TSTP Solution File: SEU895^5 by cocATP---0.2.0
View Problem
- Process Solution
%------------------------------------------------------------------------------
% File : cocATP---0.2.0
% Problem : SEU895^5 : TPTP v6.1.0. Released v4.0.0.
% Transfm : none
% Format : tptp:raw
% Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% Computer : n116.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 13:33:21 EDT 2014
% Result : Timeout 300.06s
% Output : None
% Verified :
% SZS Type : None (Parsing solution fails)
% Syntax : Number of formulae : 0
% Comments :
%------------------------------------------------------------------------------
%----NO SOLUTION OUTPUT BY SYSTEM
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% % Problem : SEU895^5 : TPTP v6.1.0. Released v4.0.0.
% % Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n116.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 11:41:16 CDT 2014
% % CPUTime : 300.06
% 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 0x2055950>, <kernel.Type object at 0x20557e8>) of role type named b_type
% Using role type
% Declaring b:Type
% FOF formula (<kernel.Constant object at 0x242acf8>, <kernel.Type object at 0x2055a28>) of role type named a_type
% Using role type
% Declaring a:Type
% FOF formula (<kernel.Constant object at 0x2055cf8>, <kernel.DependentProduct object at 0x20555a8>) of role type named f
% Using role type
% Declaring f:(b->a)
% FOF formula (<kernel.Constant object at 0x20557e8>, <kernel.DependentProduct object at 0x2055c68>) of role type named y
% Using role type
% Declaring y:(b->Prop)
% FOF formula (<kernel.Constant object at 0x2055950>, <kernel.DependentProduct object at 0x20555f0>) of role type named x
% Using role type
% Declaring x:(b->Prop)
% FOF formula (((eq (a->Prop)) (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt))))))) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) of role conjecture named cX5202_pme
% Conjecture to prove = (((eq (a->Prop)) (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt))))))) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))):Prop
% Parameter b_DUMMY:b.
% Parameter a_DUMMY:a.
% We need to prove ['(((eq (a->Prop)) (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt))))))) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt))))))))']
% Parameter b:Type.
% Parameter a:Type.
% Parameter f:(b->a).
% Parameter y:(b->Prop).
% Parameter x:(b->Prop).
% Trying to prove (((eq (a->Prop)) (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt))))))) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt))))))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt)))))))->(P (fun (x0:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))))
% Found (eta_expansion_dep000 P) as proof of (P0 (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt)))))))
% Found ((eta_expansion_dep00 (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt))))))) P) as proof of (P0 (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt)))))))
% Found (((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt))))))) P) as proof of (P0 (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt)))))))
% Found ((((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt))))))) P) as proof of (P0 (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt)))))))
% Found ((((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt))))))) P) as proof of (P0 (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt)))))))
% Found eta_expansion0000:=(eta_expansion000 P):((P (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt)))))))->(P (fun (x0:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))))
% Found (eta_expansion000 P) as proof of (P0 (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt)))))))
% Found ((eta_expansion00 (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt))))))) P) as proof of (P0 (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt)))))))
% Found (((eta_expansion0 Prop) (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt))))))) P) as proof of (P0 (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt)))))))
% Found ((((eta_expansion a) Prop) (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt))))))) P) as proof of (P0 (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt)))))))
% Found ((((eta_expansion a) Prop) (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt))))))) P) as proof of (P0 (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt)))))))
% Found eq_ref000:=(eq_ref00 P):((P (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt)))))))->(P (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt))))))))
% Found (eq_ref00 P) as proof of (P0 (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt)))))))
% Found ((eq_ref0 (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt))))))) P) as proof of (P0 (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt)))))))
% Found (((eq_ref (a->Prop)) (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt))))))) P) as proof of (P0 (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt)))))))
% Found (((eq_ref (a->Prop)) (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt))))))) P) as proof of (P0 (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt)))))))
% Found eq_ref00:=(eq_ref0 (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt))))))):(((eq (a->Prop)) (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt))))))) (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt)))))))
% Found (eq_ref0 (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt))))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt))))))) b0)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt))))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt))))))) b0)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt))))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt))))))) b0)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt))))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt))))))) b0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b0):(((eq (a->Prop)) b0) (fun (x:a)=> (b0 x)))
% Found (eta_expansion_dep00 b0) as proof of (((eq (a->Prop)) b0) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt))))))))
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) b0) as proof of (((eq (a->Prop)) b0) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt))))))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b0) as proof of (((eq (a->Prop)) b0) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt))))))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b0) as proof of (((eq (a->Prop)) b0) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt))))))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b0) as proof of (((eq (a->Prop)) b0) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt))))))))
% Found eta_expansion000:=(eta_expansion00 b0):(((eq (a->Prop)) b0) (fun (x:a)=> (b0 x)))
% Found (eta_expansion00 b0) as proof of (((eq (a->Prop)) b0) (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt)))))))
% Found ((eta_expansion0 Prop) b0) as proof of (((eq (a->Prop)) b0) (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt)))))))
% Found (((eta_expansion a) Prop) b0) as proof of (((eq (a->Prop)) b0) (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt)))))))
% Found (((eta_expansion a) Prop) b0) as proof of (((eq (a->Prop)) b0) (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt)))))))
% Found (((eta_expansion a) Prop) b0) as proof of (((eq (a->Prop)) b0) (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt)))))))
% Found eta_expansion000:=(eta_expansion00 (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))):(((eq (a->Prop)) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) (fun (x0:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))))
% Found (eta_expansion00 (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) b0)
% Found ((eta_expansion0 Prop) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) b0)
% Found (((eta_expansion a) Prop) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) b0)
% Found (((eta_expansion a) Prop) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) b0)
% Found (((eta_expansion a) Prop) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) b0)
% Found eta_expansion0000:=(eta_expansion000 P):((P (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt))))))))->(P (fun (x0:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))))
% Found (eta_expansion000 P) as proof of (P0 (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt))))))))
% Found ((eta_expansion00 (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) P) as proof of (P0 (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt))))))))
% Found (((eta_expansion0 Prop) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) P) as proof of (P0 (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt))))))))
% Found ((((eta_expansion a) Prop) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) P) as proof of (P0 (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt))))))))
% Found ((((eta_expansion a) Prop) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) P) as proof of (P0 (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt))))))))
% Found eta_expansion0000:=(eta_expansion000 P):((P (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt))))))))->(P (fun (x0:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))))
% Found (eta_expansion000 P) as proof of (P0 (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt))))))))
% Found ((eta_expansion00 (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) P) as proof of (P0 (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt))))))))
% Found (((eta_expansion0 Prop) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) P) as proof of (P0 (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt))))))))
% Found ((((eta_expansion a) Prop) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) P) as proof of (P0 (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt))))))))
% Found ((((eta_expansion a) Prop) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) P) as proof of (P0 (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt))))))))
% Found eq_ref000:=(eq_ref00 P):((P ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))->(P ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))))
% Found (eq_ref00 P) as proof of (P0 ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found ((eq_ref0 ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) P) as proof of (P0 ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found (((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) P) as proof of (P0 ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found (((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) P) as proof of (P0 ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found eq_ref000:=(eq_ref00 P):((P ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))->(P ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))))
% Found (eq_ref00 P) as proof of (P0 ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found ((eq_ref0 ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) P) as proof of (P0 ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found (((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) P) as proof of (P0 ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found (((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) P) as proof of (P0 ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found eq_ref00:=(eq_ref0 ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))):(((eq Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found (eq_ref0 ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) b0)
% Found ((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) b0)
% Found ((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) b0)
% Found ((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found eq_ref00:=(eq_ref0 ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))):(((eq Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found (eq_ref0 ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) b0)
% Found ((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) b0)
% Found ((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) b0)
% Found ((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found eq_ref000:=(eq_ref00 P):((P ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))->(P ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))))
% Found (eq_ref00 P) as proof of (P0 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found ((eq_ref0 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) P) as proof of (P0 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found (((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) P) as proof of (P0 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found (((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) P) as proof of (P0 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found eq_ref000:=(eq_ref00 P):((P ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))->(P ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))))
% Found (eq_ref00 P) as proof of (P0 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found ((eq_ref0 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) P) as proof of (P0 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found (((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) P) as proof of (P0 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found (((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) P) as proof of (P0 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found eq_ref00:=(eq_ref0 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))):(((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found (eq_ref0 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b0)
% Found ((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b0)
% Found ((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b0)
% Found ((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found eq_ref00:=(eq_ref0 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))):(((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found (eq_ref0 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b0)
% Found ((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b0)
