TSTP Solution File: SEV046^5 by cocATP---0.2.0
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- Process Solution
%------------------------------------------------------------------------------
% File : cocATP---0.2.0
% Problem : SEV046^5 : TPTP v6.1.0. Released v4.0.0.
% Transfm : none
% Format : tptp:raw
% Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% Computer : n105.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 13:33:39 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 : SEV046^5 : TPTP v6.1.0. Released v4.0.0.
% % Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n105.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 07:43:01 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 0x1a54830>, <kernel.Type object at 0x1a54d40>) of role type named a_type
% Using role type
% Declaring a:Type
% FOF formula (<kernel.Constant object at 0x1e305f0>, <kernel.Type object at 0x1a548c0>) of role type named b_type
% Using role type
% Declaring b:Type
% FOF formula (forall (Xp:(a->(a->Prop))) (Xp2:(a->(b->(b->Prop)))), (((and ((and ((and (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz))))) (forall (Xx:a), (((Xp Xx) Xx)->((and (forall (Xx0:b) (Xy:b), ((((Xp2 Xx) Xx0) Xy)->(((Xp2 Xx) Xy) Xx0)))) (forall (Xx0:b) (Xy:b) (Xz:b), (((and (((Xp2 Xx) Xx0) Xy)) (((Xp2 Xx) Xy) Xz))->(((Xp2 Xx) Xx0) Xz)))))))) (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((eq (b->(b->Prop))) (Xp2 Xx)) (Xp2 Xy)))))->((and (forall (Xx:(a->b)) (Xy:(a->b)), ((forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xx Xx0)) (Xy Xy0))))->(forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xy Xx0)) (Xx Xy0))))))) (forall (Xx:(a->b)) (Xy:(a->b)) (Xz:(a->b)), (((and (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xx Xx0)) (Xy Xy0))))) (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xy Xx0)) (Xz Xy0)))))->(forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xx Xx0)) (Xz Xy0))))))))) of role conjecture named cTHM507_pme
% Conjecture to prove = (forall (Xp:(a->(a->Prop))) (Xp2:(a->(b->(b->Prop)))), (((and ((and ((and (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz))))) (forall (Xx:a), (((Xp Xx) Xx)->((and (forall (Xx0:b) (Xy:b), ((((Xp2 Xx) Xx0) Xy)->(((Xp2 Xx) Xy) Xx0)))) (forall (Xx0:b) (Xy:b) (Xz:b), (((and (((Xp2 Xx) Xx0) Xy)) (((Xp2 Xx) Xy) Xz))->(((Xp2 Xx) Xx0) Xz)))))))) (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((eq (b->(b->Prop))) (Xp2 Xx)) (Xp2 Xy)))))->((and (forall (Xx:(a->b)) (Xy:(a->b)), ((forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xx Xx0)) (Xy Xy0))))->(forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xy Xx0)) (Xx Xy0))))))) (forall (Xx:(a->b)) (Xy:(a->b)) (Xz:(a->b)), (((and (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xx Xx0)) (Xy Xy0))))) (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xy Xx0)) (Xz Xy0)))))->(forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xx Xx0)) (Xz Xy0))))))))):Prop
% Parameter a_DUMMY:a.
% Parameter b_DUMMY:b.
% We need to prove ['(forall (Xp:(a->(a->Prop))) (Xp2:(a->(b->(b->Prop)))), (((and ((and ((and (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz))))) (forall (Xx:a), (((Xp Xx) Xx)->((and (forall (Xx0:b) (Xy:b), ((((Xp2 Xx) Xx0) Xy)->(((Xp2 Xx) Xy) Xx0)))) (forall (Xx0:b) (Xy:b) (Xz:b), (((and (((Xp2 Xx) Xx0) Xy)) (((Xp2 Xx) Xy) Xz))->(((Xp2 Xx) Xx0) Xz)))))))) (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((eq (b->(b->Prop))) (Xp2 Xx)) (Xp2 Xy)))))->((and (forall (Xx:(a->b)) (Xy:(a->b)), ((forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xx Xx0)) (Xy Xy0))))->(forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xy Xx0)) (Xx Xy0))))))) (forall (Xx:(a->b)) (Xy:(a->b)) (Xz:(a->b)), (((and (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xx Xx0)) (Xy Xy0))))) (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xy Xx0)) (Xz Xy0)))))->(forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xx Xx0)) (Xz Xy0)))))))))']
% Parameter a:Type.
% Parameter b:Type.
% Trying to prove (forall (Xp:(a->(a->Prop))) (Xp2:(a->(b->(b->Prop)))), (((and ((and ((and (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz))))) (forall (Xx:a), (((Xp Xx) Xx)->((and (forall (Xx0:b) (Xy:b), ((((Xp2 Xx) Xx0) Xy)->(((Xp2 Xx) Xy) Xx0)))) (forall (Xx0:b) (Xy:b) (Xz:b), (((and (((Xp2 Xx) Xx0) Xy)) (((Xp2 Xx) Xy) Xz))->(((Xp2 Xx) Xx0) Xz)))))))) (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((eq (b->(b->Prop))) (Xp2 Xx)) (Xp2 Xy)))))->((and (forall (Xx:(a->b)) (Xy:(a->b)), ((forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xx Xx0)) (Xy Xy0))))->(forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xy Xx0)) (Xx Xy0))))))) (forall (Xx:(a->b)) (Xy:(a->b)) (Xz:(a->b)), (((and (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xx Xx0)) (Xy Xy0))))) (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xy Xx0)) (Xz Xy0)))))->(forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xx Xx0)) (Xz Xy0)))))))))
% Found eq_ref00:=(eq_ref0 (forall (Xx:(a->b)) (Xy:(a->b)) (Xz:(a->b)), (((and (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xx Xx0)) (Xy Xy0))))) (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xy Xx0)) (Xz Xy0)))))->(forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xx Xx0)) (Xz Xy0))))))):(((eq Prop) (forall (Xx:(a->b)) (Xy:(a->b)) (Xz:(a->b)), (((and (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xx Xx0)) (Xy Xy0))))) (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xy Xx0)) (Xz Xy0)))))->(forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xx Xx0)) (Xz Xy0))))))) (forall (Xx:(a->b)) (Xy:(a->b)) (Xz:(a->b)), (((and (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xx Xx0)) (Xy Xy0))))) (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xy Xx0)) (Xz Xy0)))))->(forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xx Xx0)) (Xz Xy0)))))))
% Found (eq_ref0 (forall (Xx:(a->b)) (Xy:(a->b)) (Xz:(a->b)), (((and (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xx Xx0)) (Xy Xy0))))) (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xy Xx0)) (Xz Xy0)))))->(forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xx Xx0)) (Xz Xy0))))))) as proof of (((eq Prop) (forall (Xx:(a->b)) (Xy:(a->b)) (Xz:(a->b)), (((and (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xx Xx0)) (Xy Xy0))))) (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xy Xx0)) (Xz Xy0)))))->(forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xx Xx0)) (Xz Xy0))))))) b0)
% Found ((eq_ref Prop) (forall (Xx:(a->b)) (Xy:(a->b)) (Xz:(a->b)), (((and (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xx Xx0)) (Xy Xy0))))) (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xy Xx0)) (Xz Xy0)))))->(forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xx Xx0)) (Xz Xy0))))))) as proof of (((eq Prop) (forall (Xx:(a->b)) (Xy:(a->b)) (Xz:(a->b)), (((and (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xx Xx0)) (Xy Xy0))))) (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xy Xx0)) (Xz Xy0)))))->(forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xx Xx0)) (Xz Xy0))))))) b0)
% Found ((eq_ref Prop) (forall (Xx:(a->b)) (Xy:(a->b)) (Xz:(a->b)), (((and (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xx Xx0)) (Xy Xy0))))) (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xy Xx0)) (Xz Xy0)))))->(forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xx Xx0)) (Xz Xy0))))))) as proof of (((eq Prop) (forall (Xx:(a->b)) (Xy:(a->b)) (Xz:(a->b)), (((and (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xx Xx0)) (Xy Xy0))))) (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xy Xx0)) (Xz Xy0)))))->(forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xx Xx0)) (Xz Xy0))))))) b0)
% Found ((eq_ref Prop) (forall (Xx:(a->b)) (Xy:(a->b)) (Xz:(a->b)), (((and (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xx Xx0)) (Xy Xy0))))) (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xy Xx0)) (Xz Xy0)))))->(forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xx Xx0)) (Xz Xy0))))))) as proof of (((eq Prop) (forall (Xx:(a->b)) (Xy:(a->b)) (Xz:(a->b)), (((and (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xx Xx0)) (Xy Xy0))))) (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xy Xx0)) (Xz Xy0)))))->(forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xx Xx0)) (Xz Xy0))))))) b0)
% 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:(a->b)) (Xy:(a->b)) (Xz:(a->b)), (((and (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xx Xx0)) (Xy Xy0))))) (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xy Xx0)) (Xz Xy0)))))->(forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xx Xx0)) (Xz Xy0))))))):(((eq Prop) (forall (Xx:(a->b)) (Xy:(a->b)) (Xz:(a->b)), (((and (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xx Xx0)) (Xy Xy0))))) (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xy Xx0)) (Xz Xy0)))))->(forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xx Xx0)) (Xz Xy0))))))) (forall (Xx:(a->b)) (Xy:(a->b)) (Xz:(a->b)), (((and (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xx Xx0)) (Xy Xy0))))) (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xy Xx0)) (Xz Xy0)))))->(forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xx Xx0)) (Xz Xy0)))))))
% Found (eq_ref0 (forall (Xx:(a->b)) (Xy:(a->b)) (Xz:(a->b)), (((and (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xx Xx0)) (Xy Xy0))))) (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xy Xx0)) (Xz Xy0)))))->(forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xx Xx0)) (Xz Xy0))))))) as proof of (((eq Prop) (forall (Xx:(a->b)) (Xy:(a->b)) (Xz:(a->b)), (((and (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xx Xx0)) (Xy Xy0))))) (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xy Xx0)) (Xz Xy0)))))->(forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xx Xx0)) (Xz Xy0))))))) b0)
