%------------------------------------------------------------------------------
% File     : cocATP---0.2.0
% Problem  : SEV231^5 : TPTP v6.1.0. Released v4.0.0.
% Transfm  : none
% Format   : tptp:raw
% Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p

% Computer : n189.star.cs.uiowa.edu
% Model    : x86_64 x86_64
% CPU      : Intel(R) Xeon(R) CPU E5-2609 0 2.40GHz
% Memory   : 32286.75MB
% OS       : Linux 2.6.32-431.20.3.el6.x86_64
% CPULimit : 300s
% DateTime : Thu Jul 17 13:33:54 EDT 2014

% Result   : Timeout 300.03s
% Output   : None 
% Verified : 
% SZS Type : None (Parsing solution fails)
% Syntax   : Number of formulae    : 0

% Comments : 
%------------------------------------------------------------------------------
%----NO SOLUTION OUTPUT BY SYSTEM
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% % Problem  : SEV231^5 : TPTP v6.1.0. Released v4.0.0.
% % Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n189.star.cs.uiowa.edu
% % Model    : x86_64 x86_64
% % CPU      : Intel(R) Xeon(R) CPU E5-2609 0 @ 2.40GHz
% % Memory   : 32286.75MB
% % OS       : Linux 2.6.32-431.20.3.el6.x86_64
% % CPULimit : 300
% % DateTime : Thu Jul 17 08:34:06 CDT 2014
% % CPUTime  : 300.03 
% 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 0x2770290>, <kernel.Type object at 0x2770bd8>) of role type named a_type
% Using role type
% Declaring a:Type
% FOF formula (<kernel.Constant object at 0x250f368>, <kernel.DependentProduct object at 0x2770248>) of role type named y
% Using role type
% Declaring y:(a->Prop)
% FOF formula (<kernel.Constant object at 0x250f368>, <kernel.DependentProduct object at 0x27704d0>) of role type named x
% Using role type
% Declaring x:(a->Prop)
% FOF formula (((eq (a->Prop)) (fun (Xx0:a)=> ((and (x Xx0)) (y Xx0)))) (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))) of role conjecture named cX5201_pme
% Conjecture to prove = (((eq (a->Prop)) (fun (Xx0:a)=> ((and (x Xx0)) (y Xx0)))) (fun (Xx0:a)=> (forall (S:(a->Prop)), (((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 (Xx0:a)=> ((and (x Xx0)) (y Xx0)))) (fun (Xx0:a)=> (forall (S:(a->Prop)), (((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 (Xx0:a)=> ((and (x Xx0)) (y Xx0)))) (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0)))))
% Found eq_ref00:=(eq_ref0 (fun (Xx0:a)=> ((and (x Xx0)) (y Xx0)))):(((eq (a->Prop)) (fun (Xx0:a)=> ((and (x Xx0)) (y Xx0)))) (fun (Xx0:a)=> ((and (x Xx0)) (y Xx0))))
% Found (eq_ref0 (fun (Xx0:a)=> ((and (x Xx0)) (y Xx0)))) as proof of (((eq (a->Prop)) (fun (Xx0:a)=> ((and (x Xx0)) (y Xx0)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx0:a)=> ((and (x Xx0)) (y Xx0)))) as proof of (((eq (a->Prop)) (fun (Xx0:a)=> ((and (x Xx0)) (y Xx0)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx0:a)=> ((and (x Xx0)) (y Xx0)))) as proof of (((eq (a->Prop)) (fun (Xx0:a)=> ((and (x Xx0)) (y Xx0)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx0:a)=> ((and (x Xx0)) (y Xx0)))) as proof of (((eq (a->Prop)) (fun (Xx0:a)=> ((and (x Xx0)) (y 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 (Xx0:a)=> (forall (S:(a->Prop)), (((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)=> (forall (S:(a->Prop)), (((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)=> (forall (S:(a->Prop)), (((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)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0)))))
% Found x01:(P (fun (Xx0:a)=> ((and (x Xx0)) (y Xx0))))
% Found (fun (x01:(P (fun (Xx0:a)=> ((and (x Xx0)) (y Xx0)))))=> x01) as proof of (P (fun (Xx0:a)=> ((and (x Xx0)) (y Xx0))))
% Found (fun (x01:(P (fun (Xx0:a)=> ((and (x Xx0)) (y Xx0)))))=> x01) as proof of (P0 (fun (Xx0:a)=> ((and (x Xx0)) (y Xx0))))
% Found x01:(P (fun (Xx0:a)=> ((and (x Xx0)) (y Xx0))))
% Found (fun (x01:(P (fun (Xx0:a)=> ((and (x Xx0)) (y Xx0)))))=> x01) as proof of (P (fun (Xx0:a)=> ((and (x Xx0)) (y Xx0))))
% Found (fun (x01:(P (fun (Xx0:a)=> ((and (x Xx0)) (y Xx0)))))=> x01) as proof of (P0 (fun (Xx0:a)=> ((and (x Xx0)) (y Xx0))))
% Found x01:(P (fun (Xx0:a)=> ((and (x Xx0)) (y Xx0))))
% Found (fun (x01:(P (fun (Xx0:a)=> ((and (x Xx0)) (y Xx0)))))=> x01) as proof of (P (fun (Xx0:a)=> ((and (x Xx0)) (y Xx0))))
% Found (fun (x01:(P (fun (Xx0:a)=> ((and (x Xx0)) (y Xx0)))))=> x01) as proof of (P0 (fun (Xx0:a)=> ((and (x Xx0)) (y Xx0))))
% 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)=> ((and (x Xx0)) (y Xx0))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx0:a)=> ((and (x Xx0)) (y Xx0))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx0:a)=> ((and (x Xx0)) (y Xx0))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx0:a)=> ((and (x Xx0)) (y Xx0))))
% Found eta_expansion000:=(eta_expansion00 (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))):(((eq (a->Prop)) (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))) (fun (x0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))))
% Found (eta_expansion00 (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))) as proof of (((eq (a->Prop)) (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))) b)
% Found ((eta_expansion0 Prop) (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))) as proof of (((eq (a->Prop)) (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))) b)
% Found (((eta_expansion a) Prop) (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))) as proof of (((eq (a->Prop)) (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))) b)
% Found (((eta_expansion a) Prop) (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))) as proof of (((eq (a->Prop)) (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))) b)
% Found (((eta_expansion a) Prop) (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))) as proof of (((eq (a->Prop)) (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))) b)
% Found eta_expansion0000:=(eta_expansion000 P):((P (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0)))))->(P (fun (x0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))))
% Found (eta_expansion000 P) as proof of (P0 (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0)))))
% Found ((eta_expansion00 (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))) P) as proof of (P0 (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0)))))
% Found (((eta_expansion0 Prop) (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))) P) as proof of (P0 (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0)))))
% Found ((((eta_expansion a) Prop) (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))) P) as proof of (P0 (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0)))))
% Found ((((eta_expansion a) Prop) (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))) P) as proof of (P0 (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0)))))
% Found eq_ref000:=(eq_ref00 P):((P (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0)))))->(P (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))))
% Found (eq_ref00 P) as proof of (P0 (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0)))))
% Found ((eq_ref0 (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))) P) as proof of (P0 (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0)))))
% Found (((eq_ref (a->Prop)) (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))) P) as proof of (P0 (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0)))))
% Found (((eq_ref (a->Prop)) (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))) P) as proof of (P0 (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0)))))
% Found eq_ref00:=(eq_ref0 ((and (x x0)) (y x0))):(((eq Prop) ((and (x x0)) (y x0))) ((and (x x0)) (y x0)))
% Found (eq_ref0 ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (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) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found eq_ref00:=(eq_ref0 ((and (x x0)) (y x0))):(((eq Prop) ((and (x x0)) (y x0))) ((and (x x0)) (y x0)))
% Found (eq_ref0 ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (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) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found x11:(P ((and (x x0)) (y x0)))
% Found (fun (x11:(P ((and (x x0)) (y x0))))=> x11) as proof of (P ((and (x x0)) (y x0)))
% Found (fun (x11:(P ((and (x x0)) (y x0))))=> x11) as proof of (P0 ((and (x x0)) (y x0)))
% Found x11:(P ((and (x x0)) (y x0)))
% Found (fun (x11:(P ((and (x x0)) (y x0))))=> x11) as proof of (P ((and (x x0)) (y x0)))
% Found (fun (x11:(P ((and (x x0)) (y x0))))=> x11) as proof of (P0 ((and (x x0)) (y x0)))
% Found eq_ref000:=(eq_ref00 P):((P (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))->(P (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))))
% Found (eq_ref00 P) as proof of (P0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found ((eq_ref0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) P) as proof of (P0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found (((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) P) as proof of (P0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found (((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) P) as proof of (P0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found eq_ref000:=(eq_ref00 P):((P (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))->(P (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))))
% Found (eq_ref00 P) as proof of (P0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found ((eq_ref0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) P) as proof of (P0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found (((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) P) as proof of (P0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found (((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) P) as proof of (P0 (forall (S:(a->Prop)), (((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) ((and (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (x x0)) (y x0)))
% Found eq_ref00:=(eq_ref0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))):(((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found (eq_ref0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b)
% Found eq_ref00:=(eq_ref0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))):(((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found (eq_ref0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((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) ((and (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (x x0)) (y x0)))
% Found eq_ref00:=(eq_ref0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))):(((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found (eq_ref0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((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) ((and (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (x x0)) (y x0)))
% Found eq_ref00:=(eq_ref0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))):(((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found (eq_ref0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((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) ((and (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (x x0)) (y x0)))
% Found x0:(P (fun (Xx0:a)=> ((and (x Xx0)) (y Xx0))))
% Instantiate: b:=(fun (Xx0:a)=> ((and (x Xx0)) (y Xx0))):(a->Prop)
% Found x0 as proof of (P0 b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))):(((eq (a->Prop)) (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))) (fun (x0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))))
% Found (eta_expansion_dep00 (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))) as proof of (((eq (a->Prop)) (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))) b)
% Found ((eta_expansion_dep0 (fun (x2:a)=> Prop)) (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))) as proof of (((eq (a->Prop)) (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))) b)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))) as proof of (((eq (a->Prop)) (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))) b)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))) as proof of (((eq (a->Prop)) (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))) b)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))) as proof of (((eq (a->Prop)) (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))) b)
% Found x0:(P (fun (Xx0:a)=> ((and (x Xx0)) (y Xx0))))
