%------------------------------------------------------------------------------
% File     : cocATP---0.2.0
% Problem  : SEU905^5 : TPTP v6.1.0. Released v4.0.0.
% Transfm  : none
% Format   : tptp:raw
% Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p

% Computer : n094.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 300.10s
% Output   : None 
% Verified : 
% SZS Type : None (Parsing solution fails)
% Syntax   : Number of formulae    : 0

% Comments : 
%------------------------------------------------------------------------------
%----NO SOLUTION OUTPUT BY SYSTEM
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% % Problem  : SEU905^5 : TPTP v6.1.0. Released v4.0.0.
% % Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n094.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 11:41:41 CDT 2014
% % CPUTime  : 300.10 
% 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 0x2a10098>, <kernel.Type object at 0x29f1200>) of role type named g_type
% Using role type
% Declaring g:Type
% FOF formula (<kernel.Constant object at 0x2a10e60>, <kernel.Type object at 0x29f1200>) of role type named b_type
% Using role type
% Declaring b:Type
% FOF formula (<kernel.Constant object at 0x2a10fc8>, <kernel.Type object at 0x29f1248>) of role type named a_type
% Using role type
% Declaring a:Type
% FOF formula (forall (Xh1:(g->b)) (Xh2:(b->a)) (Xs1:(g->Prop)) (Xf1:(g->(g->g))) (Xs2:(b->Prop)) (Xf2:(b->(b->b))) (Xs3:(a->Prop)) (Xf3:(a->(a->a))), ((((and ((and ((and (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(Xs1 ((Xf1 Xx) Xy))))) (forall (Xx:a) (Xy:a), (((and (Xs3 Xx)) (Xs3 Xy))->(Xs3 ((Xf3 Xx) Xy)))))) (forall (Xx:g), ((Xs1 Xx)->(Xs3 (Xh2 (Xh1 Xx))))))) (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy)))))))->False)->(((and ((and ((and ((and ((and ((and ((and (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(Xs1 ((Xf1 Xx) Xy))))) (forall (Xx:b) (Xy:b), (((and (Xs2 Xx)) (Xs2 Xy))->(Xs2 ((Xf2 Xx) Xy)))))) (forall (Xx:g), ((Xs1 Xx)->(Xs2 (Xh1 Xx)))))) (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq b) (Xh1 ((Xf1 Xx) Xy))) ((Xf2 (Xh1 Xx)) (Xh1 Xy))))))) (forall (Xx:b) (Xy:b), (((and (Xs2 Xx)) (Xs2 Xy))->(Xs2 ((Xf2 Xx) Xy)))))) (forall (Xx:a) (Xy:a), (((and (Xs3 Xx)) (Xs3 Xy))->(Xs3 ((Xf3 Xx) Xy)))))) (forall (Xx:b), ((Xs2 Xx)->(Xs3 (Xh2 Xx)))))) (forall (Xx:b) (Xy:b), (((and (Xs2 Xx)) (Xs2 Xy))->(((eq a) (Xh2 ((Xf2 Xx) Xy))) ((Xf3 (Xh2 Xx)) (Xh2 Xy))))))->False))) of role conjecture named cTHM126A_pme
% Conjecture to prove = (forall (Xh1:(g->b)) (Xh2:(b->a)) (Xs1:(g->Prop)) (Xf1:(g->(g->g))) (Xs2:(b->Prop)) (Xf2:(b->(b->b))) (Xs3:(a->Prop)) (Xf3:(a->(a->a))), ((((and ((and ((and (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(Xs1 ((Xf1 Xx) Xy))))) (forall (Xx:a) (Xy:a), (((and (Xs3 Xx)) (Xs3 Xy))->(Xs3 ((Xf3 Xx) Xy)))))) (forall (Xx:g), ((Xs1 Xx)->(Xs3 (Xh2 (Xh1 Xx))))))) (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy)))))))->False)->(((and ((and ((and ((and ((and ((and ((and (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(Xs1 ((Xf1 Xx) Xy))))) (forall (Xx:b) (Xy:b), (((and (Xs2 Xx)) (Xs2 Xy))->(Xs2 ((Xf2 Xx) Xy)))))) (forall (Xx:g), ((Xs1 Xx)->(Xs2 (Xh1 Xx)))))) (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq b) (Xh1 ((Xf1 Xx) Xy))) ((Xf2 (Xh1 Xx)) (Xh1 Xy))))))) (forall (Xx:b) (Xy:b), (((and (Xs2 Xx)) (Xs2 Xy))->(Xs2 ((Xf2 Xx) Xy)))))) (forall (Xx:a) (Xy:a), (((and (Xs3 Xx)) (Xs3 Xy))->(Xs3 ((Xf3 Xx) Xy)))))) (forall (Xx:b), ((Xs2 Xx)->(Xs3 (Xh2 Xx)))))) (forall (Xx:b) (Xy:b), (((and (Xs2 Xx)) (Xs2 Xy))->(((eq a) (Xh2 ((Xf2 Xx) Xy))) ((Xf3 (Xh2 Xx)) (Xh2 Xy))))))->False))):Prop
% Parameter g_DUMMY:g.
% Parameter b_DUMMY:b.
% Parameter a_DUMMY:a.
% We need to prove ['(forall (Xh1:(g->b)) (Xh2:(b->a)) (Xs1:(g->Prop)) (Xf1:(g->(g->g))) (Xs2:(b->Prop)) (Xf2:(b->(b->b))) (Xs3:(a->Prop)) (Xf3:(a->(a->a))), ((((and ((and ((and (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(Xs1 ((Xf1 Xx) Xy))))) (forall (Xx:a) (Xy:a), (((and (Xs3 Xx)) (Xs3 Xy))->(Xs3 ((Xf3 Xx) Xy)))))) (forall (Xx:g), ((Xs1 Xx)->(Xs3 (Xh2 (Xh1 Xx))))))) (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy)))))))->False)->(((and ((and ((and ((and ((and ((and ((and (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(Xs1 ((Xf1 Xx) Xy))))) (forall (Xx:b) (Xy:b), (((and (Xs2 Xx)) (Xs2 Xy))->(Xs2 ((Xf2 Xx) Xy)))))) (forall (Xx:g), ((Xs1 Xx)->(Xs2 (Xh1 Xx)))))) (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq b) (Xh1 ((Xf1 Xx) Xy))) ((Xf2 (Xh1 Xx)) (Xh1 Xy))))))) (forall (Xx:b) (Xy:b), (((and (Xs2 Xx)) (Xs2 Xy))->(Xs2 ((Xf2 Xx) Xy)))))) (forall (Xx:a) (Xy:a), (((and (Xs3 Xx)) (Xs3 Xy))->(Xs3 ((Xf3 Xx) Xy)))))) (forall (Xx:b), ((Xs2 Xx)->(Xs3 (Xh2 Xx)))))) (forall (Xx:b) (Xy:b), (((and (Xs2 Xx)) (Xs2 Xy))->(((eq a) (Xh2 ((Xf2 Xx) Xy))) ((Xf3 (Xh2 Xx)) (Xh2 Xy))))))->False)))']
% Parameter g:Type.
% Parameter b:Type.
% Parameter a:Type.
% Trying to prove (forall (Xh1:(g->b)) (Xh2:(b->a)) (Xs1:(g->Prop)) (Xf1:(g->(g->g))) (Xs2:(b->Prop)) (Xf2:(b->(b->b))) (Xs3:(a->Prop)) (Xf3:(a->(a->a))), ((((and ((and ((and (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(Xs1 ((Xf1 Xx) Xy))))) (forall (Xx:a) (Xy:a), (((and (Xs3 Xx)) (Xs3 Xy))->(Xs3 ((Xf3 Xx) Xy)))))) (forall (Xx:g), ((Xs1 Xx)->(Xs3 (Xh2 (Xh1 Xx))))))) (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy)))))))->False)->(((and ((and ((and ((and ((and ((and ((and (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(Xs1 ((Xf1 Xx) Xy))))) (forall (Xx:b) (Xy:b), (((and (Xs2 Xx)) (Xs2 Xy))->(Xs2 ((Xf2 Xx) Xy)))))) (forall (Xx:g), ((Xs1 Xx)->(Xs2 (Xh1 Xx)))))) (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq b) (Xh1 ((Xf1 Xx) Xy))) ((Xf2 (Xh1 Xx)) (Xh1 Xy))))))) (forall (Xx:b) (Xy:b), (((and (Xs2 Xx)) (Xs2 Xy))->(Xs2 ((Xf2 Xx) Xy)))))) (forall (Xx:a) (Xy:a), (((and (Xs3 Xx)) (Xs3 Xy))->(Xs3 ((Xf3 Xx) Xy)))))) (forall (Xx:b), ((Xs2 Xx)->(Xs3 (Xh2 Xx)))))) (forall (Xx:b) (Xy:b), (((and (Xs2 Xx)) (Xs2 Xy))->(((eq a) (Xh2 ((Xf2 Xx) Xy))) ((Xf3 (Xh2 Xx)) (Xh2 Xy))))))->False)))
% Found x0:False
% Found (fun (x1:((and ((and ((and ((and ((and ((and ((and (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(Xs1 ((Xf1 Xx) Xy))))) (forall (Xx:b) (Xy:b), (((and (Xs2 Xx)) (Xs2 Xy))->(Xs2 ((Xf2 Xx) Xy)))))) (forall (Xx:g), ((Xs1 Xx)->(Xs2 (Xh1 Xx)))))) (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq b) (Xh1 ((Xf1 Xx) Xy))) ((Xf2 (Xh1 Xx)) (Xh1 Xy))))))) (forall (Xx:b) (Xy:b), (((and (Xs2 Xx)) (Xs2 Xy))->(Xs2 ((Xf2 Xx) Xy)))))) (forall (Xx:a) (Xy:a), (((and (Xs3 Xx)) (Xs3 Xy))->(Xs3 ((Xf3 Xx) Xy)))))) (forall (Xx:b), ((Xs2 Xx)->(Xs3 (Xh2 Xx)))))) (forall (Xx:b) (Xy:b), (((and (Xs2 Xx)) (Xs2 Xy))->(((eq a) (Xh2 ((Xf2 Xx) Xy))) ((Xf3 (Xh2 Xx)) (Xh2 Xy)))))))=> x0) as proof of False
% Found (fun (x1:((and ((and ((and ((and ((and ((and ((and (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(Xs1 ((Xf1 Xx) Xy))))) (forall (Xx:b) (Xy:b), (((and (Xs2 Xx)) (Xs2 Xy))->(Xs2 ((Xf2 Xx) Xy)))))) (forall (Xx:g), ((Xs1 Xx)->(Xs2 (Xh1 Xx)))))) (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq b) (Xh1 ((Xf1 Xx) Xy))) ((Xf2 (Xh1 Xx)) (Xh1 Xy))))))) (forall (Xx:b) (Xy:b), (((and (Xs2 Xx)) (Xs2 Xy))->(Xs2 ((Xf2 Xx) Xy)))))) (forall (Xx:a) (Xy:a), (((and (Xs3 Xx)) (Xs3 Xy))->(Xs3 ((Xf3 Xx) Xy)))))) (forall (Xx:b), ((Xs2 Xx)->(Xs3 (Xh2 Xx)))))) (forall (Xx:b) (Xy:b), (((and (Xs2 Xx)) (Xs2 Xy))->(((eq a) (Xh2 ((Xf2 Xx) Xy))) ((Xf3 (Xh2 Xx)) (Xh2 Xy)))))))=> x0) as proof of (((and ((and ((and ((and ((and ((and ((and (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(Xs1 ((Xf1 Xx) Xy))))) (forall (Xx:b) (Xy:b), (((and (Xs2 Xx)) (Xs2 Xy))->(Xs2 ((Xf2 Xx) Xy)))))) (forall (Xx:g), ((Xs1 Xx)->(Xs2 (Xh1 Xx)))))) (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq b) (Xh1 ((Xf1 Xx) Xy))) ((Xf2 (Xh1 Xx)) (Xh1 Xy))))))) (forall (Xx:b) (Xy:b), (((and (Xs2 Xx)) (Xs2 Xy))->(Xs2 ((Xf2 Xx) Xy)))))) (forall (Xx:a) (Xy:a), (((and (Xs3 Xx)) (Xs3 Xy))->(Xs3 ((Xf3 Xx) Xy)))))) (forall (Xx:b), ((Xs2 Xx)->(Xs3 (Xh2 Xx)))))) (forall (Xx:b) (Xy:b), (((and (Xs2 Xx)) (Xs2 Xy))->(((eq a) (Xh2 ((Xf2 Xx) Xy))) ((Xf3 (Xh2 Xx)) (Xh2 Xy))))))->False)
% Found x2:False
% Found (fun (x3:(forall (Xx:b) (Xy:b), (((and (Xs2 Xx)) (Xs2 Xy))->(((eq a) (Xh2 ((Xf2 Xx) Xy))) ((Xf3 (Xh2 Xx)) (Xh2 Xy))))))=> x2) as proof of False
% Found (fun (x3:(forall (Xx:b) (Xy:b), (((and (Xs2 Xx)) (Xs2 Xy))->(((eq a) (Xh2 ((Xf2 Xx) Xy))) ((Xf3 (Xh2 Xx)) (Xh2 Xy))))))=> x2) as proof of ((forall (Xx:b) (Xy:b), (((and (Xs2 Xx)) (Xs2 Xy))->(((eq a) (Xh2 ((Xf2 Xx) Xy))) ((Xf3 (Xh2 Xx)) (Xh2 Xy)))))->False)
% Found x1:False
% Found (fun (x3:(forall (Xx:b) (Xy:b), (((and (Xs2 Xx)) (Xs2 Xy))->(((eq a) (Xh2 ((Xf2 Xx) Xy))) ((Xf3 (Xh2 Xx)) (Xh2 Xy))))))=> x1) as proof of False
