%------------------------------------------------------------------------------
% File     : cocATP---0.2.0
% Problem  : SEV155^5 : TPTP v6.1.0. Released v4.0.0.
% Transfm  : none
% Format   : tptp:raw
% Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p

% Computer : n109.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 13:33:48 EDT 2014

% Result   : Timeout 300.01s
% Output   : None 
% Verified : 
% SZS Type : None (Parsing solution fails)
% Syntax   : Number of formulae    : 0

% Comments : 
%------------------------------------------------------------------------------
%----NO SOLUTION OUTPUT BY SYSTEM
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% % Problem  : SEV155^5 : TPTP v6.1.0. Released v4.0.0.
% % Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n109.star.cs.uiowa.edu
% % Model    : x86_64 x86_64
% % CPU      : Intel(R) Xeon(R) CPU E5-2609 0 @ 2.40GHz
% % Memory   : 32286.75MB
% % OS       : Linux 2.6.32-431.20.3.el6.x86_64
% % CPULimit : 300
% % DateTime : Thu Jul 17 08:15:01 CDT 2014
% % CPUTime  : 300.01 
% 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 0x10b57e8>, <kernel.Type object at 0x10b5170>) of role type named a_type
% Using role type
% Declaring a:Type
% FOF formula (forall (R:(a->(a->Prop))) (S:(a->(a->Prop))) (Xx:a) (Xy:a), ((((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp1 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp1 Xx1) Xy1)) ((Xp1 Xy1) Xz))->((Xp1 Xx1) Xz))))->((Xp1 Xx0) Xy0)))) (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp1 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp1 Xx1) Xy1)) ((Xp1 Xy1) Xz))->((Xp1 Xx1) Xz))))->((Xp1 Xx0) Xy0))))->(forall (Xp1:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((or ((R Xx1) Xy1)) ((S Xx1) Xy1))->((Xp1 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp1 Xx1) Xy1)) ((Xp1 Xy1) Xz))->((Xp1 Xx1) Xz))))->((Xp1 Xx0) Xy0)))))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((or ((R Xx1) Xy1)) ((S Xx1) Xy1))->((Xp1 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz0:a), (((and ((Xp1 Xx1) Xy1)) ((Xp1 Xy1) Xz0))->((Xp1 Xx1) Xz0))))->((Xp1 Xx0) Xy0)))) (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((or ((R Xx1) Xy1)) ((S Xx1) Xy1))->((Xp1 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz0:a), (((and ((Xp1 Xx1) Xy1)) ((Xp1 Xy1) Xz0))->((Xp1 Xx1) Xz0))))->((Xp1 Xy0) Xz))))->(forall (Xp1:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((or ((R Xx1) Xy1)) ((S Xx1) Xy1))->((Xp1 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz0:a), (((and ((Xp1 Xx1) Xy1)) ((Xp1 Xy1) Xz0))->((Xp1 Xx1) Xz0))))->((Xp1 Xx0) Xz))))))->(forall (Xp1:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp1 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz))))->((Xp1 Xx) Xy))))->(forall (Xp1:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp1 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz))))->((Xp1 Xx) Xy))))) of role conjecture named cTHM250D_pme
% Conjecture to prove = (forall (R:(a->(a->Prop))) (S:(a->(a->Prop))) (Xx:a) (Xy:a), ((((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp1 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp1 Xx1) Xy1)) ((Xp1 Xy1) Xz))->((Xp1 Xx1) Xz))))->((Xp1 Xx0) Xy0)))) (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp1 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp1 Xx1) Xy1)) ((Xp1 Xy1) Xz))->((Xp1 Xx1) Xz))))->((Xp1 Xx0) Xy0))))->(forall (Xp1:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((or ((R Xx1) Xy1)) ((S Xx1) Xy1))->((Xp1 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp1 Xx1) Xy1)) ((Xp1 Xy1) Xz))->((Xp1 Xx1) Xz))))->((Xp1 Xx0) Xy0)))))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((or ((R Xx1) Xy1)) ((S Xx1) Xy1))->((Xp1 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz0:a), (((and ((Xp1 Xx1) Xy1)) ((Xp1 Xy1) Xz0))->((Xp1 Xx1) Xz0))))->((Xp1 Xx0) Xy0)))) (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((or ((R Xx1) Xy1)) ((S Xx1) Xy1))->((Xp1 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz0:a), (((and ((Xp1 Xx1) Xy1)) ((Xp1 Xy1) Xz0))->((Xp1 Xx1) Xz0))))->((Xp1 Xy0) Xz))))->(forall (Xp1:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((or ((R Xx1) Xy1)) ((S Xx1) Xy1))->((Xp1 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz0:a), (((and ((Xp1 Xx1) Xy1)) ((Xp1 Xy1) Xz0))->((Xp1 Xx1) Xz0))))->((Xp1 Xx0) Xz))))))->(forall (Xp1:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp1 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz))))->((Xp1 Xx) Xy))))->(forall (Xp1:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp1 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz))))->((Xp1 Xx) Xy))))):Prop
% Parameter a_DUMMY:a.