% Found ((eq_ref Prop) (forall (Xx:(a->b)) (Xy:(a->b)) (Xz:(a->b)), (((and (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xx Xx0)) (Xy Xy0))))) (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xy Xx0)) (Xz Xy0)))))->(forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xx Xx0)) (Xz Xy0))))))) as proof of (((eq Prop) (forall (Xx:(a->b)) (Xy:(a->b)) (Xz:(a->b)), (((and (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xx Xx0)) (Xy Xy0))))) (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xy Xx0)) (Xz Xy0)))))->(forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xx Xx0)) (Xz Xy0))))))) b0)
% Found ((eq_ref Prop) (forall (Xx:(a->b)) (Xy:(a->b)) (Xz:(a->b)), (((and (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xx Xx0)) (Xy Xy0))))) (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xy Xx0)) (Xz Xy0)))))->(forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xx Xx0)) (Xz Xy0))))))) as proof of (((eq Prop) (forall (Xx:(a->b)) (Xy:(a->b)) (Xz:(a->b)), (((and (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xx Xx0)) (Xy Xy0))))) (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xy Xx0)) (Xz Xy0)))))->(forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xx Xx0)) (Xz Xy0))))))) b0)
% Found ((eq_ref Prop) (forall (Xx:(a->b)) (Xy:(a->b)) (Xz:(a->b)), (((and (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xx Xx0)) (Xy Xy0))))) (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xy Xx0)) (Xz Xy0)))))->(forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xx Xx0)) (Xz Xy0))))))) as proof of (((eq Prop) (forall (Xx:(a->b)) (Xy:(a->b)) (Xz:(a->b)), (((and (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xx Xx0)) (Xy Xy0))))) (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xy Xx0)) (Xz Xy0)))))->(forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xx Xx0)) (Xz Xy0))))))) b0)
% Found x1:(forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((eq (b->(b->Prop))) (Xp2 Xx)) (Xp2 Xy))))
% Instantiate: b0:=(forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((eq (b->(b->Prop))) (Xp2 Xx)) (Xp2 Xy)))):Prop
% Found x1 as proof of b0
% Found eq_ref00:=(eq_ref0 (forall (Xx:(a->b)) (Xy:(a->b)) (Xz:(a->b)), (((and (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xx Xx0)) (Xy Xy0))))) (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xy Xx0)) (Xz Xy0)))))->(forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xx Xx0)) (Xz Xy0))))))):(((eq Prop) (forall (Xx:(a->b)) (Xy:(a->b)) (Xz:(a->b)), (((and (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xx Xx0)) (Xy Xy0))))) (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xy Xx0)) (Xz Xy0)))))->(forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xx Xx0)) (Xz Xy0))))))) (forall (Xx:(a->b)) (Xy:(a->b)) (Xz:(a->b)), (((and (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xx Xx0)) (Xy Xy0))))) (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xy Xx0)) (Xz Xy0)))))->(forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xx Xx0)) (Xz Xy0)))))))
% Found (eq_ref0 (forall (Xx:(a->b)) (Xy:(a->b)) (Xz:(a->b)), (((and (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xx Xx0)) (Xy Xy0))))) (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xy Xx0)) (Xz Xy0)))))->(forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xx Xx0)) (Xz Xy0))))))) as proof of (((eq Prop) (forall (Xx:(a->b)) (Xy:(a->b)) (Xz:(a->b)), (((and (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xx Xx0)) (Xy Xy0))))) (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xy Xx0)) (Xz Xy0)))))->(forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xx Xx0)) (Xz Xy0))))))) b0)
% Found ((eq_ref Prop) (forall (Xx:(a->b)) (Xy:(a->b)) (Xz:(a->b)), (((and (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xx Xx0)) (Xy Xy0))))) (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xy Xx0)) (Xz Xy0)))))->(forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xx Xx0)) (Xz Xy0))))))) as proof of (((eq Prop) (forall (Xx:(a->b)) (Xy:(a->b)) (Xz:(a->b)), (((and (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xx Xx0)) (Xy Xy0))))) (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xy Xx0)) (Xz Xy0)))))->(forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xx Xx0)) (Xz Xy0))))))) b0)
% Found ((eq_ref Prop) (forall (Xx:(a->b)) (Xy:(a->b)) (Xz:(a->b)), (((and (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xx Xx0)) (Xy Xy0))))) (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xy Xx0)) (Xz Xy0)))))->(forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xx Xx0)) (Xz Xy0))))))) as proof of (((eq Prop) (forall (Xx:(a->b)) (Xy:(a->b)) (Xz:(a->b)), (((and (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xx Xx0)) (Xy Xy0))))) (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xy Xx0)) (Xz Xy0)))))->(forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xx Xx0)) (Xz Xy0))))))) b0)
% Found ((eq_ref Prop) (forall (Xx:(a->b)) (Xy:(a->b)) (Xz:(a->b)), (((and (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xx Xx0)) (Xy Xy0))))) (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xy Xx0)) (Xz Xy0)))))->(forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xx Xx0)) (Xz Xy0))))))) as proof of (((eq Prop) (forall (Xx:(a->b)) (Xy:(a->b)) (Xz:(a->b)), (((and (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xx Xx0)) (Xy Xy0))))) (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xy Xx0)) (Xz Xy0)))))->(forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xx Xx0)) (Xz Xy0))))))) b0)
% Found x1:(forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((eq (b->(b->Prop))) (Xp2 Xx)) (Xp2 Xy))))
% Instantiate: b0:=(forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((eq (b->(b->Prop))) (Xp2 Xx)) (Xp2 Xy)))):Prop
% Found x1 as proof of b0
% 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:(a->b)) (Xy:(a->b)) (Xz:(a->b)), (((and (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xx Xx0)) (Xy Xy0))))) (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xy Xx0)) (Xz Xy0)))))->(forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xx Xx0)) (Xz Xy0))))))):(((eq Prop) (forall (Xx:(a->b)) (Xy:(a->b)) (Xz:(a->b)), (((and (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xx Xx0)) (Xy Xy0))))) (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xy Xx0)) (Xz Xy0)))))->(forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xx Xx0)) (Xz Xy0))))))) (forall (Xx:(a->b)) (Xy:(a->b)) (Xz:(a->b)), (((and (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xx Xx0)) (Xy Xy0))))) (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xy Xx0)) (Xz Xy0)))))->(forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xx Xx0)) (Xz Xy0)))))))
% Found (eq_ref0 (forall (Xx:(a->b)) (Xy:(a->b)) (Xz:(a->b)), (((and (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xx Xx0)) (Xy Xy0))))) (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xy Xx0)) (Xz Xy0)))))->(forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xx Xx0)) (Xz Xy0))))))) as proof of (((eq Prop) (forall (Xx:(a->b)) (Xy:(a->b)) (Xz:(a->b)), (((and (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xx Xx0)) (Xy Xy0))))) (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xy Xx0)) (Xz Xy0)))))->(forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xx Xx0)) (Xz Xy0))))))) b0)
% Found ((eq_ref Prop) (forall (Xx:(a->b)) (Xy:(a->b)) (Xz:(a->b)), (((and (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xx Xx0)) (Xy Xy0))))) (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xy Xx0)) (Xz Xy0)))))->(forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xx Xx0)) (Xz Xy0))))))) as proof of (((eq Prop) (forall (Xx:(a->b)) (Xy:(a->b)) (Xz:(a->b)), (((and (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xx Xx0)) (Xy Xy0))))) (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xy Xx0)) (Xz Xy0)))))->(forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xx Xx0)) (Xz Xy0))))))) b0)
% Found ((eq_ref Prop) (forall (Xx:(a->b)) (Xy:(a->b)) (Xz:(a->b)), (((and (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xx Xx0)) (Xy Xy0))))) (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xy Xx0)) (Xz Xy0)))))->(forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xx Xx0)) (Xz Xy0))))))) as proof of (((eq Prop) (forall (Xx:(a->b)) (Xy:(a->b)) (Xz:(a->b)), (((and (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xx Xx0)) (Xy Xy0))))) (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xy Xx0)) (Xz Xy0)))))->(forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xx Xx0)) (Xz Xy0))))))) b0)
% Found ((eq_ref Prop) (forall (Xx:(a->b)) (Xy:(a->b)) (Xz:(a->b)), (((and (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xx Xx0)) (Xy Xy0))))) (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xy Xx0)) (Xz Xy0)))))->(forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xx Xx0)) (Xz Xy0))))))) as proof of (((eq Prop) (forall (Xx:(a->b)) (Xy:(a->b)) (Xz:(a->b)), (((and (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xx Xx0)) (Xy Xy0))))) (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xy Xx0)) (Xz Xy0)))))->(forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xx Xx0)) (Xz Xy0))))))) b0)
% 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_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 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 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 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 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 (Xx Xy0)):(((eq b) (Xx Xy0)) (Xx Xy0))
% Found (eq_ref0 (Xx Xy0)) as proof of (((eq b) (Xx Xy0)) b0)
% Found ((eq_ref b) (Xx Xy0)) as proof of (((eq b) (Xx Xy0)) b0)
% Found ((eq_ref b) (Xx Xy0)) as proof of (((eq b) (Xx Xy0)) b0)
% Found ((eq_ref b) (Xx Xy0)) as proof of (((eq b) (Xx Xy0)) b0)
% Found eq_ref00:=(eq_ref0 (Xz Xy0)):(((eq b) (Xz Xy0)) (Xz Xy0))
% Found (eq_ref0 (Xz Xy0)) as proof of (((eq b) (Xz Xy0)) b0)
% Found ((eq_ref b) (Xz Xy0)) as proof of (((eq b) (Xz Xy0)) b0)
% Found ((eq_ref b) (Xz Xy0)) as proof of (((eq b) (Xz Xy0)) b0)
% Found ((eq_ref b) (Xz Xy0)) as proof of (((eq b) (Xz Xy0)) b0)
% Found eq_ref00:=(eq_ref0 (Xx Xy0)):(((eq b) (Xx Xy0)) (Xx Xy0))
% Found (eq_ref0 (Xx Xy0)) as proof of (((eq b) (Xx Xy0)) b0)
% Found ((eq_ref b) (Xx Xy0)) as proof of (((eq b) (Xx Xy0)) b0)
% Found ((eq_ref b) (Xx Xy0)) as proof of (((eq b) (Xx Xy0)) b0)