% Found ((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b0)
% Found ((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b0)
% Found eq_ref00:=(eq_ref0 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))):(((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found (eq_ref0 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b0)
% Found ((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b0)
% Found ((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b0)
% Found ((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found eq_ref00:=(eq_ref0 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))):(((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found (eq_ref0 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b0)
% Found ((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b0)
% Found ((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b0)
% Found ((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found x0:(P (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt)))))))
% Instantiate: b0:=(fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt)))))):(a->Prop)
% Found x0 as proof of (P0 b0)
% Found eq_ref00:=(eq_ref0 (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))):(((eq (a->Prop)) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt))))))))
% Found (eq_ref0 (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) b0)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) b0)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) b0)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) b0)
% Found x0:(P (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt)))))))
% Instantiate: f0:=(fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt)))))):(a->Prop)
% Found x0 as proof of (P0 f0)
% Found eq_ref00:=(eq_ref0 (f0 x00)):(((eq Prop) (f0 x00)) (f0 x00))
% Found (eq_ref0 (f0 x00)) as proof of (((eq Prop) (f0 x00)) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x00) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x00) (f Xt)))))))
% Found ((eq_ref Prop) (f0 x00)) as proof of (((eq Prop) (f0 x00)) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x00) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x00) (f Xt)))))))
% Found ((eq_ref Prop) (f0 x00)) as proof of (((eq Prop) (f0 x00)) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x00) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x00) (f Xt)))))))
% Found (fun (x00:a)=> ((eq_ref Prop) (f0 x00))) as proof of (((eq Prop) (f0 x00)) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x00) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x00) (f Xt)))))))
% Found (fun (x00:a)=> ((eq_ref Prop) (f0 x00))) as proof of (forall (x0:a), (((eq Prop) (f0 x0)) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))))
% Found x0:(P (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt)))))))
% Instantiate: f0:=(fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt)))))):(a->Prop)
% Found x0 as proof of (P0 f0)
% Found eq_ref00:=(eq_ref0 (f0 x00)):(((eq Prop) (f0 x00)) (f0 x00))
% Found (eq_ref0 (f0 x00)) as proof of (((eq Prop) (f0 x00)) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x00) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x00) (f Xt)))))))
% Found ((eq_ref Prop) (f0 x00)) as proof of (((eq Prop) (f0 x00)) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x00) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x00) (f Xt)))))))
% Found ((eq_ref Prop) (f0 x00)) as proof of (((eq Prop) (f0 x00)) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x00) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x00) (f Xt)))))))
% Found (fun (x00:a)=> ((eq_ref Prop) (f0 x00))) as proof of (((eq Prop) (f0 x00)) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x00) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x00) (f Xt)))))))
% Found (fun (x00:a)=> ((eq_ref Prop) (f0 x00))) as proof of (forall (x0:a), (((eq Prop) (f0 x0)) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))))
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found eq_ref00:=(eq_ref0 ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))):(((eq Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found (eq_ref0 ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) b0)
% Found ((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) b0)
% Found ((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) b0)
% Found ((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found eq_ref00:=(eq_ref0 ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))):(((eq Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found (eq_ref0 ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) b0)
% Found ((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) b0)
% Found ((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) b0)
% Found ((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) b0)
% Found eta_expansion000:=(eta_expansion00 (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt))))))):(((eq (a->Prop)) (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt))))))) (fun (x0:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))))
% Found (eta_expansion00 (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt))))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt))))))) b0)
% Found ((eta_expansion0 Prop) (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt))))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt))))))) b0)
% Found (((eta_expansion a) Prop) (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt))))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt))))))) b0)
% Found (((eta_expansion a) Prop) (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt))))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt))))))) b0)
% Found (((eta_expansion a) Prop) (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt))))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt))))))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq (a->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (a->Prop)) b0) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt))))))))
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt))))))))
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt))))))))
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt))))))))
% Found eq_ref000:=(eq_ref00 P):((P (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt)))))))->(P (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt))))))))
% Found (eq_ref00 P) as proof of (P0 (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt)))))))
% Found ((eq_ref0 (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt))))))) P) as proof of (P0 (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt)))))))
% Found (((eq_ref (a->Prop)) (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt))))))) P) as proof of (P0 (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt)))))))
% Found (((eq_ref (a->Prop)) (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt))))))) P) as proof of (P0 (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt)))))))
% Found eq_ref00:=(eq_ref0 (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt))))))):(((eq (a->Prop)) (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt))))))) (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt)))))))
% Found (eq_ref0 (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt))))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt))))))) b0)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt))))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt))))))) b0)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt))))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt))))))) b0)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt))))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt))))))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq (a->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (a->Prop)) b0) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt))))))))
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt))))))))
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt))))))))
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt))))))))
% Found x0:(P (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt))))))))
% Instantiate: b0:=(fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt))))))):(a->Prop)
% Found x0 as proof of (P0 b0)
% Found eta_expansion000:=(eta_expansion00 (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt))))))):(((eq (a->Prop)) (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt))))))) (fun (x0:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))))
% Found (eta_expansion00 (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt))))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt))))))) b0)
% Found ((eta_expansion0 Prop) (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt))))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt))))))) b0)
% Found (((eta_expansion a) Prop) (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt))))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt))))))) b0)
% Found (((eta_expansion a) Prop) (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt))))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt))))))) b0)
% Found (((eta_expansion a) Prop) (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt))))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt))))))) b0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))):(((eq (a->Prop)) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) (fun (x0:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))))
% Found (eta_expansion_dep00 (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) b0)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) b0)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) b0)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) b0)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) b0)
% Found x0:(P0 b0)
% Instantiate: b0:=(fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt)))))):(a->Prop)
% Found (fun (x0:(P0 b0))=> x0) as proof of (P0 (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt)))))))
% Found (fun (P0:((a->Prop)->Prop)) (x0:(P0 b0))=> x0) as proof of ((P0 b0)->(P0 (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt))))))))
% Found (fun (P0:((a->Prop)->Prop)) (x0:(P0 b0))=> x0) as proof of (P b0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt))))))):(((eq (a->Prop)) (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt))))))) (fun (x0:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))))
% Found (eta_expansion_dep00 (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt))))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt))))))) b00)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt))))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt))))))) b00)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt))))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt))))))) b00)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt))))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt))))))) b00)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt))))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt))))))) b00)
% Found eq_ref00:=(eq_ref0 b00):(((eq (a->Prop)) b00) b00)
% Found (eq_ref0 b00) as proof of (((eq (a->Prop)) b00) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt))))))))
% Found ((eq_ref (a->Prop)) b00) as proof of (((eq (a->Prop)) b00) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt))))))))
% Found ((eq_ref (a->Prop)) b00) as proof of (((eq (a->Prop)) b00) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt))))))))
% Found ((eq_ref (a->Prop)) b00) as proof of (((eq (a->Prop)) b00) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt))))))))
% Found x0:(P (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt))))))))
% Instantiate: f0:=(fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt))))))):(a->Prop)
% Found x0 as proof of (P0 f0)
% Found eq_ref00:=(eq_ref0 (f0 x00)):(((eq Prop) (f0 x00)) (f0 x00))
% Found (eq_ref0 (f0 x00)) as proof of (((eq Prop) (f0 x00)) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x00) (f Xt))))))
% Found ((eq_ref Prop) (f0 x00)) as proof of (((eq Prop) (f0 x00)) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x00) (f Xt))))))
% Found ((eq_ref Prop) (f0 x00)) as proof of (((eq Prop) (f0 x00)) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x00) (f Xt))))))
% Found (fun (x00:a)=> ((eq_ref Prop) (f0 x00))) as proof of (((eq Prop) (f0 x00)) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x00) (f Xt))))))
% Found (fun (x00:a)=> ((eq_ref Prop) (f0 x00))) as proof of (forall (x0:a), (((eq Prop) (f0 x0)) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))))
% Found x0:(P (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt))))))))
% Instantiate: f0:=(fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt))))))):(a->Prop)
% Found x0 as proof of (P0 f0)
% Found eq_ref00:=(eq_ref0 (f0 x00)):(((eq Prop) (f0 x00)) (f0 x00))
% Found (eq_ref0 (f0 x00)) as proof of (((eq Prop) (f0 x00)) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x00) (f Xt))))))
% Found ((eq_ref Prop) (f0 x00)) as proof of (((eq Prop) (f0 x00)) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x00) (f Xt))))))
% Found ((eq_ref Prop) (f0 x00)) as proof of (((eq Prop) (f0 x00)) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x00) (f Xt))))))
% Found (fun (x00:a)=> ((eq_ref Prop) (f0 x00))) as proof of (((eq Prop) (f0 x00)) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x00) (f Xt))))))
% Found (fun (x00:a)=> ((eq_ref Prop) (f0 x00))) as proof of (forall (x0:a), (((eq Prop) (f0 x0)) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))))
% Found x1:(P ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Instantiate: b0:=((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))):Prop
% Found x1 as proof of (P0 b0)
% Found eq_ref00:=(eq_ref0 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))):(((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found (eq_ref0 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b0)
% Found ((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b0)
% Found ((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b0)
% Found ((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b0)