% Found (fun (x2:((and ((and ((and ((and ((and ((and (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(Xs1 ((Xf1 Xx) Xy))))) (forall (Xx:b) (Xy:b), (((and (Xs2 Xx)) (Xs2 Xy))->(Xs2 ((Xf2 Xx) Xy)))))) (forall (Xx:g), ((Xs1 Xx)->(Xs2 (Xh1 Xx)))))) (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq b) (Xh1 ((Xf1 Xx) Xy))) ((Xf2 (Xh1 Xx)) (Xh1 Xy))))))) (forall (Xx:b) (Xy:b), (((and (Xs2 Xx)) (Xs2 Xy))->(Xs2 ((Xf2 Xx) Xy)))))) (forall (Xx:a) (Xy:a), (((and (Xs3 Xx)) (Xs3 Xy))->(Xs3 ((Xf3 Xx) Xy)))))) (forall (Xx:b), ((Xs2 Xx)->(Xs3 (Xh2 Xx)))))) (x3:(forall (Xx:b) (Xy:b), (((and (Xs2 Xx)) (Xs2 Xy))->(((eq a) (Xh2 ((Xf2 Xx) Xy))) ((Xf3 (Xh2 Xx)) (Xh2 Xy))))))=> x1) as proof of ((forall (Xx:b) (Xy:b), (((and (Xs2 Xx)) (Xs2 Xy))->(((eq a) (Xh2 ((Xf2 Xx) Xy))) ((Xf3 (Xh2 Xx)) (Xh2 Xy)))))->False)
% Found (fun (x2:((and ((and ((and ((and ((and ((and (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(Xs1 ((Xf1 Xx) Xy))))) (forall (Xx:b) (Xy:b), (((and (Xs2 Xx)) (Xs2 Xy))->(Xs2 ((Xf2 Xx) Xy)))))) (forall (Xx:g), ((Xs1 Xx)->(Xs2 (Xh1 Xx)))))) (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq b) (Xh1 ((Xf1 Xx) Xy))) ((Xf2 (Xh1 Xx)) (Xh1 Xy))))))) (forall (Xx:b) (Xy:b), (((and (Xs2 Xx)) (Xs2 Xy))->(Xs2 ((Xf2 Xx) Xy)))))) (forall (Xx:a) (Xy:a), (((and (Xs3 Xx)) (Xs3 Xy))->(Xs3 ((Xf3 Xx) Xy)))))) (forall (Xx:b), ((Xs2 Xx)->(Xs3 (Xh2 Xx)))))) (x3:(forall (Xx:b) (Xy:b), (((and (Xs2 Xx)) (Xs2 Xy))->(((eq a) (Xh2 ((Xf2 Xx) Xy))) ((Xf3 (Xh2 Xx)) (Xh2 Xy))))))=> x1) as proof of (((and ((and ((and ((and ((and ((and (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(Xs1 ((Xf1 Xx) Xy))))) (forall (Xx:b) (Xy:b), (((and (Xs2 Xx)) (Xs2 Xy))->(Xs2 ((Xf2 Xx) Xy)))))) (forall (Xx:g), ((Xs1 Xx)->(Xs2 (Xh1 Xx)))))) (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq b) (Xh1 ((Xf1 Xx) Xy))) ((Xf2 (Xh1 Xx)) (Xh1 Xy))))))) (forall (Xx:b) (Xy:b), (((and (Xs2 Xx)) (Xs2 Xy))->(Xs2 ((Xf2 Xx) Xy)))))) (forall (Xx:a) (Xy:a), (((and (Xs3 Xx)) (Xs3 Xy))->(Xs3 ((Xf3 Xx) Xy)))))) (forall (Xx:b), ((Xs2 Xx)->(Xs3 (Xh2 Xx)))))->((forall (Xx:b) (Xy:b), (((and (Xs2 Xx)) (Xs2 Xy))->(((eq a) (Xh2 ((Xf2 Xx) Xy))) ((Xf3 (Xh2 Xx)) (Xh2 Xy)))))->False))
% Found x4:False
% Found (fun (x5:(forall (Xx:b), ((Xs2 Xx)->(Xs3 (Xh2 Xx)))))=> x4) as proof of False
% Found (fun (x5:(forall (Xx:b), ((Xs2 Xx)->(Xs3 (Xh2 Xx)))))=> x4) as proof of ((forall (Xx:b), ((Xs2 Xx)->(Xs3 (Xh2 Xx))))->False)
% Found x3:False
% Found (fun (x5:(forall (Xx:b), ((Xs2 Xx)->(Xs3 (Xh2 Xx)))))=> x3) as proof of False
% Found (fun (x4:((and ((and ((and ((and ((and (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(Xs1 ((Xf1 Xx) Xy))))) (forall (Xx:b) (Xy:b), (((and (Xs2 Xx)) (Xs2 Xy))->(Xs2 ((Xf2 Xx) Xy)))))) (forall (Xx:g), ((Xs1 Xx)->(Xs2 (Xh1 Xx)))))) (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq b) (Xh1 ((Xf1 Xx) Xy))) ((Xf2 (Xh1 Xx)) (Xh1 Xy))))))) (forall (Xx:b) (Xy:b), (((and (Xs2 Xx)) (Xs2 Xy))->(Xs2 ((Xf2 Xx) Xy)))))) (forall (Xx:a) (Xy:a), (((and (Xs3 Xx)) (Xs3 Xy))->(Xs3 ((Xf3 Xx) Xy)))))) (x5:(forall (Xx:b), ((Xs2 Xx)->(Xs3 (Xh2 Xx)))))=> x3) as proof of ((forall (Xx:b), ((Xs2 Xx)->(Xs3 (Xh2 Xx))))->False)
% Found (fun (x4:((and ((and ((and ((and ((and (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(Xs1 ((Xf1 Xx) Xy))))) (forall (Xx:b) (Xy:b), (((and (Xs2 Xx)) (Xs2 Xy))->(Xs2 ((Xf2 Xx) Xy)))))) (forall (Xx:g), ((Xs1 Xx)->(Xs2 (Xh1 Xx)))))) (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq b) (Xh1 ((Xf1 Xx) Xy))) ((Xf2 (Xh1 Xx)) (Xh1 Xy))))))) (forall (Xx:b) (Xy:b), (((and (Xs2 Xx)) (Xs2 Xy))->(Xs2 ((Xf2 Xx) Xy)))))) (forall (Xx:a) (Xy:a), (((and (Xs3 Xx)) (Xs3 Xy))->(Xs3 ((Xf3 Xx) Xy)))))) (x5:(forall (Xx:b), ((Xs2 Xx)->(Xs3 (Xh2 Xx)))))=> x3) as proof of (((and ((and ((and ((and ((and (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(Xs1 ((Xf1 Xx) Xy))))) (forall (Xx:b) (Xy:b), (((and (Xs2 Xx)) (Xs2 Xy))->(Xs2 ((Xf2 Xx) Xy)))))) (forall (Xx:g), ((Xs1 Xx)->(Xs2 (Xh1 Xx)))))) (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq b) (Xh1 ((Xf1 Xx) Xy))) ((Xf2 (Xh1 Xx)) (Xh1 Xy))))))) (forall (Xx:b) (Xy:b), (((and (Xs2 Xx)) (Xs2 Xy))->(Xs2 ((Xf2 Xx) Xy)))))) (forall (Xx:a) (Xy:a), (((and (Xs3 Xx)) (Xs3 Xy))->(Xs3 ((Xf3 Xx) Xy)))))->((forall (Xx:b), ((Xs2 Xx)->(Xs3 (Xh2 Xx))))->False))
% Found eq_ref00:=(eq_ref0 (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))))):(((eq Prop) (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))))) (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy)))))))
% Found (eq_ref0 (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))))) as proof of (((eq Prop) (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))))) b0)
% Found ((eq_ref Prop) (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))))) as proof of (((eq Prop) (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))))) b0)
% Found ((eq_ref Prop) (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))))) as proof of (((eq Prop) (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))))) b0)
% Found ((eq_ref Prop) (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))))) as proof of (((eq Prop) (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))))) b0)
% Found x6:False
% Found (fun (x7:(forall (Xx:a) (Xy:a), (((and (Xs3 Xx)) (Xs3 Xy))->(Xs3 ((Xf3 Xx) Xy)))))=> x6) as proof of False
% Found (fun (x7:(forall (Xx:a) (Xy:a), (((and (Xs3 Xx)) (Xs3 Xy))->(Xs3 ((Xf3 Xx) Xy)))))=> x6) as proof of ((forall (Xx:a) (Xy:a), (((and (Xs3 Xx)) (Xs3 Xy))->(Xs3 ((Xf3 Xx) Xy))))->False)
% Found eq_ref000:=(eq_ref00 P):((P (Xh2 (Xh1 ((Xf1 Xx) Xy))))->(P (Xh2 (Xh1 ((Xf1 Xx) Xy)))))
% Found (eq_ref00 P) as proof of (P0 (Xh2 (Xh1 ((Xf1 Xx) Xy))))
% Found ((eq_ref0 (Xh2 (Xh1 ((Xf1 Xx) Xy)))) P) as proof of (P0 (Xh2 (Xh1 ((Xf1 Xx) Xy))))
% Found (((eq_ref a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) P) as proof of (P0 (Xh2 (Xh1 ((Xf1 Xx) Xy))))
% Found (((eq_ref a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) P) as proof of (P0 (Xh2 (Xh1 ((Xf1 Xx) Xy))))
% Found x5:False
% Found (fun (x7:(forall (Xx:a) (Xy:a), (((and (Xs3 Xx)) (Xs3 Xy))->(Xs3 ((Xf3 Xx) Xy)))))=> x5) as proof of False
% Found (fun (x6:((and ((and ((and ((and (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(Xs1 ((Xf1 Xx) Xy))))) (forall (Xx:b) (Xy:b), (((and (Xs2 Xx)) (Xs2 Xy))->(Xs2 ((Xf2 Xx) Xy)))))) (forall (Xx:g), ((Xs1 Xx)->(Xs2 (Xh1 Xx)))))) (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq b) (Xh1 ((Xf1 Xx) Xy))) ((Xf2 (Xh1 Xx)) (Xh1 Xy))))))) (forall (Xx:b) (Xy:b), (((and (Xs2 Xx)) (Xs2 Xy))->(Xs2 ((Xf2 Xx) Xy)))))) (x7:(forall (Xx:a) (Xy:a), (((and (Xs3 Xx)) (Xs3 Xy))->(Xs3 ((Xf3 Xx) Xy)))))=> x5) as proof of ((forall (Xx:a) (Xy:a), (((and (Xs3 Xx)) (Xs3 Xy))->(Xs3 ((Xf3 Xx) Xy))))->False)
% Found (fun (x6:((and ((and ((and ((and (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(Xs1 ((Xf1 Xx) Xy))))) (forall (Xx:b) (Xy:b), (((and (Xs2 Xx)) (Xs2 Xy))->(Xs2 ((Xf2 Xx) Xy)))))) (forall (Xx:g), ((Xs1 Xx)->(Xs2 (Xh1 Xx)))))) (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq b) (Xh1 ((Xf1 Xx) Xy))) ((Xf2 (Xh1 Xx)) (Xh1 Xy))))))) (forall (Xx:b) (Xy:b), (((and (Xs2 Xx)) (Xs2 Xy))->(Xs2 ((Xf2 Xx) Xy)))))) (x7:(forall (Xx:a) (Xy:a), (((and (Xs3 Xx)) (Xs3 Xy))->(Xs3 ((Xf3 Xx) Xy)))))=> x5) as proof of (((and ((and ((and ((and (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(Xs1 ((Xf1 Xx) Xy))))) (forall (Xx:b) (Xy:b), (((and (Xs2 Xx)) (Xs2 Xy))->(Xs2 ((Xf2 Xx) Xy)))))) (forall (Xx:g), ((Xs1 Xx)->(Xs2 (Xh1 Xx)))))) (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq b) (Xh1 ((Xf1 Xx) Xy))) ((Xf2 (Xh1 Xx)) (Xh1 Xy))))))) (forall (Xx:b) (Xy:b), (((and (Xs2 Xx)) (Xs2 Xy))->(Xs2 ((Xf2 Xx) Xy)))))->((forall (Xx:a) (Xy:a), (((and (Xs3 Xx)) (Xs3 Xy))->(Xs3 ((Xf3 Xx) Xy))))->False))
% Found eq_ref00:=(eq_ref0 (Xh2 (Xh1 ((Xf1 Xx) Xy)))):(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) (Xh2 (Xh1 ((Xf1 Xx) Xy))))
% Found (eq_ref0 (Xh2 (Xh1 ((Xf1 Xx) Xy)))) as proof of (((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) b0)
% Found ((eq_ref a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) as proof of (((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) b0)
% Found ((eq_ref a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) as proof of (((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) b0)
% Found ((eq_ref a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) as proof of (((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))
% Found ((eq_ref a) b0) as proof of (((eq a) b0) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))
% Found ((eq_ref a) b0) as proof of (((eq a) b0) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))
% Found ((eq_ref a) b0) as proof of (((eq a) b0) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))