% We need to prove ['(forall (R:(a->(a->Prop))) (S:(a->(a->Prop))) (Xx:a) (Xy:a), ((((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp1 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp1 Xx1) Xy1)) ((Xp1 Xy1) Xz))->((Xp1 Xx1) Xz))))->((Xp1 Xx0) Xy0)))) (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp1 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp1 Xx1) Xy1)) ((Xp1 Xy1) Xz))->((Xp1 Xx1) Xz))))->((Xp1 Xx0) Xy0))))->(forall (Xp1:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((or ((R Xx1) Xy1)) ((S Xx1) Xy1))->((Xp1 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp1 Xx1) Xy1)) ((Xp1 Xy1) Xz))->((Xp1 Xx1) Xz))))->((Xp1 Xx0) Xy0)))))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((or ((R Xx1) Xy1)) ((S Xx1) Xy1))->((Xp1 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz0:a), (((and ((Xp1 Xx1) Xy1)) ((Xp1 Xy1) Xz0))->((Xp1 Xx1) Xz0))))->((Xp1 Xx0) Xy0)))) (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((or ((R Xx1) Xy1)) ((S Xx1) Xy1))->((Xp1 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz0:a), (((and ((Xp1 Xx1) Xy1)) ((Xp1 Xy1) Xz0))->((Xp1 Xx1) Xz0))))->((Xp1 Xy0) Xz))))->(forall (Xp1:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((or ((R Xx1) Xy1)) ((S Xx1) Xy1))->((Xp1 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz0:a), (((and ((Xp1 Xx1) Xy1)) ((Xp1 Xy1) Xz0))->((Xp1 Xx1) Xz0))))->((Xp1 Xx0) Xz))))))->(forall (Xp1:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp1 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz))))->((Xp1 Xx) Xy))))->(forall (Xp1:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp1 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz))))->((Xp1 Xx) Xy)))))']
% Parameter a:Type.
% Trying to prove (forall (R:(a->(a->Prop))) (S:(a->(a->Prop))) (Xx:a) (Xy:a), ((((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp1 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp1 Xx1) Xy1)) ((Xp1 Xy1) Xz))->((Xp1 Xx1) Xz))))->((Xp1 Xx0) Xy0)))) (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp1 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp1 Xx1) Xy1)) ((Xp1 Xy1) Xz))->((Xp1 Xx1) Xz))))->((Xp1 Xx0) Xy0))))->(forall (Xp1:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((or ((R Xx1) Xy1)) ((S Xx1) Xy1))->((Xp1 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp1 Xx1) Xy1)) ((Xp1 Xy1) Xz))->((Xp1 Xx1) Xz))))->((Xp1 Xx0) Xy0)))))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((or ((R Xx1) Xy1)) ((S Xx1) Xy1))->((Xp1 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz0:a), (((and ((Xp1 Xx1) Xy1)) ((Xp1 Xy1) Xz0))->((Xp1 Xx1) Xz0))))->((Xp1 Xx0) Xy0)))) (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((or ((R Xx1) Xy1)) ((S Xx1) Xy1))->((Xp1 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz0:a), (((and ((Xp1 Xx1) Xy1)) ((Xp1 Xy1) Xz0))->((Xp1 Xx1) Xz0))))->((Xp1 Xy0) Xz))))->(forall (Xp1:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((or ((R Xx1) Xy1)) ((S Xx1) Xy1))->((Xp1 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz0:a), (((and ((Xp1 Xx1) Xy1)) ((Xp1 Xy1) Xz0))->((Xp1 Xx1) Xz0))))->((Xp1 Xx0) Xz))))))->(forall (Xp1:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp1 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz))))->((Xp1 Xx) Xy))))->(forall (Xp1:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp1 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz))))->((Xp1 Xx) Xy)))))
% Found x0:((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp1 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz))))
% Found x0 as proof of ((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))
% Found x0:((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp1 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz))))
% Found x0 as proof of ((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))
% Found x0:((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp1 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz))))
% Found x0 as proof of ((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))
% Found x1:((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp1 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz))))