% Found x1:(P ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Instantiate: b0:=((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))):Prop
% Found x1 as proof of (P0 b0)
% Found eq_ref00:=(eq_ref0 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))):(((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found (eq_ref0 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b0)
% Found ((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b0)
% Found ((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b0)
% Found ((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b0)
% Found eta_expansion000:=(eta_expansion00 (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))):(((eq (a->Prop)) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) (fun (x0:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))))
% Found (eta_expansion00 (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) b0)
% Found ((eta_expansion0 Prop) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) b0)
% Found (((eta_expansion a) Prop) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) b0)
% Found (((eta_expansion a) Prop) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) b0)
% Found (((eta_expansion a) Prop) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) b0)
% Found eta_expansion000:=(eta_expansion00 b0):(((eq (a->Prop)) b0) (fun (x:a)=> (b0 x)))
% Found (eta_expansion00 b0) as proof of (((eq (a->Prop)) b0) (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt)))))))
% Found ((eta_expansion0 Prop) b0) as proof of (((eq (a->Prop)) b0) (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt)))))))
% Found (((eta_expansion a) Prop) b0) as proof of (((eq (a->Prop)) b0) (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt)))))))
% Found (((eta_expansion a) Prop) b0) as proof of (((eq (a->Prop)) b0) (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt)))))))
% Found (((eta_expansion a) Prop) b0) as proof of (((eq (a->Prop)) b0) (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt)))))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))):(((eq (a->Prop)) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) (fun (x0:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))))
% Found (eta_expansion_dep00 (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) b0)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) b0)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) b0)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) b0)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq (a->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (a->Prop)) b0) (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt)))))))
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt)))))))
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt)))))))
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt)))))))
% Found eq_ref00:=(eq_ref0 b0):(((eq (a->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (a->Prop)) b0) (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt)))))))
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt)))))))
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt)))))))
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt)))))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))):(((eq (a->Prop)) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) (fun (x0:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))))
% Found (eta_expansion_dep00 (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) b0)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) b0)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) b0)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) b0)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) b0)
% Found eq_ref000:=(eq_ref00 P):((P (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt))))))))->(P (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))))
% Found (eq_ref00 P) as proof of (P0 (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt))))))))
% Found ((eq_ref0 (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) P) as proof of (P0 (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt))))))))
% Found (((eq_ref (a->Prop)) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) P) as proof of (P0 (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt))))))))
% Found (((eq_ref (a->Prop)) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) P) as proof of (P0 (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt))))))))
% Found eta_expansion000:=(eta_expansion00 (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))):(((eq (a->Prop)) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) (fun (x0:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))))
% Found (eta_expansion00 (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) b0)
% Found ((eta_expansion0 Prop) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) b0)
% Found (((eta_expansion a) Prop) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) b0)
% Found (((eta_expansion a) Prop) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) b0)
% Found (((eta_expansion a) Prop) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) b0)
% Found eta_expansion000:=(eta_expansion00 b0):(((eq (a->Prop)) b0) (fun (x:a)=> (b0 x)))
% Found (eta_expansion00 b0) as proof of (((eq (a->Prop)) b0) (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt)))))))
% Found ((eta_expansion0 Prop) b0) as proof of (((eq (a->Prop)) b0) (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt)))))))
% Found (((eta_expansion a) Prop) b0) as proof of (((eq (a->Prop)) b0) (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt)))))))
% Found (((eta_expansion a) Prop) b0) as proof of (((eq (a->Prop)) b0) (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt)))))))
% Found (((eta_expansion a) Prop) b0) as proof of (((eq (a->Prop)) b0) (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt)))))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P1):((P1 (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt))))))))->(P1 (fun (x0:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))))
% Found (eta_expansion_dep000 P1) as proof of (P2 (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt))))))))
% Found ((eta_expansion_dep00 (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) P1) as proof of (P2 (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt))))))))
% Found (((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) P1) as proof of (P2 (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt))))))))
% Found ((((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) P1) as proof of (P2 (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt))))))))
% Found ((((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) P1) as proof of (P2 (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt))))))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P1):((P1 (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt))))))))->(P1 (fun (x0:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))))
% Found (eta_expansion_dep000 P1) as proof of (P2 (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt))))))))
% Found ((eta_expansion_dep00 (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) P1) as proof of (P2 (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt))))))))
% Found (((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) P1) as proof of (P2 (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt))))))))
% Found ((((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) P1) as proof of (P2 (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt))))))))
% Found ((((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) P1) as proof of (P2 (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt))))))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P1):((P1 (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt))))))))->(P1 (fun (x0:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))))
% Found (eta_expansion_dep000 P1) as proof of (P2 (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt))))))))
% Found ((eta_expansion_dep00 (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) P1) as proof of (P2 (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt))))))))
% Found (((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) P1) as proof of (P2 (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt))))))))
% Found ((((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) P1) as proof of (P2 (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt))))))))
% Found ((((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) P1) as proof of (P2 (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt))))))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P1):((P1 (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt))))))))->(P1 (fun (x0:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))))
% Found (eta_expansion_dep000 P1) as proof of (P2 (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt))))))))
% Found ((eta_expansion_dep00 (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) P1) as proof of (P2 (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt))))))))
% Found (((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) P1) as proof of (P2 (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt))))))))
% Found ((((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) P1) as proof of (P2 (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt))))))))
% Found ((((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) P1) as proof of (P2 (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt))))))))
% Found x01:(P b0)
% Found (fun (x01:(P b0))=> x01) as proof of (P b0)
% Found (fun (x01:(P b0))=> x01) as proof of (P0 b0)
% Found eq_ref000:=(eq_ref00 P1):((P1 ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))->(P1 ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))))
% Found (eq_ref00 P1) as proof of (P2 ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found ((eq_ref0 ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) P1) as proof of (P2 ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found (((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) P1) as proof of (P2 ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found (((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) P1) as proof of (P2 ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found eq_ref000:=(eq_ref00 P1):((P1 ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))->(P1 ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))))
% Found (eq_ref00 P1) as proof of (P2 ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found ((eq_ref0 ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) P1) as proof of (P2 ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found (((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) P1) as proof of (P2 ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found (((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) P1) as proof of (P2 ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found eq_ref000:=(eq_ref00 P1):((P1 ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))->(P1 ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))))
% Found (eq_ref00 P1) as proof of (P2 ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found ((eq_ref0 ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) P1) as proof of (P2 ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found (((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) P1) as proof of (P2 ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found (((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) P1) as proof of (P2 ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found eq_ref000:=(eq_ref00 P1):((P1 ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))->(P1 ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))))
% Found (eq_ref00 P1) as proof of (P2 ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found ((eq_ref0 ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) P1) as proof of (P2 ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found (((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) P1) as proof of (P2 ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found (((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) P1) as proof of (P2 ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found eq_ref00:=(eq_ref0 ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))):(((eq Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found (eq_ref0 ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) b0)
% Found ((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) b0)
% Found ((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) b0)
% Found ((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found eq_ref00:=(eq_ref0 ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))):(((eq Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found (eq_ref0 ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) b0)
% Found ((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) b0)
% Found ((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) b0)
% Found ((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found eq_ref00:=(eq_ref0 ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))):(((eq Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found (eq_ref0 ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) b0)
% Found ((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) b0)
% Found ((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) b0)
% Found ((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found eq_ref00:=(eq_ref0 ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))):(((eq Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found (eq_ref0 ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) b0)
% Found ((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) b0)
% Found ((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) b0)
% Found ((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found eq_ref00:=(eq_ref0 ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))):(((eq Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found (eq_ref0 ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) b0)
% Found ((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) b0)