% Found classical_choice:=(fun (A:Type) (B:Type) (R:(A->(B->Prop))) (b:B)=> ((fun (C:((forall (x:A), ((ex B) (fun (y:B)=> (((fun (x0:A) (y0:B)=> (((ex B) (fun (z:B)=> ((R x0) z)))->((R x0) y0))) x) y))))->((ex (A->B)) (fun (f:(A->B))=> (forall (x:A), (((fun (x0:A) (y:B)=> (((ex B) (fun (z:B)=> ((R x0) z)))->((R x0) y))) x) (f x)))))))=> (C (fun (x:A)=> ((fun (C0:((or ((ex B) (fun (z:B)=> ((R x) z)))) (not ((ex B) (fun (z:B)=> ((R x) z))))))=> ((((((or_ind ((ex B) (fun (z:B)=> ((R x) z)))) (not ((ex B) (fun (z:B)=> ((R x) z))))) ((ex B) (fun (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y))))) ((((ex_ind B) (fun (z:B)=> ((R x) z))) ((ex B) (fun (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y))))) (fun (y:B) (H:((R x) y))=> ((((ex_intro B) (fun (y0:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y0)))) y) (fun (_:((ex B) (fun (z:B)=> ((R x) z))))=> H))))) (fun (N:(not ((ex B) (fun (z:B)=> ((R x) z)))))=> ((((ex_intro B) (fun (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y)))) b) (fun (H:((ex B) (fun (z:B)=> ((R x) z))))=> ((False_rect ((R x) b)) (N H)))))) C0)) (classic ((ex B) (fun (z:B)=> ((R x) z)))))))) (((choice A) B) (fun (x:A) (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y)))))):(forall (A:Type) (B:Type) (R:(A->(B->Prop))), (B->((ex (A->B)) (fun (f:(A->B))=> (forall (x:A), (((ex B) (fun (y:B)=> ((R x) y)))->((R x) (f x))))))))
% Instantiate: b0:=(forall (A:Type) (B:Type) (R:(A->(B->Prop))), (B->((ex (A->B)) (fun (f:(A->B))=> (forall (x:A), (((ex B) (fun (y:B)=> ((R x) y)))->((R x) (f x)))))))):Prop
% Found classical_choice as proof of b0
% Found eq_ref00:=(eq_ref0 (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))))):(((eq Prop) (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))))) (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy)))))))
% Found (eq_ref0 (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))))) as proof of (((eq Prop) (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))))) b0)
% Found ((eq_ref Prop) (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))))) as proof of (((eq Prop) (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))))) b0)
% Found ((eq_ref Prop) (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))))) as proof of (((eq Prop) (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))))) b0)
% Found ((eq_ref Prop) (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))))) as proof of (((eq Prop) (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))))) b0)
% Found eq_ref00:=(eq_ref0 (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))))):(((eq Prop) (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))))) (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy)))))))
% Found (eq_ref0 (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))))) as proof of (((eq Prop) (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))))) b0)
% Found ((eq_ref Prop) (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))))) as proof of (((eq Prop) (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))))) b0)
% Found ((eq_ref Prop) (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))))) as proof of (((eq Prop) (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))))) b0)
% Found ((eq_ref Prop) (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))))) as proof of (((eq Prop) (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))))) b0)
% Found eq_ref000:=(eq_ref00 P):((P (Xh2 (Xh1 ((Xf1 Xx) Xy))))->(P (Xh2 (Xh1 ((Xf1 Xx) Xy)))))
% Found (eq_ref00 P) as proof of (P0 (Xh2 (Xh1 ((Xf1 Xx) Xy))))
% Found ((eq_ref0 (Xh2 (Xh1 ((Xf1 Xx) Xy)))) P) as proof of (P0 (Xh2 (Xh1 ((Xf1 Xx) Xy))))
% Found (((eq_ref a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) P) as proof of (P0 (Xh2 (Xh1 ((Xf1 Xx) Xy))))
% Found (((eq_ref a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) P) as proof of (P0 (Xh2 (Xh1 ((Xf1 Xx) Xy))))
% Found x8:False
% Found (fun (x9:(forall (Xx:b) (Xy:b), (((and (Xs2 Xx)) (Xs2 Xy))->(Xs2 ((Xf2 Xx) Xy)))))=> x8) as proof of False
% Found (fun (x9:(forall (Xx:b) (Xy:b), (((and (Xs2 Xx)) (Xs2 Xy))->(Xs2 ((Xf2 Xx) Xy)))))=> x8) as proof of ((forall (Xx:b) (Xy:b), (((and (Xs2 Xx)) (Xs2 Xy))->(Xs2 ((Xf2 Xx) Xy))))->False)
% Found eq_ref000:=(eq_ref00 P):((P (Xh2 (Xh1 ((Xf1 Xx) Xy))))->(P (Xh2 (Xh1 ((Xf1 Xx) Xy)))))
% Found (eq_ref00 P) as proof of (P0 (Xh2 (Xh1 ((Xf1 Xx) Xy))))
% Found ((eq_ref0 (Xh2 (Xh1 ((Xf1 Xx) Xy)))) P) as proof of (P0 (Xh2 (Xh1 ((Xf1 Xx) Xy))))
% Found (((eq_ref a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) P) as proof of (P0 (Xh2 (Xh1 ((Xf1 Xx) Xy))))
% Found (((eq_ref a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) P) as proof of (P0 (Xh2 (Xh1 ((Xf1 Xx) Xy))))
% Found x7:False
% Found (fun (x9:(forall (Xx:b) (Xy:b), (((and (Xs2 Xx)) (Xs2 Xy))->(Xs2 ((Xf2 Xx) Xy)))))=> x7) as proof of False
% Found (fun (x8:((and ((and ((and (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(Xs1 ((Xf1 Xx) Xy))))) (forall (Xx:b) (Xy:b), (((and (Xs2 Xx)) (Xs2 Xy))->(Xs2 ((Xf2 Xx) Xy)))))) (forall (Xx:g), ((Xs1 Xx)->(Xs2 (Xh1 Xx)))))) (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq b) (Xh1 ((Xf1 Xx) Xy))) ((Xf2 (Xh1 Xx)) (Xh1 Xy))))))) (x9:(forall (Xx:b) (Xy:b), (((and (Xs2 Xx)) (Xs2 Xy))->(Xs2 ((Xf2 Xx) Xy)))))=> x7) as proof of ((forall (Xx:b) (Xy:b), (((and (Xs2 Xx)) (Xs2 Xy))->(Xs2 ((Xf2 Xx) Xy))))->False)
% Found (fun (x8:((and ((and ((and (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(Xs1 ((Xf1 Xx) Xy))))) (forall (Xx:b) (Xy:b), (((and (Xs2 Xx)) (Xs2 Xy))->(Xs2 ((Xf2 Xx) Xy)))))) (forall (Xx:g), ((Xs1 Xx)->(Xs2 (Xh1 Xx)))))) (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq b) (Xh1 ((Xf1 Xx) Xy))) ((Xf2 (Xh1 Xx)) (Xh1 Xy))))))) (x9:(forall (Xx:b) (Xy:b), (((and (Xs2 Xx)) (Xs2 Xy))->(Xs2 ((Xf2 Xx) Xy)))))=> x7) as proof of (((and ((and ((and (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(Xs1 ((Xf1 Xx) Xy))))) (forall (Xx:b) (Xy:b), (((and (Xs2 Xx)) (Xs2 Xy))->(Xs2 ((Xf2 Xx) Xy)))))) (forall (Xx:g), ((Xs1 Xx)->(Xs2 (Xh1 Xx)))))) (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq b) (Xh1 ((Xf1 Xx) Xy))) ((Xf2 (Xh1 Xx)) (Xh1 Xy))))))->((forall (Xx:b) (Xy:b), (((and (Xs2 Xx)) (Xs2 Xy))->(Xs2 ((Xf2 Xx) Xy))))->False))
% Found eq_ref00:=(eq_ref0 (Xh2 (Xh1 ((Xf1 Xx) Xy)))):(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) (Xh2 (Xh1 ((Xf1 Xx) Xy))))
% Found (eq_ref0 (Xh2 (Xh1 ((Xf1 Xx) Xy)))) as proof of (((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) b0)
% Found ((eq_ref a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) as proof of (((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) b0)
% Found ((eq_ref a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) as proof of (((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) b0)
% Found ((eq_ref a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) as proof of (((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))
% Found ((eq_ref a) b0) as proof of (((eq a) b0) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))
% Found ((eq_ref a) b0) as proof of (((eq a) b0) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))
% Found ((eq_ref a) b0) as proof of (((eq a) b0) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))
% Found iff_trans:=(fun (A:Prop) (B:Prop) (C:Prop) (AB:((iff A) B)) (BC:((iff B) C))=> ((((conj (A->C)) (C->A)) (fun (x:A)=> ((((proj1 (B->C)) (C->B)) BC) ((((proj1 (A->B)) (B->A)) AB) x)))) (fun (x:C)=> ((((proj2 (A->B)) (B->A)) AB) ((((proj2 (B->C)) (C->B)) BC) x))))):(forall (A:Prop) (B:Prop) (C:Prop), (((iff A) B)->(((iff B) C)->((iff A) C))))
% Instantiate: b0:=(forall (A:Prop) (B:Prop) (C:Prop), (((iff A) B)->(((iff B) C)->((iff A) C)))):Prop
% Found iff_trans as proof of b0
% Found eq_ref00:=(eq_ref0 (Xh2 (Xh1 ((Xf1 Xx) Xy)))):(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) (Xh2 (Xh1 ((Xf1 Xx) Xy))))
% Found (eq_ref0 (Xh2 (Xh1 ((Xf1 Xx) Xy)))) as proof of (((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) b0)
% Found ((eq_ref a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) as proof of (((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) b0)
% Found ((eq_ref a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) as proof of (((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) b0)
% Found ((eq_ref a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) as proof of (((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))
% Found ((eq_ref a) b0) as proof of (((eq a) b0) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))
% Found ((eq_ref a) b0) as proof of (((eq a) b0) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))
% Found ((eq_ref a) b0) as proof of (((eq a) b0) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))