% Instantiate: f:=(fun (Xx0:a)=> ((and (x Xx0)) (y 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)) (forall (S:(a->Prop)), (((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)) (forall (S:(a->Prop)), (((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)) (forall (S:(a->Prop)), (((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)) (forall (S:(a->Prop)), (((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)) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))))
% Found x0:(P (fun (Xx0:a)=> ((and (x Xx0)) (y Xx0))))
% Instantiate: f:=(fun (Xx0:a)=> ((and (x Xx0)) (y 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)) (forall (S:(a->Prop)), (((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)) (forall (S:(a->Prop)), (((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)) (forall (S:(a->Prop)), (((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)) (forall (S:(a->Prop)), (((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)) (forall (S:(a->Prop)), (((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) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found eq_ref00:=(eq_ref0 ((and (x x0)) (y x0))):(((eq Prop) ((and (x x0)) (y x0))) ((and (x x0)) (y x0)))
% Found (eq_ref0 ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (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) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found eq_ref00:=(eq_ref0 ((and (x x0)) (y x0))):(((eq Prop) ((and (x x0)) (y x0))) ((and (x x0)) (y x0)))
% Found (eq_ref0 ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (x x0)) (y x0))) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xx0:a)=> ((and (x Xx0)) (y Xx0)))):(((eq (a->Prop)) (fun (Xx0:a)=> ((and (x Xx0)) (y Xx0)))) (fun (x0:a)=> ((and (x x0)) (y x0))))
% Found (eta_expansion_dep00 (fun (Xx0:a)=> ((and (x Xx0)) (y Xx0)))) as proof of (((eq (a->Prop)) (fun (Xx0:a)=> ((and (x Xx0)) (y Xx0)))) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xx0:a)=> ((and (x Xx0)) (y Xx0)))) as proof of (((eq (a->Prop)) (fun (Xx0:a)=> ((and (x Xx0)) (y Xx0)))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx0:a)=> ((and (x Xx0)) (y Xx0)))) as proof of (((eq (a->Prop)) (fun (Xx0:a)=> ((and (x Xx0)) (y Xx0)))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx0:a)=> ((and (x Xx0)) (y Xx0)))) as proof of (((eq (a->Prop)) (fun (Xx0:a)=> ((and (x Xx0)) (y Xx0)))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx0:a)=> ((and (x Xx0)) (y Xx0)))) as proof of (((eq (a->Prop)) (fun (Xx0:a)=> ((and (x Xx0)) (y 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 (Xx0:a)=> (forall (S:(a->Prop)), (((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)=> (forall (S:(a->Prop)), (((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)=> (forall (S:(a->Prop)), (((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)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0)))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))):(((eq (a->Prop)) (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))) (fun (x0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))))
% Found (eta_expansion_dep00 (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))) as proof of (((eq (a->Prop)) (fun (Xx0:a)=> (forall (S:(a->Prop)), (((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)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))) as proof of (((eq (a->Prop)) (fun (Xx0:a)=> (forall (S:(a->Prop)), (((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)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))) as proof of (((eq (a->Prop)) (fun (Xx0:a)=> (forall (S:(a->Prop)), (((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)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))) as proof of (((eq (a->Prop)) (fun (Xx0:a)=> (forall (S:(a->Prop)), (((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)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))) as proof of (((eq (a->Prop)) (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))) b)
% Found x0:(P0 b)
% Instantiate: b:=(fun (Xx0:a)=> ((and (x Xx0)) (y Xx0))):(a->Prop)
% Found (fun (x0:(P0 b))=> x0) as proof of (P0 (fun (Xx0:a)=> ((and (x Xx0)) (y Xx0))))
% Found (fun (P0:((a->Prop)->Prop)) (x0:(P0 b))=> x0) as proof of ((P0 b)->(P0 (fun (Xx0:a)=> ((and (x Xx0)) (y Xx0)))))
% Found (fun (P0:((a->Prop)->Prop)) (x0:(P0 b))=> x0) as proof of (P b)
% Found x0:(P (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0)))))
% Instantiate: b:=(fun (Xx0:a)=> (forall (S:(a->Prop)), (((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 (Xx0:a)=> ((and (x Xx0)) (y Xx0)))):(((eq (a->Prop)) (fun (Xx0:a)=> ((and (x Xx0)) (y Xx0)))) (fun (Xx0:a)=> ((and (x Xx0)) (y Xx0))))
% Found (eq_ref0 (fun (Xx0:a)=> ((and (x Xx0)) (y Xx0)))) as proof of (((eq (a->Prop)) (fun (Xx0:a)=> ((and (x Xx0)) (y Xx0)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx0:a)=> ((and (x Xx0)) (y Xx0)))) as proof of (((eq (a->Prop)) (fun (Xx0:a)=> ((and (x Xx0)) (y Xx0)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx0:a)=> ((and (x Xx0)) (y Xx0)))) as proof of (((eq (a->Prop)) (fun (Xx0:a)=> ((and (x Xx0)) (y Xx0)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx0:a)=> ((and (x Xx0)) (y Xx0)))) as proof of (((eq (a->Prop)) (fun (Xx0:a)=> ((and (x Xx0)) (y Xx0)))) b)
% Found x02:(P (fun (Xx0:a)=> ((and (x Xx0)) (y Xx0))))
% Found (fun (x02:(P (fun (Xx0:a)=> ((and (x Xx0)) (y Xx0)))))=> x02) as proof of (P (fun (Xx0:a)=> ((and (x Xx0)) (y Xx0))))
% Found (fun (x02:(P (fun (Xx0:a)=> ((and (x Xx0)) (y Xx0)))))=> x02) as proof of (P0 (fun (Xx0:a)=> ((and (x Xx0)) (y Xx0))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xx0:a)=> ((and (x Xx0)) (y Xx0)))):(((eq (a->Prop)) (fun (Xx0:a)=> ((and (x Xx0)) (y Xx0)))) (fun (x0:a)=> ((and (x x0)) (y x0))))
% Found (eta_expansion_dep00 (fun (Xx0:a)=> ((and (x Xx0)) (y Xx0)))) as proof of (((eq (a->Prop)) (fun (Xx0:a)=> ((and (x Xx0)) (y Xx0)))) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xx0:a)=> ((and (x Xx0)) (y Xx0)))) as proof of (((eq (a->Prop)) (fun (Xx0:a)=> ((and (x Xx0)) (y Xx0)))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx0:a)=> ((and (x Xx0)) (y Xx0)))) as proof of (((eq (a->Prop)) (fun (Xx0:a)=> ((and (x Xx0)) (y Xx0)))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx0:a)=> ((and (x Xx0)) (y Xx0)))) as proof of (((eq (a->Prop)) (fun (Xx0:a)=> ((and (x Xx0)) (y Xx0)))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx0:a)=> ((and (x Xx0)) (y Xx0)))) as proof of (((eq (a->Prop)) (fun (Xx0:a)=> ((and (x Xx0)) (y 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 (Xx0:a)=> (forall (S:(a->Prop)), (((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)=> (forall (S:(a->Prop)), (((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)=> (forall (S:(a->Prop)), (((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)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0)))))
% Found x0:(P (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0)))))
% Instantiate: f:=(fun (Xx0:a)=> (forall (S:(a->Prop)), (((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)) ((and (x x00)) (y x00)))
% Found ((eq_ref Prop) (f x00)) as proof of (((eq Prop) (f x00)) ((and (x x00)) (y x00)))
% Found ((eq_ref Prop) (f x00)) as proof of (((eq Prop) (f x00)) ((and (x x00)) (y x00)))
% Found (fun (x00:a)=> ((eq_ref Prop) (f x00))) as proof of (((eq Prop) (f x00)) ((and (x x00)) (y x00)))
% Found (fun (x00:a)=> ((eq_ref Prop) (f x00))) as proof of (forall (x0:a), (((eq Prop) (f x0)) ((and (x x0)) (y x0))))
% Found x0:(P (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0)))))
% Instantiate: f:=(fun (Xx0:a)=> (forall (S:(a->Prop)), (((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)) ((and (x x00)) (y x00)))
% Found ((eq_ref Prop) (f x00)) as proof of (((eq Prop) (f x00)) ((and (x x00)) (y x00)))
% Found ((eq_ref Prop) (f x00)) as proof of (((eq Prop) (f x00)) ((and (x x00)) (y x00)))
% Found (fun (x00:a)=> ((eq_ref Prop) (f x00))) as proof of (((eq Prop) (f x00)) ((and (x x00)) (y x00)))
% Found (fun (x00:a)=> ((eq_ref Prop) (f x00))) as proof of (forall (x0:a), (((eq Prop) (f x0)) ((and (x x0)) (y x0))))
% Found x1:(P ((and (x x0)) (y x0)))
% Instantiate: b:=((and (x x0)) (y x0)):Prop
% Found x1 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))):(((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found (eq_ref0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b)
% Found x1:(P ((and (x x0)) (y x0)))
% Instantiate: b:=((and (x x0)) (y x0)):Prop
% Found x1 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))):(((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found (eq_ref0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b)
% Found eq_ref00:=(eq_ref0 (fun (Xx0:a)=> ((and (x Xx0)) (y Xx0)))):(((eq (a->Prop)) (fun (Xx0:a)=> ((and (x Xx0)) (y Xx0)))) (fun (Xx0:a)=> ((and (x Xx0)) (y Xx0))))
% Found (eq_ref0 (fun (Xx0:a)=> ((and (x Xx0)) (y Xx0)))) as proof of (((eq (a->Prop)) (fun (Xx0:a)=> ((and (x Xx0)) (y Xx0)))) b0)
% Found ((eq_ref (a->Prop)) (fun (Xx0:a)=> ((and (x Xx0)) (y Xx0)))) as proof of (((eq (a->Prop)) (fun (Xx0:a)=> ((and (x Xx0)) (y Xx0)))) b0)
% Found ((eq_ref (a->Prop)) (fun (Xx0:a)=> ((and (x Xx0)) (y Xx0)))) as proof of (((eq (a->Prop)) (fun (Xx0:a)=> ((and (x Xx0)) (y Xx0)))) b0)
% Found ((eq_ref (a->Prop)) (fun (Xx0:a)=> ((and (x Xx0)) (y Xx0)))) as proof of (((eq (a->Prop)) (fun (Xx0:a)=> ((and (x Xx0)) (y 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) (fun (Xx0:a)=> (forall (S:(a->Prop)), (((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)=> (forall (S:(a->Prop)), (((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)=> (forall (S:(a->Prop)), (((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)=> (forall (S:(a->Prop)), (((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)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0)))))
% 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)=> ((and (x Xx0)) (y Xx0))))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx0:a)=> ((and (x Xx0)) (y Xx0))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx0:a)=> ((and (x Xx0)) (y Xx0))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx0:a)=> ((and (x Xx0)) (y Xx0))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx0:a)=> ((and (x Xx0)) (y Xx0))))
% Found eq_ref00:=(eq_ref0 (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))):(((eq (a->Prop)) (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))) (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0)))))
% Found (eq_ref0 (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))) as proof of (((eq (a->Prop)) (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))) as proof of (((eq (a->Prop)) (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))) as proof of (((eq (a->Prop)) (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))) as proof of (((eq (a->Prop)) (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))) 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)=> ((and (x Xx0)) (y Xx0))))
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx0:a)=> ((and (x Xx0)) (y Xx0))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx0:a)=> ((and (x Xx0)) (y Xx0))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx0:a)=> ((and (x Xx0)) (y Xx0))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx0:a)=> ((and (x Xx0)) (y Xx0))))
% Found eq_ref00:=(eq_ref0 (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))):(((eq (a->Prop)) (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))) (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0)))))
% Found (eq_ref0 (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))) as proof of (((eq (a->Prop)) (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))) as proof of (((eq (a->Prop)) (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))) as proof of (((eq (a->Prop)) (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))) as proof of (((eq (a->Prop)) (fun (Xx0:a)=> (forall (S:(a->Prop)), (((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 (Xx0:a)=> ((and (x Xx0)) (y Xx0))))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx0:a)=> ((and (x Xx0)) (y Xx0))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx0:a)=> ((and (x Xx0)) (y Xx0))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx0:a)=> ((and (x Xx0)) (y Xx0))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx0:a)=> ((and (x Xx0)) (y Xx0))))