% Found ((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) b0)
% Found ((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found eq_ref00:=(eq_ref0 ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))):(((eq Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found (eq_ref0 ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) b0)
% Found ((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) b0)
% Found ((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) b0)
% Found ((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found x01:(P b0)
% Found (fun (x01:(P b0))=> x01) as proof of (P b0)
% Found (fun (x01:(P b0))=> x01) as proof of (P0 b0)
% Found x01:(P b0)
% Found (fun (x01:(P b0))=> x01) as proof of (P b0)
% Found (fun (x01:(P b0))=> x01) as proof of (P0 b0)
% Found eq_ref000:=(eq_ref00 P):((P ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))->(P ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))))
% Found (eq_ref00 P) as proof of (P0 ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found ((eq_ref0 ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) P) as proof of (P0 ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found (((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) P) as proof of (P0 ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found (((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) P) as proof of (P0 ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found eq_ref00:=(eq_ref0 ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))):(((eq Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found (eq_ref0 ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) b0)
% Found ((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) b0)
% Found ((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) b0)
% Found ((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found eq_ref000:=(eq_ref00 P):((P ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))->(P ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))))
% Found (eq_ref00 P) as proof of (P0 ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found ((eq_ref0 ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) P) as proof of (P0 ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found (((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) P) as proof of (P0 ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found (((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) P) as proof of (P0 ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found eq_ref00:=(eq_ref0 ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))):(((eq Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found (eq_ref0 ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) b0)
% Found ((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) b0)
% Found ((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) b0)
% Found ((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found eq_ref00:=(eq_ref0 b00):(((eq (a->Prop)) b00) b00)
% Found (eq_ref0 b00) as proof of (((eq (a->Prop)) b00) b0)
% Found ((eq_ref (a->Prop)) b00) as proof of (((eq (a->Prop)) b00) b0)
% Found ((eq_ref (a->Prop)) b00) as proof of (((eq (a->Prop)) b00) b0)
% Found ((eq_ref (a->Prop)) b00) as proof of (((eq (a->Prop)) b00) b0)
% Found eq_ref00:=(eq_ref0 (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))):(((eq (a->Prop)) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt))))))))
% Found (eq_ref0 (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) b00)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) b00)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) b00)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) b00)
% Found eq_ref000:=(eq_ref00 P):((P (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt))))))))->(P (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))))
% Found (eq_ref00 P) as proof of (P0 (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt))))))))
% Found ((eq_ref0 (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) P) as proof of (P0 (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt))))))))
% Found (((eq_ref (a->Prop)) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) P) as proof of (P0 (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt))))))))
% Found (((eq_ref (a->Prop)) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) P) as proof of (P0 (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt))))))))
% Found eq_ref00:=(eq_ref0 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))):(((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found (eq_ref0 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b0)
% Found ((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b0)
% Found ((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b0)
% Found ((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b0)
% Found x1:(P0 b0)
% Instantiate: b0:=((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))):Prop
% Found (fun (x1:(P0 b0))=> x1) as proof of (P0 ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found (fun (P0:(Prop->Prop)) (x1:(P0 b0))=> x1) as proof of ((P0 b0)->(P0 ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))))
% Found (fun (P0:(Prop->Prop)) (x1:(P0 b0))=> x1) as proof of (P b0)
% Found eq_ref00:=(eq_ref0 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))):(((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found (eq_ref0 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b0)
% Found ((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b0)
% Found ((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b0)
% Found ((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b0)
% Found x1:(P0 b0)
% Instantiate: b0:=((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))):Prop
% Found (fun (x1:(P0 b0))=> x1) as proof of (P0 ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found (fun (P0:(Prop->Prop)) (x1:(P0 b0))=> x1) as proof of ((P0 b0)->(P0 ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))))
% Found (fun (P0:(Prop->Prop)) (x1:(P0 b0))=> x1) as proof of (P b0)
% Found x1:(P ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Instantiate: b0:=((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))):Prop
% Found x1 as proof of (P0 b0)
% Found eq_ref00:=(eq_ref0 ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))):(((eq Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found (eq_ref0 ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) b0)
% Found ((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) b0)
% Found ((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) b0)
% Found ((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) b0)
% Found x1:(P ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Instantiate: b0:=((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))):Prop
% Found x1 as proof of (P0 b0)
% Found eq_ref00:=(eq_ref0 ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))):(((eq Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found (eq_ref0 ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) b0)
% Found ((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) b0)
% Found ((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) b0)
% Found ((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))
% Found eq_ref00:=(eq_ref0 (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))):(((eq (a->Prop)) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt))))))))
% Found (eq_ref0 (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) b00)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) b00)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) b00)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) b00)
% Found eq_ref00:=(eq_ref0 b00):(((eq (a->Prop)) b00) b00)
% Found (eq_ref0 b00) as proof of (((eq (a->Prop)) b00) b0)
% Found ((eq_ref (a->Prop)) b00) as proof of (((eq (a->Prop)) b00) b0)
% Found ((eq_ref (a->Prop)) b00) as proof of (((eq (a->Prop)) b00) b0)
% Found ((eq_ref (a->Prop)) b00) as proof of (((eq (a->Prop)) b00) b0)
% Found eq_ref00:=(eq_ref0 ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))):(((eq Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found (eq_ref0 ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) b00)
% Found ((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) b00)
% Found ((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) b00)
% Found ((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) b00)
% Found eq_ref00:=(eq_ref0 b00):(((eq Prop) b00) b00)
% Found (eq_ref0 b00) as proof of (((eq Prop) b00) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found ((eq_ref Prop) b00) as proof of (((eq Prop) b00) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found ((eq_ref Prop) b00) as proof of (((eq Prop) b00) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found ((eq_ref Prop) b00) as proof of (((eq Prop) b00) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found eq_ref00:=(eq_ref0 ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))):(((eq Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found (eq_ref0 ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) b00)
% Found ((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) b00)
% Found ((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) b00)
% Found ((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) b00)
% Found eq_ref00:=(eq_ref0 b00):(((eq Prop) b00) b00)
% Found (eq_ref0 b00) as proof of (((eq Prop) b00) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found ((eq_ref Prop) b00) as proof of (((eq Prop) b00) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found ((eq_ref Prop) b00) as proof of (((eq Prop) b00) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found ((eq_ref Prop) b00) as proof of (((eq Prop) b00) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt)))))))->(P (fun (x0:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))))
% Found (eta_expansion_dep000 P) as proof of (P0 (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt)))))))
% Found ((eta_expansion_dep00 (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt))))))) P) as proof of (P0 (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt)))))))
% Found (((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt))))))) P) as proof of (P0 (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt)))))))
% Found ((((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt))))))) P) as proof of (P0 (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt)))))))
% Found ((((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt))))))) P) as proof of (P0 (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt)))))))
% Found x1:(P ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Instantiate: b0:=((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))):Prop
% Found x1 as proof of (P0 b0)
% Found eq_ref00:=(eq_ref0 ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))):(((eq Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found (eq_ref0 ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) b0)
% Found ((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) b0)
% Found ((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) b0)
% Found ((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) b0)
% Found x1:(P ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Instantiate: b0:=((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))):Prop
% Found x1 as proof of (P0 b0)
% Found eq_ref00:=(eq_ref0 ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))):(((eq Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found (eq_ref0 ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) b0)
% Found ((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) b0)
% Found ((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) b0)
% Found ((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) b0)
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt))))))))->(P (fun (x0:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))))
% Found (eta_expansion_dep000 P) as proof of (P0 (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt))))))))
% Found ((eta_expansion_dep00 (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) P) as proof of (P0 (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt))))))))
% Found (((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) P) as proof of (P0 (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt))))))))
% Found ((((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) P) as proof of (P0 (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt))))))))
% Found ((((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) P) as proof of (P0 (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt))))))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt))))))))->(P (fun (x0:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))))
% Found (eta_expansion_dep000 P) as proof of (P0 (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt))))))))
% Found ((eta_expansion_dep00 (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) P) as proof of (P0 (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt))))))))
% Found (((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) P) as proof of (P0 (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt))))))))
% Found ((((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) P) as proof of (P0 (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt))))))))
% Found ((((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) P) as proof of (P0 (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt))))))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt))))))))->(P (fun (x0:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))))
% Found (eta_expansion_dep000 P) as proof of (P0 (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt))))))))
% Found ((eta_expansion_dep00 (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) P) as proof of (P0 (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt))))))))
% Found (((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) P) as proof of (P0 (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt))))))))
% Found ((((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) P) as proof of (P0 (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt))))))))