% Found x1 as proof of ((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))
% Found x0:((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp1 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz))))
% Found x0 as proof of ((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))
% Found x100:=(x10 x0):((Xp1 Xx) Xy)
% Found (x10 x0) as proof of ((Xp1 Xx) Xy)
% Found ((x1 Xp1) x0) as proof of ((Xp1 Xx) Xy)
% Found (fun (x3:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz))))=> ((x1 Xp1) x0)) as proof of ((Xp1 Xx) Xy)
% Found (fun (x2:(forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp1 Xx0) Xy0)))) (x3:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz))))=> ((x1 Xp1) x0)) as proof of ((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))->((Xp1 Xx) Xy))
% Found (fun (x2:(forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp1 Xx0) Xy0)))) (x3:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz))))=> ((x1 Xp1) x0)) as proof of ((forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp1 Xx0) Xy0)))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))->((Xp1 Xx) Xy)))
% Found (and_rect00 (fun (x2:(forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp1 Xx0) Xy0)))) (x3:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz))))=> ((x1 Xp1) x0))) as proof of ((Xp1 Xx) Xy)
% Found ((and_rect0 ((Xp1 Xx) Xy)) (fun (x2:(forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp1 Xx0) Xy0)))) (x3:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz))))=> ((x1 Xp1) x0))) as proof of ((Xp1 Xx) Xy)
% Found (((fun (P:Type) (x2:((forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp1 Xx0) Xy0)))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))->P)))=> (((((and_rect (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp1 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))) P) x2) x0)) ((Xp1 Xx) Xy)) (fun (x2:(forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp1 Xx0) Xy0)))) (x3:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz))))=> ((x1 Xp1) x0))) as proof of ((Xp1 Xx) Xy)
% Found (((fun (P:Type) (x2:((forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp1 Xx0) Xy0)))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))->P)))=> (((((and_rect (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp1 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))) P) x2) x0)) ((Xp1 Xx) Xy)) (fun (x2:(forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp1 Xx0) Xy0)))) (x3:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz))))=> ((x1 Xp1) x0))) as proof of ((Xp1 Xx) Xy)
% Found x0:((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp1 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz))))
% Found x0 as proof of ((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))
% Found x000:=(x00 x1):((Xp1 Xx) Xy)
% Found (x00 x1) as proof of ((Xp1 Xx) Xy)
% Found ((x0 Xp1) x1) as proof of ((Xp1 Xx) Xy)
% Found (fun (x3:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz))))=> ((x0 Xp1) x1)) as proof of ((Xp1 Xx) Xy)
% Found (fun (x2:(forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp1 Xx0) Xy0)))) (x3:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz))))=> ((x0 Xp1) x1)) as proof of ((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))->((Xp1 Xx) Xy))
% Found (fun (x2:(forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp1 Xx0) Xy0)))) (x3:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz))))=> ((x0 Xp1) x1)) as proof of ((forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp1 Xx0) Xy0)))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))->((Xp1 Xx) Xy)))
% Found (and_rect00 (fun (x2:(forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp1 Xx0) Xy0)))) (x3:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz))))=> ((x0 Xp1) x1))) as proof of ((Xp1 Xx) Xy)
% Found ((and_rect0 ((Xp1 Xx) Xy)) (fun (x2:(forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp1 Xx0) Xy0)))) (x3:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz))))=> ((x0 Xp1) x1))) as proof of ((Xp1 Xx) Xy)