% Found eq_ref00:=(eq_ref0 (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))):(((eq (a->Prop)) (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))) (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0)))))
% Found (eq_ref0 (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))) as proof of (((eq (a->Prop)) (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))) as proof of (((eq (a->Prop)) (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))) as proof of (((eq (a->Prop)) (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))) as proof of (((eq (a->Prop)) (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))) b)
% Found eq_ref000:=(eq_ref00 P):((P (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0)))))->(P (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))))
% Found (eq_ref00 P) as proof of (P0 (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0)))))
% Found ((eq_ref0 (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))) P) as proof of (P0 (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0)))))
% Found (((eq_ref (a->Prop)) (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))) P) as proof of (P0 (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0)))))
% Found (((eq_ref (a->Prop)) (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))) P) as proof of (P0 (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0)))))
% 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)=> ((and (x Xx0)) (y Xx0))))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx0:a)=> ((and (x Xx0)) (y Xx0))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx0:a)=> ((and (x Xx0)) (y Xx0))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx0:a)=> ((and (x Xx0)) (y Xx0))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx0:a)=> ((and (x Xx0)) (y Xx0))))
% Found eq_ref00:=(eq_ref0 (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))):(((eq (a->Prop)) (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))) (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0)))))
% Found (eq_ref0 (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))) as proof of (((eq (a->Prop)) (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))) as proof of (((eq (a->Prop)) (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))) as proof of (((eq (a->Prop)) (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))) as proof of (((eq (a->Prop)) (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))) b)
% Found eta_expansion0000:=(eta_expansion000 P1):((P1 (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0)))))->(P1 (fun (x0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))))
% Found (eta_expansion000 P1) as proof of (P2 (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0)))))
% Found ((eta_expansion00 (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))) P1) as proof of (P2 (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0)))))
% Found (((eta_expansion0 Prop) (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))) P1) as proof of (P2 (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0)))))
% Found ((((eta_expansion a) Prop) (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))) P1) as proof of (P2 (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0)))))
% Found ((((eta_expansion a) Prop) (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))) P1) as proof of (P2 (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0)))))
% Found eta_expansion0000:=(eta_expansion000 P1):((P1 (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0)))))->(P1 (fun (x0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))))
% Found (eta_expansion000 P1) as proof of (P2 (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0)))))
% Found ((eta_expansion00 (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))) P1) as proof of (P2 (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0)))))
% Found (((eta_expansion0 Prop) (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))) P1) as proof of (P2 (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0)))))
% Found ((((eta_expansion a) Prop) (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))) P1) as proof of (P2 (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0)))))
% Found ((((eta_expansion a) Prop) (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))) P1) as proof of (P2 (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0)))))
% Found eta_expansion0000:=(eta_expansion000 P1):((P1 (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0)))))->(P1 (fun (x0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))))
% Found (eta_expansion000 P1) as proof of (P2 (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0)))))
% Found ((eta_expansion00 (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))) P1) as proof of (P2 (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0)))))
% Found (((eta_expansion0 Prop) (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))) P1) as proof of (P2 (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0)))))
% Found ((((eta_expansion a) Prop) (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))) P1) as proof of (P2 (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0)))))
% Found ((((eta_expansion a) Prop) (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))) P1) as proof of (P2 (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0)))))
% Found eta_expansion0000:=(eta_expansion000 P1):((P1 (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0)))))->(P1 (fun (x0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))))
% Found (eta_expansion000 P1) as proof of (P2 (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0)))))
% Found ((eta_expansion00 (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))) P1) as proof of (P2 (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0)))))
% Found (((eta_expansion0 Prop) (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))) P1) as proof of (P2 (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0)))))
% Found ((((eta_expansion a) Prop) (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))) P1) as proof of (P2 (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0)))))
% Found ((((eta_expansion a) Prop) (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))) P1) as proof of (P2 (fun (Xx0:a)=> (forall (S:(a->Prop)), (((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 ((and (x x0)) (y x0))):(((eq Prop) ((and (x x0)) (y x0))) ((and (x x0)) (y x0)))
% Found (eq_ref0 ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (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) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found eq_ref00:=(eq_ref0 ((and (x x0)) (y x0))):(((eq Prop) ((and (x x0)) (y x0))) ((and (x x0)) (y x0)))
% Found (eq_ref0 ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (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) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found eq_ref00:=(eq_ref0 ((and (x x0)) (y x0))):(((eq Prop) ((and (x x0)) (y x0))) ((and (x x0)) (y x0)))
% Found (eq_ref0 ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (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) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found eq_ref00:=(eq_ref0 ((and (x x0)) (y x0))):(((eq Prop) ((and (x x0)) (y x0))) ((and (x x0)) (y x0)))
% Found (eq_ref0 ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (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) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found eq_ref00:=(eq_ref0 ((and (x x0)) (y x0))):(((eq Prop) ((and (x x0)) (y x0))) ((and (x x0)) (y x0)))
% Found (eq_ref0 ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (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) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found eq_ref00:=(eq_ref0 ((and (x x0)) (y x0))):(((eq Prop) ((and (x x0)) (y x0))) ((and (x x0)) (y x0)))
% Found (eq_ref0 ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (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) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (S:(a->Prop)), (((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 ((and (x x0)) (y x0)))
% Found (fun (x11:(P1 ((and (x x0)) (y x0))))=> x11) as proof of (P1 ((and (x x0)) (y x0)))
% Found (fun (x11:(P1 ((and (x x0)) (y x0))))=> x11) as proof of (P2 ((and (x x0)) (y x0)))
% Found x11:(P1 ((and (x x0)) (y x0)))
% Found (fun (x11:(P1 ((and (x x0)) (y x0))))=> x11) as proof of (P1 ((and (x x0)) (y x0)))
% Found (fun (x11:(P1 ((and (x x0)) (y x0))))=> x11) as proof of (P2 ((and (x x0)) (y x0)))
% Found x11:(P1 ((and (x x0)) (y x0)))
% Found (fun (x11:(P1 ((and (x x0)) (y x0))))=> x11) as proof of (P1 ((and (x x0)) (y x0)))
% Found (fun (x11:(P1 ((and (x x0)) (y x0))))=> x11) as proof of (P2 ((and (x x0)) (y x0)))
% Found x11:(P1 ((and (x x0)) (y x0)))
% Found (fun (x11:(P1 ((and (x x0)) (y x0))))=> x11) as proof of (P1 ((and (x x0)) (y x0)))
% Found (fun (x11:(P1 ((and (x x0)) (y x0))))=> x11) as proof of (P2 ((and (x x0)) (y 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 eta_expansion000:=(eta_expansion00 (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))):(((eq (a->Prop)) (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))) (fun (x0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))))
% Found (eta_expansion00 (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))) as proof of (((eq (a->Prop)) (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))) b0)
% Found ((eta_expansion0 Prop) (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))) as proof of (((eq (a->Prop)) (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))) b0)
% Found (((eta_expansion a) Prop) (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))) as proof of (((eq (a->Prop)) (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))) b0)
% Found (((eta_expansion a) Prop) (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))) as proof of (((eq (a->Prop)) (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))) b0)
% Found (((eta_expansion a) Prop) (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))) as proof of (((eq (a->Prop)) (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))) b0)
% Found eta_expansion0000:=(eta_expansion000 P):((P (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0)))))->(P (fun (x0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))))
% Found (eta_expansion000 P) as proof of (P0 (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0)))))
% Found ((eta_expansion00 (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))) P) as proof of (P0 (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0)))))
% Found (((eta_expansion0 Prop) (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))) P) as proof of (P0 (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0)))))
% Found ((((eta_expansion a) Prop) (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))) P) as proof of (P0 (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0)))))
% Found ((((eta_expansion a) Prop) (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))) P) as proof of (P0 (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0)))))
% Found eq_ref00:=(eq_ref0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))):(((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found (eq_ref0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b)
% Found x1:(P0 b)
% Instantiate: b:=((and (x x0)) (y x0)):Prop
% Found (fun (x1:(P0 b))=> x1) as proof of (P0 ((and (x x0)) (y x0)))
% Found (fun (P0:(Prop->Prop)) (x1:(P0 b))=> x1) as proof of ((P0 b)->(P0 ((and (x x0)) (y x0))))
% Found (fun (P0:(Prop->Prop)) (x1:(P0 b))=> x1) as proof of (P b)
% Found eq_ref00:=(eq_ref0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))):(((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found (eq_ref0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b)
% Found x1:(P0 b)
% Instantiate: b:=((and (x x0)) (y x0)):Prop
% Found (fun (x1:(P0 b))=> x1) as proof of (P0 ((and (x x0)) (y x0)))
% Found (fun (P0:(Prop->Prop)) (x1:(P0 b))=> x1) as proof of ((P0 b)->(P0 ((and (x x0)) (y x0))))
% Found (fun (P0:(Prop->Prop)) (x1:(P0 b))=> x1) as proof of (P b)
% Found x12:(P ((and (x x0)) (y x0)))
% Found (fun (x12:(P ((and (x x0)) (y x0))))=> x12) as proof of (P ((and (x x0)) (y x0)))
% Found (fun (x12:(P ((and (x x0)) (y x0))))=> x12) as proof of (P0 ((and (x x0)) (y x0)))
% Found eq_ref00:=(eq_ref0 ((and (x x0)) (y x0))):(((eq Prop) ((and (x x0)) (y x0))) ((and (x x0)) (y x0)))