% Found ((((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) P) as proof of (P0 (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt))))))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt))))))))->(P (fun (x0:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))))
% Found (eta_expansion_dep000 P) as proof of (P0 (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt))))))))
% Found ((eta_expansion_dep00 (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) P) as proof of (P0 (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt))))))))
% Found (((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) P) as proof of (P0 (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt))))))))
% Found ((((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) P) as proof of (P0 (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt))))))))
% Found ((((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) P) as proof of (P0 (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt))))))))
% Found eta_expansion0000:=(eta_expansion000 P):((P (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt)))))))->(P (fun (x0:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))))
% Found (eta_expansion000 P) as proof of (P0 (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt)))))))
% Found ((eta_expansion00 (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt))))))) P) as proof of (P0 (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt)))))))
% Found (((eta_expansion0 Prop) (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt))))))) P) as proof of (P0 (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt)))))))
% Found ((((eta_expansion a) Prop) (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt))))))) P) as proof of (P0 (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt)))))))
% Found ((((eta_expansion a) Prop) (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt))))))) P) as proof of (P0 (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt)))))))
% Found eq_ref000:=(eq_ref00 P):((P (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt)))))))->(P (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt))))))))
% Found (eq_ref00 P) as proof of (P0 (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt)))))))
% Found ((eq_ref0 (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt))))))) P) as proof of (P0 (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt)))))))
% Found (((eq_ref (a->Prop)) (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt))))))) P) as proof of (P0 (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt)))))))
% Found (((eq_ref (a->Prop)) (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt))))))) P) as proof of (P0 (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt)))))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt))))))):(((eq (a->Prop)) (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt))))))) (fun (x0:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))))
% Found (eta_expansion_dep00 (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt))))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt))))))) b0)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt))))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt))))))) b0)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt))))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt))))))) b0)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt))))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt))))))) b0)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt))))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt))))))) b0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b0):(((eq (a->Prop)) b0) (fun (x:a)=> (b0 x)))
% Found (eta_expansion_dep00 b0) as proof of (((eq (a->Prop)) b0) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt))))))))
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) b0) as proof of (((eq (a->Prop)) b0) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt))))))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b0) as proof of (((eq (a->Prop)) b0) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt))))))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b0) as proof of (((eq (a->Prop)) b0) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt))))))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b0) as proof of (((eq (a->Prop)) b0) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt))))))))
% Found eq_ref00:=(eq_ref0 (b0 x0)):(((eq Prop) (b0 x0)) (b0 x0))
% Found (eq_ref0 (b0 x0)) as proof of (((eq Prop) (b0 x0)) b1)
% Found ((eq_ref Prop) (b0 x0)) as proof of (((eq Prop) (b0 x0)) b1)
% Found ((eq_ref Prop) (b0 x0)) as proof of (((eq Prop) (b0 x0)) b1)
% Found ((eq_ref Prop) (b0 x0)) as proof of (((eq Prop) (b0 x0)) b1)
% Found eq_ref00:=(eq_ref0 b1):(((eq Prop) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq Prop) b1) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found ((eq_ref Prop) b1) as proof of (((eq Prop) b1) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found ((eq_ref Prop) b1) as proof of (((eq Prop) b1) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found ((eq_ref Prop) b1) as proof of (((eq Prop) b1) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found eq_ref00:=(eq_ref0 (b0 x0)):(((eq Prop) (b0 x0)) (b0 x0))
% Found (eq_ref0 (b0 x0)) as proof of (((eq Prop) (b0 x0)) b1)
% Found ((eq_ref Prop) (b0 x0)) as proof of (((eq Prop) (b0 x0)) b1)
% Found ((eq_ref Prop) (b0 x0)) as proof of (((eq Prop) (b0 x0)) b1)
% Found ((eq_ref Prop) (b0 x0)) as proof of (((eq Prop) (b0 x0)) b1)
% Found eq_ref00:=(eq_ref0 b1):(((eq Prop) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq Prop) b1) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found ((eq_ref Prop) b1) as proof of (((eq Prop) b1) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found ((eq_ref Prop) b1) as proof of (((eq Prop) b1) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found ((eq_ref Prop) b1) as proof of (((eq Prop) b1) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found eq_ref000:=(eq_ref00 P1):((P1 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))->(P1 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))))
% Found (eq_ref00 P1) as proof of (P2 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found ((eq_ref0 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) P1) as proof of (P2 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found (((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) P1) as proof of (P2 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found (((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) P1) as proof of (P2 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found eq_ref000:=(eq_ref00 P1):((P1 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))->(P1 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))))
% Found (eq_ref00 P1) as proof of (P2 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found ((eq_ref0 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) P1) as proof of (P2 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found (((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) P1) as proof of (P2 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found (((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) P1) as proof of (P2 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found eq_ref000:=(eq_ref00 P1):((P1 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))->(P1 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))))
% Found (eq_ref00 P1) as proof of (P2 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found ((eq_ref0 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) P1) as proof of (P2 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found (((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) P1) as proof of (P2 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found (((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) P1) as proof of (P2 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found eq_ref000:=(eq_ref00 P1):((P1 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))->(P1 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))))
% Found (eq_ref00 P1) as proof of (P2 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found ((eq_ref0 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) P1) as proof of (P2 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found (((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) P1) as proof of (P2 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found (((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) P1) as proof of (P2 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found eq_ref000:=(eq_ref00 P1):((P1 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))->(P1 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))))
% Found (eq_ref00 P1) as proof of (P2 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found ((eq_ref0 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) P1) as proof of (P2 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found (((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) P1) as proof of (P2 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found (((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) P1) as proof of (P2 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found eq_ref000:=(eq_ref00 P1):((P1 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))->(P1 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))))
% Found (eq_ref00 P1) as proof of (P2 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found ((eq_ref0 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) P1) as proof of (P2 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found (((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) P1) as proof of (P2 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found (((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) P1) as proof of (P2 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found eq_ref000:=(eq_ref00 P1):((P1 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))->(P1 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))))
% Found (eq_ref00 P1) as proof of (P2 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found ((eq_ref0 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) P1) as proof of (P2 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found (((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) P1) as proof of (P2 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found (((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) P1) as proof of (P2 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found eq_ref000:=(eq_ref00 P1):((P1 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))->(P1 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))))
% Found (eq_ref00 P1) as proof of (P2 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found ((eq_ref0 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) P1) as proof of (P2 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found (((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) P1) as proof of (P2 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found (((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) P1) as proof of (P2 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found eq_ref000:=(eq_ref00 P):((P ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))->(P ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))))
% Found (eq_ref00 P) as proof of (P0 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found ((eq_ref0 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) P) as proof of (P0 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found (((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) P) as proof of (P0 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found (((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) P) as proof of (P0 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found eq_ref000:=(eq_ref00 P):((P ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))->(P ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))))
% Found (eq_ref00 P) as proof of (P0 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found ((eq_ref0 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) P) as proof of (P0 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found (((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) P) as proof of (P0 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found (((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) P) as proof of (P0 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found eq_ref00:=(eq_ref0 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))):(((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found (eq_ref0 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b0)
% Found ((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b0)
% Found ((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b0)
% Found ((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b0)
% Found eq_ref00:=(eq_ref0 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))):(((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found (eq_ref0 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b0)
% Found ((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b0)
% Found ((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b0)
% Found ((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found eq_ref00:=(eq_ref0 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))):(((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found (eq_ref0 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b0)
% Found ((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b0)
% Found ((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b0)
% Found ((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found eq_ref00:=(eq_ref0 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))):(((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found (eq_ref0 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b0)
% Found ((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b0)
% Found ((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b0)
% Found ((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b0)
% Found eq_ref000:=(eq_ref00 P):((P ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))->(P ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))))