% Found ((eq_ref b) (Xx Xy0)) as proof of (((eq b) (Xx Xy0)) b0)
% Found eq_ref00:=(eq_ref0 (Xz Xy0)):(((eq b) (Xz Xy0)) (Xz Xy0))
% Found (eq_ref0 (Xz Xy0)) as proof of (((eq b) (Xz Xy0)) b0)
% Found ((eq_ref b) (Xz Xy0)) as proof of (((eq b) (Xz Xy0)) b0)
% Found ((eq_ref b) (Xz Xy0)) as proof of (((eq b) (Xz Xy0)) b0)
% Found ((eq_ref b) (Xz Xy0)) as proof of (((eq b) (Xz Xy0)) b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) Xy0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) Xy0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) Xy0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) Xy0)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) Xy0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) Xy0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) Xy0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) Xy0)
% Found eq_ref00:=(eq_ref0 (Xx Xy0)):(((eq b) (Xx Xy0)) (Xx Xy0))
% Found (eq_ref0 (Xx Xy0)) as proof of (((eq b) (Xx Xy0)) b0)
% Found ((eq_ref b) (Xx Xy0)) as proof of (((eq b) (Xx Xy0)) b0)
% Found ((eq_ref b) (Xx Xy0)) as proof of (((eq b) (Xx Xy0)) b0)
% Found ((eq_ref b) (Xx Xy0)) as proof of (((eq b) (Xx Xy0)) b0)
% Found eq_ref00:=(eq_ref0 (Xz Xy0)):(((eq b) (Xz Xy0)) (Xz Xy0))
% Found (eq_ref0 (Xz Xy0)) as proof of (((eq b) (Xz Xy0)) b0)
% Found ((eq_ref b) (Xz Xy0)) as proof of (((eq b) (Xz Xy0)) b0)
% Found ((eq_ref b) (Xz Xy0)) as proof of (((eq b) (Xz Xy0)) b0)
% Found ((eq_ref b) (Xz Xy0)) as proof of (((eq b) (Xz Xy0)) b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) b0)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b0)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b0)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (forall (Xx:(a->b)) (Xy:(a->b)) (Xz:(a->b)), (((and (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xx Xx0)) (Xy Xy0))))) (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xy Xx0)) (Xz Xy0)))))->(forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xx Xx0)) (Xz Xy0)))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (forall (Xx:(a->b)) (Xy:(a->b)) (Xz:(a->b)), (((and (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xx Xx0)) (Xy Xy0))))) (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xy Xx0)) (Xz Xy0)))))->(forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xx Xx0)) (Xz Xy0)))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (forall (Xx:(a->b)) (Xy:(a->b)) (Xz:(a->b)), (((and (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xx Xx0)) (Xy Xy0))))) (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xy Xx0)) (Xz Xy0)))))->(forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xx Xx0)) (Xz Xy0)))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (forall (Xx:(a->b)) (Xy:(a->b)) (Xz:(a->b)), (((and (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xx Xx0)) (Xy Xy0))))) (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xy Xx0)) (Xz Xy0)))))->(forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xx Xx0)) (Xz Xy0)))))))
% Found eq_ref00:=(eq_ref0 (Xx Xy0)):(((eq b) (Xx Xy0)) (Xx Xy0))
% Found (eq_ref0 (Xx Xy0)) as proof of (((eq b) (Xx Xy0)) b0)
% Found ((eq_ref b) (Xx Xy0)) as proof of (((eq b) (Xx Xy0)) b0)
% Found ((eq_ref b) (Xx Xy0)) as proof of (((eq b) (Xx Xy0)) b0)
% Found ((eq_ref b) (Xx Xy0)) as proof of (((eq b) (Xx Xy0)) 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: a0:=(forall (A:Prop) (B:Prop), (((iff A) B)->((iff B) A))):Prop
% Found iff_sym as proof of a0
% Found eq_ref00:=(eq_ref0 (Xz Xy0)):(((eq b) (Xz Xy0)) (Xz Xy0))
% Found (eq_ref0 (Xz Xy0)) as proof of (((eq b) (Xz Xy0)) b0)
% Found ((eq_ref b) (Xz Xy0)) as proof of (((eq b) (Xz Xy0)) b0)
% Found ((eq_ref b) (Xz Xy0)) as proof of (((eq b) (Xz Xy0)) b0)
% Found ((eq_ref b) (Xz Xy0)) as proof of (((eq b) (Xz Xy0)) b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) Xy0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) Xy0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) Xy0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) Xy0)
% Found eq_ref00:=(eq_ref0 (Xz Xy0)):(((eq b) (Xz Xy0)) (Xz Xy0))
% Found (eq_ref0 (Xz Xy0)) as proof of (((eq b) (Xz Xy0)) b0)
% Found ((eq_ref b) (Xz Xy0)) as proof of (((eq b) (Xz Xy0)) b0)
% Found ((eq_ref b) (Xz Xy0)) as proof of (((eq b) (Xz Xy0)) b0)
% Found ((eq_ref b) (Xz Xy0)) as proof of (((eq b) (Xz Xy0)) b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) Xy0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) Xy0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) Xy0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) Xy0)
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) b0)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b0)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b0)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (forall (Xx:(a->b)) (Xy:(a->b)) (Xz:(a->b)), (((and (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xx Xx0)) (Xy Xy0))))) (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xy Xx0)) (Xz Xy0)))))->(forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xx Xx0)) (Xz Xy0)))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (forall (Xx:(a->b)) (Xy:(a->b)) (Xz:(a->b)), (((and (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xx Xx0)) (Xy Xy0))))) (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xy Xx0)) (Xz Xy0)))))->(forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xx Xx0)) (Xz Xy0)))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (forall (Xx:(a->b)) (Xy:(a->b)) (Xz:(a->b)), (((and (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xx Xx0)) (Xy Xy0))))) (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xy Xx0)) (Xz Xy0)))))->(forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xx Xx0)) (Xz Xy0)))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (forall (Xx:(a->b)) (Xy:(a->b)) (Xz:(a->b)), (((and (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xx Xx0)) (Xy Xy0))))) (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xy Xx0)) (Xz Xy0)))))->(forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xx Xx0)) (Xz Xy0)))))))
% Found conj:(forall (A:Prop) (B:Prop), (A->(B->((and A) B))))
% Instantiate: a0:=(forall (A:Prop) (B:Prop), (A->(B->((and A) B)))):Prop
% Found conj as proof of a0
% Found eq_ref00:=(eq_ref0 (Xx Xy0)):(((eq b) (Xx Xy0)) (Xx Xy0))
% Found (eq_ref0 (Xx Xy0)) as proof of (((eq b) (Xx Xy0)) b0)
% Found ((eq_ref b) (Xx Xy0)) as proof of (((eq b) (Xx Xy0)) b0)
% Found ((eq_ref b) (Xx Xy0)) as proof of (((eq b) (Xx Xy0)) b0)
% Found ((eq_ref b) (Xx Xy0)) as proof of (((eq b) (Xx Xy0)) b0)
% Found eq_ref00:=(eq_ref0 (Xz Xy0)):(((eq b) (Xz Xy0)) (Xz Xy0))
% Found (eq_ref0 (Xz Xy0)) as proof of (((eq b) (Xz Xy0)) b0)
% Found ((eq_ref b) (Xz Xy0)) as proof of (((eq b) (Xz Xy0)) b0)
% Found ((eq_ref b) (Xz Xy0)) as proof of (((eq b) (Xz Xy0)) b0)
% Found ((eq_ref b) (Xz Xy0)) as proof of (((eq b) (Xz Xy0)) b0)
% Found eq_ref00:=(eq_ref0 (Xx Xy0)):(((eq b) (Xx Xy0)) (Xx Xy0))
% Found (eq_ref0 (Xx Xy0)) as proof of (((eq b) (Xx Xy0)) b0)
% Found ((eq_ref b) (Xx Xy0)) as proof of (((eq b) (Xx Xy0)) b0)
% Found ((eq_ref b) (Xx Xy0)) as proof of (((eq b) (Xx Xy0)) b0)
% Found ((eq_ref b) (Xx Xy0)) as proof of (((eq b) (Xx Xy0)) b0)
% Found conj:(forall (A:Prop) (B:Prop), (A->(B->((and A) B))))
% Instantiate: a0:=(forall (A:Prop) (B:Prop), (A->(B->((and A) B)))):Prop
% Found conj as proof of a0
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) Xy0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) Xy0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) Xy0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) Xy0)
% Found eq_ref00:=(eq_ref0 (Xz Xy0)):(((eq b) (Xz Xy0)) (Xz Xy0))
% Found (eq_ref0 (Xz Xy0)) as proof of (((eq b) (Xz Xy0)) b0)
% Found ((eq_ref b) (Xz Xy0)) as proof of (((eq b) (Xz Xy0)) b0)
% Found ((eq_ref b) (Xz Xy0)) as proof of (((eq b) (Xz Xy0)) b0)
% Found ((eq_ref b) (Xz Xy0)) as proof of (((eq b) (Xz Xy0)) b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) Xy0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) Xy0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) Xy0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) Xy0)
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) b0)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b0)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b0)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (forall (Xx:(a->b)) (Xy:(a->b)) (Xz:(a->b)), (((and (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xx Xx0)) (Xy Xy0))))) (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xy Xx0)) (Xz Xy0)))))->(forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xx Xx0)) (Xz Xy0)))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (forall (Xx:(a->b)) (Xy:(a->b)) (Xz:(a->b)), (((and (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xx Xx0)) (Xy Xy0))))) (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xy Xx0)) (Xz Xy0)))))->(forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xx Xx0)) (Xz Xy0)))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (forall (Xx:(a->b)) (Xy:(a->b)) (Xz:(a->b)), (((and (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xx Xx0)) (Xy Xy0))))) (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xy Xx0)) (Xz Xy0)))))->(forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xx Xx0)) (Xz Xy0)))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (forall (Xx:(a->b)) (Xy:(a->b)) (Xz:(a->b)), (((and (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xx Xx0)) (Xy Xy0))))) (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xy Xx0)) (Xz Xy0)))))->(forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xx Xx0)) (Xz Xy0)))))))
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) Xy0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) Xy0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) Xy0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) Xy0)
% Found eq_ref00:=(eq_ref0 (Xz Xy0)):(((eq b) (Xz Xy0)) (Xz Xy0))
% Found (eq_ref0 (Xz Xy0)) as proof of (((eq b) (Xz Xy0)) b0)
% Found ((eq_ref b) (Xz Xy0)) as proof of (((eq b) (Xz Xy0)) b0)
% Found ((eq_ref b) (Xz Xy0)) as proof of (((eq b) (Xz Xy0)) b0)