% Found (eq_ref0 ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (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) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found x12:(P ((and (x x0)) (y x0)))
% Found (fun (x12:(P ((and (x x0)) (y x0))))=> x12) as proof of (P ((and (x x0)) (y x0)))
% Found (fun (x12:(P ((and (x x0)) (y x0))))=> x12) as proof of (P0 ((and (x x0)) (y x0)))
% Found eq_ref00:=(eq_ref0 ((and (x x0)) (y x0))):(((eq Prop) ((and (x x0)) (y x0))) ((and (x x0)) (y x0)))
% Found (eq_ref0 ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (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) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found x1:(P (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Instantiate: b:=(forall (S:(a->Prop)), (((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 ((and (x x0)) (y x0))):(((eq Prop) ((and (x x0)) (y x0))) ((and (x x0)) (y x0)))
% Found (eq_ref0 ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (x x0)) (y x0))) b)
% Found x1:(P (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Instantiate: b:=(forall (S:(a->Prop)), (((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 ((and (x x0)) (y x0))):(((eq Prop) ((and (x x0)) (y x0))) ((and (x x0)) (y x0)))
% Found (eq_ref0 ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (x x0)) (y x0))) b)
% Found eta_expansion000:=(eta_expansion00 (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))):(((eq (a->Prop)) (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))) (fun (x0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))))
% Found (eta_expansion00 (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))) as proof of (((eq (a->Prop)) (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))) b0)
% Found ((eta_expansion0 Prop) (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))) as proof of (((eq (a->Prop)) (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))) b0)
% Found (((eta_expansion a) Prop) (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))) as proof of (((eq (a->Prop)) (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))) b0)
% Found (((eta_expansion a) Prop) (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))) as proof of (((eq (a->Prop)) (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))) b0)
% Found (((eta_expansion a) Prop) (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))) as proof of (((eq (a->Prop)) (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))) b0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b0):(((eq (a->Prop)) b0) (fun (x:a)=> (b0 x)))
% Found (eta_expansion_dep00 b0) as proof of (((eq (a->Prop)) b0) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) b0) as proof of (((eq (a->Prop)) b0) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b0) as proof of (((eq (a->Prop)) b0) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b0) as proof of (((eq (a->Prop)) b0) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b0) as proof of (((eq (a->Prop)) b0) b)
% Found x1:(P (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Instantiate: b:=(forall (S:(a->Prop)), (((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 ((and (x x0)) (y x0))):(((eq Prop) ((and (x x0)) (y x0))) ((and (x x0)) (y x0)))
% Found (eq_ref0 ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (x x0)) (y x0))) b)
% Found x1:(P (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Instantiate: b:=(forall (S:(a->Prop)), (((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 ((and (x x0)) (y x0))):(((eq Prop) ((and (x x0)) (y x0))) ((and (x x0)) (y x0)))
% Found (eq_ref0 ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (x x0)) (y x0))) b)
% Found eta_expansion0000:=(eta_expansion000 P):((P (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0)))))->(P (fun (x0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))))
% Found (eta_expansion000 P) as proof of (P0 (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0)))))
% Found ((eta_expansion00 (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))) P) as proof of (P0 (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0)))))
% Found (((eta_expansion0 Prop) (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))) P) as proof of (P0 (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0)))))
% Found ((((eta_expansion a) Prop) (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))) P) as proof of (P0 (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0)))))
% Found ((((eta_expansion a) Prop) (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))) P) as proof of (P0 (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0)))))
% Found eq_ref000:=(eq_ref00 P):((P (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0)))))->(P (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))))
% Found (eq_ref00 P) as proof of (P0 (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0)))))
% Found ((eq_ref0 (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))) P) as proof of (P0 (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0)))))
% Found (((eq_ref (a->Prop)) (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))) P) as proof of (P0 (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0)))))
% Found (((eq_ref (a->Prop)) (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))) P) as proof of (P0 (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0)))))
% Found eq_ref000:=(eq_ref00 P):((P (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0)))))->(P (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))))
% Found (eq_ref00 P) as proof of (P0 (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0)))))
% Found ((eq_ref0 (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))) P) as proof of (P0 (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0)))))
% Found (((eq_ref (a->Prop)) (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))) P) as proof of (P0 (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0)))))
% Found (((eq_ref (a->Prop)) (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))) P) as proof of (P0 (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0)))))
% Found eta_expansion0000:=(eta_expansion000 P):((P (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0)))))->(P (fun (x0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))))
% Found (eta_expansion000 P) as proof of (P0 (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0)))))
% Found ((eta_expansion00 (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))) P) as proof of (P0 (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0)))))
% Found (((eta_expansion0 Prop) (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))) P) as proof of (P0 (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0)))))
% Found ((((eta_expansion a) Prop) (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))) P) as proof of (P0 (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0)))))
% Found ((((eta_expansion a) Prop) (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))) P) as proof of (P0 (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0)))))
% Found eq_ref00:=(eq_ref0 ((and (x x0)) (y x0))):(((eq Prop) ((and (x x0)) (y x0))) ((and (x x0)) (y x0)))
% Found (eq_ref0 ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (x x0)) (y x0))) b0)
% Found ((eq_ref Prop) ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (x x0)) (y x0))) b0)
% Found ((eq_ref Prop) ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (x x0)) (y x0))) b0)
% Found ((eq_ref Prop) ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (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) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found eq_ref00:=(eq_ref0 ((and (x x0)) (y x0))):(((eq Prop) ((and (x x0)) (y x0))) ((and (x x0)) (y x0)))
% Found (eq_ref0 ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (x x0)) (y x0))) b0)
% Found ((eq_ref Prop) ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (x x0)) (y x0))) b0)
% Found ((eq_ref Prop) ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (x x0)) (y x0))) b0)
% Found ((eq_ref Prop) ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (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) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found eq_ref00:=(eq_ref0 (fun (Xx0:a)=> ((and (x Xx0)) (y Xx0)))):(((eq (a->Prop)) (fun (Xx0:a)=> ((and (x Xx0)) (y Xx0)))) (fun (Xx0:a)=> ((and (x Xx0)) (y Xx0))))
% Found (eq_ref0 (fun (Xx0:a)=> ((and (x Xx0)) (y Xx0)))) as proof of (((eq (a->Prop)) (fun (Xx0:a)=> ((and (x Xx0)) (y Xx0)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx0:a)=> ((and (x Xx0)) (y Xx0)))) as proof of (((eq (a->Prop)) (fun (Xx0:a)=> ((and (x Xx0)) (y Xx0)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx0:a)=> ((and (x Xx0)) (y Xx0)))) as proof of (((eq (a->Prop)) (fun (Xx0:a)=> ((and (x Xx0)) (y Xx0)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx0:a)=> ((and (x Xx0)) (y Xx0)))) as proof of (((eq (a->Prop)) (fun (Xx0:a)=> ((and (x Xx0)) (y Xx0)))) 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)=> (forall (S:(a->Prop)), (((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)=> (forall (S:(a->Prop)), (((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)=> (forall (S:(a->Prop)), (((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)=> (forall (S:(a->Prop)), (((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)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0)))))
% Found x01:(P (fun (Xx0:a)=> ((and (x Xx0)) (y Xx0))))
% Found (fun (x01:(P (fun (Xx0:a)=> ((and (x Xx0)) (y Xx0)))))=> x01) as proof of (P (fun (Xx0:a)=> ((and (x Xx0)) (y Xx0))))
% Found (fun (x01:(P (fun (Xx0:a)=> ((and (x Xx0)) (y Xx0)))))=> x01) as proof of (P0 (fun (Xx0:a)=> ((and (x Xx0)) (y 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) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (forall (S:(a->Prop)), (((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) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found eq_ref000:=(eq_ref00 P1):((P1 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))->(P1 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))))
% Found (eq_ref00 P1) as proof of (P2 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found ((eq_ref0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) P1) as proof of (P2 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found (((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) P1) as proof of (P2 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found (((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) P1) as proof of (P2 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found eq_ref000:=(eq_ref00 P1):((P1 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))->(P1 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))))
% Found (eq_ref00 P1) as proof of (P2 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found ((eq_ref0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) P1) as proof of (P2 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found (((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) P1) as proof of (P2 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found (((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) P1) as proof of (P2 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found eq_ref000:=(eq_ref00 P1):((P1 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))->(P1 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))))
% Found (eq_ref00 P1) as proof of (P2 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found ((eq_ref0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) P1) as proof of (P2 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found (((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) P1) as proof of (P2 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found (((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) P1) as proof of (P2 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found eq_ref000:=(eq_ref00 P1):((P1 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))->(P1 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))))