% Found (eq_ref00 P) as proof of (P0 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found ((eq_ref0 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) P) as proof of (P0 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found (((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) P) as proof of (P0 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found (((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) P) as proof of (P0 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found eq_ref00:=(eq_ref0 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))):(((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found (eq_ref0 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b0)
% Found ((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b0)
% Found ((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b0)
% Found ((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found eq_ref00:=(eq_ref0 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))):(((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found (eq_ref0 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b0)
% Found ((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b0)
% Found ((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b0)
% Found ((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found eq_ref00:=(eq_ref0 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))):(((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found (eq_ref0 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b0)
% Found ((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b0)
% Found ((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b0)
% Found ((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b0)
% Found eq_ref00:=(eq_ref0 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))):(((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found (eq_ref0 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b0)
% Found ((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b0)
% Found ((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b0)
% Found ((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found eq_ref000:=(eq_ref00 P):((P ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))->(P ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))))
% Found (eq_ref00 P) as proof of (P0 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found ((eq_ref0 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) P) as proof of (P0 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found (((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) P) as proof of (P0 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found (((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) P) as proof of (P0 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found eq_ref00:=(eq_ref0 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))):(((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found (eq_ref0 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b0)
% Found ((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b0)
% Found ((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b0)
% Found ((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found eq_ref00:=(eq_ref0 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))):(((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found (eq_ref0 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b0)
% Found ((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b0)
% Found ((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b0)
% Found ((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found eq_ref00:=(eq_ref0 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))):(((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found (eq_ref0 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b0)
% Found ((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b0)
% Found ((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b0)
% Found ((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found eq_ref00:=(eq_ref0 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))):(((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found (eq_ref0 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b0)
% Found ((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b0)
% Found ((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b0)
% Found ((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found eq_ref00:=(eq_ref0 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))):(((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found (eq_ref0 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b0)
% Found ((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b0)
% Found ((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b0)
% Found ((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b0)
% Found eq_ref00:=(eq_ref0 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))):(((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found (eq_ref0 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b0)
% Found ((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b0)
% Found ((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b0)
% Found ((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found eq_ref00:=(eq_ref0 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))):(((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found (eq_ref0 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b0)
% Found ((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b0)
% Found ((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b0)
% Found ((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found eq_ref00:=(eq_ref0 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))):(((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found (eq_ref0 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b0)
% Found ((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b0)
% Found ((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b0)
% Found ((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b0)
% Found eq_ref00:=(eq_ref0 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))):(((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found (eq_ref0 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b0)
% Found ((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b0)
% Found ((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b0)
% Found ((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found eq_ref00:=(eq_ref0 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))):(((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found (eq_ref0 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b0)
% Found ((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b0)
% Found ((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b0)
% Found ((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found eq_ref00:=(eq_ref0 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))):(((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found (eq_ref0 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b0)
% Found ((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b0)
% Found ((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b0)
% Found ((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found eq_ref00:=(eq_ref0 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))):(((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found (eq_ref0 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b0)
% Found ((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b0)
% Found ((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b0)
% Found ((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found eq_ref00:=(eq_ref0 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))):(((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found (eq_ref0 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b0)
% Found ((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b0)
% Found ((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b0)
% Found ((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found eq_ref00:=(eq_ref0 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))):(((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found (eq_ref0 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b0)
% Found ((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b0)
% Found ((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b0)
% Found ((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found eq_ref00:=(eq_ref0 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))):(((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found (eq_ref0 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b0)
% Found ((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b0)
% Found ((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b0)
% Found ((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found eq_ref00:=(eq_ref0 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))):(((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found (eq_ref0 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b0)
% Found ((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b0)
% Found ((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b0)
% Found ((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found x11:(P b0)
% Found (fun (x11:(P b0))=> x11) as proof of (P b0)
% Found (fun (x11:(P b0))=> x11) as proof of (P0 b0)
% Found x11:(P b0)
% Found (fun (x11:(P b0))=> x11) as proof of (P b0)
% Found (fun (x11:(P b0))=> x11) as proof of (P0 b0)
% Found x11:(P (b0 x0))
% Found (fun (x11:(P (b0 x0)))=> x11) as proof of (P (b0 x0))
% Found (fun (x11:(P (b0 x0)))=> x11) as proof of (P0 (b0 x0))
% Found x11:(P (b0 x0))
% Found (fun (x11:(P (b0 x0)))=> x11) as proof of (P (b0 x0))
% Found (fun (x11:(P (b0 x0)))=> x11) as proof of (P0 (b0 x0))
% Found eq_ref00:=(eq_ref0 b0):(((eq (a->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (a->Prop)) b0) b1)
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) b1)
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) b1)
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) b1)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b1):(((eq (a->Prop)) b1) (fun (x:a)=> (b1 x)))
% Found (eta_expansion_dep00 b1) as proof of (((eq (a->Prop)) b1) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt))))))))
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) b1) as proof of (((eq (a->Prop)) b1) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt))))))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b1) as proof of (((eq (a->Prop)) b1) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt))))))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b1) as proof of (((eq (a->Prop)) b1) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt))))))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b1) as proof of (((eq (a->Prop)) b1) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt))))))))
% Found eq_ref00:=(eq_ref0 a0):(((eq (b->Prop)) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))
% Found ((eq_ref (b->Prop)) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))
% Found ((eq_ref (b->Prop)) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))
% Found ((eq_ref (b->Prop)) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))
% Found eta_expansion000:=(eta_expansion00 a0):(((eq (b->Prop)) a0) (fun (x:b)=> (a0 x)))
% Found (eta_expansion00 a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))
% Found ((eta_expansion0 Prop) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))
% Found (((eta_expansion b) Prop) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))
% Found (((eta_expansion b) Prop) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))
% Found (((eta_expansion b) Prop) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))
% Found eq_ref00:=(eq_ref0 a0):(((eq (b->Prop)) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))
% Found ((eq_ref (b->Prop)) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))
% Found ((eq_ref (b->Prop)) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))
% Found ((eq_ref (b->Prop)) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))
% Found eta_expansion000:=(eta_expansion00 a0):(((eq (b->Prop)) a0) (fun (x:b)=> (a0 x)))
% Found (eta_expansion00 a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))
% Found ((eta_expansion0 Prop) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))
% Found (((eta_expansion b) Prop) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))
% Found (((eta_expansion b) Prop) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))
% Found (((eta_expansion b) Prop) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))
% Found eq_ref00:=(eq_ref0 a0):(((eq (b->Prop)) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))
% Found ((eq_ref (b->Prop)) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))
% Found ((eq_ref (b->Prop)) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))
% Found ((eq_ref (b->Prop)) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))
% Found eta_expansion000:=(eta_expansion00 a0):(((eq (b->Prop)) a0) (fun (x:b)=> (a0 x)))
% Found (eta_expansion00 a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))
% Found ((eta_expansion0 Prop) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))
% Found (((eta_expansion b) Prop) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))
% Found (((eta_expansion b) Prop) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))
% Found (((eta_expansion b) Prop) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))
% Found eta_expansion000:=(eta_expansion00 a0):(((eq (b->Prop)) a0) (fun (x:b)=> (a0 x)))
% Found (eta_expansion00 a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))
% Found ((eta_expansion0 Prop) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))
% Found (((eta_expansion b) Prop) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))
% Found (((eta_expansion b) Prop) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))
% Found (((eta_expansion b) Prop) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))
% Found eta_expansion000:=(eta_expansion00 a0):(((eq (b->Prop)) a0) (fun (x:b)=> (a0 x)))
% Found (eta_expansion00 a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))