% Found ((eq_ref b) (Xz Xy0)) as proof of (((eq b) (Xz Xy0)) b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) Xy0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) Xy0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) Xy0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) Xy0)
% 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: a0:=(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 a0
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) Xy0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) Xy0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) Xy0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) Xy0)
% Found eq_ref00:=(eq_ref0 (Xx Xy0)):(((eq b) (Xx Xy0)) (Xx Xy0))
% Found (eq_ref0 (Xx Xy0)) as proof of (((eq b) (Xx Xy0)) b0)
% Found ((eq_ref b) (Xx Xy0)) as proof of (((eq b) (Xx Xy0)) b0)
% Found ((eq_ref b) (Xx Xy0)) as proof of (((eq b) (Xx Xy0)) b0)
% Found ((eq_ref b) (Xx Xy0)) as proof of (((eq b) (Xx Xy0)) b0)
% Found conj10:=(conj1 (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((eq (b->(b->Prop))) (Xp2 Xx)) (Xp2 Xy))))):(((and ((and (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz))))) (forall (Xx:a), (((Xp Xx) Xx)->((and (forall (Xx0:b) (Xy:b), ((((Xp2 Xx) Xx0) Xy)->(((Xp2 Xx) Xy) Xx0)))) (forall (Xx0:b) (Xy:b) (Xz:b), (((and (((Xp2 Xx) Xx0) Xy)) (((Xp2 Xx) Xy) Xz))->(((Xp2 Xx) Xx0) Xz)))))))->((forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((eq (b->(b->Prop))) (Xp2 Xx)) (Xp2 Xy))))->((and ((and ((and (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz))))) (forall (Xx:a), (((Xp Xx) Xx)->((and (forall (Xx0:b) (Xy:b), ((((Xp2 Xx) Xx0) Xy)->(((Xp2 Xx) Xy) Xx0)))) (forall (Xx0:b) (Xy:b) (Xz:b), (((and (((Xp2 Xx) Xx0) Xy)) (((Xp2 Xx) Xy) Xz))->(((Xp2 Xx) Xx0) Xz)))))))) (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((eq (b->(b->Prop))) (Xp2 Xx)) (Xp2 Xy)))))))
% Found (conj1 (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((eq (b->(b->Prop))) (Xp2 Xx)) (Xp2 Xy))))) as proof of (((and ((and (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz))))) (forall (Xx:a), (((Xp Xx) Xx)->((and (forall (Xx0:b) (Xy:b), ((((Xp2 Xx) Xx0) Xy)->(((Xp2 Xx) Xy) Xx0)))) (forall (Xx0:b) (Xy:b) (Xz:b), (((and (((Xp2 Xx) Xx0) Xy)) (((Xp2 Xx) Xy) Xz))->(((Xp2 Xx) Xx0) Xz)))))))->((forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((eq (b->(b->Prop))) (Xp2 Xx)) (Xp2 Xy))))->a0))
% Found ((conj ((and ((and (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz))))) (forall (Xx:a), (((Xp Xx) Xx)->((and (forall (Xx0:b) (Xy:b), ((((Xp2 Xx) Xx0) Xy)->(((Xp2 Xx) Xy) Xx0)))) (forall (Xx0:b) (Xy:b) (Xz:b), (((and (((Xp2 Xx) Xx0) Xy)) (((Xp2 Xx) Xy) Xz))->(((Xp2 Xx) Xx0) Xz)))))))) (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((eq (b->(b->Prop))) (Xp2 Xx)) (Xp2 Xy))))) as proof of (((and ((and (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz))))) (forall (Xx:a), (((Xp Xx) Xx)->((and (forall (Xx0:b) (Xy:b), ((((Xp2 Xx) Xx0) Xy)->(((Xp2 Xx) Xy) Xx0)))) (forall (Xx0:b) (Xy:b) (Xz:b), (((and (((Xp2 Xx) Xx0) Xy)) (((Xp2 Xx) Xy) Xz))->(((Xp2 Xx) Xx0) Xz)))))))->((forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((eq (b->(b->Prop))) (Xp2 Xx)) (Xp2 Xy))))->a0))
% Found ((conj ((and ((and (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz))))) (forall (Xx:a), (((Xp Xx) Xx)->((and (forall (Xx0:b) (Xy:b), ((((Xp2 Xx) Xx0) Xy)->(((Xp2 Xx) Xy) Xx0)))) (forall (Xx0:b) (Xy:b) (Xz:b), (((and (((Xp2 Xx) Xx0) Xy)) (((Xp2 Xx) Xy) Xz))->(((Xp2 Xx) Xx0) Xz)))))))) (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((eq (b->(b->Prop))) (Xp2 Xx)) (Xp2 Xy))))) as proof of (((and ((and (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz))))) (forall (Xx:a), (((Xp Xx) Xx)->((and (forall (Xx0:b) (Xy:b), ((((Xp2 Xx) Xx0) Xy)->(((Xp2 Xx) Xy) Xx0)))) (forall (Xx0:b) (Xy:b) (Xz:b), (((and (((Xp2 Xx) Xx0) Xy)) (((Xp2 Xx) Xy) Xz))->(((Xp2 Xx) Xx0) Xz)))))))->((forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((eq (b->(b->Prop))) (Xp2 Xx)) (Xp2 Xy))))->a0))
% Found ((conj ((and ((and (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz))))) (forall (Xx:a), (((Xp Xx) Xx)->((and (forall (Xx0:b) (Xy:b), ((((Xp2 Xx) Xx0) Xy)->(((Xp2 Xx) Xy) Xx0)))) (forall (Xx0:b) (Xy:b) (Xz:b), (((and (((Xp2 Xx) Xx0) Xy)) (((Xp2 Xx) Xy) Xz))->(((Xp2 Xx) Xx0) Xz)))))))) (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((eq (b->(b->Prop))) (Xp2 Xx)) (Xp2 Xy))))) as proof of (((and ((and (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz))))) (forall (Xx:a), (((Xp Xx) Xx)->((and (forall (Xx0:b) (Xy:b), ((((Xp2 Xx) Xx0) Xy)->(((Xp2 Xx) Xy) Xx0)))) (forall (Xx0:b) (Xy:b) (Xz:b), (((and (((Xp2 Xx) Xx0) Xy)) (((Xp2 Xx) Xy) Xz))->(((Xp2 Xx) Xx0) Xz)))))))->((forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((eq (b->(b->Prop))) (Xp2 Xx)) (Xp2 Xy))))->a0))
% Found (and_rect00 ((conj ((and ((and (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz))))) (forall (Xx:a), (((Xp Xx) Xx)->((and (forall (Xx0:b) (Xy:b), ((((Xp2 Xx) Xx0) Xy)->(((Xp2 Xx) Xy) Xx0)))) (forall (Xx0:b) (Xy:b) (Xz:b), (((and (((Xp2 Xx) Xx0) Xy)) (((Xp2 Xx) Xy) Xz))->(((Xp2 Xx) Xx0) Xz)))))))) (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((eq (b->(b->Prop))) (Xp2 Xx)) (Xp2 Xy)))))) as proof of a0
% Found ((and_rect0 a0) ((conj ((and ((and (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz))))) (forall (Xx:a), (((Xp Xx) Xx)->((and (forall (Xx0:b) (Xy:b), ((((Xp2 Xx) Xx0) Xy)->(((Xp2 Xx) Xy) Xx0)))) (forall (Xx0:b) (Xy:b) (Xz:b), (((and (((Xp2 Xx) Xx0) Xy)) (((Xp2 Xx) Xy) Xz))->(((Xp2 Xx) Xx0) Xz)))))))) (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((eq (b->(b->Prop))) (Xp2 Xx)) (Xp2 Xy)))))) as proof of a0
% Found (((fun (P0:Type) (x0:(((and ((and (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz))))) (forall (Xx:a), (((Xp Xx) Xx)->((and (forall (Xx0:b) (Xy:b), ((((Xp2 Xx) Xx0) Xy)->(((Xp2 Xx) Xy) Xx0)))) (forall (Xx0:b) (Xy:b) (Xz:b), (((and (((Xp2 Xx) Xx0) Xy)) (((Xp2 Xx) Xy) Xz))->(((Xp2 Xx) Xx0) Xz)))))))->((forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((eq (b->(b->Prop))) (Xp2 Xx)) (Xp2 Xy))))->P0)))=> (((((and_rect ((and ((and (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz))))) (forall (Xx:a), (((Xp Xx) Xx)->((and (forall (Xx0:b) (Xy:b), ((((Xp2 Xx) Xx0) Xy)->(((Xp2 Xx) Xy) Xx0)))) (forall (Xx0:b) (Xy:b) (Xz:b), (((and (((Xp2 Xx) Xx0) Xy)) (((Xp2 Xx) Xy) Xz))->(((Xp2 Xx) Xx0) Xz)))))))) (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((eq (b->(b->Prop))) (Xp2 Xx)) (Xp2 Xy))))) P0) x0) x)) a0) ((conj ((and ((and (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz))))) (forall (Xx:a), (((Xp Xx) Xx)->((and (forall (Xx0:b) (Xy:b), ((((Xp2 Xx) Xx0) Xy)->(((Xp2 Xx) Xy) Xx0)))) (forall (Xx0:b) (Xy:b) (Xz:b), (((and (((Xp2 Xx) Xx0) Xy)) (((Xp2 Xx) Xy) Xz))->(((Xp2 Xx) Xx0) Xz)))))))) (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((eq (b->(b->Prop))) (Xp2 Xx)) (Xp2 Xy)))))) as proof of a0
% Found (((fun (P0:Type) (x0:(((and ((and (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz))))) (forall (Xx:a), (((Xp Xx) Xx)->((and (forall (Xx0:b) (Xy:b), ((((Xp2 Xx) Xx0) Xy)->(((Xp2 Xx) Xy) Xx0)))) (forall (Xx0:b) (Xy:b) (Xz:b), (((and (((Xp2 Xx) Xx0) Xy)) (((Xp2 Xx) Xy) Xz))->(((Xp2 Xx) Xx0) Xz)))))))->((forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((eq (b->(b->Prop))) (Xp2 Xx)) (Xp2 Xy))))->P0)))=> (((((and_rect ((and ((and (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz))))) (forall (Xx:a), (((Xp Xx) Xx)->((and (forall (Xx0:b) (Xy:b), ((((Xp2 Xx) Xx0) Xy)->(((Xp2 Xx) Xy) Xx0)))) (forall (Xx0:b) (Xy:b) (Xz:b), (((and (((Xp2 Xx) Xx0) Xy)) (((Xp2 Xx) Xy) Xz))->(((Xp2 Xx) Xx0) Xz)))))))) (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((eq (b->(b->Prop))) (Xp2 Xx)) (Xp2 Xy))))) P0) x0) x)) a0) ((conj ((and ((and (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz))))) (forall (Xx:a), (((Xp Xx) Xx)->((and (forall (Xx0:b) (Xy:b), ((((Xp2 Xx) Xx0) Xy)->(((Xp2 Xx) Xy) Xx0)))) (forall (Xx0:b) (Xy:b) (Xz:b), (((and (((Xp2 Xx) Xx0) Xy)) (((Xp2 Xx) Xy) Xz))->(((Xp2 Xx) Xx0) Xz)))))))) (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->(((eq (b->(b->Prop))) (Xp2 Xx)) (Xp2 Xy)))))) as proof of a0
% 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: a0:=(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 a0
% 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: a0:=(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 a0
% Found x7:((Xp Xx0) Xy0)
% Instantiate: Xx1:=Xy0:a
% Found x7 as proof of ((Xp Xx0) Xx1)
% Found (x400 x7) as proof of ((Xp Xx1) Xx0)
% Found ((x40 Xx1) x7) as proof of ((Xp Xx1) Xx0)
% Found (((x4 Xx0) Xx1) x7) as proof of ((Xp Xx1) Xx0)
% Found (((x4 Xx0) Xx1) x7) as proof of ((Xp Xx1) Xx0)
% Found eq_ref00:=(eq_ref0 (Xz Xy0)):(((eq b) (Xz Xy0)) (Xz Xy0))
% Found (eq_ref0 (Xz Xy0)) as proof of (((eq b) (Xz Xy0)) b0)
% Found ((eq_ref b) (Xz Xy0)) as proof of (((eq b) (Xz Xy0)) b0)
% Found ((eq_ref b) (Xz Xy0)) as proof of (((eq b) (Xz Xy0)) b0)
% Found ((eq_ref b) (Xz Xy0)) as proof of (((eq b) (Xz Xy0)) b0)
% Found eq_ref00:=(eq_ref0 (Xz Xy0)):(((eq b) (Xz Xy0)) (Xz Xy0))
% Found (eq_ref0 (Xz Xy0)) as proof of (((eq b) (Xz Xy0)) b0)
% Found ((eq_ref b) (Xz Xy0)) as proof of (((eq b) (Xz Xy0)) b0)
% Found ((eq_ref b) (Xz Xy0)) as proof of (((eq b) (Xz Xy0)) b0)
% Found ((eq_ref b) (Xz Xy0)) as proof of (((eq b) (Xz Xy0)) b0)
% Found eq_ref00:=(eq_ref0 (Xx Xy0)):(((eq b) (Xx Xy0)) (Xx Xy0))
% Found (eq_ref0 (Xx Xy0)) as proof of (((eq b) (Xx Xy0)) b0)
% Found ((eq_ref b) (Xx Xy0)) as proof of (((eq b) (Xx Xy0)) b0)
% Found ((eq_ref b) (Xx Xy0)) as proof of (((eq b) (Xx Xy0)) b0)