% Found (((fun (P:Type) (x2:((forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp1 Xx0) Xy0)))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))->P)))=> (((((and_rect (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp1 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))) P) x2) x1)) ((Xp1 Xx) Xy)) (fun (x2:(forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp1 Xx0) Xy0)))) (x3:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz))))=> ((x0 Xp1) x1))) as proof of ((Xp1 Xx) Xy)
% Found (fun (x1:((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp1 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))))=> (((fun (P:Type) (x2:((forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp1 Xx0) Xy0)))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))->P)))=> (((((and_rect (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp1 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))) P) x2) x1)) ((Xp1 Xx) Xy)) (fun (x2:(forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp1 Xx0) Xy0)))) (x3:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz))))=> ((x0 Xp1) x1)))) as proof of ((Xp1 Xx) Xy)
% Found (fun (x1:((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp1 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))))=> (((fun (P:Type) (x2:((forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp1 Xx0) Xy0)))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))->P)))=> (((((and_rect (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp1 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))) P) x2) x1)) ((Xp1 Xx) Xy)) (fun (x2:(forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp1 Xx0) Xy0)))) (x3:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz))))=> ((x0 Xp1) x1)))) as proof of (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp1 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz))))->((Xp1 Xx) Xy))
% Found x0:((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp1 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz))))
% Found x0 as proof of ((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))
% Found x4:((or ((R Xx0) Xy0)) ((S Xx0) Xy0))
% Found (fun (x4:((or ((R Xx0) Xy0)) ((S Xx0) Xy0)))=> x4) as proof of ((Xp10 Xx0) Xy0)
% Found (fun (Xy0:a) (x4:((or ((R Xx0) Xy0)) ((S Xx0) Xy0)))=> x4) as proof of (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp10 Xx0) Xy0))
% Found (fun (Xx0:a) (Xy0:a) (x4:((or ((R Xx0) Xy0)) ((S Xx0) Xy0)))=> x4) as proof of (forall (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp10 Xx0) Xy0)))
% Found (fun (Xx0:a) (Xy0:a) (x4:((or ((R Xx0) Xy0)) ((S Xx0) Xy0)))=> x4) as proof of (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp10 Xx0) Xy0)))
% Found or_comm_i10:=(or_comm_i1 ((S Xx0) Xy0)):(((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((or ((S Xx0) Xy0)) ((R Xx0) Xy0)))
% Found (or_comm_i1 ((S Xx0) Xy0)) as proof of (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp10 Xx0) Xy0))
% Found ((or_comm_i ((R Xx0) Xy0)) ((S Xx0) Xy0)) as proof of (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp10 Xx0) Xy0))
% Found (fun (Xy0:a)=> ((or_comm_i ((R Xx0) Xy0)) ((S Xx0) Xy0))) as proof of (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp10 Xx0) Xy0))
% Found (fun (Xx0:a) (Xy0:a)=> ((or_comm_i ((R Xx0) Xy0)) ((S Xx0) Xy0))) as proof of (forall (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp10 Xx0) Xy0)))
% Found (fun (Xx0:a) (Xy0:a)=> ((or_comm_i ((R Xx0) Xy0)) ((S Xx0) Xy0))) as proof of (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp10 Xx0) Xy0)))
% Found x0:((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp1 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz))))
% Found x0 as proof of ((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))
% Found x0:((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp1 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz))))
% Found x0 as proof of ((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))
% Found x0:((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp1 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz))))