% Found ((eta_expansion0 Prop) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))
% Found (((eta_expansion b) Prop) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))
% Found (((eta_expansion b) Prop) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))
% Found (((eta_expansion b) Prop) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))
% Found eq_ref00:=(eq_ref0 ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))):(((eq Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found (eq_ref0 ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) b0)
% Found ((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) b0)
% Found ((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) b0)
% Found ((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) b0)
% Found x1:(P0 b0)
% Instantiate: b0:=((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))):Prop
% Found (fun (x1:(P0 b0))=> x1) as proof of (P0 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found (fun (P0:(Prop->Prop)) (x1:(P0 b0))=> x1) as proof of ((P0 b0)->(P0 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))))
% Found (fun (P0:(Prop->Prop)) (x1:(P0 b0))=> x1) as proof of (P b0)
% Found eq_ref00:=(eq_ref0 ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))):(((eq Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found (eq_ref0 ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) b0)
% Found ((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) b0)
% Found ((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) b0)
% Found ((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) b0)
% Found x1:(P0 b0)
% Instantiate: b0:=((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))):Prop
% Found (fun (x1:(P0 b0))=> x1) as proof of (P0 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found (fun (P0:(Prop->Prop)) (x1:(P0 b0))=> x1) as proof of ((P0 b0)->(P0 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))))
% Found (fun (P0:(Prop->Prop)) (x1:(P0 b0))=> x1) as proof of (P b0)
% Found x1:(P ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Instantiate: b0:=((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))):Prop
% Found x1 as proof of (P0 b0)
% Found eq_ref00:=(eq_ref0 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))):(((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found (eq_ref0 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b0)
% Found ((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b0)
% Found ((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b0)
% Found ((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b0)
% Found x1:(P ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Instantiate: b0:=((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))):Prop
% Found x1 as proof of (P0 b0)
% Found eq_ref00:=(eq_ref0 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))):(((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found (eq_ref0 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b0)
% Found ((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b0)
% Found ((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b0)
% Found ((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq (a->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (a->Prop)) b0) (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt)))))))
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt)))))))
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt)))))))
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt)))))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))):(((eq (a->Prop)) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) (fun (x0:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))))
% Found (eta_expansion_dep00 (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) b0)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) b0)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) b0)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) b0)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq (a->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (a->Prop)) b0) (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt)))))))
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt)))))))
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt)))))))
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) (fun (Xz:a)=> ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) Xz) (f Xt)))))))
% Found eq_ref00:=(eq_ref0 (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))):(((eq (a->Prop)) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt))))))))
% Found (eq_ref0 (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) b0)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) b0)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) b0)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) b0)
% Found eta_expansion0000:=(eta_expansion000 P):((P (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt))))))))->(P (fun (x0:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))))
% Found (eta_expansion000 P) as proof of (P0 (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt))))))))
% Found ((eta_expansion00 (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) P) as proof of (P0 (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt))))))))
% Found (((eta_expansion0 Prop) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) P) as proof of (P0 (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt))))))))
% Found ((((eta_expansion a) Prop) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) P) as proof of (P0 (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt))))))))
% Found ((((eta_expansion a) Prop) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) P) as proof of (P0 (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt))))))))
% Found eta_expansion0000:=(eta_expansion000 P):((P (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt))))))))->(P (fun (x0:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))))
% Found (eta_expansion000 P) as proof of (P0 (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt))))))))
% Found ((eta_expansion00 (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) P) as proof of (P0 (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt))))))))
% Found (((eta_expansion0 Prop) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) P) as proof of (P0 (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt))))))))
% Found ((((eta_expansion a) Prop) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) P) as proof of (P0 (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt))))))))
% Found ((((eta_expansion a) Prop) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) P) as proof of (P0 (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt))))))))
% Found eta_expansion0000:=(eta_expansion000 P):((P (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt))))))))->(P (fun (x0:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))))
% Found (eta_expansion000 P) as proof of (P0 (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt))))))))
% Found ((eta_expansion00 (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) P) as proof of (P0 (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt))))))))
% Found (((eta_expansion0 Prop) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) P) as proof of (P0 (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt))))))))
% Found ((((eta_expansion a) Prop) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) P) as proof of (P0 (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt))))))))
% Found ((((eta_expansion a) Prop) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) P) as proof of (P0 (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt))))))))
% Found eta_expansion0000:=(eta_expansion000 P):((P (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt))))))))->(P (fun (x0:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))))
% Found (eta_expansion000 P) as proof of (P0 (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt))))))))
% Found ((eta_expansion00 (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) P) as proof of (P0 (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt))))))))
% Found (((eta_expansion0 Prop) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) P) as proof of (P0 (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt))))))))
% Found ((((eta_expansion a) Prop) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) P) as proof of (P0 (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt))))))))
% Found ((((eta_expansion a) Prop) (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt)))))))) P) as proof of (P0 (fun (Xz:a)=> ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xz) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xz) (f Xt))))))))
% Found eq_ref00:=(eq_ref0 b00):(((eq Prop) b00) b00)
% Found (eq_ref0 b00) as proof of (((eq Prop) b00) b0)
% Found ((eq_ref Prop) b00) as proof of (((eq Prop) b00) b0)
% Found ((eq_ref Prop) b00) as proof of (((eq Prop) b00) b0)
% Found ((eq_ref Prop) b00) as proof of (((eq Prop) b00) b0)
% Found eq_ref00:=(eq_ref0 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))):(((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found (eq_ref0 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b00)
% Found ((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b00)
% Found ((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b00)
% Found ((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b00)
% Found eq_ref00:=(eq_ref0 b00):(((eq Prop) b00) b00)
% Found (eq_ref0 b00) as proof of (((eq Prop) b00) b0)
% Found ((eq_ref Prop) b00) as proof of (((eq Prop) b00) b0)
% Found ((eq_ref Prop) b00) as proof of (((eq Prop) b00) b0)
% Found ((eq_ref Prop) b00) as proof of (((eq Prop) b00) b0)
% Found eq_ref00:=(eq_ref0 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))):(((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found (eq_ref0 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b00)
% Found ((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b00)
% Found ((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b00)
% Found ((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b00)
% Found eq_ref00:=(eq_ref0 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))):(((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found (eq_ref0 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b00)
% Found ((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b00)
% Found ((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b00)
% Found ((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b00)
% Found eq_ref00:=(eq_ref0 b00):(((eq Prop) b00) b00)
% Found (eq_ref0 b00) as proof of (((eq Prop) b00) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found ((eq_ref Prop) b00) as proof of (((eq Prop) b00) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found ((eq_ref Prop) b00) as proof of (((eq Prop) b00) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found ((eq_ref Prop) b00) as proof of (((eq Prop) b00) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found eq_ref00:=(eq_ref0 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))):(((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found (eq_ref0 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b00)
% Found ((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b00)
% Found ((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b00)
% Found ((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b00)
% Found eq_ref00:=(eq_ref0 b00):(((eq Prop) b00) b00)
% Found (eq_ref0 b00) as proof of (((eq Prop) b00) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found ((eq_ref Prop) b00) as proof of (((eq Prop) b00) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found ((eq_ref Prop) b00) as proof of (((eq Prop) b00) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found ((eq_ref Prop) b00) as proof of (((eq Prop) b00) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found eq_ref000:=(eq_ref00 P):((P ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))->(P ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))))
% Found (eq_ref00 P) as proof of (P0 ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found ((eq_ref0 ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) P) as proof of (P0 ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found (((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) P) as proof of (P0 ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found (((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) P) as proof of (P0 ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found eq_ref000:=(eq_ref00 P):((P ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))->(P ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))))
% Found (eq_ref00 P) as proof of (P0 ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found ((eq_ref0 ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) P) as proof of (P0 ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found (((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) P) as proof of (P0 ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found (((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) P) as proof of (P0 ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found eq_ref000:=(eq_ref00 P):((P ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))->(P ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))))
% Found (eq_ref00 P) as proof of (P0 ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found ((eq_ref0 ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) P) as proof of (P0 ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found (((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) P) as proof of (P0 ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found (((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) P) as proof of (P0 ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found eq_ref000:=(eq_ref00 P):((P ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))->(P ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))))