% Found ((eq_ref b) (Xx Xy0)) as proof of (((eq b) (Xx Xy0)) b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) b0)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b0)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b0)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (forall (Xx:(a->b)) (Xy:(a->b)) (Xz:(a->b)), (((and (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xx Xx0)) (Xy Xy0))))) (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xy Xx0)) (Xz Xy0)))))->(forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xx Xx0)) (Xz Xy0)))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (forall (Xx:(a->b)) (Xy:(a->b)) (Xz:(a->b)), (((and (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xx Xx0)) (Xy Xy0))))) (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xy Xx0)) (Xz Xy0)))))->(forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xx Xx0)) (Xz Xy0)))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (forall (Xx:(a->b)) (Xy:(a->b)) (Xz:(a->b)), (((and (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xx Xx0)) (Xy Xy0))))) (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xy Xx0)) (Xz Xy0)))))->(forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xx Xx0)) (Xz Xy0)))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (forall (Xx:(a->b)) (Xy:(a->b)) (Xz:(a->b)), (((and (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xx Xx0)) (Xy Xy0))))) (forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xy Xx0)) (Xz Xy0)))))->(forall (Xx0:a) (Xy0:a), (((Xp Xx0) Xy0)->(((Xp2 Xx0) (Xx Xx0)) (Xz Xy0)))))))
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) Xy0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) Xy0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) Xy0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) Xy0)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) Xy0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) Xy0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) Xy0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) Xy0)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) Xy0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) Xy0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) Xy0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) Xy0)
% Found x7:((Xp Xx0) Xy0)
% Instantiate: Xx1:=Xy0:a
% Found x7 as proof of ((Xp Xx0) Xx1)
% Found (x400 x7) as proof of ((Xp Xx1) Xx0)
% Found ((x40 Xx1) x7) as proof of ((Xp Xx1) Xx0)
% Found (((x4 Xx0) Xx1) x7) as proof of ((Xp Xx1) Xx0)
% Found (((x4 Xx0) Xx1) x7) as proof of ((Xp Xx1) Xx0)
% Found eq_ref00:=(eq_ref0 (Xz Xy0)):(((eq b) (Xz Xy0)) (Xz Xy0))
% Found (eq_ref0 (Xz Xy0)) as proof of (((eq b) (Xz Xy0)) b0)
% Found ((eq_ref b) (Xz Xy0)) as proof of (((eq b) (Xz Xy0)) b0)
% Found ((eq_ref b) (Xz Xy0)) as proof of (((eq b) (Xz Xy0)) b0)
% Found ((eq_ref b) (Xz Xy0)) as proof of (((eq b) (Xz Xy0)) b0)
% Found eq_ref00:=(eq_ref0 (Xz Xy0)):(((eq b) (Xz Xy0)) (Xz Xy0))
% Found (eq_ref0 (Xz Xy0)) as proof of (((eq b) (Xz Xy0)) b0)
% Found ((eq_ref b) (Xz Xy0)) as proof of (((eq b) (Xz Xy0)) b0)
% Found ((eq_ref b) (Xz Xy0)) as proof of (((eq b) (Xz Xy0)) b0)
% Found ((eq_ref b) (Xz Xy0)) as proof of (((eq b) (Xz Xy0)) b0)
% Found x3:(forall (Xx:a), (((Xp Xx) Xx)->((and (forall (Xx0:b) (Xy:b), ((((Xp2 Xx) Xx0) Xy)->(((Xp2 Xx) Xy) Xx0)))) (forall (Xx0:b) (Xy:b) (Xz:b), (((and (((Xp2 Xx) Xx0) Xy)) (((Xp2 Xx) Xy) Xz))->(((Xp2 Xx) Xx0) Xz))))))
% Instantiate: a0:=(forall (Xx:a), (((Xp Xx) Xx)->((and (forall (Xx0:b) (Xy:b), ((((Xp2 Xx) Xx0) Xy)->(((Xp2 Xx) Xy) Xx0)))) (forall (Xx0:b) (Xy:b) (Xz:b), (((and (((Xp2 Xx) Xx0) Xy)) (((Xp2 Xx) Xy) Xz))->(((Xp2 Xx) Xx0) Xz)))))):Prop
% Found x3 as proof of a0
% Found eq_ref00:=(eq_ref0 (Xz Xy0)):(((eq b) (Xz Xy0)) (Xz Xy0))
% Found (eq_ref0 (Xz Xy0)) as proof of (((eq b) (Xz Xy0)) b0)
% Found ((eq_ref b) (Xz Xy0)) as proof of (((eq b) (Xz Xy0)) b0)
% Found ((eq_ref b) (Xz Xy0)) as proof of (((eq b) (Xz Xy0)) b0)
% Found ((eq_ref b) (Xz Xy0)) as proof of (((eq b) (Xz Xy0)) b0)
% Found conj10:=(conj1 (forall (Xx:a), (((Xp Xx) Xx)->((and (forall (Xx0:b) (Xy:b), ((((Xp2 Xx) Xx0) Xy)->(((Xp2 Xx) Xy) Xx0)))) (forall (Xx0:b) (Xy:b) (Xz:b), (((and (((Xp2 Xx) Xx0) Xy)) (((Xp2 Xx) Xy) Xz))->(((Xp2 Xx) Xx0) Xz))))))):(((and (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz))))->((forall (Xx:a), (((Xp Xx) Xx)->((and (forall (Xx0:b) (Xy:b), ((((Xp2 Xx) Xx0) Xy)->(((Xp2 Xx) Xy) Xx0)))) (forall (Xx0:b) (Xy:b) (Xz:b), (((and (((Xp2 Xx) Xx0) Xy)) (((Xp2 Xx) Xy) Xz))->(((Xp2 Xx) Xx0) Xz))))))->((and ((and (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz))))) (forall (Xx:a), (((Xp Xx) Xx)->((and (forall (Xx0:b) (Xy:b), ((((Xp2 Xx) Xx0) Xy)->(((Xp2 Xx) Xy) Xx0)))) (forall (Xx0:b) (Xy:b) (Xz:b), (((and (((Xp2 Xx) Xx0) Xy)) (((Xp2 Xx) Xy) Xz))->(((Xp2 Xx) Xx0) Xz)))))))))
% Found (conj1 (forall (Xx:a), (((Xp Xx) Xx)->((and (forall (Xx0:b) (Xy:b), ((((Xp2 Xx) Xx0) Xy)->(((Xp2 Xx) Xy) Xx0)))) (forall (Xx0:b) (Xy:b) (Xz:b), (((and (((Xp2 Xx) Xx0) Xy)) (((Xp2 Xx) Xy) Xz))->(((Xp2 Xx) Xx0) Xz))))))) as proof of (((and (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz))))->((forall (Xx:a), (((Xp Xx) Xx)->((and (forall (Xx0:b) (Xy:b), ((((Xp2 Xx) Xx0) Xy)->(((Xp2 Xx) Xy) Xx0)))) (forall (Xx0:b) (Xy:b) (Xz:b), (((and (((Xp2 Xx) Xx0) Xy)) (((Xp2 Xx) Xy) Xz))->(((Xp2 Xx) Xx0) Xz))))))->a0))
% Found ((conj ((and (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz))))) (forall (Xx:a), (((Xp Xx) Xx)->((and (forall (Xx0:b) (Xy:b), ((((Xp2 Xx) Xx0) Xy)->(((Xp2 Xx) Xy) Xx0)))) (forall (Xx0:b) (Xy:b) (Xz:b), (((and (((Xp2 Xx) Xx0) Xy)) (((Xp2 Xx) Xy) Xz))->(((Xp2 Xx) Xx0) Xz))))))) as proof of (((and (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz))))->((forall (Xx:a), (((Xp Xx) Xx)->((and (forall (Xx0:b) (Xy:b), ((((Xp2 Xx) Xx0) Xy)->(((Xp2 Xx) Xy) Xx0)))) (forall (Xx0:b) (Xy:b) (Xz:b), (((and (((Xp2 Xx) Xx0) Xy)) (((Xp2 Xx) Xy) Xz))->(((Xp2 Xx) Xx0) Xz))))))->a0))
% Found ((conj ((and (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz))))) (forall (Xx:a), (((Xp Xx) Xx)->((and (forall (Xx0:b) (Xy:b), ((((Xp2 Xx) Xx0) Xy)->(((Xp2 Xx) Xy) Xx0)))) (forall (Xx0:b) (Xy:b) (Xz:b), (((and (((Xp2 Xx) Xx0) Xy)) (((Xp2 Xx) Xy) Xz))->(((Xp2 Xx) Xx0) Xz))))))) as proof of (((and (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz))))->((forall (Xx:a), (((Xp Xx) Xx)->((and (forall (Xx0:b) (Xy:b), ((((Xp2 Xx) Xx0) Xy)->(((Xp2 Xx) Xy) Xx0)))) (forall (Xx0:b) (Xy:b) (Xz:b), (((and (((Xp2 Xx) Xx0) Xy)) (((Xp2 Xx) Xy) Xz))->(((Xp2 Xx) Xx0) Xz))))))->a0))
% Found ((conj ((and (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz))))) (forall (Xx:a), (((Xp Xx) Xx)->((and (forall (Xx0:b) (Xy:b), ((((Xp2 Xx) Xx0) Xy)->(((Xp2 Xx) Xy) Xx0)))) (forall (Xx0:b) (Xy:b) (Xz:b), (((and (((Xp2 Xx) Xx0) Xy)) (((Xp2 Xx) Xy) Xz))->(((Xp2 Xx) Xx0) Xz))))))) as proof of (((and (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz))))->((forall (Xx:a), (((Xp Xx) Xx)->((and (forall (Xx0:b) (Xy:b), ((((Xp2 Xx) Xx0) Xy)->(((Xp2 Xx) Xy) Xx0)))) (forall (Xx0:b) (Xy:b) (Xz:b), (((and (((Xp2 Xx) Xx0) Xy)) (((Xp2 Xx) Xy) Xz))->(((Xp2 Xx) Xx0) Xz))))))->a0))
% Found (and_rect10 ((conj ((and (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz))))) (forall (Xx:a), (((Xp Xx) Xx)->((and (forall (Xx0:b) (Xy:b), ((((Xp2 Xx) Xx0) Xy)->(((Xp2 Xx) Xy) Xx0)))) (forall (Xx0:b) (Xy:b) (Xz:b), (((and (((Xp2 Xx) Xx0) Xy)) (((Xp2 Xx) Xy) Xz))->(((Xp2 Xx) Xx0) Xz)))))))) as proof of a0
% Found ((and_rect1 a0) ((conj ((and (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz))))) (forall (Xx:a), (((Xp Xx) Xx)->((and (forall (Xx0:b) (Xy:b), ((((Xp2 Xx) Xx0) Xy)->(((Xp2 Xx) Xy) Xx0)))) (forall (Xx0:b) (Xy:b) (Xz:b), (((and (((Xp2 Xx) Xx0) Xy)) (((Xp2 Xx) Xy) Xz))->(((Xp2 Xx) Xx0) Xz)))))))) as proof of a0
% Found (((fun (P0:Type) (x2:(((and (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz))))->((forall (Xx:a), (((Xp Xx) Xx)->((and (forall (Xx0:b) (Xy:b), ((((Xp2 Xx) Xx0) Xy)->(((Xp2 Xx) Xy) Xx0)))) (forall (Xx0:b) (Xy:b) (Xz:b), (((and (((Xp2 Xx) Xx0) Xy)) (((Xp2 Xx) Xy) Xz))->(((Xp2 Xx) Xx0) Xz))))))->P0)))=> (((((and_rect ((and (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz))))) (forall (Xx:a), (((Xp Xx) Xx)->((and (forall (Xx0:b) (Xy:b), ((((Xp2 Xx) Xx0) Xy)->(((Xp2 Xx) Xy) Xx0)))) (forall (Xx0:b) (Xy:b) (Xz:b), (((and (((Xp2 Xx) Xx0) Xy)) (((Xp2 Xx) Xy) Xz))->(((Xp2 Xx) Xx0) Xz))))))) P0) x2) x0)) a0) ((conj ((and (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz))))) (forall (Xx:a), (((Xp Xx) Xx)->((and (forall (Xx0:b) (Xy:b), ((((Xp2 Xx) Xx0) Xy)->(((Xp2 Xx) Xy) Xx0)))) (forall (Xx0:b) (Xy:b) (Xz:b), (((and (((Xp2 Xx) Xx0) Xy)) (((Xp2 Xx) Xy) Xz))->(((Xp2 Xx) Xx0) Xz)))))))) as proof of a0
% Found (((fun (P0:Type) (x2:(((and (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz))))->((forall (Xx:a), (((Xp Xx) Xx)->((and (forall (Xx0:b) (Xy:b), ((((Xp2 Xx) Xx0) Xy)->(((Xp2 Xx) Xy) Xx0)))) (forall (Xx0:b) (Xy:b) (Xz:b), (((and (((Xp2 Xx) Xx0) Xy)) (((Xp2 Xx) Xy) Xz))->(((Xp2 Xx) Xx0) Xz))))))->P0)))=> (((((and_rect ((and (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz))))) (forall (Xx:a), (((Xp Xx) Xx)->((and (forall (Xx0:b) (Xy:b), ((((Xp2 Xx) Xx0) Xy)->(((Xp2 Xx) Xy) Xx0)))) (forall (Xx0:b) (Xy:b) (Xz:b), (((and (((Xp2 Xx) Xx0) Xy)) (((Xp2 Xx) Xy) Xz))->(((Xp2 Xx) Xx0) Xz))))))) P0) x2) x0)) a0) ((conj ((and (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz))))) (forall (Xx:a), (((Xp Xx) Xx)->((and (forall (Xx0:b) (Xy:b), ((((Xp2 Xx) Xx0) Xy)->(((Xp2 Xx) Xy) Xx0)))) (forall (Xx0:b) (Xy:b) (Xz:b), (((and (((Xp2 Xx) Xx0) Xy)) (((Xp2 Xx) Xy) Xz))->(((Xp2 Xx) Xx0) Xz)))))))) as proof of a0
% Found conj10:=(conj1 (forall (Xx:a), (((Xp Xx) Xx)->((and (forall (Xx0:b) (Xy:b), ((((Xp2 Xx) Xx0) Xy)->(((Xp2 Xx) Xy) Xx0)))) (forall (Xx0:b) (Xy:b) (Xz:b), (((and (((Xp2 Xx) Xx0) Xy)) (((Xp2 Xx) Xy) Xz))->(((Xp2 Xx) Xx0) Xz))))))):(((and (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz))))->((forall (Xx:a), (((Xp Xx) Xx)->((and (forall (Xx0:b) (Xy:b), ((((Xp2 Xx) Xx0) Xy)->(((Xp2 Xx) Xy) Xx0)))) (forall (Xx0:b) (Xy:b) (Xz:b), (((and (((Xp2 Xx) Xx0) Xy)) (((Xp2 Xx) Xy) Xz))->(((Xp2 Xx) Xx0) Xz))))))->((and ((and (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz))))) (forall (Xx:a), (((Xp Xx) Xx)->((and (forall (Xx0:b) (Xy:b), ((((Xp2 Xx) Xx0) Xy)->(((Xp2 Xx) Xy) Xx0)))) (forall (Xx0:b) (Xy:b) (Xz:b), (((and (((Xp2 Xx) Xx0) Xy)) (((Xp2 Xx) Xy) Xz))->(((Xp2 Xx) Xx0) Xz)))))))))