% Found iff_trans:=(fun (A:Prop) (B:Prop) (C:Prop) (AB:((iff A) B)) (BC:((iff B) C))=> ((((conj (A->C)) (C->A)) (fun (x:A)=> ((((proj1 (B->C)) (C->B)) BC) ((((proj1 (A->B)) (B->A)) AB) x)))) (fun (x:C)=> ((((proj2 (A->B)) (B->A)) AB) ((((proj2 (B->C)) (C->B)) BC) x))))):(forall (A:Prop) (B:Prop) (C:Prop), (((iff A) B)->(((iff B) C)->((iff A) C))))
% Instantiate: b0:=(forall (A:Prop) (B:Prop) (C:Prop), (((iff A) B)->(((iff B) C)->((iff A) C)))):Prop
% Found iff_trans as proof of b0
% Found eq_ref00:=(eq_ref0 (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))))):(((eq Prop) (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))))) (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy)))))))
% Found (eq_ref0 (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))))) as proof of (((eq Prop) (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))))) b0)
% Found ((eq_ref Prop) (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))))) as proof of (((eq Prop) (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))))) b0)
% Found ((eq_ref Prop) (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))))) as proof of (((eq Prop) (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))))) b0)
% Found ((eq_ref Prop) (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))))) as proof of (((eq Prop) (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))))) b0)
% Found iff_trans:=(fun (A:Prop) (B:Prop) (C:Prop) (AB:((iff A) B)) (BC:((iff B) C))=> ((((conj (A->C)) (C->A)) (fun (x:A)=> ((((proj1 (B->C)) (C->B)) BC) ((((proj1 (A->B)) (B->A)) AB) x)))) (fun (x:C)=> ((((proj2 (A->B)) (B->A)) AB) ((((proj2 (B->C)) (C->B)) BC) x))))):(forall (A:Prop) (B:Prop) (C:Prop), (((iff A) B)->(((iff B) C)->((iff A) C))))
% Instantiate: b0:=(forall (A:Prop) (B:Prop) (C:Prop), (((iff A) B)->(((iff B) C)->((iff A) C)))):Prop
% Found iff_trans as proof of b0
% Found eq_ref00:=(eq_ref0 (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))))):(((eq Prop) (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))))) (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy)))))))
% Found (eq_ref0 (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))))) as proof of (((eq Prop) (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))))) b0)
% Found ((eq_ref Prop) (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))))) as proof of (((eq Prop) (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))))) b0)
% Found ((eq_ref Prop) (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))))) as proof of (((eq Prop) (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))))) b0)
% Found ((eq_ref Prop) (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))))) as proof of (((eq Prop) (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))))) b0)
% Found eq_ref00:=(eq_ref0 (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))))):(((eq Prop) (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))))) (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy)))))))
% Found (eq_ref0 (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))))) as proof of (((eq Prop) (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))))) b0)
% Found ((eq_ref Prop) (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))))) as proof of (((eq Prop) (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))))) b0)
% Found ((eq_ref Prop) (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))))) as proof of (((eq Prop) (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))))) b0)
% Found ((eq_ref Prop) (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))))) as proof of (((eq Prop) (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))))) b0)
% Found eq_ref00:=(eq_ref0 (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))))):(((eq Prop) (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))))) (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy)))))))
% Found (eq_ref0 (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))))) as proof of (((eq Prop) (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))))) b0)
% Found ((eq_ref Prop) (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))))) as proof of (((eq Prop) (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))))) b0)
% Found ((eq_ref Prop) (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))))) as proof of (((eq Prop) (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))))) b0)
% Found ((eq_ref Prop) (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))))) as proof of (((eq Prop) (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))))) b0)
% Found eq_ref000:=(eq_ref00 P):((P (Xh2 (Xh1 ((Xf1 Xx) Xy))))->(P (Xh2 (Xh1 ((Xf1 Xx) Xy)))))
% Found (eq_ref00 P) as proof of (P0 (Xh2 (Xh1 ((Xf1 Xx) Xy))))
% Found ((eq_ref0 (Xh2 (Xh1 ((Xf1 Xx) Xy)))) P) as proof of (P0 (Xh2 (Xh1 ((Xf1 Xx) Xy))))
% Found (((eq_ref a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) P) as proof of (P0 (Xh2 (Xh1 ((Xf1 Xx) Xy))))
% Found (((eq_ref a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) P) as proof of (P0 (Xh2 (Xh1 ((Xf1 Xx) Xy))))
% Found eq_ref000:=(eq_ref00 P):((P (Xh2 (Xh1 ((Xf1 Xx) Xy))))->(P (Xh2 (Xh1 ((Xf1 Xx) Xy)))))
% Found (eq_ref00 P) as proof of (P0 (Xh2 (Xh1 ((Xf1 Xx) Xy))))
% Found ((eq_ref0 (Xh2 (Xh1 ((Xf1 Xx) Xy)))) P) as proof of (P0 (Xh2 (Xh1 ((Xf1 Xx) Xy))))
% Found (((eq_ref a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) P) as proof of (P0 (Xh2 (Xh1 ((Xf1 Xx) Xy))))
% Found (((eq_ref a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) P) as proof of (P0 (Xh2 (Xh1 ((Xf1 Xx) Xy))))
% Found eq_ref00:=(eq_ref0 (Xh2 (Xh1 ((Xf1 Xx) Xy)))):(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) (Xh2 (Xh1 ((Xf1 Xx) Xy))))
% Found (eq_ref0 (Xh2 (Xh1 ((Xf1 Xx) Xy)))) as proof of (((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) b0)
% Found ((eq_ref a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) as proof of (((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) b0)
% Found ((eq_ref a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) as proof of (((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) b0)
% Found ((eq_ref a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) as proof of (((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))
% Found ((eq_ref a) b0) as proof of (((eq a) b0) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))
% Found ((eq_ref a) b0) as proof of (((eq a) b0) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))
% Found ((eq_ref a) b0) as proof of (((eq a) b0) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))
% Found eq_ref00:=(eq_ref0 (forall (Xx:g), ((Xs1 Xx)->(Xs3 (Xh2 (Xh1 Xx)))))):(((eq Prop) (forall (Xx:g), ((Xs1 Xx)->(Xs3 (Xh2 (Xh1 Xx)))))) (forall (Xx:g), ((Xs1 Xx)->(Xs3 (Xh2 (Xh1 Xx))))))
% Found (eq_ref0 (forall (Xx:g), ((Xs1 Xx)->(Xs3 (Xh2 (Xh1 Xx)))))) as proof of (((eq Prop) (forall (Xx:g), ((Xs1 Xx)->(Xs3 (Xh2 (Xh1 Xx)))))) b0)
% Found ((eq_ref Prop) (forall (Xx:g), ((Xs1 Xx)->(Xs3 (Xh2 (Xh1 Xx)))))) as proof of (((eq Prop) (forall (Xx:g), ((Xs1 Xx)->(Xs3 (Xh2 (Xh1 Xx)))))) b0)
% Found ((eq_ref Prop) (forall (Xx:g), ((Xs1 Xx)->(Xs3 (Xh2 (Xh1 Xx)))))) as proof of (((eq Prop) (forall (Xx:g), ((Xs1 Xx)->(Xs3 (Xh2 (Xh1 Xx)))))) b0)
% Found ((eq_ref Prop) (forall (Xx:g), ((Xs1 Xx)->(Xs3 (Xh2 (Xh1 Xx)))))) as proof of (((eq Prop) (forall (Xx:g), ((Xs1 Xx)->(Xs3 (Xh2 (Xh1 Xx)))))) b0)
% Found eq_ref00:=(eq_ref0 (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))))):(((eq Prop) (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))))) (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy)))))))
% Found (eq_ref0 (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))))) as proof of (((eq Prop) (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))))) b0)
% Found ((eq_ref Prop) (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))))) as proof of (((eq Prop) (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))))) b0)
% Found ((eq_ref Prop) (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))))) as proof of (((eq Prop) (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))))) b0)
% Found ((eq_ref Prop) (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))))) as proof of (((eq Prop) (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) (Xh2 (Xh1 ((Xf1 Xx) Xy))))
% Found ((eq_ref a) b0) as proof of (((eq a) b0) (Xh2 (Xh1 ((Xf1 Xx) Xy))))
% Found ((eq_ref a) b0) as proof of (((eq a) b0) (Xh2 (Xh1 ((Xf1 Xx) Xy))))
% Found ((eq_ref a) b0) as proof of (((eq a) b0) (Xh2 (Xh1 ((Xf1 Xx) Xy))))
% Found eq_ref00:=(eq_ref0 ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy)))):(((eq a) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))
% Found (eq_ref0 ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy)))) as proof of (((eq a) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy)))) b0)