% Found (eq_ref00 P1) as proof of (P2 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found ((eq_ref0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) P1) as proof of (P2 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found (((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) P1) as proof of (P2 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found (((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) P1) as proof of (P2 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found eq_ref000:=(eq_ref00 P1):((P1 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))->(P1 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))))
% Found (eq_ref00 P1) as proof of (P2 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found ((eq_ref0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) P1) as proof of (P2 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found (((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) P1) as proof of (P2 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found (((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) P1) as proof of (P2 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found eq_ref000:=(eq_ref00 P1):((P1 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))->(P1 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))))
% Found (eq_ref00 P1) as proof of (P2 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found ((eq_ref0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) P1) as proof of (P2 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found (((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) P1) as proof of (P2 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found (((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) P1) as proof of (P2 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found eq_ref000:=(eq_ref00 P1):((P1 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))->(P1 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))))
% Found (eq_ref00 P1) as proof of (P2 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found ((eq_ref0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) P1) as proof of (P2 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found (((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) P1) as proof of (P2 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found (((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) P1) as proof of (P2 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found eq_ref000:=(eq_ref00 P1):((P1 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))->(P1 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))))
% Found (eq_ref00 P1) as proof of (P2 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found ((eq_ref0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) P1) as proof of (P2 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found (((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) P1) as proof of (P2 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found (((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) P1) as proof of (P2 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found x01:(P (fun (Xx0:a)=> ((and (x Xx0)) (y Xx0))))
% Found (fun (x01:(P (fun (Xx0:a)=> ((and (x Xx0)) (y Xx0)))))=> x01) as proof of (P (fun (Xx0:a)=> ((and (x Xx0)) (y Xx0))))
% Found (fun (x01:(P (fun (Xx0:a)=> ((and (x Xx0)) (y Xx0)))))=> x01) as proof of (P0 (fun (Xx0:a)=> ((and (x Xx0)) (y Xx0))))
% Found x01:(P (fun (Xx0:a)=> ((and (x Xx0)) (y Xx0))))
% Found (fun (x01:(P (fun (Xx0:a)=> ((and (x Xx0)) (y Xx0)))))=> x01) as proof of (P (fun (Xx0:a)=> ((and (x Xx0)) (y Xx0))))
% Found (fun (x01:(P (fun (Xx0:a)=> ((and (x Xx0)) (y Xx0)))))=> x01) as proof of (P0 (fun (Xx0:a)=> ((and (x Xx0)) (y Xx0))))
% Found eq_ref000:=(eq_ref00 P):((P (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))->(P (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))))
% Found (eq_ref00 P) as proof of (P0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found ((eq_ref0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) P) as proof of (P0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found (((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) P) as proof of (P0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found (((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) P) as proof of (P0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found eq_ref000:=(eq_ref00 P):((P (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))->(P (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))))
% Found (eq_ref00 P) as proof of (P0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found ((eq_ref0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) P) as proof of (P0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found (((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) P) as proof of (P0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found (((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) P) as proof of (P0 (forall (S:(a->Prop)), (((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) ((and (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (x x0)) (y x0)))
% Found eq_ref00:=(eq_ref0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))):(((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found (eq_ref0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((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) ((and (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (x x0)) (y x0)))
% Found eq_ref00:=(eq_ref0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))):(((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found (eq_ref0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((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) ((and (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (x x0)) (y x0)))
% Found eq_ref00:=(eq_ref0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))):(((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found (eq_ref0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b)
% Found eq_ref00:=(eq_ref0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))):(((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found (eq_ref0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((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) ((and (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (x x0)) (y x0)))
% Found eq_ref00:=(eq_ref0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))):(((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found (eq_ref0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((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) ((and (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (x x0)) (y x0)))
% Found eq_ref00:=(eq_ref0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))):(((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found (eq_ref0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((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) ((and (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (x x0)) (y x0)))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (x x0)) (y x0)))
% Found eq_ref00:=(eq_ref0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))):(((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found (eq_ref0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((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) ((and (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (x x0)) (y x0)))
% Found eq_ref00:=(eq_ref0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))):(((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found (eq_ref0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b)
% Found eq_ref000:=(eq_ref00 P):((P (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))->(P (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))))
% Found (eq_ref00 P) as proof of (P0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found ((eq_ref0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) P) as proof of (P0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found (((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) P) as proof of (P0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found (((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) P) as proof of (P0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found eq_ref000:=(eq_ref00 P):((P (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))->(P (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))))
% Found (eq_ref00 P) as proof of (P0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found ((eq_ref0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) P) as proof of (P0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found (((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) P) as proof of (P0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found (((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) P) as proof of (P0 (forall (S:(a->Prop)), (((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) ((and (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (x x0)) (y x0)))
% Found eq_ref00:=(eq_ref0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))):(((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found (eq_ref0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((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) ((and (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (x x0)) (y x0)))
% Found eq_ref00:=(eq_ref0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))):(((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found (eq_ref0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((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) ((and (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (x x0)) (y x0)))
% Found eq_ref00:=(eq_ref0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))):(((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found (eq_ref0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b)
% Found eq_ref00:=(eq_ref0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))):(((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found (eq_ref0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((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) ((and (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (x x0)) (y x0)))
% Found eq_ref00:=(eq_ref0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))):(((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found (eq_ref0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((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) ((and (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (x x0)) (y x0)))
% Found eq_ref00:=(eq_ref0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))):(((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found (eq_ref0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((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) ((and (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (x x0)) (y x0)))
% Found eq_ref00:=(eq_ref0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))):(((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found (eq_ref0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((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) ((and (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (x x0)) (y x0)))
% Found eq_ref00:=(eq_ref0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))):(((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found (eq_ref0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((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) ((and (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (x x0)) (y x0)))
% Found eq_ref00:=(eq_ref0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))):(((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found (eq_ref0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((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) ((and (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (x x0)) (y x0)))
% Found eq_ref00:=(eq_ref0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))):(((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found (eq_ref0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((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) ((and (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (x x0)) (y x0)))
% Found eq_ref00:=(eq_ref0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))):(((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found (eq_ref0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((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) ((and (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (x x0)) (y x0)))