% Found x0 as proof of ((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))
% Found x0:((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp1 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz))))
% Found x0 as proof of ((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))
% Found x4:((or ((R Xx0) Xy0)) ((S Xx0) Xy0))
% Found (fun (x4:((or ((R Xx0) Xy0)) ((S Xx0) Xy0)))=> x4) as proof of ((or ((R Xx0) Xy0)) ((S Xx0) Xy0))
% Found (fun (Xy0:a) (x4:((or ((R Xx0) Xy0)) ((S Xx0) Xy0)))=> x4) as proof of (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((or ((R Xx0) Xy0)) ((S Xx0) Xy0)))
% Found (fun (Xx0:a) (Xy0:a) (x4:((or ((R Xx0) Xy0)) ((S Xx0) Xy0)))=> x4) as proof of (forall (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((or ((R Xx0) Xy0)) ((S Xx0) Xy0))))
% Found (fun (Xx0:a) (Xy0:a) (x4:((or ((R Xx0) Xy0)) ((S Xx0) Xy0)))=> x4) as proof of (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((or ((R Xx0) Xy0)) ((S Xx0) Xy0))))
% Found x0:((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp1 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz))))
% Found x0 as proof of ((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))
% Found x4:((or ((R Xx0) Xy0)) ((S Xx0) Xy0))
% Found (fun (x4:((or ((R Xx0) Xy0)) ((S Xx0) Xy0)))=> x4) as proof of ((or ((R Xx0) Xy0)) ((S Xx0) Xy0))
% Found (fun (Xy0:a) (x4:((or ((R Xx0) Xy0)) ((S Xx0) Xy0)))=> x4) as proof of (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((or ((R Xx0) Xy0)) ((S Xx0) Xy0)))
% Found (fun (Xx0:a) (Xy0:a) (x4:((or ((R Xx0) Xy0)) ((S Xx0) Xy0)))=> x4) as proof of (forall (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((or ((R Xx0) Xy0)) ((S Xx0) Xy0))))
% Found (fun (Xx0:a) (Xy0:a) (x4:((or ((R Xx0) Xy0)) ((S Xx0) Xy0)))=> x4) as proof of (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((or ((R Xx0) Xy0)) ((S Xx0) Xy0))))
% Found x0:((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp1 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz))))
% Found x0 as proof of ((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp10 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp10 Xx0) Xy0)) ((Xp10 Xy0) Xz))->((Xp10 Xx0) Xz))))
% Found x4:((or ((R Xx0) Xy0)) ((S Xx0) Xy0))
% Found (fun (x4:((or ((R Xx0) Xy0)) ((S Xx0) Xy0)))=> x4) as proof of ((or ((R Xx0) Xy0)) ((S Xx0) Xy0))
% Found (fun (Xy0:a) (x4:((or ((R Xx0) Xy0)) ((S Xx0) Xy0)))=> x4) as proof of (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((or ((R Xx0) Xy0)) ((S Xx0) Xy0)))
% Found (fun (Xx0:a) (Xy0:a) (x4:((or ((R Xx0) Xy0)) ((S Xx0) Xy0)))=> x4) as proof of (forall (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((or ((R Xx0) Xy0)) ((S Xx0) Xy0))))
% Found (fun (Xx0:a) (Xy0:a) (x4:((or ((R Xx0) Xy0)) ((S Xx0) Xy0)))=> x4) as proof of (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((or ((R Xx0) Xy0)) ((S Xx0) Xy0))))
% Found x5:((S Xx0) Xy0)
% Found (fun (x5:((S Xx0) Xy0))=> x5) as proof of ((Xp10 Xx0) Xy0)
% Found (fun (x5:((S Xx0) Xy0))=> x5) as proof of (((S Xx0) Xy0)->((Xp10 Xx0) Xy0))
% Found x5:((R Xx0) Xy0)
% Found (fun (x5:((R Xx0) Xy0))=> x5) as proof of ((Xp10 Xx0) Xy0)
% Found (fun (x5:((R Xx0) Xy0))=> x5) as proof of (((R Xx0) Xy0)->((Xp10 Xx0) Xy0))
% Found x5:((or ((R Xx0) Xy0)) ((S Xx0) Xy0))
% Found (fun (x5:((or ((R Xx0) Xy0)) ((S Xx0) Xy0)))=> x5) as proof of ((Xp10 Xx0) Xy0)
% Found (fun (Xy0:a) (x5:((or ((R Xx0) Xy0)) ((S Xx0) Xy0)))=> x5) as proof of (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp10 Xx0) Xy0))
% Found (fun (Xx0:a) (Xy0:a) (x5:((or ((R Xx0) Xy0)) ((S Xx0) Xy0)))=> x5) as proof of (forall (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp10 Xx0) Xy0)))
% Found (fun (Xx0:a) (Xy0:a) (x5:((or ((R Xx0) Xy0)) ((S Xx0) Xy0)))=> x5) as proof of (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp10 Xx0) Xy0)))
% Found or_comm_i10:=(or_comm_i1 ((S Xx0) Xy0)):(((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((or ((S Xx0) Xy0)) ((R Xx0) Xy0)))
% Found (or_comm_i1 ((S Xx0) Xy0)) as proof of (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))-
% EOF
%------------------------------------------------------------------------------