% Found ((eq_ref a) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy)))) as proof of (((eq a) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy)))) b0)
% Found ((eq_ref a) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy)))) as proof of (((eq a) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy)))) b0)
% Found ((eq_ref a) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy)))) as proof of (((eq a) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy)))) b0)
% Found eq_ref00:=(eq_ref0 (Xh2 (Xh1 ((Xf1 Xx) Xy)))):(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) (Xh2 (Xh1 ((Xf1 Xx) Xy))))
% Found (eq_ref0 (Xh2 (Xh1 ((Xf1 Xx) Xy)))) as proof of (((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) b0)
% Found ((eq_ref a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) as proof of (((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) b0)
% Found ((eq_ref a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) as proof of (((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) b0)
% Found ((eq_ref a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) as proof of (((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))
% Found ((eq_ref a) b0) as proof of (((eq a) b0) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))
% Found ((eq_ref a) b0) as proof of (((eq a) b0) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))
% Found ((eq_ref a) b0) as proof of (((eq a) b0) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))
% Found eq_ref000:=(eq_ref00 P):((P (Xh2 (Xh1 ((Xf1 Xx) Xy))))->(P (Xh2 (Xh1 ((Xf1 Xx) Xy)))))
% Found (eq_ref00 P) as proof of (P0 (Xh2 (Xh1 ((Xf1 Xx) Xy))))
% Found ((eq_ref0 (Xh2 (Xh1 ((Xf1 Xx) Xy)))) P) as proof of (P0 (Xh2 (Xh1 ((Xf1 Xx) Xy))))
% Found (((eq_ref a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) P) as proof of (P0 (Xh2 (Xh1 ((Xf1 Xx) Xy))))
% Found (((eq_ref a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) P) as proof of (P0 (Xh2 (Xh1 ((Xf1 Xx) Xy))))
% Found eq_ref000:=(eq_ref00 P):((P (Xh2 (Xh1 ((Xf1 Xx) Xy))))->(P (Xh2 (Xh1 ((Xf1 Xx) Xy)))))
% Found (eq_ref00 P) as proof of (P0 (Xh2 (Xh1 ((Xf1 Xx) Xy))))
% Found ((eq_ref0 (Xh2 (Xh1 ((Xf1 Xx) Xy)))) P) as proof of (P0 (Xh2 (Xh1 ((Xf1 Xx) Xy))))
% Found (((eq_ref a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) P) as proof of (P0 (Xh2 (Xh1 ((Xf1 Xx) Xy))))
% Found (((eq_ref a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) P) as proof of (P0 (Xh2 (Xh1 ((Xf1 Xx) Xy))))
% Found eq_ref000:=(eq_ref00 P):((P (Xh2 (Xh1 ((Xf1 Xx) Xy))))->(P (Xh2 (Xh1 ((Xf1 Xx) Xy)))))
% Found (eq_ref00 P) as proof of (P0 (Xh2 (Xh1 ((Xf1 Xx) Xy))))
% Found ((eq_ref0 (Xh2 (Xh1 ((Xf1 Xx) Xy)))) P) as proof of (P0 (Xh2 (Xh1 ((Xf1 Xx) Xy))))
% Found (((eq_ref a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) P) as proof of (P0 (Xh2 (Xh1 ((Xf1 Xx) Xy))))
% Found (((eq_ref a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) P) as proof of (P0 (Xh2 (Xh1 ((Xf1 Xx) Xy))))
% Found x10:False
% Found (fun (x11:(forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq b) (Xh1 ((Xf1 Xx) Xy))) ((Xf2 (Xh1 Xx)) (Xh1 Xy))))))=> x10) as proof of False
% Found (fun (x11:(forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq b) (Xh1 ((Xf1 Xx) Xy))) ((Xf2 (Xh1 Xx)) (Xh1 Xy))))))=> x10) as proof of ((forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq b) (Xh1 ((Xf1 Xx) Xy))) ((Xf2 (Xh1 Xx)) (Xh1 Xy)))))->False)
% Found eq_sym0:=(eq_sym Prop):(forall (a:Prop) (b:Prop), ((((eq Prop) a) b)->(((eq Prop) b) a)))
% Instantiate: b0:=(forall (a:Prop) (b:Prop), ((((eq Prop) a) b)->(((eq Prop) b) a))):Prop
% Found eq_sym0 as proof of b0
% Found eq_ref000:=(eq_ref00 P):((P (Xh2 (Xh1 ((Xf1 Xx) Xy))))->(P (Xh2 (Xh1 ((Xf1 Xx) Xy)))))
% Found (eq_ref00 P) as proof of (P0 (Xh2 (Xh1 ((Xf1 Xx) Xy))))
% Found ((eq_ref0 (Xh2 (Xh1 ((Xf1 Xx) Xy)))) P) as proof of (P0 (Xh2 (Xh1 ((Xf1 Xx) Xy))))
% Found (((eq_ref a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) P) as proof of (P0 (Xh2 (Xh1 ((Xf1 Xx) Xy))))
% Found (((eq_ref a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) P) as proof of (P0 (Xh2 (Xh1 ((Xf1 Xx) Xy))))
% Found eq_ref000:=(eq_ref00 P):((P (Xh2 (Xh1 ((Xf1 Xx) Xy))))->(P (Xh2 (Xh1 ((Xf1 Xx) Xy)))))
% Found (eq_ref00 P) as proof of (P0 (Xh2 (Xh1 ((Xf1 Xx) Xy))))
% Found ((eq_ref0 (Xh2 (Xh1 ((Xf1 Xx) Xy)))) P) as proof of (P0 (Xh2 (Xh1 ((Xf1 Xx) Xy))))
% Found (((eq_ref a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) P) as proof of (P0 (Xh2 (Xh1 ((Xf1 Xx) Xy))))
% Found (((eq_ref a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) P) as proof of (P0 (Xh2 (Xh1 ((Xf1 Xx) Xy))))
% Found functional_extensionality_double:=(fun (A:Type) (B:Type) (C:Type) (f:(A->(B->C))) (g:(A->(B->C))) (x:(forall (x:A) (y:B), (((eq C) ((f x) y)) ((g x) y))))=> (((((functional_extensionality_dep A) (fun (x2:A)=> (B->C))) f) g) (fun (x0:A)=> (((((functional_extensionality_dep B) (fun (x3:B)=> C)) (f x0)) (g x0)) (x x0))))):(forall (A:Type) (B:Type) (C:Type) (f:(A->(B->C))) (g:(A->(B->C))), ((forall (x:A) (y:B), (((eq C) ((f x) y)) ((g x) y)))->(((eq (A->(B->C))) f) g)))
% Instantiate: b0:=(forall (A:Type) (B:Type) (C:Type) (f:(A->(B->C))) (g:(A->(B->C))), ((forall (x:A) (y:B), (((eq C) ((f x) y)) ((g x) y)))->(((eq (A->(B->C))) f) g))):Prop
% Found functional_extensionality_double as proof of b0
% Found eq_ref00:=(eq_ref0 (Xh2 (Xh1 ((Xf1 Xx) Xy)))):(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) (Xh2 (Xh1 ((Xf1 Xx) Xy))))
% Found (eq_ref0 (Xh2 (Xh1 ((Xf1 Xx) Xy)))) as proof of (((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) b0)
% Found ((eq_ref a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) as proof of (((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) b0)
% Found ((eq_ref a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) as proof of (((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) b0)
% Found ((eq_ref a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) as proof of (((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))
% Found ((eq_ref a) b0) as proof of (((eq a) b0) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))
% Found ((eq_ref a) b0) as proof of (((eq a) b0) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))
% Found ((eq_ref a) b0) as proof of (((eq a) b0) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))
% Found eq_ref00:=(eq_ref0 (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))))):(((eq Prop) (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))))) (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy)))))))
% Found (eq_ref0 (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))))) as proof of (((eq Prop) (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))))) b0)
% Found ((eq_ref Prop) (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))))) as proof of (((eq Prop) (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))))) b0)
% Found ((eq_ref Prop) (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))))) as proof of (((eq Prop) (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))))) b0)
% Found ((eq_ref Prop) (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))))) as proof of (((eq Prop) (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))))) b0)
% Found x9:False
% Found (fun (x11:(forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq b) (Xh1 ((Xf1 Xx) Xy))) ((Xf2 (Xh1 Xx)) (Xh1 Xy))))))=> x9) as proof of False
% Found (fun (x10:((and ((and (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(Xs1 ((Xf1 Xx) Xy))))) (forall (Xx:b) (Xy:b), (((and (Xs2 Xx)) (Xs2 Xy))->(Xs2 ((Xf2 Xx) Xy)))))) (forall (Xx:g), ((Xs1 Xx)->(Xs2 (Xh1 Xx)))))) (x11:(forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq b) (Xh1 ((Xf1 Xx) Xy))) ((Xf2 (Xh1 Xx)) (Xh1 Xy))))))=> x9) as proof of ((forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq b) (Xh1 ((Xf1 Xx) Xy))) ((Xf2 (Xh1 Xx)) (Xh1 Xy)))))->False)
% Found (fun (x10:((and ((and (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(Xs1 ((Xf1 Xx) Xy))))) (forall (Xx:b) (Xy:b), (((and (Xs2 Xx)) (Xs2 Xy))->(Xs2 ((Xf2 Xx) Xy)))))) (forall (Xx:g), ((Xs1 Xx)->(Xs2 (Xh1 Xx)))))) (x11:(forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq b) (Xh1 ((Xf1 Xx) Xy))) ((Xf2 (Xh1 Xx)) (Xh1 Xy))))))=> x9) as proof of (((and ((and (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(Xs1 ((Xf1 Xx) Xy))))) (forall (Xx:b) (Xy:b), (((and (Xs2 Xx)) (Xs2 Xy))->(Xs2 ((Xf2 Xx) Xy)))))) (forall (Xx:g), ((Xs1 Xx)->(Xs2 (Xh1 Xx)))))->((forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq b) (Xh1 ((Xf1 Xx) Xy))) ((Xf2 (Xh1 Xx)) (Xh1 Xy)))))->False))