% Found eq_ref00:=(eq_ref0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))):(((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found (eq_ref0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((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) ((and (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (x x0)) (y x0)))
% Found eq_ref00:=(eq_ref0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))):(((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found (eq_ref0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((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) ((and (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (x x0)) (y x0)))
% Found eq_ref00:=(eq_ref0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))):(((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found (eq_ref0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((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) ((and (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (x x0)) (y x0)))
% Found eq_ref00:=(eq_ref0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))):(((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found (eq_ref0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((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) ((and (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (x x0)) (y x0)))
% Found eq_ref00:=(eq_ref0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))):(((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found (eq_ref0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((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) ((and (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (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)=> (forall (S:(a->Prop)), (((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)=> (forall (S:(a->Prop)), (((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)=> (forall (S:(a->Prop)), (((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)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0)))))
% Found eq_ref00:=(eq_ref0 ((and (x x0)) (y x0))):(((eq Prop) ((and (x x0)) (y x0))) ((and (x x0)) (y x0)))
% Found (eq_ref0 ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (x x0)) (y x0))) b)
% Found x1:(P0 b)
% Instantiate: b:=(forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))):Prop
% Found (fun (x1:(P0 b))=> x1) as proof of (P0 (forall (S:(a->Prop)), (((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 (forall (S:(a->Prop)), (((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 ((and (x x0)) (y x0))):(((eq Prop) ((and (x x0)) (y x0))) ((and (x x0)) (y x0)))
% Found (eq_ref0 ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (x x0)) (y x0))) b)
% Found x1:(P0 b)
% Instantiate: b:=(forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))):Prop
% Found (fun (x1:(P0 b))=> x1) as proof of (P0 (forall (S:(a->Prop)), (((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 (forall (S:(a->Prop)), (((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 ((and (x x0)) (y x0)))
% Instantiate: b:=((and (x x0)) (y x0)):Prop
% Found x1 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))):(((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found (eq_ref0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b)
% Found x1:(P ((and (x x0)) (y x0)))
% Instantiate: b:=((and (x x0)) (y x0)):Prop
% Found x1 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))):(((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found (eq_ref0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((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 (Xx0:a)=> ((and (x Xx0)) (y Xx0))))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx0:a)=> ((and (x Xx0)) (y Xx0))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx0:a)=> ((and (x Xx0)) (y Xx0))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx0:a)=> ((and (x Xx0)) (y Xx0))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx0:a)=> ((and (x Xx0)) (y Xx0))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))):(((eq (a->Prop)) (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))) (fun (x0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))))
% Found (eta_expansion_dep00 (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))) as proof of (((eq (a->Prop)) (fun (Xx0:a)=> (forall (S:(a->Prop)), (((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)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))) as proof of (((eq (a->Prop)) (fun (Xx0:a)=> (forall (S:(a->Prop)), (((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)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))) as proof of (((eq (a->Prop)) (fun (Xx0:a)=> (forall (S:(a->Prop)), (((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)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))) as proof of (((eq (a->Prop)) (fun (Xx0:a)=> (forall (S:(a->Prop)), (((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)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))) as proof of (((eq (a->Prop)) (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))) 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)=> ((and (x Xx0)) (y Xx0))))
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx0:a)=> ((and (x Xx0)) (y Xx0))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx0:a)=> ((and (x Xx0)) (y Xx0))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx0:a)=> ((and (x Xx0)) (y Xx0))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx0:a)=> ((and (x Xx0)) (y Xx0))))
% Found eta_expansion000:=(eta_expansion00 (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))):(((eq (a->Prop)) (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))) (fun (x0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))))
% Found (eta_expansion00 (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))) as proof of (((eq (a->Prop)) (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))) b)
% Found ((eta_expansion0 Prop) (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))) as proof of (((eq (a->Prop)) (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))) b)
% Found (((eta_expansion a) Prop) (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))) as proof of (((eq (a->Prop)) (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))) b)
% Found (((eta_expansion a) Prop) (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))) as proof of (((eq (a->Prop)) (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))) b)
% Found (((eta_expansion a) Prop) (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))) as proof of (((eq (a->Prop)) (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))) b)
% Found eta_expansion0000:=(eta_expansion000 P):((P (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0)))))->(P (fun (x0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))))
% Found (eta_expansion000 P) as proof of (P0 (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0)))))
% Found ((eta_expansion00 (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))) P) as proof of (P0 (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0)))))
% Found (((eta_expansion0 Prop) (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))) P) as proof of (P0 (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0)))))
% Found ((((eta_expansion a) Prop) (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))) P) as proof of (P0 (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0)))))
% Found ((((eta_expansion a) Prop) (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))) P) as proof of (P0 (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0)))))
% Found eq_ref000:=(eq_ref00 P):((P (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0)))))->(P (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))))
% Found (eq_ref00 P) as proof of (P0 (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0)))))
% Found ((eq_ref0 (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))) P) as proof of (P0 (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0)))))
% Found (((eq_ref (a->Prop)) (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))) P) as proof of (P0 (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0)))))
% Found (((eq_ref (a->Prop)) (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))) P) as proof of (P0 (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0)))))
% Found eq_ref000:=(eq_ref00 P):((P (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0)))))->(P (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))))
% Found (eq_ref00 P) as proof of (P0 (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0)))))
% Found ((eq_ref0 (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))) P) as proof of (P0 (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0)))))
% Found (((eq_ref (a->Prop)) (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))) P) as proof of (P0 (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0)))))
% Found (((eq_ref (a->Prop)) (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))) P) as proof of (P0 (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0)))))
% Found eta_expansion0000:=(eta_expansion000 P):((P (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0)))))->(P (fun (x0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))))
% Found (eta_expansion000 P) as proof of (P0 (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0)))))
% Found ((eta_expansion00 (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))) P) as proof of (P0 (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0)))))
% Found (((eta_expansion0 Prop) (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))) P) as proof of (P0 (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0)))))
% Found ((((eta_expansion a) Prop) (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))) P) as proof of (P0 (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0)))))
% Found ((((eta_expansion a) Prop) (fun (Xx0:a)=> (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S Xx0))))) P) as proof of (P0 (fun (Xx0:a)=> (forall (S:(a->Prop)), (((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 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))):(((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found (eq_ref0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b0)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b0)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b0)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b0)
% Found eq_ref000:=(eq_ref00 P):((P (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))->(P (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))))
% Found (eq_ref00 P) as proof of (P0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found ((eq_ref0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) P) as proof of (P0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found (((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) P) as proof of (P0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found (((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) P) as proof of (P0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found eq_ref00:=(eq_ref0 ((and (x x0)) (y x0))):(((eq Prop) ((and (x x0)) (y x0))) ((and (x x0)) (y x0)))
% Found (eq_ref0 ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (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) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found eq_ref00:=(eq_ref0 ((and (x x0)) (y x0))):(((eq Prop) ((and (x x0)) (y x0))) ((and (x x0)) (y x0)))
% Found (eq_ref0 ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (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) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found eq_ref00:=(eq_ref0 ((and (x x0)) (y x0))):(((eq Prop) ((and (x x0)) (y x0))) ((and (x x0)) (y x0)))
% Found (eq_ref0 ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (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) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found eq_ref00:=(eq_ref0 ((and (x x0)) (y x0))):(((eq Prop) ((and (x x0)) (y x0))) ((and (x x0)) (y x0)))
