) (y x0))) 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 ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found eq_ref00:=(eq_ref0 ((or (x x0)) (y x0))):(((eq Prop) ((or (x x0)) (y x0))) ((or (x x0)) (y x0)))
% Found (eq_ref0 ((or (x x0)) (y x0))) as proof of (((eq Prop) ((or (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((or (x x0)) (y x0))) as proof of (((eq Prop) ((or (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((or (x x0)) (y x0))) as proof of (((eq Prop) ((or (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((or (x x0)) (y x0))) as proof of (((eq Prop) ((or (x x0)) (y x0))) 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 ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found eq_ref00:=(eq_ref0 ((or (x x0)) (y x0))):(((eq Prop) ((or (x x0)) (y x0))) ((or (x x0)) (y x0)))
% Found (eq_ref0 ((or (x x0)) (y x0))) as proof of (((eq Prop) ((or (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((or (x x0)) (y x0))) as proof of (((eq Prop) ((or (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((or (x x0)) (y x0))) as proof of (((eq Prop) ((or (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((or (x x0)) (y x0))) as proof of (((eq Prop) ((or (x x0)) (y x0))) 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 ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found eq_ref00:=(eq_ref0 ((or (x x0)) (y x0))):(((eq Prop) ((or (x x0)) (y x0))) ((or (x x0)) (y x0)))
% Found (eq_ref0 ((or (x x0)) (y x0))) as proof of (((eq Prop) ((or (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((or (x x0)) (y x0))) as proof of (((eq Prop) ((or (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((or (x x0)) (y x0))) as proof of (((eq Prop) ((or (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((or (x x0)) (y x0))) as proof of (((eq Prop) ((or (x x0)) (y x0))) b)
% Found eq_ref00:=(eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))):(((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found (eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b0)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b0)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b0)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) ((or (x x0)) (y x0)))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((or (x x0)) (y x0)))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((or (x x0)) (y x0)))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((or (x x0)) (y x0)))
% Found eq_ref00:=(eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))):(((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found (eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b0)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b0)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b0)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) ((or (x x0)) (y x0)))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((or (x x0)) (y x0)))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((or (x x0)) (y x0)))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((or (x x0)) (y x0)))
% Found eq_ref00:=(eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))):(((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found (eq_ref0 ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b0)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b0)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b0)
% Found ((eq_ref Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) as proof of (((eq Prop) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0))))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) ((or (x x0)) (y x0)))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((or (x x0)) (y x0)))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((or (x x0)) (y x0)))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((or (x x0)) (y x0)))
% Found x11:(P b)
% Found (fun (x11:(P b))=> x11) as proof of (P b)
% Found (fun (x11:(P b))=> x11) as proof of (P0 b)
% Found x11:(P b)
% Found (fun (x11:(P b))=> x11) as proof of (P b)
% Found (fun (x11:(P b))=> x11) as proof of (P0 b)
% Found x12:(P ((or (x x0)) (y x0)))
% Found (fun (x12:(P ((or (x x0)) (y x0))))=> x12) as proof of (P ((or (x x0)) (y x0)))
% Found (fun (x12:(P ((or (x x0)) (y x0))))=> x12) as proof of (P0 ((or (x x0)) (y x0)))
% Found eq_ref00:=(eq_ref0 ((or (x x0)) (y x0))):(((eq Prop) ((or (x x0)) (y x0))) ((or (x x0)) (y x0)))
% Found (eq_ref0 ((or (x x0)) (y x0))) as proof of (((eq Prop) ((or (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((or (x x0)) (y x0))) as proof of (((eq Prop) ((or (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((or (x x0)) (y x0))) as proof of (((eq Prop) ((or (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((or (x x0)) (y x0))) as proof of (((eq Prop) ((or (x x0)) (y x0))) 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 ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))) (S x0)))))
% Found x12:(P ((or (x x0)) (y x0)))
% Found (fun (x12:(P ((or (x x0)) (y x0))))=> x12) as proof of (P ((or (x x0)) (y x0)))
% Found (fun (x12:(P ((or (x x0)) (y x0))))=> x12) as proof of (P0 ((or (x x0)) (y x0)))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop)
% EOF
%------------------------------------------------------------------------------