% Found functional_extensionality_double:=(fun (A:Type) (B:Type) (C:Type) (f:(A->(B->C))) (g:(A->(B->C))) (x:(forall (x:A) (y:B), (((eq C) ((f x) y)) ((g x) y))))=> (((((functional_extensionality_dep A) (fun (x2:A)=> (B->C))) f) g) (fun (x0:A)=> (((((functional_extensionality_dep B) (fun (x3:B)=> C)) (f x0)) (g x0)) (x x0))))):(forall (A:Type) (B:Type) (C:Type) (f:(A->(B->C))) (g:(A->(B->C))), ((forall (x:A) (y:B), (((eq C) ((f x) y)) ((g x) y)))->(((eq (A->(B->C))) f) g)))
% Instantiate: b0:=(forall (A:Type) (B:Type) (C:Type) (f:(A->(B->C))) (g:(A->(B->C))), ((forall (x:A) (y:B), (((eq C) ((f x) y)) ((g x) y)))->(((eq (A->(B->C))) f) g))):Prop
% Found functional_extensionality_double as proof of b0
% Found functional_extensionality_double:=(fun (A:Type) (B:Type) (C:Type) (f:(A->(B->C))) (g:(A->(B->C))) (x:(forall (x:A) (y:B), (((eq C) ((f x) y)) ((g x) y))))=> (((((functional_extensionality_dep A) (fun (x2:A)=> (B->C))) f) g) (fun (x0:A)=> (((((functional_extensionality_dep B) (fun (x3:B)=> C)) (f x0)) (g x0)) (x x0))))):(forall (A:Type) (B:Type) (C:Type) (f:(A->(B->C))) (g:(A->(B->C))), ((forall (x:A) (y:B), (((eq C) ((f x) y)) ((g x) y)))->(((eq (A->(B->C))) f) g)))
% Instantiate: b0:=(forall (A:Type) (B:Type) (C:Type) (f:(A->(B->C))) (g:(A->(B->C))), ((forall (x:A) (y:B), (((eq C) ((f x) y)) ((g x) y)))->(((eq (A->(B->C))) f) g))):Prop
% Found functional_extensionality_double as proof of b0
% Found eq_ref00:=(eq_ref0 (Xh2 (Xh1 ((Xf1 Xx) Xy)))):(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) (Xh2 (Xh1 ((Xf1 Xx) Xy))))
% Found (eq_ref0 (Xh2 (Xh1 ((Xf1 Xx) Xy)))) as proof of (((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) b0)
% Found ((eq_ref a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) as proof of (((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) b0)
% Found ((eq_ref a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) as proof of (((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) b0)
% Found ((eq_ref a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) as proof of (((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))
% Found ((eq_ref a) b0) as proof of (((eq a) b0) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))
% Found ((eq_ref a) b0) as proof of (((eq a) b0) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))
% Found ((eq_ref a) b0) as proof of (((eq a) b0) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))
% Found eq_ref00:=(eq_ref0 (Xh2 (Xh1 ((Xf1 Xx) Xy)))):(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) (Xh2 (Xh1 ((Xf1 Xx) Xy))))
% Found (eq_ref0 (Xh2 (Xh1 ((Xf1 Xx) Xy)))) as proof of (((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) b0)
% Found ((eq_ref a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) as proof of (((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) b0)
% Found ((eq_ref a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) as proof of (((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) b0)
% Found ((eq_ref a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) as proof of (((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))
% Found ((eq_ref a) b0) as proof of (((eq a) b0) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))
% Found ((eq_ref a) b0) as proof of (((eq a) b0) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))
% Found ((eq_ref a) b0) as proof of (((eq a) b0) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))
% Found eq_ref00:=(eq_ref0 (Xh2 (Xh1 ((Xf1 Xx) Xy)))):(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) (Xh2 (Xh1 ((Xf1 Xx) Xy))))
% Found (eq_ref0 (Xh2 (Xh1 ((Xf1 Xx) Xy)))) as proof of (((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) b0)
% Found ((eq_ref a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) as proof of (((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) b0)
% Found ((eq_ref a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) as proof of (((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) b0)
% Found ((eq_ref a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) as proof of (((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))
% Found ((eq_ref a) b0) as proof of (((eq a) b0) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))
% Found ((eq_ref a) b0) as proof of (((eq a) b0) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))
% Found ((eq_ref a) b0) as proof of (((eq a) b0) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))
% Found eq_ref00:=(eq_ref0 (Xh2 (Xh1 ((Xf1 Xx) Xy)))):(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) (Xh2 (Xh1 ((Xf1 Xx) Xy))))
% Found (eq_ref0 (Xh2 (Xh1 ((Xf1 Xx) Xy)))) as proof of (((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) b0)
% Found ((eq_ref a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) as proof of (((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) b0)
% Found ((eq_ref a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) as proof of (((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) b0)
% Found ((eq_ref a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) as proof of (((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))
% Found ((eq_ref a) b0) as proof of (((eq a) b0) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))
% Found ((eq_ref a) b0) as proof of (((eq a) b0) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))
% Found ((eq_ref a) b0) as proof of (((eq a) b0) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))
% Found eq_ref00:=(eq_ref0 (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))))):(((eq Prop) (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))))) (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy)))))))
% Found (eq_ref0 (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))))) as proof of (((eq Prop) (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))))) b0)
% Found ((eq_ref Prop) (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))))) as proof of (((eq Prop) (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))))) b0)
% Found ((eq_ref Prop) (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))))) as proof of (((eq Prop) (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))))) b0)
% Found ((eq_ref Prop) (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))))) as proof of (((eq Prop) (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))))) b0)
% Found classical_choice:=(fun (A:Type) (B:Type) (R:(A->(B->Prop))) (b:B)=> ((fun (C:((forall (x:A), ((ex B) (fun (y:B)=> (((fun (x0:A) (y0:B)=> (((ex B) (fun (z:B)=> ((R x0) z)))->((R x0) y0))) x) y))))->((ex (A->B)) (fun (f:(A->B))=> (forall (x:A), (((fun (x0:A) (y:B)=> (((ex B) (fun (z:B)=> ((R x0) z)))->((R x0) y))) x) (f x)))))))=> (C (fun (x:A)=> ((fun (C0:((or ((ex B) (fun (z:B)=> ((R x) z)))) (not ((ex B) (fun (z:B)=> ((R x) z))))))=> ((((((or_ind ((ex B) (fun (z:B)=> ((R x) z)))) (not ((ex B) (fun (z:B)=> ((R x) z))))) ((ex B) (fun (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y))))) ((((ex_ind B) (fun (z:B)=> ((R x) z))) ((ex B) (fun (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y))))) (fun (y:B) (H:((R x) y))=> ((((ex_intro B) (fun (y0:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y0)))) y) (fun (_:((ex B) (fun (z:B)=> ((R x) z))))=> H))))) (fun (N:(not ((ex B) (fun (z:B)=> ((R x) z)))))=> ((((ex_intro B) (fun (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y)))) b) (fun (H:((ex B) (fun (z:B)=> ((R x) z))))=> ((False_rect ((R x) b)) (N H)))))) C0)) (classic ((ex B) (fun (z:B)=> ((R x) z)))))))) (((choice A) B) (fun (x:A) (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y)))))):(forall (A:Type) (B:Type) (R:(A->(B->Prop))), (B->((ex (A->B)) (fun (f:(A->B))=> (forall (x:A), (((ex B) (fun (y:B)=> ((R x) y)))->((R x) (f x))))))))
% Instantiate: b0:=(forall (A:Type) (B:Type) (R:(A->(B->Prop))), (B->((ex (A->B)) (fun (f:(A->B))=> (forall (x:A), (((ex B) (fun (y:B)=> ((R x) y)))->((R x) (f x)))))))):Prop
% Found classical_choice as proof of b0
% Found eq_ref000:=(eq_ref00 P):((P (Xh2 (Xh1 ((Xf1 Xx) Xy))))->(P (Xh2 (Xh1 ((Xf1 Xx) Xy)))))
% Found (eq_ref00 P) as proof of (P0 (Xh2 (Xh1 ((Xf1 Xx) Xy))))
% Found ((eq_ref0 (Xh2 (Xh1 ((Xf1 Xx) Xy)))) P) as proof of (P0 (Xh2 (Xh1 ((Xf1 Xx) Xy))))
% Found (((eq_ref a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) P) as proof of (P0 (Xh2 (Xh1 ((Xf1 Xx) Xy))))
% Found (((eq_ref a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) P) as proof of (P0 (Xh2 (Xh1 ((Xf1 Xx) Xy))))
% Found functional_extensionality_double:=(fun (A:Type) (B:Type) (C:Type) (f:(A->(B->C))) (g:(A->(B->C))) (x:(forall (x:A) (y:B), (((eq C) ((f x) y)) ((g x) y))))=> (((((functional_extensionality_dep A) (fun (x2:A)=> (B->C))) f) g) (fun (x0:A)=> (((((functional_extensionality_dep B) (fun (x3:B)=> C)) (f x0)) (g x0)) (x x0))))):(forall (A:Type) (B:Type) (C:Type) (f:(A->(B->C))) (g:(A->(B->C))), ((forall (x:A) (y:B), (((eq C) ((f x) y)) ((g x) y)))->(((eq (A->(B->C))) f) g)))
