TSTP Solution File: SEU904^5 by cocATP---0.2.0
View Problem
- Process Solution
%------------------------------------------------------------------------------
% File : cocATP---0.2.0
% Problem : SEU904^5 : TPTP v6.1.0. Released v4.0.0.
% Transfm : none
% Format : tptp:raw
% Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% Computer : n106.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 300.04s
% Output : None
% Verified :
% SZS Type : None (Parsing solution fails)
% Syntax : Number of formulae : 0
% Comments :
%------------------------------------------------------------------------------
%----NO SOLUTION OUTPUT BY SYSTEM
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% % Problem : SEU904^5 : TPTP v6.1.0. Released v4.0.0.
% % Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n106.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 : 300.04
% 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 0x17e07a0>, <kernel.Type object at 0x17e0878>) of role type named g_type
% Using role type
% Declaring g:Type
% FOF formula (<kernel.Constant object at 0x17e2290>, <kernel.Type object at 0x17e0758>) of role type named b_type
% Using role type
% Declaring b:Type
% FOF formula (<kernel.Constant object at 0x17e2290>, <kernel.Type object at 0x17e0f38>) 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 ((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))))))->((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))))))))) of role conjecture named cTHM126_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 ((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))))))->((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))))))))):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 ((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))))))->((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)))))))))']
% 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 ((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))))))->((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)))))))))
% 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 (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_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_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 (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 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 (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 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_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 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_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 (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 (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 x3:(forall (Xx:b), ((Xs2 Xx)->(Xs3 (Xh2 Xx))))
% Instantiate: b0:=(forall (Xx:b), ((Xs2 Xx)->(Xs3 (Xh2 Xx)))):Prop
% Found x3 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 x3:(forall (Xx:b), ((Xs2 Xx)->(Xs3 (Xh2 Xx))))
% Instantiate: b0:=(forall (Xx:b), ((Xs2 Xx)->(Xs3 (Xh2 Xx)))):Prop
% Found x3 as proof of b0
% Found x3:(forall (Xx:b), ((Xs2 Xx)->(Xs3 (Xh2 Xx))))
% Instantiate: b0:=(forall (Xx:b), ((Xs2 Xx)->(Xs3 (Xh2 Xx)))):Prop
% Found x3 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), ((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 x1:(P (Xh2 (Xh1 ((Xf1 Xx) Xy))))
% Instantiate: b0:=(Xh2 (Xh1 ((Xf1 Xx) Xy))):a
% Found x1 as proof of (P0 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 x5:(forall (Xx:a) (Xy:a), (((and (Xs3 Xx)) (Xs3 Xy))->(Xs3 ((Xf3 Xx) Xy))))
% Found x5 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_substitution:=(fun (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)) (H:(((eq T) a) b))=> ((H (fun (x:T)=> (((eq U) (f a)) (f x)))) ((eq_ref U) (f a)))):(forall (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)), ((((eq T) a) b)->(((eq U) (f a)) (f b))))
% Instantiate: b0:=(forall (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)), ((((eq T) a) b)->(((eq U) (f a)) (f b)))):Prop
% Found eq_substitution as proof of 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_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_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 iff_sym:=(fun (A:Prop) (B:Prop) (H:((iff A) B))=> ((((conj (B->A)) (A->B)) (((proj2 (A->B)) (B->A)) H)) (((proj1 (A->B)) (B->A)) H))):(forall (A:Prop) (B:Prop), (((iff A) B)->((iff B) A)))