% Found (eq_ref00 P) as proof of (P0 ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found ((eq_ref0 ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) P) as proof of (P0 ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found (((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) P) as proof of (P0 ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found (((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) P) as proof of (P0 ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found eq_ref000:=(eq_ref00 P):((P ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))->(P ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))))
% Found (eq_ref00 P) as proof of (P0 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found ((eq_ref0 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) P) as proof of (P0 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found (((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) P) as proof of (P0 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found (((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) P) as proof of (P0 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found eq_ref000:=(eq_ref00 P):((P ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))->(P ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))))
% Found (eq_ref00 P) as proof of (P0 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found ((eq_ref0 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) P) as proof of (P0 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found (((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) P) as proof of (P0 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found (((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) P) as proof of (P0 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found eq_ref00:=(eq_ref0 ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))):(((eq Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found (eq_ref0 ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) b0)
% Found ((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) b0)
% Found ((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) b0)
% Found ((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found eq_ref00:=(eq_ref0 ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))):(((eq Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found (eq_ref0 ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) b0)
% Found ((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) b0)
% Found ((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) b0)
% Found ((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found eq_ref00:=(eq_ref0 ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))):(((eq Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found (eq_ref0 ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) b0)
% Found ((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) b0)
% Found ((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) b0)
% Found ((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found eq_ref00:=(eq_ref0 ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))):(((eq Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found (eq_ref0 ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) b0)
% Found ((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) b0)
% Found ((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) b0)
% Found ((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found eq_ref00:=(eq_ref0 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))):(((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found (eq_ref0 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b00)
% Found ((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b00)
% Found ((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b00)
% Found ((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b00)
% Found eq_ref00:=(eq_ref0 b00):(((eq Prop) b00) b00)
% Found (eq_ref0 b00) as proof of (((eq Prop) b00) b0)
% Found ((eq_ref Prop) b00) as proof of (((eq Prop) b00) b0)
% Found ((eq_ref Prop) b00) as proof of (((eq Prop) b00) b0)
% Found ((eq_ref Prop) b00) as proof of (((eq Prop) b00) b0)
% Found eq_ref00:=(eq_ref0 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))):(((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found (eq_ref0 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b00)
% Found ((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b00)
% Found ((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b00)
% Found ((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b00)
% Found eq_ref00:=(eq_ref0 b00):(((eq Prop) b00) b00)
% Found (eq_ref0 b00) as proof of (((eq Prop) b00) b0)
% Found ((eq_ref Prop) b00) as proof of (((eq Prop) b00) b0)
% Found ((eq_ref Prop) b00) as proof of (((eq Prop) b00) b0)
% Found ((eq_ref Prop) b00) as proof of (((eq Prop) b00) b0)
% Found eq_ref000:=(eq_ref00 P):((P ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))->(P ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))))
% Found (eq_ref00 P) as proof of (P0 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found ((eq_ref0 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) P) as proof of (P0 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found (((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) P) as proof of (P0 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found (((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) P) as proof of (P0 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found eq_ref000:=(eq_ref00 P):((P ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))->(P ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))))
% Found (eq_ref00 P) as proof of (P0 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found ((eq_ref0 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) P) as proof of (P0 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found (((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) P) as proof of (P0 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found (((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) P) as proof of (P0 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found eq_ref000:=(eq_ref00 P):((P ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))->(P ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))))
% Found (eq_ref00 P) as proof of (P0 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found ((eq_ref0 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) P) as proof of (P0 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found (((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) P) as proof of (P0 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found (((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) P) as proof of (P0 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found eq_ref000:=(eq_ref00 P):((P ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))->(P ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))))
% Found (eq_ref00 P) as proof of (P0 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found ((eq_ref0 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) P) as proof of (P0 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found (((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) P) as proof of (P0 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found (((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) P) as proof of (P0 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found x01:(P b0)
% Found (fun (x01:(P b0))=> x01) as proof of (P b0)
% Found (fun (x01:(P b0))=> x01) as proof of (P0 b0)
% Found x01:(P b0)
% Found (fun (x01:(P b0))=> x01) as proof of (P b0)
% Found (fun (x01:(P b0))=> x01) as proof of (P0 b0)
% Found eq_ref00:=(eq_ref0 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))):(((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found (eq_ref0 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b00)
% Found ((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b00)
% Found ((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b00)
% Found ((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b00)
% Found eq_ref00:=(eq_ref0 b00):(((eq Prop) b00) b00)
% Found (eq_ref0 b00) as proof of (((eq Prop) b00) (b0 x0))
% Found ((eq_ref Prop) b00) as proof of (((eq Prop) b00) (b0 x0))
% Found ((eq_ref Prop) b00) as proof of (((eq Prop) b00) (b0 x0))
% Found ((eq_ref Prop) b00) as proof of (((eq Prop) b00) (b0 x0))
% Found eq_ref00:=(eq_ref0 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))):(((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found (eq_ref0 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b00)
% Found ((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b00)
% Found ((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b00)
% Found ((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b00)
% Found eq_ref00:=(eq_ref0 b00):(((eq Prop) b00) b00)
% Found (eq_ref0 b00) as proof of (((eq Prop) b00) (b0 x0))
% Found ((eq_ref Prop) b00) as proof of (((eq Prop) b00) (b0 x0))
% Found ((eq_ref Prop) b00) as proof of (((eq Prop) b00) (b0 x0))
% Found ((eq_ref Prop) b00) as proof of (((eq Prop) b00) (b0 x0))
% Found eq_ref00:=(eq_ref0 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))):(((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found (eq_ref0 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b00)
% Found ((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b00)
% Found ((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b00)
% Found ((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b00)
% Found eq_ref00:=(eq_ref0 b00):(((eq Prop) b00) b00)
% Found (eq_ref0 b00) as proof of (((eq Prop) b00) (b0 x0))
% Found ((eq_ref Prop) b00) as proof of (((eq Prop) b00) (b0 x0))
% Found ((eq_ref Prop) b00) as proof of (((eq Prop) b00) (b0 x0))
% Found ((eq_ref Prop) b00) as proof of (((eq Prop) b00) (b0 x0))
% Found eq_ref00:=(eq_ref0 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))):(((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found (eq_ref0 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b00)
% Found ((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b00)
% Found ((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b00)
% Found ((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b00)
% Found eq_ref00:=(eq_ref0 b00):(((eq Prop) b00) b00)
% Found (eq_ref0 b00) as proof of (((eq Prop) b00) (b0 x0))
% Found ((eq_ref Prop) b00) as proof of (((eq Prop) b00) (b0 x0))
% Found ((eq_ref Prop) b00) as proof of (((eq Prop) b00) (b0 x0))
% Found ((eq_ref Prop) b00) as proof of (((eq Prop) b00) (b0 x0))
% Found eq_ref00:=(eq_ref0 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))):(((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found (eq_ref0 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b00)
% Found ((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b00)
% Found ((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b00)
% Found ((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b00)
% Found eq_ref00:=(eq_ref0 b00):(((eq Prop) b00) b00)
% Found (eq_ref0 b00) as proof of (((eq Prop) b00) (b0 x0))
% Found ((eq_ref Prop) b00) as proof of (((eq Prop) b00) (b0 x0))
% Found ((eq_ref Prop) b00) as proof of (((eq Prop) b00) (b0 x0))
% Found ((eq_ref Prop) b00) as proof of (((eq Prop) b00) (b0 x0))
% Found eq_ref00:=(eq_ref0 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))):(((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found (eq_ref0 ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b00)
% Found ((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b00)
% Found ((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b00)
% Found ((eq_ref Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) as proof of (((eq Prop) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt))))))) b00)
% Found eq_ref00:=(eq_ref0 b00):(((eq Prop) b00) b00)
% Found (eq_ref0 b00) as proof of (((eq Prop) b00) (b0 x0))
% Found ((eq_ref Prop) b00) as proof of (((eq Prop) b00) (b0 x0))
% Found ((eq_ref Prop) b00) as proof of (((eq Prop) b00) (b0 x0))
% Found ((eq_ref Prop) b00) as proof of (((eq Prop) b00) (b0 x0))
% Found eq_ref000:=(eq_ref00 P):((P ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))->(P ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))))
% Found (eq_ref00 P) as proof of (P0 ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found ((eq_ref0 ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) P) as proof of (P0 ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found (((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) P) as proof of (P0 ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found (((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) P) as proof of (P0 ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found eq_ref000:=(eq_ref00 P):((P ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))->(P ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))))
% Found (eq_ref00 P) as proof of (P0 ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found ((eq_ref0 ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) P) as proof of (P0 ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found (((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) P) as proof of (P0 ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found (((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) P) as proof of (P0 ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((or ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) x0) (f Xt)))))))
% Found eq_ref00:=(eq_ref0 ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))):(((eq Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found (eq_ref0 ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) b0)
% Found ((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) b0)
% Found ((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) b0)
% Found ((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) b0)
% Found eq_ref00:=(eq_ref0 ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))):(((eq Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt))))))
% Found (eq_ref0 ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) b0)
% Found ((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) b0)
% Found ((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and ((or (x Xt)) (y Xt))) (((eq a) x0) (f Xt)))))) b0)
% Found ((eq
% EOF
%------------------------------------------------------------------------------