% Instantiate: b0:=(forall (A:Type) (B:Type) (C:Type) (f:(A->(B->C))) (g:(A->(B->C))), ((forall (x:A) (y:B), (((eq C) ((f x) y)) ((g x) y)))->(((eq (A->(B->C))) f) g))):Prop
% Found functional_extensionality_double as proof of b0
% Found functional_extensionality_double:=(fun (A:Type) (B:Type) (C:Type) (f:(A->(B->C))) (g:(A->(B->C))) (x:(forall (x:A) (y:B), (((eq C) ((f x) y)) ((g x) y))))=> (((((functional_extensionality_dep A) (fun (x2:A)=> (B->C))) f) g) (fun (x0:A)=> (((((functional_extensionality_dep B) (fun (x3:B)=> C)) (f x0)) (g x0)) (x x0))))):(forall (A:Type) (B:Type) (C:Type) (f:(A->(B->C))) (g:(A->(B->C))), ((forall (x:A) (y:B), (((eq C) ((f x) y)) ((g x) y)))->(((eq (A->(B->C))) f) g)))
% Instantiate: b0:=(forall (A:Type) (B:Type) (C:Type) (f:(A->(B->C))) (g:(A->(B->C))), ((forall (x:A) (y:B), (((eq C) ((f x) y)) ((g x) y)))->(((eq (A->(B->C))) f) g))):Prop
% Found functional_extensionality_double as proof of b0
% Found functional_extensionality_double:=(fun (A:Type) (B:Type) (C:Type) (f:(A->(B->C))) (g:(A->(B->C))) (x:(forall (x:A) (y:B), (((eq C) ((f x) y)) ((g x) y))))=> (((((functional_extensionality_dep A) (fun (x2:A)=> (B->C))) f) g) (fun (x0:A)=> (((((functional_extensionality_dep B) (fun (x3:B)=> C)) (f x0)) (g x0)) (x x0))))):(forall (A:Type) (B:Type) (C:Type) (f:(A->(B->C))) (g:(A->(B->C))), ((forall (x:A) (y:B), (((eq C) ((f x) y)) ((g x) y)))->(((eq (A->(B->C))) f) g)))
% Instantiate: b0:=(forall (A:Type) (B:Type) (C:Type) (f:(A->(B->C))) (g:(A->(B->C))), ((forall (x:A) (y:B), (((eq C) ((f x) y)) ((g x) y)))->(((eq (A->(B->C))) f) g))):Prop
% Found functional_extensionality_double as proof of b0
% Found eq_ref00:=(eq_ref0 (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))))):(((eq Prop) (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))))) (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy)))))))
% Found (eq_ref0 (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))))) as proof of (((eq Prop) (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))))) b0)
% Found ((eq_ref Prop) (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))))) as proof of (((eq Prop) (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))))) b0)
% Found ((eq_ref Prop) (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))))) as proof of (((eq Prop) (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))))) b0)
% Found ((eq_ref Prop) (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))))) as proof of (((eq Prop) (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))))) b0)
% Found eq_ref000:=(eq_ref00 P):((P (Xh2 (Xh1 ((Xf1 Xx) Xy))))->(P (Xh2 (Xh1 ((Xf1 Xx) Xy)))))
% Found (eq_ref00 P) as proof of (P0 (Xh2 (Xh1 ((Xf1 Xx) Xy))))
% Found ((eq_ref0 (Xh2 (Xh1 ((Xf1 Xx) Xy)))) P) as proof of (P0 (Xh2 (Xh1 ((Xf1 Xx) Xy))))
% Found (((eq_ref a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) P) as proof of (P0 (Xh2 (Xh1 ((Xf1 Xx) Xy))))
% Found (((eq_ref a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) P) as proof of (P0 (Xh2 (Xh1 ((Xf1 Xx) Xy))))
% Found eq_ref00:=(eq_ref0 (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))))):(((eq Prop) (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))))) (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy)))))))
% Found (eq_ref0 (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))))) as proof of (((eq Prop) (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))))) b0)
% Found ((eq_ref Prop) (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))))) as proof of (((eq Prop) (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))))) b0)
% Found ((eq_ref Prop) (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))))) as proof of (((eq Prop) (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))))) b0)
% Found ((eq_ref Prop) (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))))) as proof of (((eq Prop) (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))))) b0)
% Found eq_ref00:=(eq_ref0 (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))))):(((eq Prop) (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))))) (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy)))))))
% Found (eq_ref0 (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))))) as proof of (((eq Prop) (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))))) b0)
% Found ((eq_ref Prop) (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))))) as proof of (((eq Prop) (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))))) b0)
% Found ((eq_ref Prop) (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))))) as proof of (((eq Prop) (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))))) b0)
% Found ((eq_ref Prop) (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))))) as proof of (((eq Prop) (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))))) b0)
% Found eq_ref00:=(eq_ref0 (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))))):(((eq Prop) (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))))) (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy)))))))
% Found (eq_ref0 (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))))) as proof of (((eq Prop) (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))))) b0)
% Found ((eq_ref Prop) (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))))) as proof of (((eq Prop) (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))))) b0)
% Found ((eq_ref Prop) (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))))) as proof of (((eq Prop) (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))))) b0)
% Found ((eq_ref Prop) (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))))) as proof of (((eq Prop) (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))))) b0)
% Found eq_ref00:=(eq_ref0 (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))))):(((eq Prop) (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))))) (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy)))))))
% Found (eq_ref0 (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))))) as proof of (((eq Prop) (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))))) b0)
% Found ((eq_ref Prop) (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))))) as proof of (((eq Prop) (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))))) b0)
% Found ((eq_ref Prop) (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))))) as proof of (((eq Prop) (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))))) b0)
% Found ((eq_ref Prop) (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))))) as proof of (((eq Prop) (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))))) b0)
% Found eq_ref00:=(eq_ref0 (Xh2 (Xh1 ((Xf1 Xx) Xy)))):(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) (Xh2 (Xh1 ((Xf1 Xx) Xy))))
% Found (eq_ref0 (Xh2 (Xh1 ((Xf1 Xx) Xy)))) as proof of (((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) b0)
% Found ((eq_ref a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) as proof of (((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) b0)
% Found ((eq_ref a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) as proof of (((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) b0)
% Found ((eq_ref a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) as proof of (((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))
% Found ((eq_ref a) b0) as proof of (((eq a) b0) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))
% Found ((eq_ref a) b0) as proof of (((eq a) b0) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))
% Found ((eq_ref a) b0) as proof of (((eq a) b0) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))
% Found x2:(P (Xh2 (Xh1 ((Xf1 Xx) Xy))))
% Instantiate: b0:=(Xh2 (Xh1 ((Xf1 Xx) Xy))):a
% Found x2 as proof of (P0 b0)
% Found eq_ref00:=(eq_ref0 (forall (Xx:g), ((Xs1 Xx)->(Xs3 (Xh2 (Xh1 Xx)))))):(((eq Prop) (forall (Xx:g), ((Xs1 Xx)->(Xs3 (Xh2 (Xh1 Xx)))))) (forall (Xx:g), ((Xs1 Xx)->(Xs3 (Xh2 (Xh1 Xx))))))
% Found (eq_ref0 (forall (Xx:g), ((Xs1 Xx)->(Xs3 (Xh2 (Xh1 Xx)))))) as proof of (((eq Prop) (forall (Xx:g), ((Xs1 Xx)->(Xs3 (Xh2 (Xh1 Xx)))))) b0)
% Found ((eq_ref Prop) (forall (Xx:g), ((Xs1 Xx)->(Xs3 (Xh2 (Xh1 Xx)))))) as proof of (((eq Prop) (forall (Xx:g), ((Xs1 Xx)->(Xs3 (Xh2 (Xh1 Xx)))))) b0)
% Found ((eq_ref Prop) (forall (Xx:g), ((Xs1 Xx)->(Xs3 (Xh2 (Xh1 Xx)))))) as proof of (((eq Prop) (forall (Xx:g), ((Xs1 Xx)->(Xs3 (Xh2 (Xh1 Xx)))))) b0)
% Found ((eq_ref Prop) (forall (Xx:g), ((Xs1 Xx)->(Xs3 (Xh2 (Xh1 Xx)))))) as proof of (((eq Prop) (forall (Xx:g), ((Xs1 Xx)->(Xs3 (Xh2 (Xh1 Xx)))))) b0)