% Found (conj1 (forall (Xx:a), (((Xp Xx) Xx)->((and (forall (Xx0:b) (Xy:b), ((((Xp2 Xx) Xx0) Xy)->(((Xp2 Xx) Xy) Xx0)))) (forall (Xx0:b) (Xy:b) (Xz:b), (((and (((Xp2 Xx) Xx0) Xy)) (((Xp2 Xx) Xy) Xz))->(((Xp2 Xx) Xx0) Xz))))))) as proof of (((and (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz))))->((forall (Xx:a), (((Xp Xx) Xx)->((and (forall (Xx0:b) (Xy:b), ((((Xp2 Xx) Xx0) Xy)->(((Xp2 Xx) Xy) Xx0)))) (forall (Xx0:b) (Xy:b) (Xz:b), (((and (((Xp2 Xx) Xx0) Xy)) (((Xp2 Xx) Xy) Xz))->(((Xp2 Xx) Xx0) Xz))))))->a0))
% Found ((conj ((and (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz))))) (forall (Xx:a), (((Xp Xx) Xx)->((and (forall (Xx0:b) (Xy:b), ((((Xp2 Xx) Xx0) Xy)->(((Xp2 Xx) Xy) Xx0)))) (forall (Xx0:b) (Xy:b) (Xz:b), (((and (((Xp2 Xx) Xx0) Xy)) (((Xp2 Xx) Xy) Xz))->(((Xp2 Xx) Xx0) Xz))))))) as proof of (((and (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz))))->((forall (Xx:a), (((Xp Xx) Xx)->((and (forall (Xx0:b) (Xy:b), ((((Xp2 Xx) Xx0) Xy)->(((Xp2 Xx) Xy) Xx0)))) (forall (Xx0:b) (Xy:b) (Xz:b), (((and (((Xp2 Xx) Xx0) Xy)) (((Xp2 Xx) Xy) Xz))->(((Xp2 Xx) Xx0) Xz))))))->a0))
% Found ((conj ((and (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz))))) (forall (Xx:a), (((Xp Xx) Xx)->((and (forall (Xx0:b) (Xy:b), ((((Xp2 Xx) Xx0) Xy)->(((Xp2 Xx) Xy) Xx0)))) (forall (Xx0:b) (Xy:b) (Xz:b), (((and (((Xp2 Xx) Xx0) Xy)) (((Xp2 Xx) Xy) Xz))->(((Xp2 Xx) Xx0) Xz))))))) as proof of (((and (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz))))->((forall (Xx:a), (((Xp Xx) Xx)->((and (forall (Xx0:b) (Xy:b), ((((Xp2 Xx) Xx0) Xy)->(((Xp2 Xx) Xy) Xx0)))) (forall (Xx0:b) (Xy:b) (Xz:b), (((and (((Xp2 Xx) Xx0) Xy)) (((Xp2 Xx) Xy) Xz))->(((Xp2 Xx) Xx0) Xz))))))->a0))
% Found ((conj ((and (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz))))) (forall (Xx:a), (((Xp Xx) Xx)->((and (forall (Xx0:b) (Xy:b), ((((Xp2 Xx) Xx0) Xy)->(((Xp2 Xx) Xy) Xx0)))) (forall (Xx0:b) (Xy:b) (Xz:b), (((and (((Xp2 Xx) Xx0) Xy)) (((Xp2 Xx) Xy) Xz))->(((Xp2 Xx) Xx0) Xz))))))) as proof of (((and (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz))))->((forall (Xx:a), (((Xp Xx) Xx)->((and (forall (Xx0:b) (Xy:b), ((((Xp2 Xx) Xx0) Xy)->(((Xp2 Xx) Xy) Xx0)))) (forall (Xx0:b) (Xy:b) (Xz:b), (((and (((Xp2 Xx) Xx0) Xy)) (((Xp2 Xx) Xy) Xz))->(((Xp2 Xx) Xx0) Xz))))))->a0))
% Found (and_rect10 ((conj ((and (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz))))) (forall (Xx:a), (((Xp Xx) Xx)->((and (forall (Xx0:b) (Xy:b), ((((Xp2 Xx) Xx0) Xy)->(((Xp2 Xx) Xy) Xx0)))) (forall (Xx0:b) (Xy:b) (Xz:b), (((and (((Xp2 Xx) Xx0) Xy)) (((Xp2 Xx) Xy) Xz))->(((Xp2 Xx) Xx0) Xz)))))))) as proof of a0
% Found ((and_rect1 a0) ((conj ((and (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz))))) (forall (Xx:a), (((Xp Xx) Xx)->((and (forall (Xx0:b) (Xy:b), ((((Xp2 Xx) Xx0) Xy)->(((Xp2 Xx) Xy) Xx0)))) (forall (Xx0:b) (Xy:b) (Xz:b), (((and (((Xp2 Xx) Xx0) Xy)) (((Xp2 Xx) Xy) Xz))->(((Xp2 Xx) Xx0) Xz)))))))) as proof of a0
% Found (((fun (P0:Type) (x2:(((and (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz))))->((forall (Xx:a), (((Xp Xx) Xx)->((and (forall (Xx0:b) (Xy:b), ((((Xp2 Xx) Xx0) Xy)->(((Xp2 Xx) Xy) Xx0)))) (forall (Xx0:b) (Xy:b) (Xz:b), (((and (((Xp2 Xx) Xx0) Xy)) (((Xp2 Xx) Xy) Xz))->(((Xp2 Xx) Xx0) Xz))))))->P0)))=> (((((and_rect ((and (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz))))) (forall (Xx:a), (((Xp Xx) Xx)->((and (forall (Xx0:b) (Xy:b), ((((Xp2 Xx) Xx0) Xy)->(((Xp2 Xx) Xy) Xx0)))) (forall (Xx0:b) (Xy:b) (Xz:b), (((and (((Xp2 Xx) Xx0) Xy)) (((Xp2 Xx) Xy) Xz))->(((Xp2 Xx) Xx0) Xz))))))) P0) x2) x0)) a0) ((conj ((and (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz))))) (forall (Xx:a), (((Xp Xx) Xx)->((and (forall (Xx0:b) (Xy:b), ((((Xp2 Xx) Xx0) Xy)->(((Xp2 Xx) Xy) Xx0)))) (forall (Xx0:b) (Xy:b) (Xz:b), (((and (((Xp2 Xx) Xx0) Xy)) (((Xp2 Xx) Xy) Xz))->(((Xp2 Xx) Xx0) Xz)))))))) as proof of a0
% Found (((fun (P0:Type) (x2:(((and (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz))))->((forall (Xx:a), (((Xp Xx) Xx)->((and (forall (Xx0:b) (Xy:b), ((((Xp2 Xx) Xx0) Xy)->(((Xp2 Xx) Xy) Xx0)))) (forall (Xx0:b) (Xy:b) (Xz:b), (((and (((Xp2 Xx) Xx0) Xy)) (((Xp2 Xx) Xy) Xz))->(((Xp2 Xx) Xx0) Xz))))))->P0)))=> (((((and_rect ((and (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz))))) (forall (Xx:a), (((Xp Xx) Xx)->((and (forall (Xx0:b) (Xy:b), ((((Xp2 Xx) Xx0) Xy)->(((Xp2 Xx) Xy) Xx0)))) (forall (Xx0:b) (Xy:b) (Xz:b), (((and (((Xp2 Xx) Xx0) Xy)) (((Xp2 Xx) Xy) Xz))->(((Xp2 Xx) Xx0) Xz))))))) P0) x2) x0)) a0) ((conj ((and (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz))))) (forall (Xx:a), (((Xp Xx) Xx)->((and (forall (Xx0:b) (Xy:b), ((((Xp2 Xx) Xx0) Xy)->(((Xp2 Xx) Xy) Xx0)))) (forall (Xx0:b) (Xy:b) (Xz:b), (((and (((Xp2 Xx) Xx0) Xy)) (((Xp2 Xx) Xy) Xz))->(((Xp2 Xx) Xx0) Xz)))))))) as proof of a0
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) Xy0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) Xy0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) Xy0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) Xy0)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) Xy0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) Xy0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) Xy0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) Xy0)
% Found x3:(forall (Xx:a), (((Xp Xx) Xx)->((and (forall (Xx0:b) (Xy:b), ((((Xp2 Xx) Xx0) Xy)->(((Xp2 Xx) Xy) Xx0)))) (forall (Xx0:b) (Xy:b) (Xz:b), (((and (((Xp2 Xx) Xx0) Xy)) (((Xp2 Xx) Xy) Xz))->(((Xp2 Xx) Xx0) Xz))))))
% Instantiate: a0:=(forall (Xx:a), (((Xp Xx) Xx)->((and (forall (Xx0:b) (Xy:b), ((((Xp2 Xx) Xx0) Xy)->(((Xp2 Xx) Xy) Xx0)))) (forall (Xx0:b) (Xy:b) (Xz:b), (((and (((Xp2 Xx) Xx0) Xy)) (((Xp2 Xx) Xy) Xz))->(((Xp2 Xx) Xx0) Xz)))))):Prop
% Found x3 as proof of a0
% Found x3:(forall (Xx:a), (((Xp Xx) Xx)->((and (forall (Xx0:b) (Xy:b), ((((Xp2 Xx) Xx0) Xy)->(((Xp2 Xx) Xy) Xx0)))) (forall (Xx0:b) (Xy:b) (Xz:b), (((and (((Xp2 Xx) Xx0) Xy)) (((Xp2 Xx) Xy) Xz))->(((Xp2 Xx) Xx0) Xz))))))
% Instantiate: a0:=(forall (Xx:a), (((Xp Xx) Xx)->((and (forall (Xx0:b) (Xy:b), ((((Xp2 Xx) Xx0) Xy)->(((Xp2 Xx) Xy) Xx0)))) (forall (Xx0:b) (Xy:b) (Xz:b), (((and (((Xp2 Xx) Xx0) Xy)) (((Xp2 Xx) Xy) Xz))->(((Xp2 Xx) Xx0) Xz)))))):Prop
% Found x3 as proof of a0
% Found x3:(forall (Xx:a), (((Xp Xx) Xx)->((and (forall (Xx0:b) (Xy:b), ((((Xp2 Xx) Xx0) Xy)->(((Xp2 Xx) Xy) Xx0)))) (forall (Xx0:b) (Xy:b) (Xz:b), (((and (((Xp2 Xx) Xx0) Xy)) (((Xp2 Xx) Xy) Xz))->(((Xp2 Xx) Xx0) Xz))))))
% Instantiate: a0:=(forall (Xx:a), (((Xp Xx) Xx)->((and (forall (Xx0:b) (Xy:b), ((((Xp2 Xx) Xx0) Xy)->(((Xp2 Xx) Xy) Xx0)))) (forall (Xx0:b) (Xy:b) (Xz:b), (((and (((Xp2 Xx) Xx0) Xy)) (((Xp2 Xx) Xy) Xz))->(((Xp2 Xx) Xx0) Xz)))))):Prop
% Found x3 as proof of a0
% Found x5:((Xp Xx0) Xy0)
% Instantiate: Xx1:=Xy0:a
% Found x5 as proof of ((Xp Xx0) Xx1)
% Found (x600 x5) as proof of ((Xp Xx1) Xx0)
% Found ((x60 Xx1) x5) as proof of ((Xp Xx1) Xx0)
% Found (((x6 Xx0) Xx1) x5) as proof of ((Xp Xx1) Xx0)
% Found (((x6 Xx0) Xx1) x5) as proof of ((Xp Xx1) Xx0)
% Found eq_ref00:=(eq_ref0 (Xx Xy0)):(((eq b) (Xx Xy0)) (Xx Xy0))
% Found (eq_ref0 (Xx Xy0)) as proof of (((eq b) (Xx Xy0)) b0)
% Found ((eq_ref b) (Xx Xy0)) as proof of (((eq b) (Xx Xy0)) b0)
% Found ((eq_ref b) (Xx Xy0)) as proof of (((eq b) (Xx Xy0)) b0)
% Found ((eq_ref b) (Xx Xy0)) as proof of (((eq b) (Xx Xy0)) b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) Xy0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) Xy0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) Xy0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) Xy0)
% Found x5:((Xp Xx0) Xy0)
% Instantiate: Xx1:=Xy0:a
% Found x5 as proof of ((Xp Xx0) Xx1)
% Found (x600 x5) as proof of ((Xp Xx1) Xx0)
% Found ((x60 Xx1) x5) as proof of ((Xp Xx1) Xx0)
% Found (((x6 Xx0) Xx1) x5) as proof of ((Xp Xx1) Xx0)
% Found (((x6 Xx0) Xx1) x5) as proof of ((Xp Xx1) Xx0)
% Found x5:((Xp Xx0) Xy0)
% Instantiate: Xx1:=Xy0:a
% Found x5 as proof of ((Xp Xx0) Xx1)
% Found (x600 x5) as proof of ((Xp Xx1) Xx0)
% Found ((x60 Xx1) x5) as proof of ((Xp Xx1) Xx0)
% Found (((x6 Xx0) Xx1) x5) as proof of ((Xp Xx1) Xx0)
% Found (((x6 Xx0) Xx1) x5) as proof of ((Xp Xx1) Xx0)
% Found eq_ref00:=(eq_ref0 (Xz Xy0)):(((eq b) (Xz Xy0)) (Xz Xy0))
% Found (eq_ref0 (Xz Xy0)) as proof of (((eq b) (Xz Xy0)) b0)
% Found ((eq_ref b) (Xz Xy0)) as proof of (((eq b) (Xz Xy0)) b0)
% Found ((eq_ref b) (Xz Xy0)) as proof of (((eq b) (Xz Xy0)) b0)
% Found ((eq_ref b) (Xz Xy0)) as proof of (((eq b) (Xz Xy0)) b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) Xy0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) Xy0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) Xy0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) Xy0)
% Found eq_ref00:=(eq_ref0 (Xz Xy0)):(((eq b) (Xz Xy0)) (Xz Xy0))
% Found (eq_ref0 (Xz Xy0)) as proof of (((eq b) (Xz Xy0)) b0)
% Found ((eq_ref b) (Xz Xy0)) as proof of (((eq b) (Xz Xy0)) b0)
% Found ((eq_ref b) (Xz Xy0)) as proof of (((eq b) (Xz Xy0)) b0)
% Found ((eq_ref b) (Xz Xy0)) as proof of (((eq b) (Xz Xy0)) b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) Xy0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) Xy0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) Xy0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) Xy0)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) Xy0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) Xy0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) Xy0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) Xy0)
% Found conj10:=(conj1 (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz)))):((forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))->((forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz)))->((and (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz))))))
% Found (conj1 (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz)))) as proof of ((forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))->((forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz)))->a0))
% Found ((conj (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz)))) as proof of ((forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))->((forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz)))->a0))