% Instantiate: b0:=(forall (A:Prop) (B:Prop), (((iff A) B)->((iff B) A))):Prop
% Found iff_sym 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_sym:=(fun (A:Prop) (B:Prop) (H:((iff A) B))=> ((((conj (B->A)) (A->B)) (((proj2 (A->B)) (B->A)) H)) (((proj1 (A->B)) (B->A)) H))):(forall (A:Prop) (B:Prop), (((iff A) B)->((iff B) A)))
% Instantiate: b0:=(forall (A:Prop) (B:Prop), (((iff A) B)->((iff B) A))):Prop
% Found iff_sym 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_sym:=(fun (A:Prop) (B:Prop) (H:((iff A) B))=> ((((conj (B->A)) (A->B)) (((proj2 (A->B)) (B->A)) H)) (((proj1 (A->B)) (B->A)) H))):(forall (A:Prop) (B:Prop), (((iff A) B)->((iff B) A)))
% Instantiate: b0:=(forall (A:Prop) (B:Prop), (((iff A) B)->((iff B) A))):Prop
% Found iff_sym as proof of b0
% Found iff_sym:=(fun (A:Prop) (B:Prop) (H:((iff A) B))=> ((((conj (B->A)) (A->B)) (((proj2 (A->B)) (B->A)) H)) (((proj1 (A->B)) (B->A)) H))):(forall (A:Prop) (B:Prop), (((iff A) B)->((iff B) A)))
% Instantiate: b0:=(forall (A:Prop) (B:Prop), (((iff A) B)->((iff B) A))):Prop
% Found iff_sym as proof of b0
% Found x3:(P (Xh2 (Xh1 ((Xf1 Xx) Xy))))
% Instantiate: b0:=(Xh2 (Xh1 ((Xf1 Xx) Xy))):a
% Found x3 as proof of (P0 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_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 ((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), ((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 choice_operator:=(fun (A:Type) (a:A)=> ((((classical_choice (A->Prop)) A) (fun (x3:(A->Prop))=> x3)) a)):(forall (A:Type), (A->((ex ((A->Prop)->A)) (fun (co:((A->Prop)->A))=> (forall (P:(A->Prop)), (((ex A) (fun (x:A)=> (P x)))->(P (co P))))))))
% Instantiate: b0:=(forall (A:Type), (A->((ex ((A->Prop)->A)) (fun (co:((A->Prop)->A))=> (forall (P:(A->Prop)), (((ex A) (fun (x:A)=> (P x)))->(P (co P)))))))):Prop
% Found choice_operator as proof of b0
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (P b0)
% Found ((eq_ref a) b0) as proof of (P b0)
% Found ((eq_ref a) b0) as proof of (P b0)
% Found ((eq_ref a) b0) as proof of (P 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 x5:(forall (Xx:a) (Xy:a), (((and (Xs3 Xx)) (Xs3 Xy))->(Xs3 ((Xf3 Xx) Xy))))
% Found x5 as proof of (forall (Xx:a) (Xy:a), (((and (Xs3 Xx)) (Xs3 Xy))->(Xs3 ((Xf3 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 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_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 ((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 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 (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_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)))) b00)
% Found ((eq_ref a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) as proof of (((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) b00)
% Found ((eq_ref a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) as proof of (((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) b00)
% Found ((eq_ref a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) as proof of (((eq a) (Xh2 (Xh1 ((Xf1 Xx) Xy)))) b00)
% Found eq_ref00:=(eq_ref0 b00):(((eq a) b00) b00)
% Found (eq_ref0 b00) as proof of (((eq a) b00) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))
% Found ((eq_ref a) b00) as proof of (((eq a) b00) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))
% Found ((eq_ref a) b00) as proof of (((eq a) b00) ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))
% Found ((eq_ref a) b00) as proof of (((eq a) b00) ((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_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 x5:(forall (Xx:a) (Xy:a), (((and (Xs3 Xx)) (Xs3 Xy))->(Xs3 ((Xf3 Xx) Xy))))
% Found x5 as proof of (forall (Xx:a) (Xy:a), (((and (Xs3 Xx)) (Xs3 Xy))->(Xs3 ((Xf3 Xx) Xy))))
% Found x1:(P ((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))))
% Instantiate: b0:=((Xf3 (Xh2 (Xh1 Xx))) (Xh2 (Xh1 Xy))):a
% Found x1 as proof of (P0 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 ((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 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 (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)))
% EOF
%------------------------------------------------------------------------------