% Found eq_ref00:=(eq_ref0 ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy)))):(((eq a) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))
% Found (eq_ref0 ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy)))) as proof of (((eq a) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy)))) b0)
% Found ((eq_ref a) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy)))) as proof of (((eq a) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy)))) b0)
% Found ((eq_ref a) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy)))) as proof of (((eq a) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy)))) b0)
% Found ((eq_ref a) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy)))) as proof of (((eq a) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy)))) b0)
% Found eq_ref00:=(eq_ref0 (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))))):(((eq Prop) (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))))) (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy)))))))
% Found (eq_ref0 (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))))) as proof of (((eq Prop) (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))))) b0)
% Found ((eq_ref Prop) (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))))) as proof of (((eq Prop) (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))))) b0)
% Found ((eq_ref Prop) (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))))) as proof of (((eq Prop) (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))))) b0)
% Found ((eq_ref Prop) (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))))) as proof of (((eq Prop) (forall (Xx:g) (Xy:g), (((and (Xs1 Xx)) (Xs1 Xy))->(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))))) b0)
% Found proj1:(forall (A:Prop) (B:Prop), (((and A) B)->A))
% Instantiate: b0:=(forall (A:Prop) (B:Prop), (((and A) B)->A)):Prop
% Found proj1 as proof of b0
% Found eq_ref00:=(eq_ref0 (forall (Xx:g), ((Xs1 Xx)->(Xs3 (Xh2 (Xh1 Xx)))))):(((eq Prop) (forall (Xx:g), ((Xs1 Xx)->(Xs3 (Xh2 (Xh1 Xx)))))) (forall (Xx:g), ((Xs1 Xx)->(Xs3 (Xh2 (Xh1 Xx))))))
% Found (eq_ref0 (forall (Xx:g), ((Xs1 Xx)->(Xs3 (Xh2 (Xh1 Xx)))))) as proof of (((eq Prop) (forall (Xx:g), ((Xs1 Xx)->(Xs3 (Xh2 (Xh1 Xx)))))) b0)
% Found ((eq_ref Prop) (forall (Xx:g), ((Xs1 Xx)->(Xs3 (Xh2 (Xh1 Xx)))))) as proof of (((eq Prop) (forall (Xx:g), ((Xs1 Xx)->(Xs3 (Xh2 (Xh1 Xx)))))) b0)
% Found ((eq_ref Prop) (forall (Xx:g), ((Xs1 Xx)->(Xs3 (Xh2 (Xh1 Xx)))))) as proof of (((eq Prop) (forall (Xx:g), ((Xs1 Xx)->(Xs3 (Xh2 (Xh1 Xx)))))) b0)
% Found ((eq_ref Prop) (forall (Xx:g), ((Xs1 Xx)->(Xs3 (Xh2 (Xh1 Xx)))))) as proof of (((eq Prop) (forall (Xx:g), ((Xs1 Xx)->(Xs3 (Xh2 (Xh1 Xx)))))) b0)
% Found eq_ref000:=(eq_ref00 P):((P (Xh2 (Xh1 ((Xf1 Xx) Xy))))->(P (Xh2 (Xh1 ((Xf1 Xx) Xy)))))
% Found (eq_ref00 P) as proof of (P0 (Xh2 (Xh1 ((Xf1 Xx) Xy))))
% Found ((eq_ref0 (Xh2 (Xh1 ((Xf1 Xx) Xy)))) P) as proof of (P0 (Xh2 (Xh1 ((Xf1 Xx) Xy))))
% Found (((eq_ref a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) P) as proof of (P0 (Xh2 (Xh1 ((Xf1 Xx) Xy))))
% Found (((eq_ref a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) P) as proof of (P0 (Xh2 (Xh1 ((Xf1 Xx) Xy))))
% Found proj1:(forall (A:Prop) (B:Prop), (((and A) B)->A))
% Instantiate: b0:=(forall (A:Prop) (B:Prop), (((and A) B)->A)):Prop
% Found proj1 as proof of b0
% Found eq_ref00:=(eq_ref0 (Xh2 (Xh1 ((Xf1 Xx) Xy)))):(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) (Xh2 (Xh1 ((Xf1 Xx) Xy))))
% Found (eq_ref0 (Xh2 (Xh1 ((Xf1 Xx) Xy)))) as proof of (((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) b0)
% Found ((eq_ref a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) as proof of (((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) b0)
% Found ((eq_ref a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) as proof of (((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) b0)
% Found ((eq_ref a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) as proof of (((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))
% Found ((eq_ref a) b0) as proof of (((eq a) b0) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))
% Found ((eq_ref a) b0) as proof of (((eq a) b0) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))
% Found ((eq_ref a) b0) as proof of (((eq a) b0) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))
% Found eq_ref000:=(eq_ref00 P):((P (Xh2 (Xh1 ((Xf1 Xx) Xy))))->(P (Xh2 (Xh1 ((Xf1 Xx) Xy)))))
% Found (eq_ref00 P) as proof of (P0 (Xh2 (Xh1 ((Xf1 Xx) Xy))))
% Found ((eq_ref0 (Xh2 (Xh1 ((Xf1 Xx) Xy)))) P) as proof of (P0 (Xh2 (Xh1 ((Xf1 Xx) Xy))))
% Found (((eq_ref a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) P) as proof of (P0 (Xh2 (Xh1 ((Xf1 Xx) Xy))))
% Found (((eq_ref a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) P) as proof of (P0 (Xh2 (Xh1 ((Xf1 Xx) Xy))))
% Found or_comm_i:=(fun (A:Prop) (B:Prop) (H:((or A) B))=> ((((((or_ind A) B) ((or B) A)) ((or_intror B) A)) ((or_introl B) A)) H)):(forall (A:Prop) (B:Prop), (((or A) B)->((or B) A)))
% Instantiate: b0:=(forall (A:Prop) (B:Prop), (((or A) B)->((or B) A))):Prop
% Found or_comm_i as proof of b0
% Found eq_ref00:=(eq_ref0 (Xh2 (Xh1 ((Xf1 Xx) Xy)))):(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) (Xh2 (Xh1 ((Xf1 Xx) Xy))))
% Found (eq_ref0 (Xh2 (Xh1 ((Xf1 Xx) Xy)))) as proof of (((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) b0)
% Found ((eq_ref a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) as proof of (((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) b0)
% Found ((eq_ref a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) as proof of (((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) b0)
% Found ((eq_ref a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) as proof of (((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))
% Found ((eq_ref a) b0) as proof of (((eq a) b0) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))
% Found ((eq_ref a) b0) as proof of (((eq a) b0) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))
% Found ((eq_ref a) b0) as proof of (((eq a) b0) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) (Xh2 (Xh1 ((Xf1 Xx) Xy))))
% Found ((eq_ref a) b0) as proof of (((eq a) b0) (Xh2 (Xh1 ((Xf1 Xx) Xy))))
% Found ((eq_ref a) b0) as proof of (((eq a) b0) (Xh2 (Xh1 ((Xf1 Xx) Xy))))
% Found ((eq_ref a) b0) as proof of (((eq a) b0) (Xh2 (Xh1 ((Xf1 Xx) Xy))))
% Found eq_ref00:=(eq_ref0 ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy)))):(((eq a) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy)))) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))
% Found (eq_ref0 ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy)))) as proof of (((eq a) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy)))) b0)
% Found ((eq_ref a) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy)))) as proof of (((eq a) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy)))) b0)
% Found ((eq_ref a) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy)))) as proof of (((eq a) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy)))) b0)
% Found ((eq_ref a) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy)))) as proof of (((eq a) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy)))) b0)
% Found x6:(forall (Xx:a) (Xy:a), (((and (Xs3 Xx)) (Xs3 Xy))->(Xs3 ((Xf3 Xx) Xy))))
% Found x6 as proof of (forall (Xx:a) (Xy:a), (((and (Xs3 Xx)) (Xs3 Xy))->(Xs3 ((Xf3 Xx) Xy))))
% Found eq_ref00:=(eq_ref0 (Xh2 (Xh1 ((Xf1 Xx) Xy)))):(((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) (Xh2 (Xh1 ((Xf1 Xx) Xy))))
% Found (eq_ref0 (Xh2 (Xh1 ((Xf1 Xx) Xy)))) as proof of (((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) b0)
% Found ((eq_ref a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) as proof of (((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) b0)
% Found ((eq_ref a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) as proof of (((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) b0)
% Found ((eq_ref a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) as proof of (((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))
% Found ((eq_ref a) b0) as proof of (((eq a) b0) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))
% Found ((eq_ref a) b0) as proof of (((eq a) b0) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))
% Found ((eq_ref a) b0) as proof of (((eq a) b0) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) (Xh2 (Xh1 ((Xf1 Xx) Xy))))
% Found ((eq_ref a) b0) as proof of (((eq a) b0) (Xh2 (Xh1 ((Xf1 Xx) Xy))))
% Found ((eq_ref a) b0) as proof of (((eq a) b0) (Xh2 (Xh1 ((Xf1 Xx) Xy))))
% Found ((eq_ref a) b0) a
% EOF
%------------------------------------------------------------------------------