% Found ((conj (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz)))) as proof of ((forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))->((forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz)))->a0))
% Found ((conj (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz)))) as proof of ((forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))->((forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz)))->a0))
% Found (and_rect20 ((conj (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz))))) as proof of a0
% Found ((and_rect2 a0) ((conj (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz))))) as proof of a0
% Found (((fun (P0:Type) (x4:((forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))->((forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz)))->P0)))=> (((((and_rect (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz)))) P0) x4) x2)) a0) ((conj (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz))))) as proof of a0
% Found (((fun (P0:Type) (x4:((forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))->((forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz)))->P0)))=> (((((and_rect (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz)))) P0) x4) x2)) a0) ((conj (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz))))) as proof of a0
% Found conj10:=(conj1 (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz)))):((forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))->((forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz)))->((and (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz))))))
% Found (conj1 (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz)))) as proof of ((forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))->((forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz)))->a0))
% Found ((conj (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz)))) as proof of ((forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))->((forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz)))->a0))
% Found ((conj (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz)))) as proof of ((forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))->((forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz)))->a0))
% Found ((conj (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz)))) as proof of ((forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))->((forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz)))->a0))
% Found (and_rect20 ((conj (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz))))) as proof of a0
% Found ((and_rect2 a0) ((conj (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz))))) as proof of a0
% Found (((fun (P0:Type) (x4:((forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))->((forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz)))->P0)))=> (((((and_rect (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz)))) P0) x4) x2)) a0) ((conj (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz))))) as proof of a0
% Found (((fun (P0:Type) (x4:((forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))->((forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz)))->P0)))=> (((((and_rect (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz)))) P0) x4) x2)) a0) ((conj (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz))))) as proof of a0
% Found conj10:=(conj1 (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz)))):((forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))->((forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz)))->((and (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz))))))
% Found (conj1 (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz)))) as proof of ((forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))->((forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz)))->a0))
% Found ((conj (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz)))) as proof of ((forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))->((forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz)))->a0))
% Found ((conj (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz)))) as proof of ((forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))->((forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz)))->a0))
% Found ((conj (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz)))) as proof of ((forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))->((forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz)))->a0))
% Found (and_rect20 ((conj (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz))))) as proof of a0
% Found ((and_rect2 a0) ((conj (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz))))) as proof of a0
% Found (((fun (P0:Type) (x4:((forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))->((forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz)))->P0)))=> (((((and_rect (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz)))) P0) x4) x2)) a0) ((conj (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz))))) as proof of a0
% Found (((fun (P0:Type) (x4:((forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))->((forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz)))->P0)))=> (((((and_rect (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz)))) P0) x4) x2)) a0) ((conj (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz))))) as proof of a0
% Found x5:((Xp Xx0) Xy0)
% Instantiate: Xx1:=Xy0:a
% Found x5 as proof of ((Xp Xx0) Xx1)
% Found (x600 x5) as proof of ((Xp Xx1) Xx0)
% Found ((x60 Xx1) x5) as proof of ((Xp Xx1) Xx0)
% Found (((x6 Xx0) Xx1) x5) as proof of ((Xp Xx1) Xx0)
% Found (((x6 Xx0) Xx1) x5) as proof of ((Xp Xx1) Xx0)
% Found eq_ref00:=(eq_ref0 (Xz Xy0)):(((eq b) (Xz Xy0)) (Xz Xy0))
% Found (eq_ref0 (Xz Xy0)) as proof of (((eq b) (Xz Xy0)) b0)
% Found ((eq_ref b) (Xz Xy0)) as proof of (((eq b) (Xz Xy0)) b0)
% Found ((eq_ref b) (Xz Xy0)) as proof of (((eq b) (Xz Xy0)) b0)
% Found ((eq_ref b) (Xz Xy0)) as proof of (((eq b) (Xz Xy0)) b0)
% Found eq_ref00:=(eq_ref0 (Xz Xy0)):(((eq b) (Xz Xy0)) (Xz Xy0))
% Found (eq_ref0 (Xz Xy0)) as proof of (((eq b) (Xz Xy0)) b0)
% Found ((eq_ref b) (Xz Xy0)) as proof of (((eq b) (Xz Xy0)) b0)
% Found ((eq_ref b) (Xz Xy0)) as proof of (((eq b) (Xz Xy0)) b0)
% Found ((eq_ref b) (Xz Xy0)) as proof of (((eq b) (Xz Xy0)) b0)
% Found eq_ref00:=(eq_ref0 (Xx Xy0)):(((eq b) (Xx Xy0)) (Xx Xy0))
% Found (eq_ref0 (Xx Xy0)) as proof of (((eq b) (Xx Xy0)) b0)
% Found ((eq_ref b) (Xx Xy0)) as proof of (((eq b) (Xx Xy0)) b0)
% Found ((eq_ref b) (Xx Xy0)) as proof of (((eq b) (Xx Xy0)) b0)
% Found ((eq_ref b) (Xx Xy0)) as proof of (((eq b) (Xx Xy0)) b0)
% Found x3:((Xp Xx0) Xy0)
% Instantiate: Xx1:=Xy0:a
% Found x3 as proof of ((Xp Xx0) Xx1)
% Found (x600 x3) as proof of ((Xp Xx1) Xx0)
% Found ((x60 Xx1) x3) as proof of ((Xp Xx1) Xx0)
% Found (((x6 Xx0) Xx1) x3) as proof of ((Xp Xx1) Xx0)
% Found (((x6 Xx0) Xx1) x3) as proof of ((Xp Xx1) Xx0)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) Xy0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) Xy0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) Xy0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) Xy0)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) Xy0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) Xy0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) Xy0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) Xy0)
% Found eq_ref00:=(eq_ref0 b0):(((eq b) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq b) b0) (Xx Xy0))
% Found ((eq_ref b) b0) as proof of (((eq b) b0) (Xx Xy0))
% Found ((eq_ref b) b0) as proof of (((eq b) b0) (Xx Xy0))
% Found ((eq_ref b) b0) as proof of (((eq b) b0) (Xx Xy0))
% Found eq_ref00:=(eq_ref0 a0):(((eq b) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq b) a0) b0)
% Found ((eq_ref b) a0) as proof of (((eq b) a0) b0)
% Found ((eq_ref b) a0) as proof of (((eq b) a0) b0)
% Found ((eq_ref b) a0) as proof of (((eq b) a0) b0)
% Found x7:((Xp Xx0) Xy0)
% Instantiate: Xx1:=Xy0:a
% Found x7 as proof of ((Xp Xx0) Xx1)
% Found (x400 x7) as proof of ((Xp Xx1) Xx0)
% Found ((x40 Xx1) x7) as proof of ((Xp Xx1) Xx0)
% Found (((x4 Xx0) Xx1) x7) as proof of ((Xp Xx1) Xx0)
% Found (((x4 Xx0) Xx1) x7) as proof of ((Xp Xx1) Xx0)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) Xy0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) Xy0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) Xy0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) Xy0)
% Found x3:((Xp Xx0) Xy0)
% Instantiate: Xx1:=Xy0:a
% Found x3 as proof of ((Xp Xx0) Xx1)
% Found (x600 x3) as proof of ((Xp Xx1) Xx0)
% Found ((x60 Xx1) x3) as proof of ((Xp Xx1) Xx0)
% Found (((x6 Xx0) Xx1) x3) as proof of ((Xp Xx1) Xx0)
% Found (((x6 Xx0) Xx1) x3) as proof of ((Xp Xx1) Xx0)
% Found eq_ref00:=(eq_ref0 (Xz Xy0)):(((eq b) (Xz Xy0)) (Xz Xy0))
% Found (eq_ref0 (Xz Xy0)) as proof of (((eq b) (Xz Xy0)) b0)
% Found ((eq_ref b) (Xz Xy0)) as proof of (((eq b) (Xz Xy0)) b0)
% Found ((eq_ref b) (Xz Xy0)) as proof of (((eq b) (Xz Xy0)) b0)
% Found ((eq_ref b) (Xz Xy0)) as proof of (((eq b) (Xz Xy0)) b0)
% Found x3:((Xp Xx0) Xy0)
% Instantiate: Xx1:=Xy0:a
% Found x3 as proof of ((Xp Xx0) Xx1)
% Found (x600 x3) as proof of ((Xp Xx1) Xx0)
% Found ((x60 Xx1) x3) as proof of ((Xp Xx1) Xx0)
% Found (((x6 Xx0) Xx1) x3) as proof of ((Xp Xx1) Xx0)
% Found (((x6 Xx0) Xx1) x3) as proof of ((Xp Xx1) Xx0)
% Found eq_ref00:=(eq_ref0 b0):(((eq b) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq b) b0) (Xz Xy0))
% Found ((eq_ref b) b0) as proof of (((eq b) b0) (Xz Xy0))
% Found ((eq_ref b) b0) as proof of (((eq b) b0) (Xz Xy0))
% Found ((eq_ref b) b0) as proof of (((eq b) b0) (Xz Xy0))
% Found eq_ref00:=(eq_ref0 a0):(((eq b) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq b) a0) b0)
% Found ((eq_ref b) a0) as proof of (((eq b) a0) b0)
% Found ((eq_ref b) a0) as proof of (((eq b) a0) b0)
% Found ((eq_ref b) a0) as proof of (((eq b) a0) b0)
% Found x7:((Xp Xx0) Xy0)
% Instantiate: Xx1:=Xy0:a
% Found x7 as proof of ((Xp Xx0) Xx1)
% Found (x400 x7) as proof of ((Xp Xx1) Xx0)
% Found ((x40 Xx1) x7) as proof of ((Xp Xx1) Xx0)
% Found (((x4 Xx0) Xx1) x7) as proof of ((Xp Xx1) Xx0)