% Found (eq_ref0 ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (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) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found eq_ref00:=(eq_ref0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))):(((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found (eq_ref0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b0)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b0)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b0)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((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 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))):(((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found (eq_ref0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b0)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b0)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b0)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((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 ((and (x x0)) (y x0)))
% Found (fun (x11:(P ((and (x x0)) (y x0))))=> x11) as proof of (P ((and (x x0)) (y x0)))
% Found (fun (x11:(P ((and (x x0)) (y x0))))=> x11) as proof of (P0 ((and (x x0)) (y x0)))
% Found x11:(P ((and (x x0)) (y x0)))
% Found (fun (x11:(P ((and (x x0)) (y x0))))=> x11) as proof of (P ((and (x x0)) (y x0)))
% Found (fun (x11:(P ((and (x x0)) (y x0))))=> x11) as proof of (P0 ((and (x x0)) (y x0)))
% Found x11:(P ((and (x x0)) (y x0)))
% Found (fun (x11:(P ((and (x x0)) (y x0))))=> x11) as proof of (P ((and (x x0)) (y x0)))
% Found (fun (x11:(P ((and (x x0)) (y x0))))=> x11) as proof of (P0 ((and (x x0)) (y x0)))
% Found x11:(P ((and (x x0)) (y x0)))
% Found (fun (x11:(P ((and (x x0)) (y x0))))=> x11) as proof of (P ((and (x x0)) (y x0)))
% Found (fun (x11:(P ((and (x x0)) (y x0))))=> x11) as proof of (P0 ((and (x x0)) (y x0)))
% Found eq_ref000:=(eq_ref00 P):((P (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))->(P (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))))
% Found (eq_ref00 P) as proof of (P0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found ((eq_ref0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) P) as proof of (P0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found (((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) P) as proof of (P0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found (((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) P) as proof of (P0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found eq_ref000:=(eq_ref00 P):((P (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))->(P (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))))
% Found (eq_ref00 P) as proof of (P0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found ((eq_ref0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) P) as proof of (P0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found (((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) P) as proof of (P0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found (((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) P) as proof of (P0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found eq_ref000:=(eq_ref00 P):((P (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))->(P (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))))
% Found (eq_ref00 P) as proof of (P0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found ((eq_ref0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) P) as proof of (P0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found (((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) P) as proof of (P0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found (((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) P) as proof of (P0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found eq_ref000:=(eq_ref00 P):((P (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))->(P (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))))
% Found (eq_ref00 P) as proof of (P0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found ((eq_ref0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) P) as proof of (P0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found (((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) P) as proof of (P0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found (((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) P) as proof of (P0 (forall (S:(a->Prop)), (((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 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 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))):(((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found (eq_ref0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b0)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b0)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b0)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((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 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))):(((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found (eq_ref0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b0)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b0)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b0)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b0)
% Found eq_ref00:=(eq_ref0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))):(((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found (eq_ref0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b0)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b0)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b0)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((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 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))):(((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found (eq_ref0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b0)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b0)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b0)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((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 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))):(((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found (eq_ref0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b0)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b0)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b0)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((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 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))):(((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found (eq_ref0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b0)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b0)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b0)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((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 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) ((and (x x0)) (y x0)))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and (x x0)) (y x0)))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and (x x0)) (y x0)))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and (x x0)) (y x0)))
% Found eq_ref00:=(eq_ref0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))):(((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found (eq_ref0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b0)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b0)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b0)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b0)
% Found eq_ref00:=(eq_ref0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))):(((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found (eq_ref0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b0)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b0)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b0)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((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) ((and (x x0)) (y x0)))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and (x x0)) (y x0)))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and (x x0)) (y x0)))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and (x x0)) (y x0)))
% Found eq_ref00:=(eq_ref0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))):(((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found (eq_ref0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b0)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b0)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b0)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((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) ((and (x x0)) (y x0)))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and (x x0)) (y x0)))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and (x x0)) (y x0)))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and (x x0)) (y x0)))
% Found eq_ref00:=(eq_ref0 ((and (x x0)) (y x0))):(((eq Prop) ((and (x x0)) (y x0))) ((and (x x0)) (y x0)))
% Found (eq_ref0 ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (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) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (S:(a->Prop)), (((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) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found eq_ref00:=(eq_ref0 ((and (x x0)) (y x0))):(((eq Prop) ((and (x x0)) (y x0))) ((and (x x0)) (y x0)))
% Found (eq_ref0 ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (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) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found eq_ref00:=(eq_ref0 ((and (x x0)) (y x0))):(((eq Prop) ((and (x x0)) (y x0))) ((and (x x0)) (y x0)))
% Found (eq_ref0 ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (x x0)) (y x0))) b)
% Found eq_ref00:=(eq_ref0 ((and (x x0)) (y x0))):(((eq Prop) ((and (x x0)) (y x0))) ((and (x x0)) (y x0)))
% Found (eq_ref0 ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (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) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found eq_ref00:=(eq_ref0 ((and (x x0)) (y x0))):(((eq Prop) ((and (x x0)) (y x0))) ((and (x x0)) (y x0)))
% Found (eq_ref0 ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (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) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (S:(a->Prop)), (((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) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found eq_ref00:=(eq_ref0 ((and (x x0)) (y x0))):(((eq Prop) ((and (x x0)) (y x0))) ((and (x x0)) (y x0)))
% Found (eq_ref0 ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (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) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found eq_ref00:=(eq_ref0 ((and (x x0)) (y x0))):(((eq Prop) ((and (x x0)) (y x0))) ((and (x x0)) (y x0)))
% Found (eq_ref0 ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (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) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found eq_ref00:=(eq_ref0 ((and (x x0)) (y x0))):(((eq Prop) ((and (x x0)) (y x0))) ((and (x x0)) (y x0)))
% Found (eq_ref0 ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (x x0)) (y x0))) b)
% Found eq_ref00:=(eq_ref0 ((and (x x0)) (y x0))):(((eq Prop) ((and (x x0)) (y x0))) ((and (x x0)) (y x0)))
% Found (eq_ref0 ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (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) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (S:(a->Prop)), (((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) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found eq_ref00:=(eq_ref0 ((and (x x0)) (y x0))):(((eq Prop) ((and (x x0)) (y x0))) ((and (x x0)) (y x0)))
% Found (eq_ref0 ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (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) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found eq_ref00:=(eq_ref0 ((and (x x0)) (y x0))):(((eq Prop) ((and (x x0)) (y x0))) ((and (x x0)) (y x0)))
% Found (eq_ref0 ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (x x0)) (y x0))) b)
% Found eq_ref00:=(eq_ref0 ((and (x x0)) (y x0))):(((eq Prop) ((and (x x0)) (y x0))) ((and (x x0)) (y x0)))
% Found (eq_ref0 ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (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) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found eq_ref00:=(eq_ref0 ((and (x x0)) (y x0))):(((eq Prop) ((and (x x0)) (y x0))) ((and (x x0)) (y x0)))