% Found (((x4 Xx0) Xx1) x7) as proof of ((Xp Xx1) Xx0)
% Found x5:((Xp Xx0) Xy0)
% Instantiate: Xx1:=Xy0:a
% Found x5 as proof of ((Xp Xx0) Xx1)
% Found (x600 x5) as proof of ((Xp Xx1) Xx0)
% Found ((x60 Xx1) x5) as proof of ((Xp Xx1) Xx0)
% Found (((x6 Xx0) Xx1) x5) as proof of ((Xp Xx1) Xx0)
% Found (fun (x7:(forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz))))=> (((x6 Xx0) Xx1) x5)) as proof of ((Xp Xx1) Xx0)
% Found (fun (x6:(forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))) (x7:(forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz))))=> (((x6 Xx0) Xx1) x5)) as proof of ((forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz)))->((Xp Xx1) Xx0))
% Found (fun (x6:(forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))) (x7:(forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz))))=> (((x6 Xx0) Xx1) x5)) as proof of ((forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))->((forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz)))->((Xp Xx1) Xx0)))
% Found (and_rect20 (fun (x6:(forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))) (x7:(forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz))))=> (((x6 Xx0) Xx1) x5))) as proof of ((Xp Xx1) Xx0)
% Found ((and_rect2 ((Xp Xx1) Xx0)) (fun (x6:(forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))) (x7:(forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz))))=> (((x6 Xx0) Xx1) x5))) as proof of ((Xp Xx1) Xx0)
% Found (((fun (P0:Type) (x6:((forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))->((forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz)))->P0)))=> (((((and_rect (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz)))) P0) x6) x2)) ((Xp Xx1) Xx0)) (fun (x6:(forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))) (x7:(forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz))))=> (((x6 Xx0) Xx1) x5))) as proof of ((Xp Xx1) Xx0)
% Found (((fun (P0:Type) (x6:((forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))->((forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz)))->P0)))=> (((((and_rect (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz)))) P0) x6) x2)) ((Xp Xx1) Xx0)) (fun (x6:(forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))) (x7:(forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz))))=> (((x6 Xx0) Xx1) x5))) as proof of ((Xp Xx1) Xx0)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) Xy0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) Xy0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) Xy0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) Xy0)
% Found eq_ref00:=(eq_ref0 (Xz Xy0)):(((eq b) (Xz Xy0)) (Xz Xy0))
% Found (eq_ref0 (Xz Xy0)) as proof of (((eq b) (Xz Xy0)) b0)
% Found ((eq_ref b) (Xz Xy0)) as proof of (((eq b) (Xz Xy0)) b0)
% Found ((eq_ref b) (Xz Xy0)) as proof of (((eq b) (Xz Xy0)) b0)
% Found ((eq_ref b) (Xz Xy0)) as proof of (((eq b) (Xz Xy0)) b0)
% Found x3:((Xp Xx0) Xy0)
% Instantiate: Xx1:=Xy0:a
% Found x3 as proof of ((Xp Xx0) Xx1)
% Found (x600 x3) as proof of ((Xp Xx1) Xx0)
% Found ((x60 Xx1) x3) as proof of ((Xp Xx1) Xx0)
% Found (((x6 Xx0) Xx1) x3) as proof of ((Xp Xx1) Xx0)
% Found (((x6 Xx0) Xx1) x3) as proof of ((Xp Xx1) Xx0)
% Found x1:((Xp Xx0) Xy0)
% Instantiate: Xx1:=Xy0:a
% Found x1 as proof of ((Xp Xx0) Xx1)
% Found (x600 x1) as proof of ((Xp Xx1) Xx0)
% Found ((x60 Xx1) x1) as proof of ((Xp Xx1) Xx0)
% Found (((x6 Xx0) Xx1) x1) as proof of ((Xp Xx1) Xx0)
% Found (((x6 Xx0) Xx1) x1) as proof of ((Xp Xx1) Xx0)
% Found x7:((Xp Xx0) Xy0)
% Instantiate: Xx1:=Xy0:a
% Found x7 as proof of ((Xp Xx0) Xx1)
% Found (x400 x7) as proof of ((Xp Xx1) Xx0)
% Found ((x40 Xx1) x7) as proof of ((Xp Xx1) Xx0)
% Found (((x4 Xx0) Xx1) x7) as proof of ((Xp Xx1) Xx0)
% Found (((x4 Xx0) Xx1) x7) as proof of ((Xp Xx1) Xx0)
% Found x5:((Xp Xx0) Xy0)
% Instantiate: Xx1:=Xy0:a
% Found x5 as proof of ((Xp Xx0) Xx1)
% Found (x600 x5) as proof of ((Xp Xx1) Xx0)
% Found ((x60 Xx1) x5) as proof of ((Xp Xx1) Xx0)
% Found (((x6 Xx0) Xx1) x5) as proof of ((Xp Xx1) Xx0)
% Found (fun (x7:(forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz))))=> (((x6 Xx0) Xx1) x5)) as proof of ((Xp Xx1) Xx0)
% Found (fun (x6:(forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))) (x7:(forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz))))=> (((x6 Xx0) Xx1) x5)) as proof of ((forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz)))->((Xp Xx1) Xx0))
% Found (fun (x6:(forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))) (x7:(forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz))))=> (((x6 Xx0) Xx1) x5)) as proof of ((forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))->((forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz)))->((Xp Xx1) Xx0)))
% Found (and_rect20 (fun (x6:(forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))) (x7:(forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz))))=> (((x6 Xx0) Xx1) x5))) as proof of ((Xp Xx1) Xx0)
% Found ((and_rect2 ((Xp Xx1) Xx0)) (fun (x6:(forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))) (x7:(forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz))))=> (((x6 Xx0) Xx1) x5))) as proof of ((Xp Xx1) Xx0)
% Found (((fun (P0:Type) (x6:((forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))->((forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz)))->P0)))=> (((((and_rect (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz)))) P0) x6) x2)) ((Xp Xx1) Xx0)) (fun (x6:(forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))) (x7:(forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz))))=> (((x6 Xx0) Xx1) x5))) as proof of ((Xp Xx1) Xx0)
% Found (((fun (P0:Type) (x6:((forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))->((forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz)))->P0)))=> (((((and_rect (forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz)))) P0) x6) x2)) ((Xp Xx1) Xx0)) (fun (x6:(forall (Xx:a) (Xy:a), (((Xp Xx) Xy)->((Xp Xy) Xx)))) (x7:(forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz))))=> (((x6 Xx0) Xx1) x5))) as proof of ((Xp Xx1) Xx0)
% Found eq_ref00:=(eq_ref0 (Xz Xy0)):(((eq b) (Xz Xy0)) (Xz Xy0))
% Found (eq_ref0 (Xz Xy0)) as proof of (((eq b) (Xz Xy0)) b0)
% Found ((eq_ref b) (Xz Xy0)) as proof of (((eq b) (Xz Xy0)) b0)
% Found ((eq_ref b) (Xz Xy0)) as proof of (((eq b) (Xz Xy0)) b0)
% Found ((eq_ref b) (Xz Xy0)) as proof of (((eq b) (Xz Xy0)) b0)
% Found eq_ref00:=(eq_ref0 (Xz Xy0)):(((eq b) (Xz Xy0)) (Xz Xy0))
% Found (eq_ref0 (Xz Xy0)) as proof of (((eq b) (Xz Xy0)) b0)
% Found ((eq_ref b) (Xz Xy0)) as proof of (((eq b) (Xz Xy0)) b0)
% Found ((eq_ref b) (Xz Xy0)) as proof of (((eq b) (Xz Xy0)) b0)
% Found ((eq_ref b) (Xz Xy0)) as proof of (((eq b) (Xz Xy0)) b0)
% Found x3:((Xp Xx0) Xy0)
% Instantiate: Xx1:=Xy0:a
% Found x3 as proof of ((Xp Xx0) Xx1)
% Found (x500 x3) as proof of ((Xp Xx1) Xx0)
% Found ((x50 Xx1) x3) as proof of ((Xp Xx1) Xx0)
% Found (((x5 Xx0) Xx1) x3) as proof of ((Xp Xx1) Xx0)
% Found (fun (x6:(forall (Xx10:a) (Xy1:a) (Xz:a), (((and ((Xp Xx10) Xy1)) ((Xp Xy1) Xz))->((Xp Xx10) Xz))))=> (((x5 Xx0) Xx1) x3)) as proof of ((Xp Xx1) Xx0)
% Found (fun (x5:(forall (Xx10:a) (Xy1:a), (((Xp Xx10) Xy1)->((Xp Xy1) Xx10)))) (x6:(forall (Xx10:a) (Xy1:a) (Xz:a), (((and ((Xp Xx10) Xy1)) ((Xp Xy1) Xz))->((Xp Xx10) Xz))))=> (((x5 Xx0) Xx1) x3)) as proof of ((forall (Xx10:a) (Xy1:a) (Xz:a), (((and ((Xp Xx10) Xy1)) ((Xp Xy1) Xz))->((Xp Xx10) Xz)))->((Xp Xx1) Xx0))
% Found (fun (x5:(forall (Xx10:a) (Xy1:a), (((Xp Xx10) Xy1)->((Xp Xy1) Xx10)))) (x6:(forall (Xx10:a) (Xy1:a) (Xz:a), (((and ((Xp Xx10) Xy1)) ((Xp Xy1) Xz))->((Xp Xx10) Xz))))=> (((x5 Xx0) Xx1) x3)) as proof of ((forall (Xx10:a) (Xy1:a), (((Xp Xx10) Xy1)->((Xp Xy1) Xx10)))->((forall (Xx10:a) (Xy1:a) (Xz:a), (((and ((Xp Xx10) Xy1)) ((Xp Xy1) Xz))->((Xp Xx10) Xz)))->((Xp Xx1) Xx0)))
% Found (and_rect20 (fun (x5:(forall (Xx10:a) (Xy1:a), (((Xp Xx10) Xy1)->((Xp Xy1) Xx10)))) (x6:(forall (Xx10:a) (Xy1:a) (Xz:a), (((and ((Xp Xx10) Xy1)) ((Xp Xy1) Xz))->((Xp Xx10) Xz))))=> (((x5 Xx0) Xx1) x3))) as proof of ((Xp Xx1) Xx0)
% Found ((and_rect2 ((Xp Xx1) Xx0)) (fun (x5:(forall (Xx10:a) (Xy1:a), (((Xp Xx10) Xy1)->((Xp Xy1) Xx10)))) (x6:(forall (Xx10:a) (Xy1:a) (Xz:a), (((and ((Xp Xx10) Xy1)) ((Xp Xy1) Xz))->((Xp Xx10) Xz))))=> (((x5 Xx0) Xx1) x3))) as proof of ((Xp Xx1) Xx0)
% Found (((fun (P0:Type) (x5:((forall (Xx1:a) (Xy1:a), (((Xp Xx1) Xy1)->((Xp Xy1) Xx1)))->((forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp Xx1) Xy1)) ((Xp Xy1) Xz))->((Xp Xx1) Xz)))->P0)))=> (((((and_rect (forall (Xx1:a) (Xy1:a), (((Xp Xx1) Xy1)->((Xp Xy1) Xx1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp Xx1) Xy1)) ((Xp Xy1) Xz))->((Xp Xx1) Xz)))) P0) x5) x4)) ((Xp Xx1) Xx0)) (fun (x5:(forall (Xx10:a) (Xy1:a), (((Xp Xx10) Xy1)->((Xp Xy1) Xx10)))) (x6:(forall (Xx10:a) (Xy1:a) (Xz:a), (((and ((Xp Xx10) Xy1)) ((Xp Xy1) Xz))->((Xp Xx10) Xz))))=> (((x5 Xx0) Xx1) x3))) as proof of ((Xp Xx1) Xx0)
% Found (((fun (P0:Type) (x5:((forall (Xx1:a) (Xy1:a), (((Xp Xx1) Xy1)->((Xp Xy1) Xx1)))->((forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp Xx1) Xy1)) ((Xp Xy1) Xz))->((Xp Xx1) Xz)))->P0)))=> (((((and_rect (forall (Xx1:a) (Xy1:a), (((Xp Xx1) Xy1)->((Xp Xy1) Xx1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp Xx1) Xy1)) ((Xp Xy1) Xz))->((Xp Xx1) Xz)))) P0) x5) x4)) ((Xp Xx1) Xx0)) (fun (x5:(forall (Xx10:a) (Xy1:a), (((Xp Xx10) Xy1)->((Xp Xy1) Xx10)))) (x6:(forall (Xx10:a) (Xy1:a) (Xz:a), (((and ((Xp Xx10) Xy1)) ((Xp Xy1) Xz))->((Xp Xx10) Xz))))=> (((x5 Xx0) Xx1) x3))) as proof of ((Xp Xx1) Xx0)
% Found eq_ref00:=(eq_ref0 b0):(((eq b) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq b) b0) (Xx Xy0))
% Found ((eq_ref b) b0) as proof of (((eq b) b0) (Xx Xy0))
% Found ((eq_ref b) b0) as proof of (((eq b) b0) (Xx Xy0))
% Found ((eq_ref b) b0) as proof of (((eq b) b0) (Xx Xy0))
% Found eq_ref00:=(eq_ref0 a0):(((eq b) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq b) a0) b0)
% Found ((eq_ref b) a0) as proof of (((eq b) a0) b0)
% Found ((eq_ref b) a0) as proof of (((eq b) a0) b0)
% Found ((eq
% EOF
%------------------------------------------------------------------------------