% Found (eq_ref0 ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (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) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (S:(a->Prop)), (((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) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found eq_ref00:=(eq_ref0 ((and (x x0)) (y x0))):(((eq Prop) ((and (x x0)) (y x0))) ((and (x x0)) (y x0)))
% Found (eq_ref0 ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (x x0)) (y x0))) 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)
% 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 ((and (x x0)) (y x0)))
% Found (fun (x12:(P ((and (x x0)) (y x0))))=> x12) as proof of (P ((and (x x0)) (y x0)))
% Found (fun (x12:(P ((and (x x0)) (y x0))))=> x12) as proof of (P0 ((and (x x0)) (y x0)))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found eq_ref00:=(eq_ref0 ((and (x x0)) (y x0))):(((eq Prop) ((and (x x0)) (y x0))) ((and (x x0)) (y x0)))
% Found (eq_ref0 ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (x x0)) (y x0))) b)
% Found x12:(P ((and (x x0)) (y x0)))
% Found (fun (x12:(P ((and (x x0)) (y x0))))=> x12) as proof of (P ((and (x x0)) (y x0)))
% Found (fun (x12:(P ((and (x x0)) (y x0))))=> x12) as proof of (P0 ((and (x x0)) (y x0)))
% Found eq_ref00:=(eq_ref0 ((and (x x0)) (y x0))):(((eq Prop) ((and (x x0)) (y x0))) ((and (x x0)) (y x0)))
% Found (eq_ref0 ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (x x0)) (y x0))) b)
% Found ((eq_ref Prop) ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (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) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found eq_ref000:=(eq_ref00 P):((P (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))->(P (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))))
% Found (eq_ref00 P) as proof of (P0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found ((eq_ref0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) P) as proof of (P0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found (((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) P) as proof of (P0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found (((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) P) as proof of (P0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found eq_ref000:=(eq_ref00 P):((P (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))->(P (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))))
% Found (eq_ref00 P) as proof of (P0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found ((eq_ref0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) P) as proof of (P0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found (((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) P) as proof of (P0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found (((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) P) as proof of (P0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found eq_ref000:=(eq_ref00 P):((P (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))->(P (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))))
% Found (eq_ref00 P) as proof of (P0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found ((eq_ref0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) P) as proof of (P0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found (((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) P) as proof of (P0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found (((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) P) as proof of (P0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found eq_ref000:=(eq_ref00 P):((P (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))->(P (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))))
% Found (eq_ref00 P) as proof of (P0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found ((eq_ref0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) P) as proof of (P0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found (((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) P) as proof of (P0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found (((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) P) as proof of (P0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found eq_ref000:=(eq_ref00 P):((P (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))->(P (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))))
% Found (eq_ref00 P) as proof of (P0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found ((eq_ref0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) P) as proof of (P0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found (((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) P) as proof of (P0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found (((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) P) as proof of (P0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found eq_ref000:=(eq_ref00 P):((P (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))->(P (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))))
% Found (eq_ref00 P) as proof of (P0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found ((eq_ref0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) P) as proof of (P0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found (((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) P) as proof of (P0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found (((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) P) as proof of (P0 (forall (S:(a->Prop)), (((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) ((and (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (x x0)) (y x0)))
% Found eq_ref00:=(eq_ref0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))):(((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found (eq_ref0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((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) ((and (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (x x0)) (y x0)))
% Found eq_ref00:=(eq_ref0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))):(((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found (eq_ref0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b)
% Found eq_ref00:=(eq_ref0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))):(((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found (eq_ref0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((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) ((and (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (x x0)) (y x0)))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (x x0)) (y x0)))
% Found eq_ref00:=(eq_ref0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))):(((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found (eq_ref0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b)
% Found eq_ref00:=(eq_ref0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))):(((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found (eq_ref0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((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) ((and (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (x x0)) (y x0)))
% Found eq_ref00:=(eq_ref0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))):(((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found (eq_ref0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((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) ((and (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (x x0)) (y x0)))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (x x0)) (y x0)))
% Found eq_ref00:=(eq_ref0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))):(((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found (eq_ref0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b)
% Found eq_ref00:=(eq_ref0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))):(((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found (eq_ref0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((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) ((and (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (x x0)) (y x0)))
% Found eq_ref00:=(eq_ref0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))):(((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found (eq_ref0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((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) ((and (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (x x0)) (y x0)))
% Found eq_ref00:=(eq_ref0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))):(((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found (eq_ref0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((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) ((and (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (x x0)) (y x0)))
% Found eq_ref00:=(eq_ref0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))):(((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found (eq_ref0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((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) ((and (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (x x0)) (y x0)))
% Found eq_ref00:=(eq_ref0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))):(((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found (eq_ref0 (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) b)
% Found ((eq_ref Prop) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0)))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((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) ((and (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (x x0)) (y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (x x0)) (y x0)))
% 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 ((and (x x0)) (y x0))):(((eq Prop) ((and (x x0)) (y x0))) ((and (x x0)) (y x0)))
% Found (eq_ref0 ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (x x0)) (y x0))) b0)
% Found ((eq_ref Prop) ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (x x0)) (y x0))) b0)
% Found ((eq_ref Prop) ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (x x0)) (y x0))) b0)
% Found ((eq_ref Prop) ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (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) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (forall (S:(a->Prop)), (((or (((eq (a->Prop)) S) x)) (((eq (a->Prop)) S) y))->(S x0))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (forall (S:(a->Prop)), (((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) b0)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b0)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b0)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) 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 ((and (x x0)) (y x0))):(((eq Prop) ((and (x x0)) (y x0))) ((and (x x0)) (y x0)))
% Found (eq_ref0 ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (x x0)) (y x0))) b0)
% Found ((eq_ref Prop) ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (x x0)) (y x0))) b0)
% Found ((eq_ref Prop) ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (x x0)) (y x0))) b0)
% Found ((eq_ref Prop) ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (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) 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 ((and (x x0)) (y x0))):(((eq Prop) ((and (x x0)) (y x0))) ((and (x x0)) (y x0)))
% Found (eq_ref0 ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (x x0)) (y x0))) b0)
% Found ((eq_ref Prop) ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (x x0)) (y x0))) b0)
% Found ((eq_ref Prop) ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (x x0)) (y x0))) b0)
% Found ((eq_ref Prop) ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (x x0)) (y x0))) b0)
% Found eq_ref00:=(eq_ref0 ((and (x x0)) (y x0))):(((eq Prop) ((and (x x0)) (y x0))) ((and (x x0)) (y x0)))
% Found (eq_ref0 ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (x x0)) (y x0))) b0)
% Found ((eq_ref Prop) ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (x x0)) (y x0))) b0)
% Found ((eq_ref Prop) ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (x x0)) (y x0))) b0)
% Found ((eq_ref Prop) ((and (x x0)) (y x0))) as proof of (((eq Prop) ((and (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) 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)
% EOF
%------------------------------------------------------------------------------